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Projection-Free Online Convex Optimization with Time-Varying Constraints

Dan Garber, Ben Kretzu

TL;DR

This work studies projection-free online convex optimization under adversarial, time-varying constraints by restricting hard feasibility to a fixed convex set $\mathcal{K}$ and enforcing soft, time-varying constraints on average. It develops two main projection-free algorithms: (i) a drift-plus-penalty method that uses a linear optimization oracle to perform approximate projections and achieves interval sublinear regret $\tilde{O}(T^{3/4})$ and constraint violation $O(T^{7/8})$ with $T$ LOO calls, and (ii) a more efficient Lagrangian-based primal-dual method requiring only first-order information that attains similar full-sequence guarantees; both extend to a bandit setting with bounds stated in expectation. The results show that sublinear performance is achievable without costly projections, even under adversarial, time-varying constraints, and they provide practical algorithms for high-dimensional domains where projection is expensive. The bandit extension broadens applicability to settings with limited feedback, important for online decision-making in constrained environments.

Abstract

We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length $T$ and using overall $T$ calls to the LOO, guarantees $\tilde{O}(T^{3/4})$ regret w.r.t. the losses and $O(T^{7/8})$ constraints violation (ignoring all quantities except for $T$) . In particular, these bounds hold w.r.t. any interval of the sequence. We also present a more efficient algorithm that requires only first-order oracle access to the soft constraints and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of $T$) in expectation.

Projection-Free Online Convex Optimization with Time-Varying Constraints

TL;DR

This work studies projection-free online convex optimization under adversarial, time-varying constraints by restricting hard feasibility to a fixed convex set and enforcing soft, time-varying constraints on average. It develops two main projection-free algorithms: (i) a drift-plus-penalty method that uses a linear optimization oracle to perform approximate projections and achieves interval sublinear regret and constraint violation with LOO calls, and (ii) a more efficient Lagrangian-based primal-dual method requiring only first-order information that attains similar full-sequence guarantees; both extend to a bandit setting with bounds stated in expectation. The results show that sublinear performance is achievable without costly projections, even under adversarial, time-varying constraints, and they provide practical algorithms for high-dimensional domains where projection is expensive. The bandit extension broadens applicability to settings with limited feedback, important for online decision-making in constrained environments.

Abstract

We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length and using overall calls to the LOO, guarantees regret w.r.t. the losses and constraints violation (ignoring all quantities except for ) . In particular, these bounds hold w.r.t. any interval of the sequence. We also present a more efficient algorithm that requires only first-order oracle access to the soft constraints and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of ) in expectation.
Paper Structure (9 sections, 10 theorems, 90 equations, 5 algorithms)

This paper contains 9 sections, 10 theorems, 90 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions (informally stated)
  3. Preliminaries
  4. Assumptions and notation
  5. Fast approximately-feasible projections via linear optimization
  6. Drift-plus-Penalty-inspired Algorithm with Adaptive Regret Guarantees
  7. Lagrangian-based Primal-Dual Algorithm
  8. Bandit Feedback Setting
  9. Bandit Feedback Setting

Key Result

Lemma 2.2

Let $\mathcal{K}\subset\mathbb{R}^n$ be convex and compact such that $\mathbf{0}\in\mathcal{K}\subseteq{}R\mathcal{B}$ for some $R>0$. Algorithm alg:CIP-FW guarantees that it returns $(\x,\y) \in \mathcal{K}\times \mathbb{R}^n$ such that the following three conditions hold: I. $\forall \z \in \mathc

Theorems & Definitions (18)

  • Definition 2.1: Approximately-feasible Projection Oracle
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:d+p']}
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • proof : Proof of Theorem \ref{['thm:LF']}
  • ...and 8 more