Online Learning with Low Rank Experts

We consider the problem of prediction with expert advice when the losses of the experts have low-dimensional structure: they are restricted to an unknown $d$-dimensional subspace. We devise algorithms with regret bounds that are independent of the number of experts and depend only on the rank $d$. For the stochastic model we show a tight bound of $\Theta(\sqrt{dT})$, and extend it to a setting of an approximate $d$ subspace. For the adversarial model we show an upper bound of $O(d\sqrt{T})$ and a lower bound of $\Omega(\sqrt{dT})$.

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

[2]  Nathan Srebro,et al.  Concentration-Based Guarantees for Low-Rank Matrix Reconstruction , 2011, COLT.

[3]  Ambuj Tewari,et al.  On the Complexity of Linear Prediction: Risk Bounds, Margin Bounds, and Regularization , 2008, NIPS.

[4]  Wouter M. Koolen,et al.  Follow the leader if you can, hedge if you must , 2013, J. Mach. Learn. Res..

[5]  Claudio Gentile,et al.  Regret Minimization for Branching Experts , 2022 .

[6]  K. Ball.  An Elementary Introduction to Modern Convex Geometry , 1997 .

[7]  Shai Ben-David,et al.  Agnostic Online Learning , 2009, COLT.

[8]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[9]  Elad Hazan,et al.  Better Algorithms for Benign Bandits , 2009, J. Mach. Learn. Res..

[10]  Yoram Singer,et al.  Adaptive Subgradient Methods for Online Learning and Stochastic Optimization , 2011, J. Mach. Learn. Res..

[11]  Adi Shraibman,et al.  Rank, Trace-Norm and Max-Norm , 2005, COLT.

[12]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[13]  Ohad Shamir,et al.  Relax and Randomize : From Value to Algorithms , 2012, NIPS.

[14]  Noga Alon,et al.  The approximate rank of a matrix and its algorithmic applications: approximate rank , 2013, STOC '13.

[15]  Elad Hazan,et al.  On Stochastic and Worst-case Models for Investing , 2009, NIPS.

[16]  Ohad Shamir,et al.  Localization and Adaptation in Online Learning , 2013, AISTATS.

[17]  Roi Livni,et al.  Classification with Low Rank and Missing Data , 2015, ICML.

[18]  Ohad Shamir,et al.  Large-Scale Convex Minimization with a Low-Rank Constraint , 2011, ICML.

[19]  Robert D. Nowak,et al.  Transduction with Matrix Completion: Three Birds with One Stone , 2010, NIPS.

[20]  N. Littlestone.  Learning Quickly When Irrelevant Attributes Abound: A New Linear-Threshold Algorithm , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

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

[22]  Elad Hazan,et al.  Extracting certainty from uncertainty: regret bounded by variation in costs , 2008, Machine Learning.

[23]  Tommi S. Jaakkola,et al.  Maximum-Margin Matrix Factorization , 2004, NIPS.

[24]  Ambuj Tewari,et al.  Online Learning: Random Averages, Combinatorial Parameters, and Learnability , 2010, NIPS.

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

[26]  Yishay Mansour,et al.  Improved second-order bounds for prediction with expert advice , 2006, Machine Learning.

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

[28]  Rong Jin,et al.  25th Annual Conference on Learning Theory Online Optimization with Gradual Variations , 2022 .

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

[30]  K. Ball.  An elementary introduction to modern convex geometry, in flavors of geometry , 1997 .