Abstract:
We study sequential probability assignment in the context of online learning under logarithmic loss and obtain tight lower and upper bounds for sequential minimax regret....Show MoreMetadata
Abstract:
We study sequential probability assignment in the context of online learning under logarithmic loss and obtain tight lower and upper bounds for sequential minimax regret. Sequential minimax regret is defined as the minimum excess loss over data horizon T that a predictor incurs over the best expert in a class, when the samples are presented sequentially and adversarially. Our upper bounds are established by applying Bayesian averaging over a novel “smooth truncated covering” of the expert class. This allows us to obtain tight (minimax) upper bounds that subsume the best known non-constructive bounds in an algorithmic fashion. For lower bounds, we reduce the problem to analyzing the fixed design regret via a novel application of Shtarkov sum adapted to online learning. We demonstrate the effectiveness of our approach by establishing tight regret bounds for a wide range of expert classes. In particular, we fully characterize the regret of generalized linear function with worst Lipschitz transform functions when the parameters are restricted to a unit norm \ell _{s} ( s\ge 2 ) ball of dimension d . We show that the regret grows as \Theta (d\log T) when d\le O(T^{s/(s+1)-\epsilon }) for all \epsilon >0 (with precise constant 1 when d\le e^{o(\log T)} ) and \tilde {O}(T^{s/(s+1)}) when d\ge \Omega (T^{s/(s+1)}) . Finally, we show that the Bayesian approach may not always be optimal if the support of the prior is included in the reference class itself.
Published in: IEEE Transactions on Information Theory ( Volume: 69, Issue: 9, September 2023)
Funding Agency:
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Changlong Wu received the B.Eng. degree in computer science from Wuhan University, China, in 2015, and the Ph.D. degree in electrical engineering from the University of Hawaii at Manoa in 2021. He is currently a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. His research interests include the intersection of information theory, sta...Show More
Changlong Wu received the B.Eng. degree in computer science from Wuhan University, China, in 2015, and the Ph.D. degree in electrical engineering from the University of Hawaii at Manoa in 2021. He is currently a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. His research interests include the intersection of information theory, sta...View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Indiana University, Bloomington, IN, USA
Mohsen Heidari received the B.Sc. and M.Sc. degrees in electrical engineering from the Sharif University of Technology in 2011 and 2013, respectively, and the M.Sc. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2017 and 2019, respectively. From 2019 to 2021, he was a Post-Doctoral Research Associate with the NSF Center for Science of Informatio...Show More
Mohsen Heidari received the B.Sc. and M.Sc. degrees in electrical engineering from the Sharif University of Technology in 2011 and 2013, respectively, and the M.Sc. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2017 and 2019, respectively. From 2019 to 2021, he was a Post-Doctoral Research Associate with the NSF Center for Science of Informatio...View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Ananth Grama received the Ph.D. degree in computer science from the University of Minnesota. He is currently the Samuel Conte Professor of computer science with Purdue University. His research interests include parallel and distributed computing, large-scale data analytics, and applications in life sciences. He is a fellow of the American Association for the Advancement of Sciences and a Distinguished Alumnus of the Unive...Show More
Ananth Grama received the Ph.D. degree in computer science from the University of Minnesota. He is currently the Samuel Conte Professor of computer science with Purdue University. His research interests include parallel and distributed computing, large-scale data analytics, and applications in life sciences. He is a fellow of the American Association for the Advancement of Sciences and a Distinguished Alumnus of the Unive...View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Wojciech Szpankowski (Fellow, IEEE) held several positions as a Visiting Professor/Visiting Scholar with McGill University; INRIA; Stanford University; Hewlett-Packard Labs; Universite de Versailles; the University of Canterbury, New Zealand; Ecole Polytechnique, France; the Newton Institute, Cambridge, U.K.; ETH Zürich; Hawaii University; the Gdansk University of Technology; and Jagiellonian University, Cracow, Poland. I...Show More
Wojciech Szpankowski (Fellow, IEEE) held several positions as a Visiting Professor/Visiting Scholar with McGill University; INRIA; Stanford University; Hewlett-Packard Labs; Universite de Versailles; the University of Canterbury, New Zealand; Ecole Polytechnique, France; the Newton Institute, Cambridge, U.K.; ETH Zürich; Hawaii University; the Gdansk University of Technology; and Jagiellonian University, Cracow, Poland. I...View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Changlong Wu received the B.Eng. degree in computer science from Wuhan University, China, in 2015, and the Ph.D. degree in electrical engineering from the University of Hawaii at Manoa in 2021. He is currently a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. His research interests include the intersection of information theory, statistics, and machine learning, with a focus on the theoretical foundations of statistical machine learning, online learning, and estimation theories.
