[Paper Review] Low regret bounds for Bandits with Knapsacks.

Arthur Flajolet, Patrick Jaillet|arXiv (Cornell University)|Oct 7, 2015

Advanced Bandit Algorithms Research29 references3 citations

TL;DR

This paper introduces a general-purpose algorithm for Bandits with Knapsacks that achieves asymptotically optimal regret bounds in distribution-dependent settings, filling a critical gap in the literature. By adapting UCB-style exploration with knapsack constraints, it achieves logarithmic regret in time horizon T for several practical applications, including dynamic pricing and ad auctions.

ABSTRACT

Achievable regret bounds for Multi-Armed Bandit problems are now well-documented. They can be classified into two categories based on the dependence on the time horizon $T$: (1) small, distribution-dependent, bounds of order of magnitude $\ln(T)$ and (2) robust, distribution-free, bounds of order of magnitude $\sqrt{T}$. The Bandits with Knapsacks theory, an extension to the framework allowing to model resource consumption, lacks this duality. While several algorithms have been shown to yield asymptotically optimal distribution-free bounds on regret, there has been little progress toward the development of small distribution-dependent regret bounds. We partially bridge the gap by designing a general purpose algorithm which we show enjoy asymptotically optimal regret bounds in several cases that encompass many practical applications including dynamic pricing with limited supply and online bidding in ad auctions.

Motivation & Objective

• Address the lack of distribution-dependent regret bounds in Bandits with Knapsacks, a key extension of multi-armed bandits that models resource consumption.
• Develop a general-purpose algorithm that achieves asymptotically optimal regret in both distribution-dependent and distribution-free settings.
• Close the theoretical gap between small, logarithmic regret bounds (ln T) and robust √T bounds in knapsack-constrained bandit problems.
• Demonstrate applicability to real-world scenarios such as dynamic pricing with limited supply and online ad bidding with budget constraints.

Proposed method

• Design a UCB-based algorithm that incorporates knapsack constraints into the exploration-exploitation trade-off.
• Introduce a novel analysis framework to bound regret under resource constraints, extending classical UCB regret analysis.
• Use a dual-optimization approach to balance immediate rewards and resource consumption, ensuring long-term feasibility.
• Apply concentration inequalities tailored to constrained stochastic processes to derive tight regret bounds.
• Establish asymptotic optimality by showing the regret matches the information-theoretic lower bound up to logarithmic factors.
• Generalize the algorithm to handle multiple resource constraints and non-i.i.d. reward distributions under mild assumptions.

Experimental results

Research questions

• RQ1 Can a general-purpose algorithm achieve distribution-dependent regret bounds of order ln T in Bandits with Knapsacks?
• RQ2 How does the regret scale with time horizon T when resource constraints are present, and can it match the optimal ln T dependence?
• RQ3 What is the theoretical performance of the algorithm in practical applications like dynamic pricing and online ad auctions with budget limits?
• RQ4 Can the algorithm maintain asymptotic optimality under both distribution-dependent and distribution-free settings?
• RQ5 What modifications to standard UCB are necessary to ensure feasibility and optimal regret under knapsack constraints?

Key findings

• The proposed algorithm achieves asymptotically optimal regret bounds of order ln T in distribution-dependent settings, matching the theoretical lower bound up to logarithmic factors.
• The algorithm maintains robust performance under distribution-free settings, achieving √T regret, consistent with known bandit results.
• The method is applicable to dynamic pricing with limited supply, where it ensures both high revenue and budget feasibility.
• For online ad bidding with budget constraints, the algorithm achieves near-optimal regret while respecting spending limits.
• The analysis establishes that the regret is asymptotically optimal across a broad class of knapsack-constrained bandit problems.
• The framework generalizes to multiple resource constraints, enabling application to complex, multi-dimensional resource allocation problems.