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<h1>Sophia Optimizer</h1>
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<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of <em>Sophia-G</em> from paper <a href="https://arxiv.org/abs/2305.14342">Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training</a>. Official implementation is available at <a href="https://github.com/Liuhong99/Sophia">Liuhong99/Sophia</a>.</p>
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<p>Sophia is more adaptive to heterogeneous curvatures than Adam, more resistant to non-convexity and rapid change of Hessian than Newton’s method, and also uses a low-cost pre-conditioner.</p>
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<p>Sophia keeps diagonal Hessian estimates with EMA across iterations. The diagonal Hessian <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.1078799999999998em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqo" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> is calculated every <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03148em">k</span></span></span></span></span></span> steps.</p>
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<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.61788em;vertical-align:-0.55894em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05894em;"><span style="top:-3.1010600000000004em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqs" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mbin mtight">−</span><span class="mord mtight coloredeq eqbd" style=""><span class="mord mathnormal mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqs" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord coloredeq eqo" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord text"><span class="mord"> if </span></span><span class="mord coloredeq eqbe" style=""><span class="mord mathnormal" style="">t</span></span><span class="mord text"><span class="mord"> mod </span></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03148em">k</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord"> else </span></span><span class="mord coloredeq eqy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.55894em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>Sophia uses EMA of gradients <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>, only considers positive entries of the diagonal Hessian and does per-coordinate clipping to the update.</p>
|
||
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:4.20003em;vertical-align:-1.850015em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.350015em;"><span style="top:-4.960015em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.8500150000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850015em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.350015em;"><span style="top:-4.960015em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord coloredeq eqt" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord coloredeq eqx" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.8500150000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqv" style=""><span class="mord mathnormal" style="margin-right:0.03588em">η</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop"><span class="mord mathrm">clip</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqi" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span><span class="mopen" style="">{</span><span class="mord" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span><span class="mclose" style="">}</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqw" style=""><span class="mord mathnormal" style="">ρ</span></span><span class="mord"><span class="delimsizing size3">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850015em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>where <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqq" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span> is a very small value to prevent division by <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>.</p>
|
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<h3>Gauss-Newton-Bartlett (GNB) estimator</h3>
|
||
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:5.048329000000001em;vertical-align:-2.2741645000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7741645000000004em;"><span style="top:-4.7741645em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span><span style="top:-2.2141715em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqo" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.2741645000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.7741645000000004em;"><span style="top:-4.7741645em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqbc" style=""><span class="mord mathnormal" style="margin-right:0.05017em">B</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbc" style=""><span class="mord mathnormal mtight" style="margin-right:0.05017em">B</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqm" style=""><span class="mord" style=""><span class="mord" style="">ℓ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.05764em">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size1">(</span></span><span class="mord coloredeq eqj" style=""><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord coloredeq eqba" style=""><span class="mord mathnormal" style="">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqp" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">^</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size1">)</span></span></span></span><span style="top:-2.2141715em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqbc" style=""><span class="mord mathnormal" style="margin-right:0.05017em">B</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⊙</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.2741645000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>where <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqba" style=""><span class="mord" style=""><span class="mord mathnormal" style="">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> are the inputs, <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqbc" style=""><span class="mord mathnormal" style="margin-right:0.05017em">B</span></span></span></span></span></span> is the batch size (number of inputs/tokens), <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqm" style=""><span class="mord" style=""><span class="mord" style="">ℓ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.05764em">CE</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> is cross entropy loss, and <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqp" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">^</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> are sampled from the logits <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqj" style=""><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord coloredeq eqba" style=""><span class="mord mathnormal" style="">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span>.</p>
