# “How hard is my MDP?” Distribution-norm to the rescue.

Odalric-Ambrym Maillard, Timothy A. Mann, Shie Mannor.
In advances in Neural Information Processing Systems, 2014.

 Abstract: In Reinforcement Learning (RL), state-of-the-art algorithms require a large number of samples per state-action pair to estimate the transition kernel p. In many problems, a good approximation of p is not needed. For instance, if from one state-action pair (s,a), one can only transit to states with the same value, learning p(⋅|s,a) accurately is irrelevant (only its support matters). This paper aims at capturing such behavior by defining a novel hardness measure for Markov Decision Processes (MDPs) we call the distribution-norm. The distribution-norm w.r.t. a measure ν is defined on zero ν-mean functions f by the standard variation of f with respect to ν. We first provide a concentration inequality for the dual of the distribution-norm. This allows us to replace the generic but loose ||⋅||1 concentration inequalities used in most previous analysis of RL algorithms, to benefit from this new hardness measure. We then show that several common RL benchmarks have low hardness when measured using the new norm. The distribution-norm captures finer properties than the number of states or the diameter and can be used to assess the difficulty of MDPs.

You can dowload the paper from the NIPS website (here) or from the HAL online open depository* (soon).

 Bibtex: @incollection{MaiManMan14, title = {How hard is my MDP?” The distribution-norm to the rescue”}, author = {Maillard, Odalric-Ambrym and Mann, Timothy A and Mannor, Shie}, booktitle = {Advances in Neural Information Processing Systems 27}, editor = {Z. Ghahramani and M. Welling and C. Cortes and N.D. Lawrence and K.Q. Weinberger}, pages = {1835–1843}, year = {2014}, publisher = {Curran Associates, Inc.}, url = {http://papers.nips.cc/paper/5441-how-hard-is-my-mdp-the-distribution-norm-to-the-rescue.pdf} }

# Selecting Near-Optimal Approximate State Representations in Reinforcement Learning.

Ronald Ortner, Odalric-Ambrym Maillard, Daniil Ryabko.
In Algorithmic Learning Theory, 2014.

 Abstract: We consider a reinforcement learning setting introduced in (Maillard et al., NIPS 2011) where the learner does not have explicit access to the states of the underlying Markov decision process (MDP). Instead, she has access to several models that map histories of past interactions to states. Here we improve over known regret bounds in this setting, and more importantly generalize to the case where the models given to the learner do not contain a true model resulting in an MDP representation but only approximations of it. We also give improved error bounds for state aggregation

You can dowload the paper from the  ALT Website (here) or from the HAL online open depository* (here).

 Bibtex: @inproceedings{ortner2014selecting, title={Selecting near-optimal approximate state representations in reinforcement learning}, author={Ortner, Ronald and Maillard, Odalric-Ambrym and Ryabko, Daniil}, booktitle={Algorithmic Learning Theory}, pages={140--154}, year={2014}, organization={Springer} }
 Related Publications: Competing with an infinite set of models in reinforcement learning, Phuong Nguyen, Odalric-Ambrym Maillard, Daniil Ryabko, and Ronald Ortner, in Proceedings of the International Conference on Artificial Intelligence and Statistics (AI&STATS), volume 31 of JMLR W&CP , pages 463–471, Arizona, USA, 2013. Optimal regret bounds for selecting the state representation in reinforcement learning. Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, Daniil Ryabko. In Proceedings of the 30th international conference on machine learning, ICML 2013, 2013. Selecting the State-Representation in Reinforcement Learning. Odalric-Ambrym Maillard, Daniil Ryabko, Rémi Munos. In Proceedings of the 24th conference on advances in Neural Information Processing Systems, NIPS ’11, pages 2627–2635, 2011.

# Sub-sampling for multi-armed bandits.

Discussing articles

Akram Baransi, Odalric-Ambrym Maillard, Shie Mannor.
In European conference on Machine Learning, 2014.

 Abstract: The stochastic multi-armed bandit problem is a popular model of the exploration/exploitation trade-off in sequential decision problems. We introduce a novel algorithm that is based on sub-sampling. Despite its simplicity, we show that the algorithm demonstrates excellent empirical performances against state-of-the-art algorithms, including Thompson sampling and KL-UCB. The algorithm is very flexible, it does need to know a set of reward distributions in advance nor the range of the rewards. It is not restricted to Bernoulli distributions and is also invariant under rescaling of the rewards. We provide a detailed experimental study comparing the algorithm to the state of the art, the main intuition that explains the striking results, and conclude with a finite-time regret analysis for this algorithm in the simplified two-arm bandit setting.

You can dowload the paper from the ECML website (here) or from the HAL online open depository* (here).

