Custom Gradient Estimators are Straight-Through Estimators in Disguise

TL;DR

This paper demonstrates that many gradient estimators used in quantization-aware training are essentially equivalent to straight-through estimators, revealing a fundamental connection and unifying perspective.

Contribution

It proves the equivalence of various gradient estimators to STE under certain conditions and shows this holds for different optimizers and models.

Findings

Gradient estimators are equivalent to STE with small learning rates.

The equivalence holds for adaptive optimizers like Adam.

Experimental results on MNIST and ImageNet support the theory.

Abstract

Quantization-aware training comes with a fundamental challenge: the derivative of quantization functions such as rounding are zero almost everywhere and nonexistent elsewhere. Various differentiable approximations of quantization functions have been proposed to address this issue. In this paper, we prove that when the learning rate is sufficiently small, a large class of weight gradient estimators is equivalent with the straight through estimator (STE). Specifically, after swapping in the STE and adjusting both the weight initialization and the learning rate in SGD, the model will train in almost exactly the same way as it did with the original gradient estimator. Moreover, we show that for adaptive learning rate algorithms like Adam, the same result can be seen without any modifications to the weight initialization and learning rate. We experimentally show that these results hold for both a small convolutional model trained on the MNIST dataset and for a ResNet50 model trained on ImageNet.