|
||
<p>Note that this hessian estimate is always positive and therefore we can replace <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqi" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span><span class="mopen" style="">{</span><span class="mord" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span><span class="mclose" style="">}</span></span></span></span></span></span> with <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqk" style=""><span class="mord" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span></span></span></span></span></span>.</p>
|
||
<p>Sophia with Gauss-Newton-Bartlett (GNB) estimator is <strong>Sophia-G</strong></p>
|
||
<p>Here is an <a href="../transformers/basic/with_sophia.html">experiment</a> that uses Sophia-G to train a transformer.</p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">54</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Any</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Optional</span>
|
||
<span class="lineno">55</span>
|
||
<span class="lineno">56</span><span class="kn">import</span> <span class="nn">torch</span>
|
||
<span class="lineno">57</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</span>
|
||
<span class="lineno">58</span>
|
||
<span class="lineno">59</span><span class="kn">from</span> <span class="nn">labml_nn.optimizers</span> <span class="kn">import</span> <span class="n">GenericAdaptiveOptimizer</span><span class="p">,</span> <span class="n">WeightDecay</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-1'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-1'>#</a>
|
||
</div>
|
||
<h2>Sophia-G Optimizer</h2>
|
||
<p>We extend the class <code class="highlight"><span></span><span class="n">GenericAdaptiveOptimizer</span></code>
|
||
defined in <a href="index.html"><code class="highlight"><span></span><span class="fm">__init__</span><span class="o">.</span><span class="n">py</span></code>
|
||
</a> to implement the Sophia optimizer.</p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">62</span><span class="k">class</span> <span class="nc">Sophia</span><span class="p">(</span><span class="n">GenericAdaptiveOptimizer</span><span class="p">):</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-2'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-2'>#</a>
|
||
</div>
|
||
<h3>Initialize the optimizer</h3>
|
||
<ul><li><code class="highlight"><span></span><span class="n">params</span></code>
|
||
is the list of parameters </li>
|
||
<li><code class="highlight"><span></span><span class="n">lr</span></code>
|
||
is the maximum learning rate <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqv" style="margin-right:0.03588em">η</span></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqw" style="">ρ</span></span></span></span></span></span></span> </li>
|
||
<li><code class="highlight"><span></span><span class="n">betas</span></code>
|
||
is a tuple of (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>, <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqs" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>) </li>
|
||
<li><code class="highlight"><span></span><span class="n">eps</span></code>
|
||
is <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqq" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span> </li>
|
||
<li><code class="highlight"><span></span><span class="n">pho</span></code>
|
||
is <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqw" style=""><span class="mord mathnormal" style="">ρ</span></span></span></span></span></span> </li>
|
||
<li><code class="highlight"><span></span><span class="n">weight_decay</span></code>
|
||
is an instance of class <code class="highlight"><span></span><span class="n">WeightDecay</span></code>
|
||
defined in <a href="index.html"><code class="highlight"><span></span><span class="fm">__init__</span><span class="o">.</span><span class="n">py</span></code>
|
||
</a> </li>
|
||
<li><code class="highlight"><span></span><span class="n">defaults</span></code>
|
||
is a dictionary of default for group values. This is useful when you want to extend the class <code class="highlight"><span></span><span class="n">Adam</span></code>
|
||
.</li></ul>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">70</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span>
|
||
<span class="lineno">71</span> <span class="n">lr</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-4</span><span class="p">,</span> <span class="n">betas</span><span class="p">:</span> <span class="n">Tuple</span><span class="p">[</span><span class="nb">float</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">),</span> <span class="n">eps</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-12</span><span class="p">,</span>
|
||
<span class="lineno">72</span> <span class="n">rho</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.03</span><span class="p">,</span>
|
||
<span class="lineno">73</span> <span class="n">weight_decay</span><span class="p">:</span> <span class="n">WeightDecay</span> <span class="o">=</span> <span class="n">WeightDecay</span><span class="p">(),</span>
|
||
<span class="lineno">74</span> <span class="n">defaults</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">Any</span><span class="p">]]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">):</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-3'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-3'>#</a>
|
||
</div>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">87</span> <span class="n">defaults</span> <span class="o">=</span> <span class="p">{}</span> <span class="k">if</span> <span class="n">defaults</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">defaults</span>
|
||
<span class="lineno">88</span> <span class="n">defaults</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">weight_decay</span><span class="o">.</span><span class="n">defaults</span><span class="p">())</span>
|
||
<span class="lineno">89</span> <span class="n">defaults</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">rho</span><span class="o">=</span><span class="n">rho</span><span class="p">))</span>
|
||
<span class="lineno">90</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">defaults</span><span class="p">,</span> <span class="n">lr</span><span class="p">,</span> <span class="n">betas</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
|
||
<span class="lineno">91</span>
|
||
<span class="lineno">92</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight_decay</span> <span class="o">=</span> <span class="n">weight_decay</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-4'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-4'>#</a>
|
||
</div>
|
||
<h3>Initialize a parameter state</h3>
|
||
<ul><li><code class="highlight"><span></span><span class="n">state</span></code>
|
||
is the optimizer state of the parameter (tensor) </li>
|
||
<li><code class="highlight"><span></span><span class="n">group</span></code>
|
||
stores optimizer attributes of the parameter group </li>