 Bibtex: @incollection{baransi2014sub, title={Sub-sampling for Multi-armed Bandits}, author={Baransi, Akram and Maillard, Odalric-Ambrym and Mannor, Shie}, booktitle={Machine Learning and Knowledge Discovery in Databases}, pages={115–131}, year={2014}, publisher={Springer} }

# Concentration inequalities for sampling without replacement.

Rémi Bardenet, Odalric-Ambrym Maillard.
In Bernoulli Journal, 2014.

 Abstract: Concentration inequalities quantify the deviation of a random variable from a fixed value. In spite of numerous applications, such as opinion surveys or ecological counting procedures, few concentration results are known for the setting of sampling without replacement from a finite population. Until now, the best general concentration inequality has been a Hoeffding inequality due to Serfling (1974). In this paper, we first improve on the fundamental result of Serfling (1974), and further extend it to obtain a Bernstein concentration bound for sampling without replacement. We then derive an empirical version of our bound that does not require the variance to be known to the user.

You can dowload the paper from the Bernoulli website (here) or from the HAL online open depository* (here).

 Bibtex: (soon)

# Latent Bandits.

Odalric-Ambrym Maillard, Shie Mannor
In International Conference on Machine Learning, 2014.

 Abstract: We consider a multi-armed bandit problem where the reward distributions are indexed by two sets –one for arms, one for type– and can be partitioned into a small number of clusters according to the type. First,we consider the setting where all reward distributions are known and all types have the same underlying cluster, the type’s identity is, however, unknown. Second, we study the case where types may come from different classes, which is significantly more challenging. Finally, we tackle the case where the reward distributions are completely unknown. In each setting, we introduce specific algorithms and derive non-trivial regret performance. Numerical experiments show that, in the most challenging agnostic case, the proposed algorithm achieves excellent performance in several difficult scenarios.

You can dowload the paper from the JMLR website (here) or from the HAL online open depository* (here).

You can download the Java code used to generate the experiments here.

 Bibtex: @inproceedings{maillard2014latent, title={Latent Bandits.}, author={Maillard, Odalric-Ambrym and Mannor, Shie}, booktitle={Proceedings of The 31st International Conference on Machine Learning}, pages={136--144}, year={2014} }

# Robust Risk-averse Multi-armed Bandits.

Odalric-Ambrym Maillard.
In Algorithmic Learning Theory, 2013.

 Abstract: We study a variant of the standard stochastic multi-armed bandit problem when one is not interested in the arm with the best mean, but instead in the arm maximizing some coherent risk measure criterion. Further, we are studying the deviations of the regret instead of the less informative expected regret. We provide an algorithm, called RA-UCB to solve this problem, together with a high probability bound on its regret.

You can dowload the paper from the ALT website (here) or from the HAL online open depository* (here).

 Bibtex: @incollection{Maillard2013, year={2013}, isbn={978-3-642-40934-9}, booktitle={Algorithmic Learning Theory}, volume={8139}, series={Lecture Notes in Computer Science}, editor={Jain, Sanjay and Munos, Rémi and Stephan, Frank and Zeugmann, Thomas}, title={Robust Risk-Averse Stochastic Multi-armed Bandits}, publisher={Springer Berlin Heidelberg}, author={Maillard, Odalric-Ambrym}, pages={218-233} }
 Related Publications: Kullback-Leibler Upper Confidence Bounds for  Optimal Sequential Allocation. Olivier Cappé, Aurelien Garivier, Odalric-Ambrym Maillard, Rémi Munos, Gilles Stoltz. In The Annals of Statistics, 2013. Finite-Time Analysis of Multi-armed Bandits Problems with Kullback-Leibler Divergences. Odalric-Ambrym Maillard, Gilles Stoltz, Rémi Munos. In Proceedings of the 24th annual Conference On Learning Theory, COLT, 2011.

# Kullback-Leibler Upper Confidence Bounds for Optimal Sequential Allocation.

Olivier Cappé, Aurélien Garivier,
Odalric-Ambrym Maillard,
Rémi Munos, Gilles Stoltz.
In The Annals of Statistics, 2013.

 Abstract: We consider optimal sequential allocation in the context of the so-called stochastic multi-armed bandit model. We describe a generic index policy, in the sense of  Gittins (1979), based on upper confidence bounds of the arm payoffs computed using the Kullback-Leibler divergence. We consider two classes of distributions for which instances of this general idea are analyzed: The kl-UCB algorithm is designed for one-parameter exponential families and the empirical KL-UCB algorithm for bounded and finitely supported distributions. Our main contribution is a unified finite-time analysis of the regret of these algorithms that asymptotically matches the lower bounds of Lai and Robbins (1985) and Burnetas et Katehakis (1996), respectively. We also investigate the behavior of these algorithms when used with general bounded rewards, showing in particular that they provide significant improvements over the state-of-the-art.

You can dowload the paper from the Annals of Statistics (here) or from the HAL online open depository* (here).

