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Projection-free Online Learning over Strongly Convex Sets

Yuanyu Wan, Lijun Zhang

TL;DR

This paper proves that OFW can enjoy a better regret bound of O(T^{2/3}) for general convex losses and proposes a strongly convex variant of OFW by redefining the surrogate loss function in OFW.

Abstract

To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of projection-based algorithms, where $T$ is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of $O(T^{2/3})$ for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of $O(T^{2/3})$ over general convex sets and a better regret bound of $O(\sqrt{T})$ over strongly convex sets.

Projection-free Online Learning over Strongly Convex Sets

TL;DR

This paper proves that OFW can enjoy a better regret bound of O(T^{2/3}) for general convex losses and proposes a strongly convex variant of OFW by redefining the surrogate loss function in OFW.

Abstract

To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of , which is worse than the regret of projection-based algorithms, where is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of over general convex sets and a better regret bound of over strongly convex sets.

Paper Structure

This paper contains 16 sections, 9 theorems, 69 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Results
  4. Preliminaries
  5. OFW with Line Search
  6. SC-OFW for Strongly Convex Losses
  7. Theoretical Analysis
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Theorem \ref{['thm2']}
  10. Proof of Lemma \ref{['thm1_lem1']}
  11. Conclusion and Future Work
  12. Acknowledgments
  13. Proof of Lemma \ref{['thm2_lem1']}
  14. Proof of Lemma \ref{['lem_1_thm2_lem1']}
  15. Proof of Theorem \ref{['thm3']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathcal{K}$ be an $\alpha_K$-strongly convex set with respect to the $\ell_2$ norm. Let $\mathbf{x}_{t}^\ast=\mathop{\mathrm{argmin}}\limits_{\mathbf{x}\in \mathcal{K}}F_{t-1}(\mathbf{x})$ for any $t\in[T+1]$, where $F_t(\mathbf{x})$ is defined in (eq_F1). Then, for any $t\in[T+1]$, Algorithm where $C=\max\left(4D^2,\frac{4096}{3\alpha_K^2}\right).$

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Lemma 3
  • Lemma 4
  • ...and 2 more