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<h1>Prioritized Experience Replay Buffer</h1>
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<p>This implements paper <a href="https://arxiv.org/abs/1511.05952">Prioritized experience replay</a>,
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using a binary segment tree.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">13</span><span></span><span class="kn">import</span> <span class="nn">random</span>
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<span class="lineno">14</span>
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<span class="lineno">15</span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>Buffer for Prioritized Experience Replay</h2>
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<p><a href="https://arxiv.org/abs/1511.05952">Prioritized experience replay</a>
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samples important transitions more frequently.
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The transitions are prioritized by the Temporal Difference error (td error), $\delta$.</p>
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<p>We sample transition $i$ with probability,
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<script type="math/tex; mode=display">P(i) = \frac{p_i^\alpha}{\sum_k p_k^\alpha}</script>
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where $\alpha$ is a hyper-parameter that determines how much
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prioritization is used, with $\alpha = 0$ corresponding to uniform case.
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$p_i$ is the priority.</p>
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<p>We use proportional prioritization $p_i = |\delta_i| + \epsilon$ where
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$\delta_i$ is the temporal difference for transition $i$.</p>
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<p>We correct the bias introduced by prioritized replay using
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importance-sampling (IS) weights
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<script type="math/tex; mode=display">w_i = \bigg(\frac{1}{N} \frac{1}{P(i)}\bigg)^\beta</script> in the loss function.
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This fully compensates when $\beta = 1$.
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We normalize weights by $\frac{1}{\max_i w_i}$ for stability.
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Unbiased nature is most important towards the convergence at end of training.
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Therefore we increase $\beta$ towards end of training.</p>
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<h3>Binary Segment Tree</h3>
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<p>We use a binary segment tree to efficiently calculate
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$\sum_k^i p_k^\alpha$, the cumulative probability,
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which is needed to sample.
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We also use a binary segment tree to find $\min p_i^\alpha$,
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which is needed for $\frac{1}{\max_i w_i}$.
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We can also use a min-heap for this.
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Binary Segment Tree lets us calculate these in $\mathcal{O}(\log n)$
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time, which is way more efficient that the naive $\mathcal{O}(n)$
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approach.</p>
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<p>This is how a binary segment tree works for sum;
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it is similar for minimum.
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Let $x_i$ be the list of $N$ values we want to represent.
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Let $b_{i,j}$ be the $j^{\mathop{th}}$ node of the $i^{\mathop{th}}$ row
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in the binary tree.
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That is two children of node $b_{i,j}$ are $b_{i+1,2j}$ and $b_{i+1,2j + 1}$.</p>
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<p>The leaf nodes on row $D = \left\lceil {1 + \log_2 N} \right\rceil$
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will have values of $x$.
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Every node keeps the sum of the two child nodes.
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That is, the root node keeps the sum of the entire array of values.
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The left and right children of the root node keep
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the sum of the first half of the array and
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the sum of the second half of the array, respectively.
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And so on…</p>
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<p>
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<script type="math/tex; mode=display">b_{i,j} = \sum_{k = (j -1) * 2^{D - i} + 1}^{j * 2^{D - i}} x_k</script>
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</p>
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<p>Number of nodes in row $i$,
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<script type="math/tex; mode=display">N_i = \left\lceil{\frac{N}{D - i + 1}} \right\rceil</script>
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This is equal to the sum of nodes in all rows above $i$.
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So we can use a single array $a$ to store the tree, where,
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<script type="math/tex; mode=display">b_{i,j} \rightarrow a_{N_i + j}</script>
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</p>
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<p>Then child nodes of $a_i$ are $a_{2i}$ and $a_{2i + 1}$.
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That is,
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<script type="math/tex; mode=display">a_i = a_{2i} + a_{2i + 1}</script>
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</p>
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<p>This way of maintaining binary trees is very easy to program.