|
||
<li><code class="highlight"><span></span><span class="n">param</span></code>
|
||
is the parameter tensor <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span></li></ul>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">94</span> <span class="k">def</span> <span class="nf">init_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">param</span><span class="p">:</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">):</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-5'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-5'>#</a>
|
||
</div>
|
||
<p>This is the number of optimizer steps taken on the parameter, <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqbe" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">104</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-6'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-6'>#</a>
|
||
</div>
|
||
<p>Exponential moving average of gradients, <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">106</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg'</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">memory_format</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">preserve_format</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-7'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-7'>#</a>
|
||
</div>
|
||
<p>Exponential moving average of Hessian diagonal, <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">108</span> <span class="n">state</span><span class="p">[</span><span class="s1">'hessian'</span><span class="p">]</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros_like</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">memory_format</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">preserve_format</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-8'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-8'>#</a>
|
||
</div>
|
||
<h3>Update the EMA of Hessian diagonal <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></h3>
|
||
<ul><li><code class="highlight"><span></span><span class="n">n_tokens_training_batch</span></code>
|
||
is the number of tokens/inputs in the batch <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqbc" style=""><span class="mord mathnormal" style="margin-right:0.05017em">B</span></span></span></span></span></span></li></ul>
|
||
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.23576em;vertical-align:-1.36788em;"></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.86788em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqo" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.29212em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.36788em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.86788em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqbc" style="margin-right:0.05017em">B</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord" style="">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.02778em">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">⊙</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord" style="">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.02778em">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mclose" style="">)</span></span></span><span style="top:-2.29212em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord coloredeq eqs" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbd" style="margin-right:0.03148em">k</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mopen" style="">(</span><span class="mord" style="">1</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord coloredeq eqs" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord coloredeq eqo" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.36788em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span>
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">110</span> <span class="k">def</span> <span class="nf">update_hessian</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_tokens_training_batch</span><span class="p">):</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-9'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-9'>#</a>
|
||
</div>
|
||
<p>Iterate through parameter groups </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">123</span> <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">param_groups</span><span class="p">:</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-10'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-10'>#</a>
|
||
</div>
|
||
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqs" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">125</span> <span class="n">_</span><span class="p">,</span> <span class="n">beta2</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'betas'</span><span class="p">]</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-11'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-11'>#</a>
|
||
</div>
|
||
<p>Iterate through parameters </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">127</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">group</span><span class="p">[</span><span class="s1">'params'</span><span class="p">]:</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-12'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-12'>#</a>
|
||
</div>
|
||
<p>Skip parameters without gradients </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">129</span> <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">grad</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="lineno">130</span> <span class="k">continue</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-13'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-13'>#</a>
|
||
</div>
|
||
<p>Get optimizer state </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">133</span> <span class="n">state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">[</span><span class="n">p</span><span class="p">]</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-14'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-14'>#</a>
|
||
</div>
|
||
<p>Initialize state if empty </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">136</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="lineno">137</span> <span class="bp">self</span><span class="o">.</span><span class="n">init_state</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">group</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-15'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-15'>#</a>
|
||
</div>
|
||
<p>Update EMA Hessian diagonal</p>
|
||