 Bibtex: @article{CaGaMaMuSt2013, AUTHOR = {Olivier Capp\'{e}, Aur\'{e}lien Garivier, Odalric-Ambrym Maillard, R\'{e}mi Munos, Gilles Stoltz}, TITLE = {Kullback–Leibler upper confidence bounds for optimal sequential allocation}, JOURNAL = {Ann. Statist.}, FJOURNAL = {Annals of Statistics}, YEAR = {2013}, VOLUME = {41}, NUMBER = {3}, PAGES = {1516-1541}, ISSN = {0090-5364}, DOI = {10.1214/13-AOS1119} }
 Related Publications: Finite-Time Analysis of Multi-armed Bandits Problems with Kullback-Leibler Divergences. Odalric-Ambrym Maillard, Gilles Stoltz, Rémi Munos. In Proceedings of the 24th annual Conference On Learning Theory, COLT, 2011.

# Competing with an Infinite Set of Models in Reinforcement Learning.

Phuong Nguyen, Odalric-Ambrym Maillard, Daniil Ryabko,Ronald Ortner.
In International Conference on Artificial Intelligence and Statistics, 2013.

 Abstract: We consider a reinforcement learning setting where the learner also has to deal with the problem of finding a suitable state-representation function from a given set of models. This has to be done while interacting with the environment in an online fashion (no resets), and the goal is to have small regret with respect to any Markov model in the set. For this setting, recently the BLB algorithm has been proposed, which achieves regret of order T^{2/3}, provided that the given set of models is finite. Our first contribution is to extend this result to a countably infinite set of models. Moreover, the BLB regret bound suffers from an additive term that can be exponential in the diameter of the MDP involved, since the diameter has to be guessed. The algorithm we propose avoids guessing the diameter, thus improving the regret bound.

You can dowload the paper from the JMLR website (here) or from the HAL online open depository* (soon).

 Bibtex: @InProceedings{Nguyen13, author = “Nguyen, P. and Maillard, O. and Ryabko, D. and Ortner, R. “, title = “Competing with an Infinite Set of Models in Reinforcement Learning”, booktitle = “AISTATS”, series = {JMLR W\&CP 31}, address = “Arizona, USA”, year = “2013”, pages = “463–471” }
 Related Publications: Optimal regret bounds for selecting the state representation in reinforcement learning. Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, Daniil Ryabko. In Proceedings of the 30th international conference on machine learning, ICML 2013, 2013. Selecting the State-Representation in Reinforcement Learning. Odalric-Ambrym Maillard, Daniil Ryabko, Rémi Munos. In Proceedings of the 24th conference on advances in Neural Information Processing Systems, NIPS ’11, pages 2627–2635, 2011.

# Optimal regret bounds for selecting the state representation in reinforcement learning.

Odalric-Ambrym Maillard, Phuong Nguyen, Ronald Ortner, Daniil Ryabko.
In Proceedings of the 30th international conference on machine learning, ICML 2013, 2013.

 Abstract: We consider an agent interacting with an environment in a single stream of actions, observations, and rewards, with no reset. This process is not assumed to be a Markov Decision Process (MDP). Rather, the agent has several representations (mapping histories of past interactions to a discrete state space) of the environment with unknown dynamics, only some of which result in an MDP. The goal is to minimize the average regret criterion against an agent who knows an MDP representation giving the highest optimal reward, and acts optimally in it. Recent regret bounds for this setting are of order O(T^{2/3}) with an additive term constant yet exponential in some characteristics of the optimal MDP. We propose an algorithm whose regret after T time steps is O(\sqrt{T}), with all constants reasonably small. This is optimal in T since O(\sqrt{T}) is the optimal regret in the setting of learning in a (single discrete) MDP.

You can dowload the paper from the ICML website (here), and a corrected version from the HAL online open depository* (here), or from arXiv (here) (the correction is minor and changes only a constant 2 into 2\sqrt{2}). See also a talk presenting this work here.

 Bibtex: @inproceedings{MaillardNguyenOrtnerRyabko13, author = “Maillard, O. and Nguyen, P. and Ortner, R. and Ryabko, D.”, title = “Optimal Regret Bounds for Selecting the State Representation in Reinforcement Learning”, booktitle = “International conference on Machine Learning”, series = {JMLR W\&CP 28(1)}, address = “Atlanta, USA”, year = “2013”, pages = ” 543-551″ }
 Related Publications: Selecting the State-Representation in Reinforcement Learning. Odalric-Ambrym Maillard, Daniil Ryabko, Rémi Munos. In Proceedings of the 24th conference on advances in Neural Information Processing Systems, NIPS ’11, pages 2627–2635, 2011.

# Hierarchical Optimistic Region Selection driven by Curiosity.

Odalric-Ambrym Maillard.
In Proceedings of the 25th conference on advances in Neural Information Processing Systems, NIPS ’12, 2012.