Lipschitz and Comparator-Norm Adaptivity in Online Learning

COLT, pp.2858-2887, (2020)

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We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained

Abstract:

We study Online Convex Optimization in the unbounded setting where neither predictions nor gradient are constrained. The goal is to simultaneously adapt to both the sequence of gradients and the comparator. We first develop parameter-free and scale-free algorithms for a simplified setting with hints. We present two versions: the first a...More

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Introduction
  • The authors consider the setting of online convex optimization where the goal is to make sequential predictions to minimize a certain notion of regret.
  • At the beginning of each round t ě 1, a learner predicts wpt in some convex set W Ď Rd in dimension d P N.
  • The authors are interested in online algorithms which can guarantee a good regret bound against any comparator vector w P W, even when W is unbounded, and without prior knowledge of the maximum norm L :“ maxtďT }gt} of the observed sub-gradients.
  • The authors refer to L as the Lipschitz constant
Highlights
  • We consider the setting of online convex optimization where the goal is to make sequential predictions to minimize a certain notion of regret
  • We show how our algorithms can be applied to learn linear models
  • As we saw in Corollary 3, given a base algorithm A, which takes a sequence of hints phtq such that ht ě Lt, for all t ě 1, and suffers regret RTApwq against comparator w P W, there exists an algorithm B for the setting without hints which suffers the same regret against w up to an additive pptheeanntaailltttyyisLLnToTt}}wpwo}}s33siibsleLnotTotairmempplaarocxvetPatrbhTlesepBieftn,oawnltehyeiLnresTiBsatst mo“nařxatPrtser“Tg1sr}eBgt tsb}bo{yuLntthd.eIotnfytpohircidsaelsrleyOcrstipmo?naT,llweqr.eqWsuheaonawtlisttyoha?sthVtohTwe, where
  • As in Section 3, we focus on algorithms which make predictions based on observed sub-gradients; in this case, gt “ xtδt P xt Bp0,1q pyt, yptq “ Bftpuptq, t ě 1, where ftpwq “ pyt, w xtq
Results
  • The authors improve on current state-of-the-art algorithms in two ways; First, the authors provide a scale-invariant algorithm which guarantees regret bound, against any w P Rd, of order db ÿ |wi|.
Conclusion
  • The algorithms designed can be used in the role of algorithm A in the reductions presented in Section 2.2.
  • This will yield algorithms which achieve the goal; they adapt to the norm of the comparator and the Lipschitz constant and are completely scale-free, for both bounded and unbounded sets, without requiring hints.
  • By restricting gt to be aligned with wpt, the problem reduces to finding x “ }gt} such that x}wpt} ́ |}wpt} ̈ pLt1 _ xq{2 ́ St1| ě 0
Summary
  • Introduction:

    The authors consider the setting of online convex optimization where the goal is to make sequential predictions to minimize a certain notion of regret.
  • At the beginning of each round t ě 1, a learner predicts wpt in some convex set W Ď Rd in dimension d P N.
  • The authors are interested in online algorithms which can guarantee a good regret bound against any comparator vector w P W, even when W is unbounded, and without prior knowledge of the maximum norm L :“ maxtďT }gt} of the observed sub-gradients.
  • The authors refer to L as the Lipschitz constant
  • Results:

    The authors improve on current state-of-the-art algorithms in two ways; First, the authors provide a scale-invariant algorithm which guarantees regret bound, against any w P Rd, of order db ÿ |wi|.
  • Conclusion:

    The algorithms designed can be used in the role of algorithm A in the reductions presented in Section 2.2.
  • This will yield algorithms which achieve the goal; they adapt to the norm of the comparator and the Lipschitz constant and are completely scale-free, for both bounded and unbounded sets, without requiring hints.
  • By restricting gt to be aligned with wpt, the problem reduces to finding x “ }gt} such that x}wpt} ́ |}wpt} ̈ pLt1 _ xq{2 ́ St1| ě 0
Related work
  • Related Work For an overview of Online

    Convex Optimization in the bounded setting, we refer to the textbook (Hazan, 2016). The unconstrained case was first studied by McMahan and Streeter (2010). A powerful methodology for the unbounded case is Coin Betting by Orabona and Pal (2016a). Even though not always visible, our potential functions are inspired by this style of thinking. We build our unbounded OCO learner by targeting a specific other constrained problem. We further employ several general reductions from the literature, including gradient clipping Cutkosky (2019), the constrained-to-unconstrained reduction Cutkosky and Orabona (2018), and the restart wrapper to pacify the final-vs-initial scale ratio appearing inside logarithms by Mhammedi et al (2019). Our analysis is, at its core, proving a certain minimax result about sufficient-statisticbased potentials reminiscent of the Burkholder approach pioneered by Foster et al (2017, 2018). Applications for scale-invariant learning in linear models were studied by Kempka et al (2019). For our multidimensional learner we took inspiration from the Gaussian Exp-concavity step in the analysis of the MetaGrad algorithm by Van Erven and Koolen (2016).
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