Model Evolution Under Zeroth-Order Optimization: A Neural Tangent Kernel Perspective
This work addresses a theoretical gap for researchers in optimization and machine learning, offering insights into zeroth-order methods, though it is incremental as it builds on existing NTK theory.
The paper tackled the challenge of understanding training dynamics in zeroth-order optimization for neural networks, which lacks clear theoretical characterization, by introducing the Neural Zeroth-order Kernel (NZK) to model evolution in function space. It proved invariance of the expected NZK for linear models, providing closed-form expressions, and demonstrated acceleration in experiments on datasets like MNIST and CIFAR-10 with a single shared random vector.
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.