Online Budget Allocation with Censored Semi-Bandit Feedback

TL;DR

This paper introduces an online budget allocation algorithm for tasks with censored semi-bandit feedback, achieving near-optimal regret bounds in diminishing-returns regimes and a worst-case bound of  O(K\u221a T) for general cases.

Contribution

It proposes an optimism-based algorithm for censored semi-bandit feedback in budget allocation, with regret bounds that are polylogarithmic in T for diminishing returns and optimal  O(K T) in the worst case.

Findings

Regret scales polylogarithmically with T in diminishing-returns regimes.

Achieves worst-case regret of  O(K T) for general nondecreasing curves.

Establishes a matching lower bound of   K T, even for full-feedback algorithms.

Abstract

We study a stochastic budget-allocation problem over $K$ tasks. At each round $t$, the learner chooses an allocation $X_t \in \Delta_K$. Task $k$ succeeds with probability $F_k(X_{t,k})$, where $F_1,\dots,F_K$ are nondecreasing budget-to-success curves, and upon success yields a random reward with unknown mean $\mu_k$. The learner observes which tasks succeed, and observes a task's reward only upon success (censored semi-bandit feedback). This model captures, for instance, splitting payments across crowdsourcing workers or distributing bids across simultaneous auctions, and subsumes stochastic multi-armed bandits and semi-bandits. We design an optimism-based algorithm that operates under censored semi-bandit feedback. Our main result shows that in diminishing-returns regimes, the regret of this algorithm scales polylogarithmically with the horizon $T$ without any ad hoc tuning. For general nondecreasing curves, we prove that the same algorithm (with the same tuning) achieves a worst-case regret upper bound of $\tilde O(K\sqrt{T})$. Finally, we establish a matching worst-case regret lower bound of $\Omega(K\sqrt{T})$ that holds even for full-feedback algorithms, highlighting the intrinsic hardness of our problem outside diminishing returns.