No-Regret is not enough! Bandits with General Constraints through Adaptive Regret Minimization (2405.06575v1)

Abstract: In the bandits with knapsacks framework (BwK) the learner has $m$ resource-consumption (packing) constraints. We focus on the generalization of BwK in which the learner has a set of general long-term constraints. The goal of the learner is to maximize their cumulative reward, while at the same time achieving small cumulative constraints violations. In this scenario, there exist simple instances where conventional methods for BwK fail to yield sublinear violations of constraints. We show that it is possible to circumvent this issue by requiring the primal and dual algorithm to be weakly adaptive. Indeed, even in absence on any information on the Slater's parameter $\rho$ characterizing the problem, the interplay between weakly adaptive primal and dual regret minimizers yields a "self-bounding" property of dual variables. In particular, their norm remains suitably upper bounded across the entire time horizon even without explicit projection steps. By exploiting this property, we provide best-of-both-worlds guarantees for stochastic and adversarial inputs. In the first case, we show that the algorithm guarantees sublinear regret. In the latter case, we establish a tight competitive ratio of $\rho/(1+\rho)$. In both settings, constraints violations are guaranteed to be sublinear in time. Finally, this results allow us to obtain new result for the problem of contextual bandits with linear constraints, providing the first no-$\alpha$-regret guarantees for adversarial contexts.

• The paper demonstrates that weakly adaptive primal-dual strategies naturally bound dual variables and ensure sublinear regret in stochastic settings.
• It achieves competitive ratios near the theoretical optimum in adversarial scenarios, tightening performance bounds compared to traditional methods.
• The results suggest promising extensions to broader online learning challenges, notably improving dynamic resource allocation in practical applications.

Understanding Primal-Dual Regret Minimization in Bandits with Knapsacks under Weak Adaptivity

The Core Problem and Motivation

Bandits with Knapsacks (BwK) models, where a learner must balance between maximizing rewards and managing multiple resources, have proven to be challenging, especially under varying constraint types beyond simple resource consumption. The fundamental issue tackled by the highlighted paper revolves around achieving small constraint violations while securing maximum rewards, even when the system constraints change unpredictably over time. This generalization poses two primary problems: ensuring that the constraints are not overly violated and that the competitive ratio — the ratio comparing the performance of the algorithm against the optimal decision in hindsight — remains as tight as possible.

Technical Insights and Challenges

The paper introduces a notion that standard approaches to handling primal-dual algorithms under the BwK framework often fail when faced with non-standard, dynamic constraints. Traditional methods rely on static duals or presume knowledge of the environment's complexity, expressed as the "Slater’s parameter" — a metric of feasibility. However, the unpredictability in adversarial environments where constraints and rewards could significantly shift makes predefined strategies fallible.

Adaptive Strategies

To address these limitations, the researchers innovate by employing weakly adaptive primal and dual algorithms. These algorithms adjust continuously, refining their strategies within any sub-interval of the decision horizon. The key breakthrough here is demonstrating that dual variables, crucial in adjusting the balance between resource consumption and reward maximization, can remain bounded without prior knowledge of constraints' tightness. This property, termed "self-bounding," means that the dual adjustments inherently do not spiral out of control, thus maintaining an effective check on violations.

The Dual Variable Insight

In typical scenarios, bounding dual variables is necessary to prevent them from excessively penalizing the reward function. The leap made here is that even without explicitly constraining these variables (via projection), they naturally remain within a reasonable range due to the interaction between the adaptive primal and dual algorithms.

Main Contributions and Practical Implications

The results derived from employing these weakly adaptive strategies are striking. For stochastic inputs (where data follows some probability distribution), the algorithms robustly yield sublinear regrets — that is, the missed rewards diminish over time as compared to the best possible strategy determined in hindsight. For adversarial inputs, where the worst-case scenario dictates the dynamics, the proposed method achieves a competitive ratio very close to the theoretically optimal, confirming the robustness of weak adaptivity in general BwK frameworks.

1. Stochastic Inputs: The algorithm effectively approximates the best fixed strategy without the need for preliminary rounds typically used for estimating unknown parameters.
2. Adversarial Settings: Achieves competitive ratios significantly tightening the bounds on performance relative to the best unconstrained strategy, an essential feature in environments with high variability.

Future Directions

The advancements suggest several intriguing avenues for further research:

• Extending the weak adaptivity principle to other types of online learning problems where the environment's dynamics are poorly understood.
• Exploring whether more aggressive adaptivity than weak adaptivity could yield even tighter control over dual variables and further improve performance metrics.
• Applying these methods in real-world scenarios, such as dynamic resource allocation in networks or adaptive budget management in advertising campaigns, could test the practical utility and robustness of these theoretical advances.

Conclusion

By ensuring that both primal and dual elements of the learner's strategy are weakly adaptive, this work significantly enhances the capability of BwK models to handle environments with complex, long-term constraints. This is a substantial stride in making adaptive algorithms both theoretically sound and practically applicable, particularly in adversarially volatile environments where maintaining constraint compliance and decision optimality is crucial. The inclusion of a self-bounding property for dual variables eliminates the need for detailed prior knowledge of the environment, paving the way for more autonomous, robust decision-making frameworks in the face of uncertainty.