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<em>Note that we are indexing starting from 1</em>.</p>
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<p>We use the same structure to compute the minimum.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">18</span><span class="k">class</span> <span class="nc">ReplayBuffer</span><span class="p">:</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<h3>Initialize</h3>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">88</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">capacity</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-3'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-3'>#</a>
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</div>
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<p>We use a power of $2$ for capacity because it simplifies the code and debugging</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">93</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">=</span> <span class="n">capacity</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<p>$\alpha$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">95</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-5'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-5'>#</a>
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</div>
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<p>Maintain segment binary trees to take sum and find minimum over a range</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">98</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">)]</span>
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<span class="lineno">99</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">)]</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-6'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-6'>#</a>
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</div>
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<p>Current max priority, $p$, to be assigned to new transitions</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">102</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_priority</span> <span class="o">=</span> <span class="mf">1.</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-7'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-7'>#</a>
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</div>
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<p>Arrays for buffer</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">105</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="p">{</span>
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<span class="lineno">106</span> <span class="s1">'obs'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">capacity</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">84</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint8</span><span class="p">),</span>
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<span class="lineno">107</span> <span class="s1">'action'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">capacity</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">),</span>
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<span class="lineno">108</span> <span class="s1">'reward'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">capacity</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span>
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<span class="lineno">109</span> <span class="s1">'next_obs'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">capacity</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">84</span><span class="p">,</span> <span class="mi">84</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">uint8</span><span class="p">),</span>
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<span class="lineno">110</span> <span class="s1">'done'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">capacity</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">bool</span><span class="p">)</span>
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<span class="lineno">111</span> <span class="p">}</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-8'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-8'>#</a>
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</div>
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<p>We use cyclic buffers to store data, and <code>next_idx</code> keeps the index of the next empty
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slot</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">114</span> <span class="bp">self</span><span class="o">.</span><span class="n">next_idx</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-9'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-9'>#</a>
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</div>
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<p>Size of the buffer</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">117</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-10'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-10'>#</a>
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</div>
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<h3>Add sample to queue</h3>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">119</span> <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obs</span><span class="p">,</span> <span class="n">action</span><span class="p">,</span> <span class="n">reward</span><span class="p">,</span> <span class="n">next_obs</span><span class="p">,</span> <span class="n">done</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-11'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-11'>#</a>
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</div>
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<p>Get next available slot</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">125</span> <span class="n">idx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">next_idx</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-12'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-12'>#</a>
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</div>
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<p>store in the queue</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">128</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="s1">'obs'</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">obs</span>
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<span class="lineno">129</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="s1">'action'</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">action</span>
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<span class="lineno">130</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="s1">'reward'</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">reward</span>
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<span class="lineno">131</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="s1">'next_obs'</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">next_obs</span>
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<span class="lineno">132</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="s1">'done'</span><span class="p">][</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">done</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-13'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-13'>#</a>
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</div>
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<p>Increment next available slot</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">135</span> <span class="bp">self</span><span class="o">.</span><span class="n">next_idx</span> <span class="o">=</span> <span class="p">(</span><span class="n">idx</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span></pre></div>
|
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</div>
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</div>
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<div class='section' id='section-14'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-14'>#</a>
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</div>
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<p>Calculate the size</p>
|
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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">137</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
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</div>
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</div>
|
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<div class='section' id='section-15'>
|
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<div class='docs'>
|
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<div class='section-link'>
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<a href='#section-15'>#</a>
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</div>
|
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<p>$p_i^\alpha$, new samples get <code>max_priority</code></p>
|
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</div>
|
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<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">140</span> <span class="n">priority_alpha</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_priority</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-16'>
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<div class='docs'>
|
|
<div class='section-link'>
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<a href='#section-16'>#</a>
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</div>
|
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<p>Update the two segment trees for sum and minimum</p>
|
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</div>
|
|
<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">142</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_priority_min</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">priority_alpha</span><span class="p">)</span>
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<span class="lineno">143</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_priority_sum</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">priority_alpha</span><span class="p">)</span></pre></div>
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</div>
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</div>
|
|
<div class='section' id='section-17'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-17'>#</a>
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|
</div>
|
|
<h4>Set priority in binary segment tree for minimum</h4>
|
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</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">145</span> <span class="k">def</span> <span class="nf">_set_priority_min</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">,</span> <span class="n">priority_alpha</span><span class="p">):</span></pre></div>
|
|
</div>
|
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</div>
|
|
<div class='section' id='section-18'>
|
|
<div class='docs'>
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<div class='section-link'>
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|
<a href='#section-18'>#</a>
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</div>
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<p>Leaf of the binary tree</p>
|
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">151</span> <span class="n">idx</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>
|
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<span class="lineno">152</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">priority_alpha</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-19'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-19'>#</a>
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|
</div>
|
|
<p>Update tree, by traversing along ancestors.
