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CLASP: An online learning algorithm for Convex Losses And Squared Penalties

Ricardo N. Ferreira, João Xavier, Cláudia Soares

TL;DR

This work addresses Constrained Online Convex Optimization with dynamic constraints by introducing CLASP, a simple algorithm that performs a gradient step on the latest loss followed by a projection onto the current constraint set, exploiting the firm non-expansiveness of projections. For general convex losses, CLASP delivers a tunable trade-off between regret $Regret_T$ and squared constraint violations $CCV_{T,2}$, achieving $Regret_T = O\Big(T^{\max\{\beta,1-\beta\}}\Big)$ and $CCV_{T,2} = O\Big(T^{1-\beta}\Big)$ for any $\beta\in(0,1)$; for strongly convex losses, it attains logarithmic bounds: $Regret_T = O(\log T)$ and $CCV_{T,2} = O(\log T)$. The analysis hinges on the firmly non-expansive property of projection operators, enabling a modular treatment of regret and constraint violations and enabling straightforward extensions to multiple or persistent constraints. Empirically, CLASP competes effectively with state-of-the-art COCO algorithms on online linear regression and SVM-style tasks, offering strong theoretical guarantees alongside practical memory efficiency. The results underscore the practical relevance of sharp penalties on constraint violations and demonstrate that strong convexity yields fast, logarithmic convergence in both objective and constraint-violation measures.

Abstract

We study Constrained Online Convex Optimization (COCO), where a learner chooses actions iteratively, observes both unanticipated convex loss and convex constraint, and accumulates loss while incurring penalties for constraint violations. We introduce CLASP (Convex Losses And Squared Penalties), an algorithm that minimizes cumulative loss together with squared constraint violations. Our analysis departs from prior work by fully leveraging the firm non-expansiveness of convex projectors, a proof strategy not previously applied in this setting. For convex losses, CLASP achieves regret $O\left(T^{\max\{β,1-β\}}\right)$ and cumulative squared penalty $O\left(T^{1-β}\right)$ for any $β\in (0,1)$. Most importantly, for strongly convex problems, CLASP provides the first logarithmic guarantees on both regret and cumulative squared penalty. In the strongly convex case, the regret is upper bounded by $O( \log T )$ and the cumulative squared penalty is also upper bounded by $O( \log T )$.

CLASP: An online learning algorithm for Convex Losses And Squared Penalties

TL;DR

This work addresses Constrained Online Convex Optimization with dynamic constraints by introducing CLASP, a simple algorithm that performs a gradient step on the latest loss followed by a projection onto the current constraint set, exploiting the firm non-expansiveness of projections. For general convex losses, CLASP delivers a tunable trade-off between regret and squared constraint violations , achieving and for any ; for strongly convex losses, it attains logarithmic bounds: and . The analysis hinges on the firmly non-expansive property of projection operators, enabling a modular treatment of regret and constraint violations and enabling straightforward extensions to multiple or persistent constraints. Empirically, CLASP competes effectively with state-of-the-art COCO algorithms on online linear regression and SVM-style tasks, offering strong theoretical guarantees alongside practical memory efficiency. The results underscore the practical relevance of sharp penalties on constraint violations and demonstrate that strong convexity yields fast, logarithmic convergence in both objective and constraint-violation measures.

Abstract

We study Constrained Online Convex Optimization (COCO), where a learner chooses actions iteratively, observes both unanticipated convex loss and convex constraint, and accumulates loss while incurring penalties for constraint violations. We introduce CLASP (Convex Losses And Squared Penalties), an algorithm that minimizes cumulative loss together with squared constraint violations. Our analysis departs from prior work by fully leveraging the firm non-expansiveness of convex projectors, a proof strategy not previously applied in this setting. For convex losses, CLASP achieves regret and cumulative squared penalty for any . Most importantly, for strongly convex problems, CLASP provides the first logarithmic guarantees on both regret and cumulative squared penalty. In the strongly convex case, the regret is upper bounded by and the cumulative squared penalty is also upper bounded by .
Paper Structure (29 sections, 5 theorems, 39 equations, 3 figures, 1 algorithm)

This paper contains 29 sections, 5 theorems, 39 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Static constraints.
  4. Dynamic constraints.
  5. Contributions
  6. Preliminaries
  7. Projection operators and distance functions.
  8. Convex and strongly convex functions.
  9. Assumptions.
  10. The CLASP algorithm
  11. Bounding $\text{CCV}_{T,2}$
  12. Analysis of CLASP
  13. Convex losses.
  14. Strongly convex losses.
  15. Extensions
  16. ...and 14 more sections

Key Result

Lemma 1

There holds $\sum_{t=1}^{T} d_{{\mathcal{K}}_t}(\widehat{x}_{t+1})^2 \leq O\left( \sum_{t=1}^T \eta_t \right)$.

Figures (3)

  • Figure 1: Online linear regression with adversarial constraints. We report (a) cumulative loss, (b) linear violation $\mathrm{CCV}_{T,1}$, and (c) squared violation $\mathrm{CCV}_{T,2}$. AdaGrad achieves the lowest loss but at the cost of very large constraint violations. CLASP controls both violation metrics competitively with RECOO, while remaining more memory-efficient than Switch. Switch attains the smallest violations overall but with higher per-iteration complexity. All $\mathrm{CCV}_{T,2}$ values are reported a posteriori.
  • Figure 2: Online Linear Regression with additional rounds.
  • Figure 3: Online Support Vector Machine

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1: Convex losses
  • proof
  • Theorem 2: Strongly convex losses
  • proof