Interactive Learning of Single-Index Models via Stochastic Gradient Descent

TL;DR

This paper analyzes the learning dynamics of stochastic gradient descent (SGD) in high-dimensional single-index models, revealing a two-phase process and demonstrating near-optimal sample complexity and regret guarantees with proper learning rate schedules.

Contribution

It provides the first detailed analysis of SGD's behavior in sequential learning of single-index models, including phase transition and optimality guarantees.

Findings

SGD exhibits a burn-in and learning phase in single-index models.

Proper learning rate schedules enable near-optimal sample complexity.

SGD achieves competitive regret bounds in adaptive data settings.

Abstract

Stochastic gradient descent (SGD) is a cornerstone algorithm for high-dimensional optimization, renowned for its empirical successes. Recent theoretical advances have provided a deep understanding of how SGD enables feature learning in high-dimensional nonlinear models, most notably the \textit{single-index model} with i.i.d. data. In this work, we study the sequential learning problem for single-index models, also known as generalized linear bandits or ridge bandits, where SGD is a simple and natural solution, yet its learning dynamics remain largely unexplored. We show that, similar to the optimal interactive learner, SGD undergoes a distinct ``burn-in'' phase before entering the ``learning'' phase in this setting. Moreover, with an appropriately chosen learning rate schedule, a single SGD procedure simultaneously achieves near-optimal (or best-known) sample complexity and regret guarantees across both phases, for a broad class of link functions. Our results demonstrate that SGD remains highly competitive for learning single-index models under adaptive data.