A Regularized Online Newton Method for Stochastic Convex Bandits with Linear Vanishing Noise

TL;DR

This paper introduces a Regularized Online Newton Method for stochastic convex bandits with linearly decreasing noise, achieving polylogarithmic regret and extending to new models with scaled and multiplicative noise.

Contribution

It proposes a novel RONM algorithm for convex bandits with decreasing noise, improving regret bounds and analyzing new noise-scaled bandit models.

Findings

Achieves polylogarithmic regret in quadratic growth scenarios.

Linear growth of the precision matrix is optimal for analysis.

Provides better convergence rates for faster-growing loss functions.

Abstract

We study a stochastic convex bandit problem where the subgaussian noise parameter is assumed to decrease linearly as the learner selects actions closer and closer to the minimizer of the convex loss function. Accordingly, we propose a Regularized Online Newton Method (RONM) for solving the problem, based on the Online Newton Method (ONM) of arXiv:2406.06506. Our RONM reaches a polylogarithmic regret in the time horizon $n$ when the loss function grows quadratically in the constraint set, which recovers the results of arXiv:2402.12042 in linear bandits. Our analyses rely on the growth rate of the precision matrix $\Sigma_t^{-1}$ in ONM and we find that linear growth solves the question exactly. These analyses also help us obtain better convergence rates when the loss function grows faster. We also study and analyze two new bandit models: stochastic convex bandits with noise scaled to a subgaussian parameter function and convex bandits with stochastic multiplicative noise.