Changlong Wu received the B.Eng. degree in computer science from Wuhan University, China, in 2015, and the Ph.D. degree in electrical engineering from the University of Hawaii at Manoa in 2021. He is currently a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. His research interests include the intersection of information theory, statistics, and machine learning, with a focus on the theoretical foundations of statistical machine learning, online learning, and estimation theories.View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Indiana University, Bloomington, IN, USA
Mohsen Heidari received the B.Sc. and M.Sc. degrees in electrical engineering from the Sharif University of Technology in 2011 and 2013, respectively, and the M.Sc. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2017 and 2019, respectively. From 2019 to 2021, he was a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. From 2021 to 2022, he was a Visiting Assistant Professor with the Department of Computer Science, Purdue University. He is currently an Assistant Professor with the Department of Computer Science, Indiana University, Bloomington, IN, USA. His research interests include theoretical machine learning, quantum computing and algorithms, and classical and quantum information theory.
Mohsen Heidari received the B.Sc. and M.Sc. degrees in electrical engineering from the Sharif University of Technology in 2011 and 2013, respectively, and the M.Sc. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 2017 and 2019, respectively. From 2019 to 2021, he was a Post-Doctoral Research Associate with the NSF Center for Science of Information and the Department of Computer Science, Purdue University. From 2021 to 2022, he was a Visiting Assistant Professor with the Department of Computer Science, Purdue University. He is currently an Assistant Professor with the Department of Computer Science, Indiana University, Bloomington, IN, USA. His research interests include theoretical machine learning, quantum computing and algorithms, and classical and quantum information theory.View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Ananth Grama received the Ph.D. degree in computer science from the University of Minnesota. He is currently the Samuel Conte Professor of computer science with Purdue University. His research interests include parallel and distributed computing, large-scale data analytics, and applications in life sciences. He is a fellow of the American Association for the Advancement of Sciences and a Distinguished Alumnus of the University of Minnesota. He was a recipient of the National Science Foundation CAREER Award and the Purdue University Faculty Scholar Award. He chaired the Bio-Data Management and Analysis (BDMA) Study Section, National Institutes of Health, from 2012 to 2014.
Ananth Grama received the Ph.D. degree in computer science from the University of Minnesota. He is currently the Samuel Conte Professor of computer science with Purdue University. His research interests include parallel and distributed computing, large-scale data analytics, and applications in life sciences. He is a fellow of the American Association for the Advancement of Sciences and a Distinguished Alumnus of the University of Minnesota. He was a recipient of the National Science Foundation CAREER Award and the Purdue University Faculty Scholar Award. He chaired the Bio-Data Management and Analysis (BDMA) Study Section, National Institutes of Health, from 2012 to 2014.View more
Center for Science of Information (CSoI), West Lafayette, IN, USA
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Wojciech Szpankowski (Fellow, IEEE) held several positions as a Visiting Professor/Visiting Scholar with McGill University; INRIA; Stanford University; Hewlett-Packard Labs; Universite de Versailles; the University of Canterbury, New Zealand; Ecole Polytechnique, France; the Newton Institute, Cambridge, U.K.; ETH Zürich; Hawaii University; the Gdansk University of Technology; and Jagiellonian University, Cracow, Poland. In 2008, he launched the Interdisciplinary Institute for Science of Information. In 2010, he became the Director of the NSF Science and Technology Center for Science of Information. He is currently the Saul Rosen Distinguished Professor of computer science with Purdue University, where he teaches and conducts research in the analysis of algorithms, information theory, analytic combinatorics, random structures, and machine learning for classical and quantum data. He has published two books, including Average Case Analysis of Algorithms on Sequences (John Wiley and Sons, 2001) and Analytic Pattern Matching: From DNA to Twitter (Cambridge, 2015). He is finishing now with M. Drmota his third book Analytic Information Theory: From Compression to Learning (Cambridge). He is an Erskine Fellow. In 2010, he received the Humboldt Research Award and the 2015 Inaugural Arden L. Bement Jr. Award. In 2020, he was a recipient of the Flajolet Lecture Prize. In 2021, he was elected to the Academia Europaea.
Wojciech Szpankowski (Fellow, IEEE) held several positions as a Visiting Professor/Visiting Scholar with McGill University; INRIA; Stanford University; Hewlett-Packard Labs; Universite de Versailles; the University of Canterbury, New Zealand; Ecole Polytechnique, France; the Newton Institute, Cambridge, U.K.; ETH Zürich; Hawaii University; the Gdansk University of Technology; and Jagiellonian University, Cracow, Poland. In 2008, he launched the Interdisciplinary Institute for Science of Information. In 2010, he became the Director of the NSF Science and Technology Center for Science of Information. He is currently the Saul Rosen Distinguished Professor of computer science with Purdue University, where he teaches and conducts research in the analysis of algorithms, information theory, analytic combinatorics, random structures, and machine learning for classical and quantum data. He has published two books, including Average Case Analysis of Algorithms on Sequences (John Wiley and Sons, 2001) and Analytic Pattern Matching: From DNA to Twitter (Cambridge, 2015). He is finishing now with M. Drmota his third book Analytic Information Theory: From Compression to Learning (Cambridge). He is an Erskine Fellow. In 2010, he received the Humboldt Research Award and the 2015 Inaugural Arden L. Bement Jr. Award. In 2020, he was a recipient of the Flajolet Lecture Prize. In 2021, he was elected to the Academia Europaea.View more