Rectified Adam (RAdam) optimizer

This implementation is based on the official implementation of the paper On the Variance of the Adaptive Learning Rate and Beyond.

We have implemented it as an extension to our AMSGrad implementation thus requiring only the modifications to be implemented.

Adam optimizer sometimes converges to a bad local optima during the initial stages of the training; especially when training transformers. Researches use warmups to counter this; for the the initial training steps (warm-up stage) they use a low learning rate. This paper identifies the problem to be the high variance of adaptive learning rate during initial stages of training, and counters it using a new rectification term to reduce variance.

The paper also evaluates two variance reduction mechanisms: * Adam-2k: Only compute the adaptive learning rate ($v_t$ in Adam) during the first 2k steps, without changing parameters or calculating momentum ($m_t$). * Adam-eps: Adam with large $\epsilon \approx 10^{-4}$.

Rectified Adam

Let $\sigma(g_1, …, g_t)$ and $\psi(g_1, …, g_t)$ be the functions to calculate momentum and adaptive learning rate. For Adam, they are $$\begin{align} \sigma(g_1, ..., g_t) &= \frac{(1 - \beta_1)\sum_{i=1}^t \beta_1^{t-i} g_i}{1 - \beta_1^t} \\ \psi(g_1, ..., g_t) &= \sqrt \frac{1 - \beta_2^t}{(1 - \beta_2)\sum_{i=1}^t \beta_2^{t-i} g_i^2} \end{align}$$

Exponential moving average as simple moving average

The distribution of exponential moving average can be approximated as a simple moving average. $$\begin{align} p\Bigg(\frac{(1-\beta_2) \sum_{i=1}^t \beta_2^{t-i} g_i^2}{1 - \beta_2^t} \Bigg) \approx p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg) \end{align}$$ Here we are taking the simple moving average of the last $f(t,\beta_2)$ gradients. $f(t,\beta_2)$ satisfies the following, $$\begin{align} \frac{(1-\beta_2) \sum_{i=1}^t \beta_2^{t-i} \cdot i}{1 - \beta_2^t} = \frac{\sum_{i=1}^{f(t,\beta_2)} (t+1-i)}{f(t,\beta_2)} \end{align}$$ which gives, $$f(t,\beta_2) = \frac{2}{1-\beta_2} - 1 - \frac{2 t \beta_2^t}{1 - \beta_2^t}$$

Scaled inverse chi-squared

From above we have $$p\Big( \psi^2(g_1, ..., g_t) \Big) \approx p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg)$$ where $g_i \sim \mathcal{N}(0, \sigma^2)$. Note that $sigma$ here is the standard deviation and different from $\sigma(.)$ for momentum.

Scaled inverse chi-squared is the distribution of squared inverse of mean of $p$ normal distributions. $$p\Bigg(\frac{\sum_{i=1}^{f(t,\beta_2)} g_{t+1-i}^2}{f(t,\beta_2)} \Bigg) \sim \text{Scale-inv} \mathcal{X}^2(\rho,\frac{1}{\sigma^2})$$ where $\rho = f(t,\beta_2)$.

Rectification

They prove that variance of $\psi(.)$ decreases with $\rho$ when $\psi^2(.) \sim \text{Scale-inv} \mathcal{X}^2(\rho,\frac{1}{\sigma^2})$.

Therefore the variance is minimized at maximal $\rho$ which is $\rho_{\infty} = \frac{2}{1-\beta_2} - 1$. Let the minimum variance be $C_{\text{var}}$

In order to ensure that the adaptive learning rate $\psi(.)$ has consistent variance, we rectify the variance with $r$ $$\begin{align} r = \sqrt{\frac{C_{\text{var}}}{Var\big[\psi(.)\big]}} \end{align}$$

Approximating $Var[\psi(.)]$

They estimate $Var[\psi(.)] \approx \frac{Var[\psi^2(.)]}{4 \mathbb{E}[\psi^2(.)}$ based on first order expansion of $\sqrt{\psi^2(.)}$ 🤪 I didn’t get how it was derived.

From $\text{Scale-inv} \mathcal{X}^2$ distribution we have, $$\begin{align} \mathbb{E}\big[\psi^2(.)\big] &= \frac{\rho / \sigma^2}{\rho-2} \\ Var\big[\psi^2(.)\big] &= \frac{2 \rho / \sigma^4}{(\rho-2)^2 (\rho - 2)} \end{align}$$ which gives, $$Var[\psi(.)] \approx \frac{\rho}{2(\rho-2)(\rho-4)\sigma^2}$$

Rectification term

We have $$\begin{align} r &= \sqrt{\frac{C_{\text{var}}}{Var\big[\psi(.)\big]}} \\ Var[\psi(.)] &\approx \frac{\rho}{2(\rho-2)(\rho-4)\sigma^2} \end{align}$$

where $C_{\text{var}}$ is $Var\big[\psi(.)\big]$ for $\rho_\infty$. Lt $\rho$ and step $t$ be $\rho_t$, and $r_t$ be the rectification term at step $t$.

$$\begin{align} C_{\text{var}} &\approx \frac{\rho_\infty}{2(\rho_\infty-2)(\rho_\infty-4)\sigma^2} \\ Var[\psi(g_1,...,g_t)] &\approx \frac{\rho_t}{2(\rho_t-2)(\rho_t-4)\sigma^2} \end{align}$$

This gives, $$\begin{align} r_t &= \sqrt{\frac{(\rho_t-2)(\rho_t-4)\rho_\infty}{(\rho_\infty-2)(\rho_\infty-4)\rho_t}} \end{align}$$

Rectified Adam Optimizer

This class extends from AMSAdam optimizer defined in amsadam.py.

Initialize the optimizer

• params is the list of parameters
• lr is the learning rate $\alpha$
• betas is a tuple of ($\beta_1$, $\beta_2$)
• eps is $\hat{\epsilon}$ or $\epsilon$ based on optimized_update
• weight_decay is an instance of class WeightDecay defined in __init__.py
• ‘optimized_update’ is a flag whether to optimize the bias correction of the second moment by doing it after adding $\epsilon$
• amsgrad is a flag indicating whether to use AMSGrad or fallback to plain Adam
• degenerate_to_sgd whether to use sgd when the rectification term $r_t is intractable.
• defaults is a dictionary of default for group values. This is useful when you want to extend the class RAdam.

Take an update step for a given parameter tensor

• state is the optimizer state of the parameter (tensor)
• group stores optimizer attributes of the parameter group
• grad is the current gradient tensor $g_t$ for the parameter $\theta_{t-1}$
• param is the parameter tensor $\theta_{t-1}$

Calculate rectification term $r_t$

Do the RAdam parameter update

• state is the optimizer state of the parameter (tensor)
• group stores optimizer attributes of the parameter group
• param is the parameter tensor $\theta_{t-1}$
• m and v are the uncorrected first and second moments $m_t$ and $v_t$; i.e. $\sigma(.)$ and $\psi(.)$ without bias correction

Plot $r_t$ against $t$ for various $\beta_2$