A Regularized Online Newton Method for Stochastic Convex Bandits with Linear Vanishing Noise
This work addresses bandit optimization for scenarios with decreasing noise, offering incremental improvements in regret bounds for specific convex loss functions.
The authors tackled the stochastic convex bandit problem with linearly vanishing noise by proposing a Regularized Online Newton Method (RONM), achieving polylogarithmic regret in the time horizon when the loss function is quadratic, which matches prior results in linear bandits.
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 $Σ_t^{-1}$ 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.