Title: Interactive Learning of Single-Index Models via Stochastic Gradient Descent
Primary area: learning theory
Keywords: single-index model, stochastic gradient descent, nonlinear bandit

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 \emph{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.

TL;DR: We show that SGD, applied to single-index models (ridge bandits), naturally transitions from a burn-in to a learning phase. With the right step-size schedule, SGD achieves near-optimal regret across both phases for a wide range of link functions.