Adafactor

Author: Aolei Cao (ac3237), Ziyang Li (zl986), Junjia Liang (jl4439) (ChemE 6800 Fall 2024)

Stewards: Nathan Preuss, Wei-Han Chen, Tianqi Xiao, Guoqing Hu

Problem formulation

1. Objective

Minimize the loss function $f(x)$, where $x\in R^{n}$ and $x$ is the weight vector to be optimized.

2. Parameters

• Gradient:

$G_{t}=\nabla f(x_{t-1})$

• Second moment estimate:

${\hat {V}}_{t}={\hat {\beta }}_{2t}{\hat {V}}_{t-1}+(1-{\hat {\beta }}_{2t})(G_{t}^{2}+\epsilon _{1}1_{n})$

• Where:
  • ${\hat {V}}_{t}$ is the running average of the squared gradient.
  • ${\hat {\beta }}_{2t}$ is the corrected decay parameter.
  • $\epsilon _{1}$ is a regularization constant.

• Step size:

$\alpha _{t}=\max(\epsilon _{2},{\text{RMS}}(x_{t-1}))\rho _{t}$

• Where:
  • $\rho _{t}$ is the relative step size.
  • $\epsilon _{2}$ is a regularization constant.
  • ${\text{RMS}}$ is the root mean square, defined as:
    • $u_{xt}={\frac {-g_{xt}}{\sqrt {{\hat {v}}_{xt}}}}$
    • ${\text{RMS}}(U_{t})={\text{RMS}}_{x\in X}(u_{xt})={\sqrt {{\text{Mean}}_{x\in X}\left({\frac {(g_{xt})^{2}}{{\hat {v}}_{xt}}}\right)}}$

3. Algorithms

Adafactor for Weighted Vectors

Inputs:

• Initial point: $X_{0}\in \mathbb {R} ^{n}$
• Relative step sizes: $\rho _{t}$ for $t=1$ to $T$
• Second moment decay: ${\hat {\beta }}_{2t}$ for $t=1$ to $T$, with ${\hat {\beta }}_{21}=0$
• Regularization constants: $\epsilon _{1},\epsilon _{2}$
• Clipping threshold: $d$

Algorithm:

• For $t=1$ to $T$:
  • Compute adaptive step size: $\alpha _{t}=\max(\epsilon _{2},{\text{RMS}}(X_{t-1}))\rho _{t}$
  • Compute gradient: $G_{t}=\nabla f_{t}(X_{t-1})$
  • Update second moment estimate: ${\hat {V}}_{t}={\hat {\beta }}_{2t}{\hat {V}}_{t-1}+(1-{\hat {\beta }}_{2t})(G_{t}^{2}+\epsilon _{1}1_{n})$
  • Compute normalized gradient: $U_{t}={\frac {G_{t}}{\sqrt {{\hat {V}}_{t}}}}$
  • Apply clipping: ${\hat {U}}_{t}={\frac {U_{t}}{\max(1,{\text{RMS}}(U_{t})/d)}}$
  • Update parameter: $X_{t}=X_{t-1}-\alpha _{t}{\hat {U}}_{t}$
• End for

Adafactor for Weighted Matrices

Inputs:

• Initial point: $X_{0}\in \mathbb {R} ^{n\times m}$
• Relative step sizes: $\rho _{t}$ for $t=1$ to $T$
• Second moment decay: ${\hat {\beta }}_{2t}$ for $t=1$ to $T$, with ${\hat {\beta }}_{21}=0$
• Regularization constants: $\epsilon _{1},\epsilon _{2}$
• Clipping threshold: $d$

Algorithm:

