Projection-free Online Learning over Strongly Convex Sets

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We introduce a strongly convex variant of online frank-wolfe, and prove that it achieves a regret bound of O(T 2/3√) over general convex sets and a better regret bound of O( T ) over strongly convex sets

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 projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of...More

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Introduction
Highlights
  • Online convex optimization (OCO) is a powerful framework that has been used to model and solve problems from diverse domains such as online routing (Awerbuch and Kleinberg 2004, 2008), online portfolio selection (Blum and Kalai

    1999; Agarwal et al 2006) and prediction from expert advice (Cesa-Bianchi et al 1997; Freund et al 1997)
  • We study OCO with strongly convex losses, and propose a strongly convex variant of online frank-wolfe (OFW) (SC-OFW)
  • We present an improved regret bound for OFW over strongly convex sets
  • Since this paper considers OCO over strongly convex sets, our SC-OFW adopts the linear optimization step utilized in the original OFW, and simplifies Ft(x) in (4) as t
  • For strongly convex losses, we introduce a strongly convex variant of OFW, and prove that it achieves a regret bound of O(T 2/3√) over general convex sets and a better regret bound of O( T ) over strongly convex sets
  • An open question is whether the regret of OFW and its strongly convex variant over strongly convex sets can be further improved if the losses are smooth
Results
  • The authors first introduce necessary preliminaries including common notations, definitions and assumptions.
  • The authors present an improved regret bound for OFW over strongly convex sets.
  • The authors introduce the SC-OFW algorithm for strongly convex OCO as well as its theoretical guarantees.
  • The convex set K belongs to a finite vector space E, and the authors denote the l2 norm of any vector x ∈ K by x.
  • The authors recall two standard definitions for smooth and strongly convex functions (Boyd and Vandenberghe 2004), respectively
Conclusion
  • The authors first prove that the classical OFW algorithm attains an O(T 2/3) regret bound for OCO over strongly convex sets, which is better than the O(T 3/4) regret bound for the general OCO.
  • An open question is whether the regret of OFW and its strongly convex variant over strongly convex sets can be further improved if the losses are smooth.
  • The authors note that Hazan and Minasyan (2020) have proposed a projectionfree algorithm for OCO over general convex sets, and established an improved regret bound of O(T 2/3) by taking advantage of the smoothness
Summary
  • Introduction:

    Online convex optimization (OCO) is a powerful framework that has been used to model and solve problems from diverse domains such as online routing (Awerbuch and Kleinberg 2004, 2008), online portfolio selection (Blum and Kalai

    1999; Agarwal et al 2006) and prediction from expert advice (Cesa-Bianchi et al 1997; Freund et al 1997).
  • The player chooses a decision xt from a convex set K.
  • The goal of the player is to choose decisions so that the regret defined as T R(T ) = t=1 ft(xt) −.
  • Various algorithms such as online gradient descent (OGD) (Zinkevich 2003), online Newton step (Hazan, Agarwal, and Kale 2007) and follow-the-regularized-leader (Shalev-Shwartz 2007; Shalev-Shwartz and Singer 2007) have been proposed to yield optimal regret bounds under different scenarios
  • Results:

    The authors first introduce necessary preliminaries including common notations, definitions and assumptions.
  • The authors present an improved regret bound for OFW over strongly convex sets.
  • The authors introduce the SC-OFW algorithm for strongly convex OCO as well as its theoretical guarantees.
  • The convex set K belongs to a finite vector space E, and the authors denote the l2 norm of any vector x ∈ K by x.
  • The authors recall two standard definitions for smooth and strongly convex functions (Boyd and Vandenberghe 2004), respectively
  • Conclusion:

    The authors first prove that the classical OFW algorithm attains an O(T 2/3) regret bound for OCO over strongly convex sets, which is better than the O(T 3/4) regret bound for the general OCO.
  • An open question is whether the regret of OFW and its strongly convex variant over strongly convex sets can be further improved if the losses are smooth.
  • The authors note that Hazan and Minasyan (2020) have proposed a projectionfree algorithm for OCO over general convex sets, and established an improved regret bound of O(T 2/3) by taking advantage of the smoothness
Related work
  • In this section, we briefly review the related work about projection-free algorithms for OCO.

    OFW (Hazan and Kale 2012; Hazan 2016) is the first projection-free algorithms for OCO, which attains a regret bound of O(T 3/4). Recently, some studies have proposed projection-free online algorithms which attain better regret bounds, for special cases of OCO. Specifically, if the decision set is a polytope, Garber and Haza√n (2016) proposed variants of OFW, which enjoy an O( T ) regret bound for convex losses and an O(log T ) regret bound for strongly convex losses. For OCO over smooth sets, Levy and Krause (2019) proposed a projection-free variant of OGD via devising a fa√st approximate projection for such sets, and established O( T ) and O(log T ) regret bounds for convex and strongly convex losses, respectively. Besides these improvements for OCO over special decision sets, Hazan and Minasyan (2020) proposed a randomized projection-free algorithm for OCO with smooth losses, and achieved an expected regret bound of O(T 2/3).
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