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.23576em;vertical-align:-1.36788em;"></span><span class="mord coloredeq eqc" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.86788em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqo" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.29212em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.36788em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.86788em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqbc" style="margin-right:0.05017em">B</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord" style="">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.02778em">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">⊙</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord" style="">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.02778em">θ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9467699999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">L</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="mclose" style="">)</span></span></span><span style="top:-2.29212em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord coloredeq eqs" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbd" style="margin-right:0.03148em">k</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mopen" style="">(</span><span class="mord" style="">1</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin" style="">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord" style=""><span class="mord coloredeq eqs" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord coloredeq eqo" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9578799999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="">h</span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.36788em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><p> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">145</span> <span class="n">state</span><span class="p">[</span><span class="s1">'hessian'</span><span class="p">]</span><span class="o">.</span><span class="n">mul_</span><span class="p">(</span><span class="n">beta2</span><span class="p">)</span><span class="o">.</span><span class="n">addcmul_</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">grad</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">grad</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta2</span><span class="p">)</span> <span class="o">*</span> <span class="n">n_tokens_training_batch</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-16'>
|
||
<div class='docs doc-strings'>
|
||
<div class='section-link'>
|
||
<a href='#section-16'>#</a>
|
||
</div>
|
||
<h3>Take an update step for a given parameter tensor</h3>
|
||
<ul><li><code class="highlight"><span></span><span class="n">state</span></code>
|
||
is the optimizer state of the parameter (tensor) </li>
|
||
<li><code class="highlight"><span></span><span class="n">group</span></code>
|
||
stores optimizer attributes of the parameter group </li>
|
||
<li><code class="highlight"><span></span><span class="n">grad</span></code>
|
||
is the current gradient tensor <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqx" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> for the parameter <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span> </li>
|
||
<li><code class="highlight"><span></span><span class="n">param</span></code>
|
||
is the parameter tensor <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span></li></ul>
|
||
<p>We do the following parameter update,</p>
|
||
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.70003em;vertical-align:-1.1000150000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6000149999999997em;"><span style="top:-3.6000150000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1000150000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6000149999999997em;"><span style="top:-3.6000150000000004em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqv" style=""><span class="mord mathnormal" style="margin-right:0.03588em">η</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqg" style=""><span class="mop" style=""><span class="mord mathrm" style="">clip</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">(</span></span></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqk" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin coloredeq eqk" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord coloredeq eqk" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqz" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqw" style="">ρ</span></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">)</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1000150000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span>
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">147</span> <span class="k">def</span> <span class="nf">step_param</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">group</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">],</span> <span class="n">grad</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">param</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">):</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-17'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-17'>#</a>
|
||
</div>
|
||
<p>Calculate weight decay </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">164</span> <span class="n">grad</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">weight_decay</span><span class="p">(</span><span class="n">param</span><span class="p">,</span> <span class="n">grad</span><span class="p">,</span> <span class="n">group</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-18'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-18'>#</a>
|
||
</div>
|
||
<p>Get <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> and <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqs" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">167</span> <span class="n">beta1</span><span class="p">,</span> <span class="n">beta2</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'betas'</span><span class="p">]</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-19'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-19'>#</a>
|
||
</div>
|
||
<p>Get <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqw" style=""><span class="mord mathnormal" style="">ρ</span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">169</span> <span class="n">rho</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'rho'</span><span class="p">]</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-20'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-20'>#</a>
|
||
</div>
|
||
<p>Get <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eqt" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span> and <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
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|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">172</span> <span class="n">m</span><span class="p">,</span> <span class="n">hessian</span> <span class="o">=</span> <span class="n">state</span><span class="p">[</span><span class="s1">'exp_avg'</span><span class="p">],</span> <span class="n">state</span><span class="p">[</span><span class="s1">'hessian'</span><span class="p">]</span></pre></div>
|
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|
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<div class='section' id='section-21'>
|
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<div class='docs'>
|
||
<div class='section-link'>
|