• For $t=1$ to $T$:
  • Compute adaptive step size: $\alpha _{t}=\max(\epsilon _{2},{\text{RMS}}(X_{t-1}))\rho _{t}$
  • Compute gradient: $G_{t}=\nabla f_{t}(X_{t-1})$
  • Update row-wise second moment: $R_{t}={\hat {\beta }}_{2t}R_{t-1}+(1-{\hat {\beta }}_{2t})(G_{t}^{2}+\epsilon _{1}1_{n}1_{m}^{T})1_{m}$
  • Update column-wise second moment: $C_{t}={\hat {\beta }}_{2t}C_{t-1}+(1-{\hat {\beta }}_{2t})1_{n}^{T}(G_{t}^{2}+\epsilon _{1}1_{n}1_{m}^{T})$
  • Update overall second moment estimate: ${\hat {V}}_{t}={\frac {R_{t}C_{t}}{1_{n}^{T}R_{t}}}$
  • Compute normalized gradient: $U_{t}={\frac {G_{t}}{\sqrt {{\hat {V}}_{t}}}}$
  • Apply clipping: ${\hat {U}}_{t}={\frac {U_{t}}{\max(1,{\text{RMS}}(U_{t})/d)}}$
  • Update parameter: $X_{t}=X_{t-1}-\alpha _{t}{\hat {U}}_{t}$
• End for

Why Clipping

Adafactor employs clipping to maintain numerical stability, especially since it is designed for use with very large models and often works with unscaled learning rates.

• Clipping prevents the update step from becoming very large, which would destabilize training
• Clipping mitigates the effects of very large gradients preventing numerical instability

Therefore, implementing clipping helps ensure stability and efficient training without requiring per-parameter scaling like Adam.

Why Adafactor is more memory efficient, compared to Adam

Row-wise and Column-wise Second Moment Updates

• $R_{t}={\hat {\beta }}_{2t}R_{t-1}+(1-{\hat {\beta }}_{2t})(G_{t}^{2}+\epsilon _{1}1_{n}1_{m}^{T})1_{m}$
• $C_{t}={\hat {\beta }}_{2t}C_{t-1}+(1-{\hat {\beta }}_{2t})1_{n}^{T}(G_{t}^{2}+\epsilon _{1}1_{n}1_{m}^{T})$

Instead of storing the full $G_{t}^{2}$, Adafactor computes the row and column respectively, which reduces the memory requirements from $O(n\times m)$ to $O(n+m)$

Factored Representation of the Second Moment

• ${\hat {V}}_{t}={\frac {R_{t}C_{t}}{1_{n}^{T}R_{t}}}$

This updates the second momentum based on the outer product $R_{t}C_{t}$.

• However, this is not $O(n\times m)$ since
  • The operation is performed element-wise, so it actually never materializes ${\hat {V_{t}}}$ as a $n\times n$ matrix
  • It also only storing $R_{t}$and $C_{t}$ instead of storage the full second-moment matrix

4. Proposed Hyperparameters for Adafactor

• Regularization constant 1: $\epsilon _{1}=10^{-30}$
• Ensures numerical stability by preventing division by zero in the calculation of second-moment estimates, so the numerical value should be very close to zero
• Regularization constant 2: $\epsilon _{2}=10^{-3}$
• Help to stabilize parameter updates by controlling the effect of second-moment scaling in low-magnitude scenarios. Compared to $\epsilon _{2}$, a relatively larger value ensures the stability of noise and low-magnitude scenarios.
• Clipping threshold: $d=1$
• A threshold of 1 balances stability and learning efficiency. It avoids excessive suppression of large gradients, which could hinder learning, while still protecting against extreme updates that could destabilize the model.
• Relative step size: $\rho _{t}=\min(10^{-2},1/{\sqrt {t}})$
  • $min(10^{-}2,...)$ can caps the learning rate at 10^-2, which is a empirical found for upper bound
  • ${\frac {1}{\sqrt {t}}}$ This step size promote convergence of the model. This rate ensures a balance between sufficient exploration in early iteration and stability in later iterations
• Second moment decay: ${\hat {\beta }}_{2t}=1-t^{-0.8}$
  • 1-...: ensures the decay factor remains close to 1
  • $t^{-0,8}$ the power 0.8 ensures a balance between rapid adaptation in early training and later iterations