Model Evolution Under Zeroth-Order Optimization: A Neural Tangent Kernel Perspective

TL;DR

This paper introduces the Neural Zeroth-order Kernel (NZK) to analyze the training dynamics of zeroth-order optimization in neural networks, providing theoretical insights and empirical validation for model evolution and convergence acceleration.

Contribution

It develops the NZK framework to characterize ZO training dynamics, extending NTK theory to zeroth-order methods and demonstrating potential for faster convergence.

Findings

Expected NZK remains constant during training for linear models

Explicit formula for model evolution under squared loss

Empirical results show acceleration with shared random vectors

Abstract

Zeroth-order (ZO) optimization enables memory-efficient training of neural networks by estimating gradients via forward passes only, eliminating the need for backpropagation. However, the stochastic nature of gradient estimation significantly obscures the training dynamics, in contrast to the well-characterized behavior of first-order methods under Neural Tangent Kernel (NTK) theory. To address this, we introduce the Neural Zeroth-order Kernel (NZK) to describe model evolution in function space under ZO updates. For linear models, we prove that the expected NZK remains constant throughout training and depends explicitly on the first and second moments of the random perturbation directions. This invariance yields a closed-form expression for model evolution under squared loss. We further extend the analysis to linearized neural networks. Interpreting ZO updates as kernel gradient descent via NZK provides a novel perspective for potentially accelerating convergence. Extensive experiments across synthetic and real-world datasets (including MNIST, CIFAR-10, and Tiny ImageNet) validate our theoretical results and demonstrate acceleration when using a single shared random vector.