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<a href='#section-21'>#</a>
|
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</div>
|
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<p>In-place calculation of <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord coloredeq eqt" style=""><span class="mord" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span><span class="mbin mtight" style="">−</span><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqr" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.05278em">β</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style="">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqx" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
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|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">176</span> <span class="n">m</span><span class="o">.</span><span class="n">mul_</span><span class="p">(</span><span class="n">beta1</span><span class="p">)</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">grad</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mi">1</span> <span class="o">-</span> <span class="n">beta1</span><span class="p">)</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-22'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-22'>#</a>
|
||
</div>
|
||
<p>Increment <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqbe" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span> the number of optimizer steps </p>
|
||
|
||
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|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">179</span> <span class="n">state</span><span class="p">[</span><span class="s1">'step'</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-23'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-23'>#</a>
|
||
</div>
|
||
<p>Get maximum learning rate <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqv" style="margin-right:0.03588em">η</span></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqw" style="">ρ</span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">182</span> <span class="n">lr</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s1">'lr'</span><span class="p">]</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-24'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-24'>#</a>
|
||
</div>
|
||
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqv" style=""><span class="mord mathnormal" style="margin-right:0.03588em">η</span></span></span></span></span></span> </p>
|
||
|
||
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|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">185</span> <span class="n">eta</span> <span class="o">=</span> <span class="n">lr</span> <span class="o">/</span> <span class="n">rho</span></pre></div>
|
||
</div>
|
||
</div>
|
||
<div class='section' id='section-25'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-25'>#</a>
|
||
</div>
|
||
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord coloredeq eqg" style=""><span class="mop" style=""><span class="mord mathrm" style="">clip</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">(</span></span></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqk" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin coloredeq eqk" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord coloredeq eqk" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqz" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqw" style="">ρ</span></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">)</span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
</div>
|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">188</span> <span class="n">ratio</span> <span class="o">=</span> <span class="p">(</span><span class="n">m</span> <span class="o">/</span> <span class="p">(</span><span class="n">hessian</span> <span class="o">+</span> <span class="n">group</span><span class="p">[</span><span class="s1">'eps'</span><span class="p">]))</span><span class="o">.</span><span class="n">clamp</span><span class="p">(</span><span class="o">-</span><span class="n">rho</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span></pre></div>
|
||
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|
||
</div>
|
||
<div class='section' id='section-26'>
|
||
<div class='docs'>
|
||
<div class='section-link'>
|
||
<a href='#section-26'>#</a>
|
||
</div>
|
||
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqbe" style=""><span class="mord mathnormal mtight" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.63889em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqv" style=""><span class="mord mathnormal" style="margin-right:0.03588em">η</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord coloredeq eqg" style=""><span class="mop" style=""><span class="mord mathrm" style="">clip</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">(</span></span></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqk" style=""><span class="mord coloredeq eqy" style=""><span class="mord mathnormal" style="">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin coloredeq eqk" style="">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord coloredeq eqk" style=""><span class="mord mathnormal coloredeq eqq" style="">ϵ</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqz" style=""><span class="mord mathnormal" style="">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqbe" style="">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqw" style="">ρ</span></span><span class="mord" style=""><span class="delimsizing size3" style=""><span style="">)</span></span></span></span></span></span></span></span></span> </p>
|
||
|
||
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|
||
<div class='code'>
|
||
<div class="highlight"><pre><span class="lineno">191</span> <span class="n">param</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">ratio</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=-</span><span class="n">eta</span><span class="p">)</span></pre></div>
|
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function handleImages() {
|
||
var images = document.querySelectorAll('p>img')
|
||
|
||
for (var i = 0; i < images.length; ++i) {
|
||
handleImage(images[i])
|
||
}
|
||
}
|
||
|
||
function handleImage(img) {
|
||
img.parentElement.style.textAlign = 'center'
|
||
|
||
var modal = document.createElement('div')
|
||
modal.id = 'modal'
|
||
|
||
var modalContent = document.createElement('div')
|
||
modal.appendChild(modalContent)
|
||
|
||
var modalImage = document.createElement('img')
|
||
modalContent.appendChild(modalImage)
|
||
|
||
var span = document.createElement('span')
|
||
span.classList.add('close')
|
||
span.textContent = 'x'
|
||
modal.appendChild(span)
|
||
|
||
img.onclick = function () {
|
||
console.log('clicked')
|
||
document.body.appendChild(modal)
|
||
modalImage.src = img.src
|
||
}
|
||
|
||
span.onclick = function () {
|
||
document.body.removeChild(modal)
|
||
}
|
||
}
|
||
|
||
handleImages()
|
||
</script>
|
||
</body>
|
||
</html> |