|
|
Continue until the root of the tree.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">156</span> <span class="k">while</span> <span class="n">idx</span> <span class="o">>=</span> <span class="mi">2</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 the index of the parent node</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">158</span> <span class="n">idx</span> <span class="o">//=</span> <span class="mi">2</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-21'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-21'>#</a>
|
|
</div>
|
|
<p>Value of the parent node is the minimum of it’s two children</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">160</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-22'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-22'>#</a>
|
|
</div>
|
|
<h4>Set priority in binary segment tree for sum</h4>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">162</span> <span class="k">def</span> <span class="nf">_set_priority_sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">,</span> <span class="n">priority</span><span class="p">):</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>Leaf of the binary tree</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">168</span> <span class="n">idx</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</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>Set the priority at the leaf</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">170</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="n">priority</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>Update tree, by traversing along ancestors.
|
|
Continue until the root of the tree.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">174</span> <span class="k">while</span> <span class="n">idx</span> <span class="o">>=</span> <span class="mi">2</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-26'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-26'>#</a>
|
|
</div>
|
|
<p>Get the index of the parent node</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">176</span> <span class="n">idx</span> <span class="o">//=</span> <span class="mi">2</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-27'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-27'>#</a>
|
|
</div>
|
|
<p>Value of the parent node is the sum of it’s two children</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">178</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-28'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-28'>#</a>
|
|
</div>
|
|
<h4>$\sum_k p_k^\alpha$</h4>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">180</span> <span class="k">def</span> <span class="nf">_sum</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-29'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-29'>#</a>
|
|
</div>
|
|
<p>The root node keeps the sum of all values</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">186</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-30'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-30'>#</a>
|
|
</div>
|
|
<h4>$\min_k p_k^\alpha$</h4>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">188</span> <span class="k">def</span> <span class="nf">_min</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-31'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-31'>#</a>
|
|
</div>
|
|
<p>The root node keeps the minimum of all values</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">194</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_min</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-32'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-32'>#</a>
|
|
</div>
|
|
<h4>Find largest $i$ such that $\sum_{k=1}^{i} p_k^\alpha \le P$</h4>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">196</span> <span class="k">def</span> <span class="nf">find_prefix_sum_idx</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prefix_sum</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-33'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-33'>#</a>
|
|
</div>
|
|
<p>Start from the root</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">202</span> <span class="n">idx</span> <span class="o">=</span> <span class="mi">1</span>
|
|
<span class="lineno">203</span> <span class="k">while</span> <span class="n">idx</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-34'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-34'>#</a>
|
|
</div>
|
|
<p>If the sum of the left branch is higher than required sum</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">205</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="n">idx</span> <span class="o">*</span> <span class="mi">2</span><span class="p">]</span> <span class="o">></span> <span class="n">prefix_sum</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-35'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-35'>#</a>
|
|
</div>
|
|
<p>Go to left branch of the tree</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">207</span> <span class="n">idx</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span>
|
|
<span class="lineno">208</span> <span class="k">else</span><span class="p">:</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-36'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-36'>#</a>
|
|
</div>
|
|
<p>Otherwise go to right branch and reduce the sum of left
|
|
branch from required sum</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">211</span> <span class="n">prefix_sum</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="n">idx</span> <span class="o">*</span> <span class="mi">2</span><span class="p">]</span>
|
|
<span class="lineno">212</span> <span class="n">idx</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">idx</span> <span class="o">+</span> <span class="mi">1</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-37'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-37'>#</a>
|
|
</div>
|
|
<p>We are at the leaf node. Subtract the capacity by the index in the tree
|
|
to get the index of actual value</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">216</span> <span class="k">return</span> <span class="n">idx</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-38'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-38'>#</a>
|
|
</div>
|
|
<h3>Sample from buffer</h3>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">218</span> <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-39'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-39'>#</a>
|
|
</div>
|
|
<p>Initialize samples</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">224</span> <span class="n">samples</span> <span class="o">=</span> <span class="p">{</span>
|
|
<span class="lineno">225</span> <span class="s1">'weights'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">),</span>
|
|
<span class="lineno">226</span> <span class="s1">'indexes'</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">shape</span><span class="o">=</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span>
|
|
<span class="lineno">227</span> <span class="p">}</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-40'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-40'>#</a>
|
|
</div>
|
|
<p>Get sample indexes</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">230</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">batch_size</span><span class="p">):</span>
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<span class="lineno">231</span> <span class="n">p</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sum</span><span class="p">()</span>
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<span class="lineno">232</span> <span class="n">idx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find_prefix_sum_idx</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
