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Safe and Efficient Online Convex Optimization with Linear Budget Constraints and Partial Feedback

Shanqi Liu, Xin Liu

TL;DR

This work tackles online convex optimization with unknown linear budget constraints and partial feedback, formalizing a budgeted loss minimization problem where only gradient information and budget usage are observed. The authors introduce SELO, a Lyapunov-optimization–based primal–dual algorithm that uses gradient-based loss estimation, pessimistic budget estimates, and virtual queues to balance regret and budget adherence; its surrogate objective is $V\hat{f}_t(\mathbf x) + \langle \mathbf Q_t, \hat{g}_t(\mathbf x) \rangle + \frac{1}{2\eta}\|\mathbf x - \mathbf x_{t-1}\|^2$. They prove that SELO achieves $\tilde{O}(\sqrt{T})$ regret with $O(1)$ constraint violation under soft budgets and $\tilde{O}(\sqrt{T})$ regret with no violation under hard budgets, via a multi-step Lyapunov-drift analysis and a key regret+drift lemma. The approach is computationally efficient (nearly unconstrained primal update) and is validated on energy-aware task processing in distributed data centers, where SELO outperforms baselines while maintaining safety, highlighting its practical impact for safe, efficient online decision-making with partial feedback.

Abstract

This paper studies online convex optimization with unknown linear budget constraints, where only the gradient information of the objective and the bandit feedback of constraint functions are observed. We propose a safe and efficient Lyapunov-optimization algorithm (SELO) that can achieve an $O(\sqrt{T})$ regret and zero cumulative constraint violation. The result also implies SELO achieves $O(\sqrt{T})$ regret when the budget is hard and not allowed to be violated. The proposed algorithm is computationally efficient as it resembles a primal-dual algorithm where the primal problem is an unconstrained, strongly convex and smooth problem, and the dual problem has a simple gradient-type update. The algorithm and theory are further justified in a simulated application of energy-efficient task processing in distributed data centers.

Safe and Efficient Online Convex Optimization with Linear Budget Constraints and Partial Feedback

TL;DR

This work tackles online convex optimization with unknown linear budget constraints and partial feedback, formalizing a budgeted loss minimization problem where only gradient information and budget usage are observed. The authors introduce SELO, a Lyapunov-optimization–based primal–dual algorithm that uses gradient-based loss estimation, pessimistic budget estimates, and virtual queues to balance regret and budget adherence; its surrogate objective is . They prove that SELO achieves regret with constraint violation under soft budgets and regret with no violation under hard budgets, via a multi-step Lyapunov-drift analysis and a key regret+drift lemma. The approach is computationally efficient (nearly unconstrained primal update) and is validated on energy-aware task processing in distributed data centers, where SELO outperforms baselines while maintaining safety, highlighting its practical impact for safe, efficient online decision-making with partial feedback.

Abstract

This paper studies online convex optimization with unknown linear budget constraints, where only the gradient information of the objective and the bandit feedback of constraint functions are observed. We propose a safe and efficient Lyapunov-optimization algorithm (SELO) that can achieve an regret and zero cumulative constraint violation. The result also implies SELO achieves regret when the budget is hard and not allowed to be violated. The proposed algorithm is computationally efficient as it resembles a primal-dual algorithm where the primal problem is an unconstrained, strongly convex and smooth problem, and the dual problem has a simple gradient-type update. The algorithm and theory are further justified in a simulated application of energy-efficient task processing in distributed data centers.

Paper Structure

This paper contains 8 sections, 3 theorems, 31 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Model
  4. Safe and efficient Lyapunov optimization algorithm
  5. Theoretical Results
  6. Theoretical Analysis of the key "Regret + Lyapunov Drift" lemma
  7. Experiments
  8. Conclusions

Key Result

Theorem 1

Lett $V = \sqrt{T},$$\eta = 1/T,$$\xi = \log^2 T/\sqrt{T}$ and $T_0=\log T/\beta.$ Under Assumptions assumption: set--assumption: slater, SELO algorithm achieves the following theoretical results for OCOwLB 1) under soft budget constraints such that 2) under hard budget constraints such that

Figures (1)

  • Figure 1: Performance comparison: SELO v.s. "AnytimeSafe".

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2