Instance-dependent Stochastic Lipschitz bandit

Adaptive learning needs both a state that preserves what observations imply and opportunities to act on that state. We study this width--depth tradeoff in stochastic Lipschitz bandits. After each pull, the learner retains at most $W$ bits of live reward-dependent state and organizes its pulls into at most $B$ committed batches. For $W\gtrsim_d\log(eT)$, we characterize minimax expected pseudo-regret up to logarithmic factors; the lower bounds hold for every $W$. Besides the classical sequential and unrestricted-memory batch costs, the frontier contains the new penalty $$T^{\frac{d+2}{d+3}} \bigl(1+(B-1)W\bigr)^{-\frac1{d(d+3)}},$$ proving that state width and update depth are not interchangeable. The interaction is an information-routing constraint: at regional scale $s$, low regret forces the committed action transcript to encode $Θ_d(s^{-d})$ regional decisions, while the collected boundary states carry at most $(B-1)W$ bits of entropy. Matching policies stream and erase verification statistics while retaining a mask of a safe active set, either in memory or fragment by fragment. The theorem recovers the full-dimensional worst-case batch-only frontier and logarithmic-memory achievability in the fully sequential specialization; static batch boundaries match predictable adaptive ones.

We study the $\textit{single-index bandit}$ problem, where rewards depend on an unknown one-dimensional projection of high-dimensional contexts through an unknown reward function. This model extends linear and generalized linear bandits to a nonparametric setting, and is particularly relevant when the reward function is not known in advance. While optimal regret guarantees are known for monotone reward functions, the general non-monotone case remains poorly understood, with the best known bound being $\tilde{\mathcal{O}}(T^{3/4})$ (under standard boundedness and Lipschitz assumptions on the reward function [Kang et al., 2025]). We close this gap by establishing the optimal regret for general single-index bandits. We propose a simple two-phase algorithm, namely, Zoomed Single Index Bandit with Upper Confidence Bound ($\texttt{ZoomSIB-UCB}$), that first estimates the projection direction via a normalized Stein estimator, and then reduces the problem to a one-dimensional bandit using discretization and finally use UCB. This approach achieves a regret of $\tilde{\mathcal{O}}(T^{2/3})$, and improves significantly upon prior work without any additional assumptions. We also prove a matching minimax lower bound of $\tildeΩ(T^{2/3})$, showing that the upper bound is essentially tight. Our upper and lower bounds together provide a sharp characterization of the regret in single-index bandits. Moreover, the empirical results further demonstrate the effectiveness and robustness of our approach.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.