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<span class="lineno">233</span> <span class="n">samples</span><span class="p">[</span><span class="s1">'indexes'</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">idx</span></pre></div>
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</div>
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<div class='section' id='section-41'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-41'>#</a>
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<p>$\min_i P(i) = \frac{\min_i p_i^\alpha}{\sum_k p_k^\alpha}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">236</span> <span class="n">prob_min</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_min</span><span class="p">()</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sum</span><span class="p">()</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-42'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-42'>#</a>
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</div>
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<p>$\max_i w_i = \bigg(\frac{1}{N} \frac{1}{\min_i P(i)}\bigg)^\beta$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">238</span> <span class="n">max_weight</span> <span class="o">=</span> <span class="p">(</span><span class="n">prob_min</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span>
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<span class="lineno">239</span>
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<span class="lineno">240</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">batch_size</span><span class="p">):</span>
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<span class="lineno">241</span> <span class="n">idx</span> <span class="o">=</span> <span class="n">samples</span><span class="p">[</span><span class="s1">'indexes'</span><span class="p">][</span><span class="n">i</span><span class="p">]</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-43'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-43'>#</a>
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</div>
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<p>$P(i) = \frac{p_i^\alpha}{\sum_k p_k^\alpha}$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">243</span> <span class="n">prob</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">priority_sum</span><span class="p">[</span><span class="n">idx</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">]</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sum</span><span class="p">()</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-44'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-44'>#</a>
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</div>
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<p>$w_i = \bigg(\frac{1}{N} \frac{1}{P(i)}\bigg)^\beta$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">245</span> <span class="n">weight</span> <span class="o">=</span> <span class="p">(</span><span class="n">prob</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-45'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-45'>#</a>
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</div>
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<p>Normalize by $\frac{1}{\max_i w_i}$,
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which also cancels off the $\frac{1}{N}$ term</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">248</span> <span class="n">samples</span><span class="p">[</span><span class="s1">'weights'</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">weight</span> <span class="o">/</span> <span class="n">max_weight</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-46'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-46'>#</a>
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</div>
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<p>Get samples data</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">251</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
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<span class="lineno">252</span> <span class="n">samples</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="n">samples</span><span class="p">[</span><span class="s1">'indexes'</span><span class="p">]]</span>
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<span class="lineno">253</span>
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<span class="lineno">254</span> <span class="k">return</span> <span class="n">samples</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-47'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-47'>#</a>
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</div>
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<h3>Update priorities</h3>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">256</span> <span class="k">def</span> <span class="nf">update_priorities</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indexes</span><span class="p">,</span> <span class="n">priorities</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-48'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-48'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">261</span> <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">priority</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">indexes</span><span class="p">,</span> <span class="n">priorities</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-49'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-49'>#</a>
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</div>
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<p>Set current max priority</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">263</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_priority</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">max_priority</span><span class="p">,</span> <span class="n">priority</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-50'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-50'>#</a>
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</div>
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<p>Calculate $p_i^\alpha$</p>
|
|
</div>
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<div class='code'>
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|
<div class="highlight"><pre><span class="lineno">266</span> <span class="n">priority_alpha</span> <span class="o">=</span> <span class="n">priority</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-51'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-51'>#</a>
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</div>
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<p>Update the trees</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">268</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_priority_min</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">priority_alpha</span><span class="p">)</span>
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<span class="lineno">269</span> <span class="bp">self</span><span class="o">.</span><span class="n">_set_priority_sum</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">priority_alpha</span><span class="p">)</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-52'>
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|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-52'>#</a>
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</div>
|
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<h3>Whether the buffer is full</h3>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">271</span> <span class="k">def</span> <span class="nf">is_full</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-53'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-53'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">275</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span></pre></div>
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