Risk-aware linear bandits with convex loss

TL;DR

This paper introduces a new framework for risk-aware contextual bandits using convex loss functions, deriving confidence sequences and proposing algorithms with regret guarantees for critical applications.

Contribution

It extends risk-aware bandit algorithms to contextual settings with convex loss functions, providing confidence sequences and practical algorithms with theoretical guarantees.

Findings

Proposed a convex loss-based risk measure estimation method.

Developed an optimistic UCB algorithm with regret guarantees.

Validated algorithms through numerical experiments.

Abstract

In decision-making problems such as the multi-armed bandit, an agent learns sequentially by optimizing a certain feedback. While the mean reward criterion has been extensively studied, other measures that reflect an aversion to adverse outcomes, such as mean-variance or conditional value-at-risk (CVaR), can be of interest for critical applications (healthcare, agriculture). Algorithms have been proposed for such risk-aware measures under bandit feedback without contextual information. In this work, we study contextual bandits where such risk measures can be elicited as linear functions of the contexts through the minimization of a convex loss. A typical example that fits within this framework is the expectile measure, which is obtained as the solution of an asymmetric least-square problem. Using the method of mixtures for supermartingales, we derive confidence sequences for the estimation of such risk measures. We then propose an optimistic UCB algorithm to learn optimal risk-aware actions, with regret guarantees similar to those of generalized linear bandits. This approach requires solving a convex problem at each round of the algorithm, which we can relax by allowing only approximated solution obtained by online gradient descent, at the cost of slightly higher regret. We conclude by evaluating the resulting algorithms on numerical experiments.