Beyond $\tilde{O}(\sqrt{T})$ Constraint Violation for Online Convex Optimization with Adversarial Constraints (2505.06709v1)

Abstract: We revisit the Online Convex Optimization problem with adversarial constraints (COCO) where, in each round, a learner is presented with a convex cost function and a convex constraint function, both of which may be chosen adversarially. The learner selects actions from a convex decision set in an online fashion, with the goal of minimizing both regret and the cumulative constraint violation (CCV) over a horizon of $T$ rounds. The best-known policy for this problem achieves $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV. In this paper, we present a surprising improvement that achieves a significantly smaller CCV by trading it off with regret. Specifically, for any bounded convex cost and constraint functions, we propose an online policy that achieves $\tilde{O}(\sqrt{dT}+ T^\beta)$ regret and $\tilde{O}(dT^{1-\beta})$ CCV, where $d$ is the dimension of the decision set and $\beta \in [0,1]$ is a tunable parameter. We achieve this result by first considering the special case of $\textsf{Constrained Expert}$ problem where the decision set is a probability simplex and the cost and constraint functions are linear. Leveraging a new adaptive small-loss regret bound, we propose an efficient policy for the $\textsf{Constrained Expert}$ problem, that attains $O(\sqrt{T\ln N}+T^{\beta})$ regret and $\tilde{O}(T^{1-\beta} \ln N)$ CCV, where $N$ is the number of experts. The original problem is then reduced to the $\textsf{Constrained Expert}$ problem via a covering argument. Finally, with an additional smoothness assumption, we propose an efficient gradient-based policy attaining $O(T^{\max(\frac{1}{2},\beta)})$ regret and $\tilde{O}(T^{1-\beta})$ CCV.

Insightful Overview of "Beyond $\tilde{O}(\sqrt{T})$ Constraint Violation for Online Convex Optimization with Adversarial Constraints"

The paper under discussion, authored by Abhishek Sinha and Rahul Vaze, revisits the domain of Online Convex Optimization (OCO) with adversarial constraints, often abbreviated as COCO. The central objective in this setting is to minimize both the regret and the cumulative constraint violation (CCV) over a time horizon $T$ in scenarios where constraint functions, in addition to cost functions, can be selected adversarially. The innovation introduced by the authors is a set of policies that achieve improved CCV beyond the best-known $\tilde{O}(\sqrt{T})$ bound, by manipulating the trade-off between regret and CCV.

Theoretical Contributions

The authors propose a novel online policy adaptable to bounded convex cost and constraint functions. A tunable parameter $\beta \in [0,1]$ is crucial here, facilitating a continuous trade-off between regret and CCV. The main results can be succinctly summarized as follows:

1. General Policy Design: The proposed policy attains $\tilde{O}(\sqrt{dT}+ T^\beta)$ regret and $\tilde{O}(dT^{1-\beta})$ CCV, where $d$ represents the dimensionality of the decision set. The adaptability of the algorithm is anchored in the parameter $\beta$, offering a spectrum of options from prioritizing lower regret to minimizing CCV.
2. Constrained Expert Problem as a Special Case: The authors initially focus on the Constrained Expert problem, where the decision space is a probability simplex, facilitating a simpler analytical frame. In this setup, a new adaptive policy that yields a small-loss regret bound is introduced. This forms a grounding case for their broader arguments.
3. Gradient-Based Policy: For smooth convex functions, a gradient-based online policy is proposed. This alternative offers improvement across any dimensional set, demonstrating $\tilde{O}(\sqrt{T^\beta})$ regret and $\tilde{O}(T^{1-\beta})$ CCV, assuming additional smoothness in functions.

Practical Implications and Future Directions

This work has significant theoretical and application implications, notably in environments where ensuring safety (low CCV) is more critical than achieving minimal regret. These situations are prevalent in adaptive control systems, safe reinforcement learning, and autonomous vehicles. Practically, the policy design can be extended to scenarios characterized by varying adversarial constraints, emphasizing safety over long-term performance metrics.

The paper leaves some open ends for future research, especially concerning the development of more computationally efficient algorithms that match or improve upon these theoretical bounds in fixed-dimensional settings. Furthermore, it would be insightful to delve into establishing lower bounds for CCV, akin to existing bounds on regret, under these new conditions.

Conclusion

In conclusion, this paper contributes significantly to the literature on online optimization under adversarial contexts, proposing policies that extend beyond current CCV limits through controlled trade-offs with regret. By innovatively utilizing a tunable parameter to achieve this balance, the authors pave the way for further exploration into minimizing violations in adversarial online learning tasks.