Statistical Complexity and Optimal Algorithms for Non-linear Ridge Bandits

TL;DR

This paper investigates non-linear bandit problems, revealing unique phenomena like a burn-in period and proposing optimal algorithms for ridge functions, with theoretical bounds and practical strategies.

Contribution

It introduces the concept of burn-in costs in non-linear bandits, derives bounds for ridge functions, and proposes a two-stage optimal algorithm.

Findings

Two-stage algorithm achieves optimal burn-in cost.

Classical algorithms like UCB are suboptimal in this setting.

Differential equations characterize the learning trajectory.

Abstract

We consider the sequential decision-making problem where the mean outcome is a non-linear function of the chosen action. Compared with the linear model, two curious phenomena arise in non-linear models: first, in addition to the "learning phase" with a standard parametric rate for estimation or regret, there is an "burn-in period" with a fixed cost determined by the non-linear function; second, achieving the smallest burn-in cost requires new exploration algorithms. For a special family of non-linear functions named ridge functions in the literature, we derive upper and lower bounds on the optimal burn-in cost, and in addition, on the entire learning trajectory during the burn-in period via differential equations. In particular, a two-stage algorithm that first finds a good initial action and then treats the problem as locally linear is statistically optimal. In contrast, several classical algorithms, such as UCB and algorithms relying on regression oracles, are provably suboptimal.