Online Convex Optimization with Stochastic Constraints (1708.03741v1)

Abstract: This paper considers online convex optimization (OCO) with stochastic constraints, which generalizes Zinkevich's OCO over a known simple fixed set by introducing multiple stochastic functional constraints that are i.i.d. generated at each round and are disclosed to the decision maker only after the decision is made. This formulation arises naturally when decisions are restricted by stochastic environments or deterministic environments with noisy observations. It also includes many important problems as special cases, such as OCO with long term constraints, stochastic constrained convex optimization, and deterministic constrained convex optimization. To solve this problem, this paper proposes a new algorithm that achieves $O(\sqrt{T})$ expected regret and constraint violations and $O(\sqrt{T}\log(T))$ high probability regret and constraint violations. Experiments on a real-world data center scheduling problem further verify the performance of the new algorithm.

• The paper introduces a novel algorithm that integrates stochastic constraints into online convex optimization, achieving O(√T) expected regret and constraint violation bounds.
• It employs a Lyapunov-drift-plus-penalty framework with virtual queues to dynamically adjust decisions based on observed loss gradients and stochastic feedback.
• The approach offers practical improvements for real-world applications like online scheduling and network routing in environments characterized by uncertainty.

Online Convex Optimization with Stochastic Constraints: A Detailed Analysis

"Online Convex Optimization with Stochastic Constraints" by Hao Yu, Michael J. Neely, and Xiaohan Wei, addresses the challenges associated with decision-making in environments characterized by uncertainty and variability. This research extends upon the canonical framework of Online Convex Optimization (OCO) by incorporating independently and identically distributed (i.i.d.) stochastic functional constraints that arise naturally in many practical scenarios such as stochastic environments or deterministic environments influenced by noise.

Problem Context and Contributions

Online convex optimization is a strategic decision-making process where a decision maker is tasked with choosing actions iteratively without full knowledge of the subsequent loss functions. The principled goal of OCO is to develop algorithms that ensure the regret, defined as the cumulative difference between the algorithm's performance and the best fixed decision in hindsight, grows sub-linearly with respect to the time horizon, $T$.

This paper broadens the OCO framework by integrating stochastic constraints, where each constraint function is drawn i.i.d. from an unknown distribution. The stochastic constraints define a dynamic feasible set, presented initially as unknown to the decision maker, thus providing a more realistic model for applications like online data center scheduling.

The paper's central contribution is the development of a novel algorithm for handling OCO with stochastic constraints, achieving $O(\sqrt{T})$ expected regret and constraint violations. The algorithm also assures $O(\sqrt{T}\log(T))$ regret and constraint violation bounds with high probability, an improvement over traditional methods which either disregard constraints or are limited to deterministic scenarios.

Algorithmic Insights

The proposed algorithm operates by dynamically adjusting decisions based on observed loss gradients and stochastic constraints after the decision. It introduces virtual queues to track constraint violations, leveraging a Lyapunov-drift-plus-penalty framework to manage both regret minimization and constraint satisfaction. The algorithm's essence lies in its ability to make near-optimal decisions under uncertainty, guided by historical observations and intrinsic system randomness.

One of the paper's technical contributions is the introduction of a new drift lemma for stochastic processes. This lemma provides the foundational analysis technique allowing the derivation of queue bounds essential for controlling constraint violations, significantly easing the handling of stochastic convex programs under uncertainty.

Theoretical and Practical Implications

Theoretically, this work contributes to the understanding of OCO with stochastic constraints, substantiating that simultaneous sublinear regret and constraint violation is possible under realistic conditions. Practically, it addresses complex decision-making problems in real-world systems like network routing in time-varying channels and job scheduling in distributed data centers with variable power costs. These applications benefit from the algorithm's ability to adapt in scenarios where environmental uncertainties play a critical role in achieving operational efficiency.

Future Directions

The research opens avenues for extending OCO framework analysis in broader stochastic contexts. Future advancements may explore even more relaxed assumptions about the stochastic processes involved, potentially combining this methodology with reinforcement learning or adaptive control for enhanced decision-making in more unpredictable environments. Moreover, scaling the algorithm for high-dimensional applications and improving computational efficiency while maintaining strong performance guarantees represents a promising research trajectory.

In conclusion, the paper tactfully integrates stochastic variability into the OCO framework, offering robust solutions for complex optimization problems faced in dynamic settings, highlighting its theoretical robustness and direct applicability in modern technological challenges.