Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T23:50:38.118Z Has data issue: false hasContentIssue false
  • English
  • Français

EXPLORATION–EXPLOITATION POLICIES WITH ALMOST SURE, ARBITRARILY SLOW GROWING ASYMPTOTIC REGRET

Part of: Nonparametric inference

Published online by Cambridge University Press:  26 January 2019

Wesley Cowan  and
Michael N. Katehakis
Wesley Cowan
Affiliation:
Department of Computer Science, Rutgers University, Piscataway, NJ08854, USA E-mail: cwcowan@math.rutgers.edu
Michael N. Katehakis
Affiliation:
Department of Management Science and Information Systems, Rutgers University, Piscataway, NJ08854, USA E-mail: mnk@rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to provide further understanding into the structure of the sequential allocation (“stochastic multi-armed bandit”) problem by establishing probability one finite horizon bounds and convergence rates for the sample regret associated with two simple classes of allocation policies. For any slowly increasing function g, subject to mild regularity constraints, we construct two policies (the g-Forcing, and the g-Inflated Sample Mean) that achieve a measure of regret of order O(g(n)) almost surely as n → ∞, bound from above and below. Additionally, almost sure upper and lower bounds on the remainder term are established. In the constructions herein, the function g effectively controls the “exploration” of the classical “exploration/exploitation” tradeoff.

Keywords

banditsforcing actionsinflated sample meansmulti-armedonline learningsequential allocationupper confidence bounds

MSC classification

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Special Issue 3: Analytic Methods in Health Care and in Clinical Trials In Honor and Memory of Lycourgos (Lee) Papayanopoulos , July 2020 , pp. 406 - 428
Copyright
Copyright © Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Audibert, J-Y, Munos, R., & Szepesvári, C. (2009). Exploration - exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science 410: 1876–1902.CrossRefGoogle Scholar
2.Auer, P., Cesa-Bianchi, N., & Fischer, P. (2002). Finite-time analysis of the multiarmed bandit problem. Machine Learning 47: 235–256.CrossRefGoogle Scholar
3.Bubeck, S. & Cesa-Bianchi, N. (2012). Regret analysis of stochastic and nonstochastic multi-armed bandit problems. arXiv preprint arXiv:1204.5721.CrossRefGoogle Scholar
4.Burnetas, A.N. & Katehakis, M.N. (1996). Optimal adaptive policies for sequential allocation problems. Advances in Applied Mathematics 17: 122–142.CrossRefGoogle Scholar
5.Cowan, W. & Katehakis, M.N. (2015a). An Asymptotically Optimal Policy for Uniform Bandits of Unknown Support. arXiv preprint: arXiv:1505.01918.Google Scholar
6.Cowan, W. & Katehakis, M.N. (2015b). Multi-armed bandits under general depreciation and commitment. Probability in the Engineering and Informational Sciences 29(1): 51–76.CrossRefGoogle Scholar
7.Cowan, W. & Katehakis, M.N. (2015c). Asymptotically Optimal Sequential Experimentation Under Generalized Ranking. arXiv preprint arXiv:1510.02041.Google Scholar
8.Cowan, W., Honda, J., & Katehakis, M.N. (2018). Normal bandits of unknown means and variances. Journal of Machine Learning Research 18(154): 1–28.Google Scholar
9.Garivier, A., Ménard, P., & Stoltz, G. (2018). Explore first, exploit next: the true shape of regret in bandit problems. Mathematics of Operations Research. doi: 10.1287/moor.2017.0928.Google Scholar
10.Honda, J. & Takemura, A (2010) An asymptotically optimal bandit algorithm for bounded support models. In COLT, 67–79, Citeseer.Google Scholar
11.Honda, J. & Takemura, A. (2011). An asymptotically optimal policy for finite support models in the multiarmed bandit problem. Machine Learning 85: 361–391.CrossRefGoogle Scholar
12.Honda, J. & Takemura, A. (2013) Optimality of Thompson sampling for Gaussian bandits depends on priors. arXiv preprint arXiv:1311.1894.Google Scholar
13.Lai, T.L. & Robbins, H.E. (1985). Asymptotically efficient adaptive allocation rules. Advances in Applied Mathematics 6: 4–2.CrossRefGoogle Scholar
14.Lattimore, L. (2018). Refining the confidence level for optimistic bandit strategies. Journal of Machine Learning Research 19: 765–796.Google Scholar
15.Orabona, F., Pál, D. (2016) Open problem: Parameter-free and scale-free online algorithms. Conference on Learning Theory, 1659–1664.Google Scholar
16.Ortner, R. (2018). Regret Bounds for Reinforcement Learning via Markov Chain Concentration. arXiv preprint arXiv:1808.01813.Google Scholar
17.Robbins, H.E. (1952). Some aspects of the sequential design of experiments. Bulletin of the American Mathematical Monthly 58: 527–536.CrossRefGoogle Scholar