Probably Anytime-Safe Stochastic Combinatorial Semi-Bandits

TL;DR

This paper introduces a new stochastic combinatorial semi-bandit problem with a safety constraint on cumulative variance, proposing an optimal algorithm that balances reward maximization and risk control over time.

Contribution

It formulates the probably anytime-safe constraint, develops the PASCombUCB algorithm, and provides theoretical analysis showing near-optimal regret bounds.

Findings

PASCombUCB achieves near-optimal regret bounds.

The safety constraint effectively controls risk over the horizon.

Experimental results support theoretical claims.

Abstract

Motivated by concerns about making online decisions that incur undue amount of risk at each time step, in this paper, we formulate the probably anytime-safe stochastic combinatorial semi-bandits problem. In this problem, the agent is given the option to select a subset of size at most $K$ from a set of $L$ ground items. Each item is associated to a certain mean reward as well as a variance that represents its risk. To mitigate the risk that the agent incurs, we require that with probability at least $1-\delta$, over the entire horizon of time $T$, each of the choices that the agent makes should contain items whose sum of variances does not exceed a certain variance budget. We call this probably anytime-safe constraint. Under this constraint, we design and analyze an algorithm {\sc PASCombUCB} that minimizes the regret over the horizon of time $T$. By developing accompanying information-theoretic lower bounds, we show that under both the problem-dependent and problem-independent paradigms, {\sc PASCombUCB} is almost asymptotically optimal. Experiments are conducted to corroborate our theoretical findings. Our problem setup, the proposed {\sc PASCombUCB} algorithm, and novel analyses are applicable to domains such as recommendation systems and transportation in which an agent is allowed to choose multiple items at a single time step and wishes to control the risk over the whole time horizon.