Following the Leader and Fast Rates in Online Linear Prediction: Curved Constraint Sets and Other Regularities

Follow the leader (FTL) is a simple online learning algorithm that is known to perform well when the loss functions are convex and positively curved. In this paper we ask whether there are other settings when FTL achieves low regret. In particular, we study the fundamental problem of linear prediction over a convex, compact domain with non-empty interior. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys logarithmic regret, while for polytope domains and stochastic data it enjoys finite expected regret. The former result is also extended to strongly convex domains by establishing an equivalence between the strong convexity of sets and the minimum curvature of their boundary, which may be of independent interest. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the smaller regret of FTL when the data is ‘easy’. Finally, we show that such guarantees are achievable directly (e.g., by the follow the regularized leader algorithm or by a shrinkage-based variant of FTL) when the constraint set is an ellipsoid.

[1]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

[2]  Neri Merhav,et al.  Universal sequential learning and decision from individual data sequences , 1992, COLT '92.

[3]  R. Schneider Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[4]  Yoav Freund,et al.  A decision-theoretic generalization of on-line learning and an application to boosting , 1995, EuroCOLT.

[5]  E. Polovinkin Strongly convex analysis , 1996 .

[6]  Fabio Stella,et al.  Stochastic Nonstationary Optimization for Finding Universal Portfolios , 2000, Ann. Oper. Res..

[7]  A. Pressley Elementary Differential Geometry , 2000 .

[8]  Claudio Gentile,et al.  On the generalization ability of on-line learning algorithms , 2001, IEEE Transactions on Information Theory.

[9]  Gábor Lugosi,et al.  Prediction, learning, and games , 2006 .

[10]  Elad Hazan,et al.  Logarithmic regret algorithms for online convex optimization , 2006, Machine Learning.

[11]  Peter L. Bartlett,et al.  Adaptive Online Gradient Descent , 2007, NIPS.

[12]  Ambuj Tewari,et al.  Optimal Stragies and Minimax Lower Bounds for Online Convex Games , 2008, COLT.

[13]  Sham M. Kakade,et al.  Mind the Duality Gap: Logarithmic regret algorithms for online optimization , 2008, NIPS.

[14]  H. Brendan McMahan,et al.  Follow-the-Regularized-Leader and Mirror Descent: Equivalence Theorems and Implicit Updates , 2010, arXiv.org.

[15]  Yurii Nesterov,et al.  Generalized Power Method for Sparse Principal Component Analysis , 2008, J. Mach. Learn. Res..

[16]  Shai Shalev-Shwartz,et al.  Online Learning and Online Convex Optimization , 2012, Found. Trends Mach. Learn..

[17]  Claudio Gentile,et al.  Beyond Logarithmic Bounds in Online Learning , 2012, AISTATS.

[18]  Karthik Sridharan,et al.  Online Learning with Predictable Sequences , 2012, COLT.

[19]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[20]  Alessandro Lazaric,et al.  Exploiting easy data in online optimization , 2014, NIPS.

[21]  Ambuj Tewari,et al.  Online Linear Optimization via Smoothing , 2014, COLT.

[22]  Shai Ben-David,et al.  Understanding Machine Learning - From Theory to Algorithms , 2014 .

[23]  Karthik Sridharan,et al.  Adaptive Online Learning , 2015, NIPS.

[24]  Elad Hazan,et al.  Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets , 2014, ICML.

[25]  Wojciech Kot lowski Minimax strategy for prediction with expert advice under stochastic assumptions , 2015 .

[26]  Mark D. Reid,et al.  Fast rates in statistical and online learning , 2015, J. Mach. Learn. Res..

[27]  Tor Lattimore,et al.  Following the Leader and Fast Rates in Linear Prediction: Curved Constraint Sets and Other Regularities , 2016, NIPS.

[28]  Wouter M. Koolen,et al.  Combining Adversarial Guarantees and Stochastic Fast Rates in Online Learning , 2016, NIPS.

[29]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .