arXiv subjects

Rémi Munos

Publications and source records attributed to Rémi Munos.

At least 91 records · Page 5Linked to original sources

arXiv2017

Minimax Regret Bounds for Reinforcement Learning

We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of $\tilde{O}( \sqrt{HSAT} + H^2S^2A+H\sqrt{T})$ where $H$ is the time horizon, $S$ the number of states, $A$ the number of actions and $T$ the number of time-steps. This result improves over the best previous known bound $\tilde{O}(HS \sqrt{AT})$ achieved by the UCRL2 algorithm of Jaksch et al., 2010. The key significance of our new results is that when $T\geq H^3S^3A$ and $SA\geq H$, it leads to a regret of $\tilde{O}(\sqrt{HSAT})$ that matches the established lower bound of $Ω(\sqrt{HSAT})$ up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in $S$), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in $H$).

stat.ML↗

arXiv2017

Observational Learning by Reinforcement Learning

Observational learning is a type of learning that occurs as a function of observing, retaining and possibly replicating or imitating the behaviour of another agent. It is a core mechanism appearing in various instances of social learning and has been found to be employed in several intelligent species, including humans. In this paper, we investigate to what extent the explicit modelling of other agents is necessary to achieve observational learning through machine learning. Especially, we argue that observational learning can emerge from pure Reinforcement Learning (RL), potentially coupled with memory. Through simple scenarios, we demonstrate that an RL agent can leverage the information provided by the observations of an other agent performing a task in a shared environment. The other agent is only observed through the effect of its actions on the environment and never explicitly modeled. Two key aspects are borrowed from observational learning: i) the observer behaviour needs to change as a result of viewing a 'teacher' (another agent) and ii) the observer needs to be motivated somehow to engage in making use of the other agent's behaviour. The later is naturally modeled by RL, by correlating the learning agent's reward with the teacher agent's behaviour.

cs.LG↗

arXiv2017

The Cramer Distance as a Solution to Biased Wasserstein Gradients

The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demonstrated, among others, in ordinal regression and generative modelling. In this paper we describe three natural properties of probability divergences that reflect requirements from machine learning: sum invariance, scale sensitivity, and unbiased sample gradients. The Wasserstein metric possesses the first two properties but, unlike the Kullback-Leibler divergence, does not possess the third. We provide empirical evidence suggesting that this is a serious issue in practice. Leveraging insights from probabilistic forecasting we propose an alternative to the Wasserstein metric, the Cramér distance. We show that the Cramér distance possesses all three desired properties, combining the best of the Wasserstein and Kullback-Leibler divergences. To illustrate the relevance of the Cramér distance in practice we design a new algorithm, the Cramér Generative Adversarial Network (GAN), and show that it performs significantly better than the related Wasserstein GAN.

cs.LG↗

arXiv2016

Safe and Efficient Off-Policy Reinforcement Learning

In this work, we take a fresh look at some old and new algorithms for off-policy, return-based reinforcement learning. Expressing these in a common form, we derive a novel algorithm, Retrace($λ$), with three desired properties: (1) it has low variance; (2) it safely uses samples collected from any behaviour policy, whatever its degree of "off-policyness"; and (3) it is efficient as it makes the best use of samples collected from near on-policy behaviour policies. We analyze the contractive nature of the related operator under both off-policy policy evaluation and control settings and derive online sample-based algorithms. We believe this is the first return-based off-policy control algorithm converging a.s. to $Q^*$ without the GLIE assumption (Greedy in the Limit with Infinite Exploration). As a corollary, we prove the convergence of Watkins' Q($λ$), which was an open problem since 1989. We illustrate the benefits of Retrace($λ$) on a standard suite of Atari 2600 games.

cs.LG↗

arXiv2015

Increasing the Action Gap: New Operators for Reinforcement Learning

This paper introduces new optimality-preserving operators on Q-functions. We first describe an operator for tabular representations, the consistent Bellman operator, which incorporates a notion of local policy consistency. We show that this local consistency leads to an increase in the action gap at each state; increasing this gap, we argue, mitigates the undesirable effects of approximation and estimation errors on the induced greedy policies. This operator can also be applied to discretized continuous space and time problems, and we provide empirical results evidencing superior performance in this context. Extending the idea of a locally consistent operator, we then derive sufficient conditions for an operator to preserve optimality, leading to a family of operators which includes our consistent Bellman operator. As corollaries we provide a proof of optimality for Baird's advantage learning algorithm and derive other gap-increasing operators with interesting properties. We conclude with an empirical study on 60 Atari 2600 games illustrating the strong potential of these new operators.

cs.AI↗

arXiv2015

Upper-Confidence-Bound Algorithms for Active Learning in Multi-Armed Bandits

In this paper, we study the problem of estimating uniformly well the mean values of several distributions given a finite budget of samples. If the variance of the distributions were known, one could design an optimal sampling strategy by collecting a number of independent samples per distribution that is proportional to their variance. However, in the more realistic case where the distributions are not known in advance, one needs to design adaptive sampling strategies in order to select which distribution to sample from according to the previously observed samples. We describe two strategies based on pulling the distributions a number of times that is proportional to a high-probability upper-confidence-bound on their variance (built from previous observed samples) and report a finite-sample performance analysis on the excess estimation error compared to the optimal allocation. We show that the performance of these allocation strategies depends not only on the variances but also on the full shape of the distributions.

cs.LG↗

arXiv2014

Fast gradient descent for drifting least squares regression, with application to bandits

Online learning algorithms require to often recompute least squares regression estimates of parameters. We study improving the computational complexity of such algorithms by using stochastic gradient descent (SGD) type schemes in place of classic regression solvers. We show that SGD schemes efficiently track the true solutions of the regression problems, even in the presence of a drift. This finding coupled with an $O(d)$ improvement in complexity, where $d$ is the dimension of the data, make them attractive for implementation in the big data settings. In the case when strong convexity in the regression problem is guaranteed, we provide bounds on the error both in expectation and high probability (the latter is often needed to provide theoretical guarantees for higher level algorithms), despite the drifting least squares solution. As an example of this case we prove that the regret performance of an SGD version of the PEGE linear bandit algorithm [Rusmevichientong and Tsitsiklis 2010] is worse that that of PEGE itself only by a factor of $O(\log^4 n)$. When strong convexity of the regression problem cannot be guaranteed, we investigate using an adaptive regularisation. We make an empirical study of an adaptively regularised, SGD version of LinUCB [Li et al. 2010] in a news article recommendation application, which uses the large scale news recommendation dataset from Yahoo! front page. These experiments show a large gain in computational complexity, with a consistently low tracking error and click-through-rate (CTR) performance that is $75\%$ close.

cs.LG↗

arXiv2014

Best-Arm Identification in Linear Bandits

We study the best-arm identification problem in linear bandit, where the rewards of the arms depend linearly on an unknown parameter $θ^*$ and the objective is to return the arm with the largest reward. We characterize the complexity of the problem and introduce sample allocation strategies that pull arms to identify the best arm with a fixed confidence, while minimizing the sample budget. In particular, we show the importance of exploiting the global linear structure to improve the estimate of the reward of near-optimal arms. We analyze the proposed strategies and compare their empirical performance. Finally, as a by-product of our analysis, we point out the connection to the $G$-optimality criterion used in optimal experimental design.

cs.LG↗

arXiv2013

Kullback-Leibler upper confidence bounds for optimal sequential allocation

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 [J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 148-177], 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 [Adv. in Appl. Math. 6 (1985) 4-22] and Burnetas and Katehakis [Adv. in Appl. Math. 17 (1996) 122-142], 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.

math.PR↗

arXiv2013

Selecting the State-Representation in Reinforcement Learning

The problem of selecting the right state-representation in a reinforcement learning problem is considered. Several models (functions mapping past observations to a finite set) of the observations are given, and it is known that for at least one of these models the resulting state dynamics are indeed Markovian. Without knowing neither which of the models is the correct one, nor what are the probabilistic characteristics of the resulting MDP, it is required to obtain as much reward as the optimal policy for the correct model (or for the best of the correct models, if there are several). We propose an algorithm that achieves that, with a regret of order T^{2/3} where T is the horizon time.

cs.LG↗

arXiv2013

Risk-Aversion in Multi-armed Bandits

Stochastic multi-armed bandits solve the Exploration-Exploitation dilemma and ultimately maximize the expected reward. Nonetheless, in many practical problems, maximizing the expected reward is not the most desirable objective. In this paper, we introduce a novel setting based on the principle of risk-aversion where the objective is to compete against the arm with the best risk-return trade-off. This setting proves to be intrinsically more difficult than the standard multi-arm bandit setting due in part to an exploration risk which introduces a regret associated to the variability of an algorithm. Using variance as a measure of risk, we introduce two new algorithms, investigate their theoretical guarantees, and report preliminary empirical results.

cs.LG↗

arXiv2012

Adaptive Stratified Sampling for Monte-Carlo integration of Differentiable functions

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are well spread on the domain (this notion shares similitude with low discrepancy). We prove that the estimate returned by the algorithm is almost similarly accurate as the estimate that an optimal oracle strategy (that would know the variations of the function everywhere) would return, and provide a finite-sample analysis.

stat.ML↗

arXiv2012

Regret Bounds for Restless Markov Bandits

We consider the restless Markov bandit problem, in which the state of each arm evolves according to a Markov process independently of the learner's actions. We suggest an algorithm that after $T$ steps achieves $\tilde{O}(\sqrt{T})$ regret with respect to the best policy that knows the distributions of all arms. No assumptions on the Markov chains are made except that they are irreducible. In addition, we show that index-based policies are necessarily suboptimal for the considered problem.

cs.LG↗

arXiv2012

Thompson Sampling: An Asymptotically Optimal Finite Time Analysis

The question of the optimality of Thompson Sampling for solving the stochastic multi-armed bandit problem had been open since 1933. In this paper we answer it positively for the case of Bernoulli rewards by providing the first finite-time analysis that matches the asymptotic rate given in the Lai and Robbins lower bound for the cumulative regret. The proof is accompanied by a numerical comparison with other optimal policies, experiments that have been lacking in the literature until now for the Bernoulli case.

stat.ML↗

arXiv2012

Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit

We consider a linear stochastic bandit problem where the dimension $K$ of the unknown parameter $θ$ is larger than the sampling budget $n$. In such cases, it is in general impossible to derive sub-linear regret bounds since usual linear bandit algorithms have a regret in $O(K\sqrt{n})$. In this paper we assume that $θ$ is $S-$sparse, i.e. has at most $S-$non-zero components, and that the space of arms is the unit ball for the $||.||_2$ norm. We combine ideas from Compressed Sensing and Bandit Theory and derive algorithms with regret bounds in $O(S\sqrt{n})$.

math.ST↗

arXiv2012

Minimax Number of Strata for Online Stratified Sampling given Noisy Samples

We consider the problem of online stratified sampling for Monte Carlo integration of a function given a finite budget of $n$ noisy evaluations to the function. More precisely we focus on the problem of choosing the number of strata $K$ as a function of the budget $n$. We provide asymptotic and finite-time results on how an oracle that has access to the function would choose the partition optimally. In addition we prove a \textit{lower bound} on the learning rate for the problem of stratified Monte-Carlo. As a result, we are able to state, by improving the bound on its performance, that algorithm MC-UCB, defined in \citep{MC-UCB}, is minimax optimal both in terms of the number of samples n and the number of strata K, up to a $\sqrt{\log(nK)}$. This enables to deduce a minimax optimal bound on the difference between the performance of the estimate outputted by MC-UCB, and the performance of the estimate outputted by the best oracle static strategy, on the class of Hölder continuous functions, and upt to a $\sqrt{\log(n)}$.

math.ST↗

arXiv2011

A Finite-Time Analysis of Multi-armed Bandits Problems with Kullback-Leibler Divergences

We consider a Kullback-Leibler-based algorithm for the stochastic multi-armed bandit problem in the case of distributions with finite supports (not necessarily known beforehand), whose asymptotic regret matches the lower bound of \cite{Burnetas96}. Our contribution is to provide a finite-time analysis of this algorithm; we get bounds whose main terms are smaller than the ones of previously known algorithms with finite-time analyses (like UCB-type algorithms).

math.ST↗

arXiv2011

X-Armed Bandits

We consider a generalization of stochastic bandits where the set of arms, $\cX$, is allowed to be a generic measurable space and the mean-payoff function is "locally Lipschitz" with respect to a dissimilarity function that is known to the decision maker. Under this condition we construct an arm selection policy, called HOO (hierarchical optimistic optimization), with improved regret bounds compared to previous results for a large class of problems. In particular, our results imply that if $\cX$ is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally continuous with a known smoothness degree, then the expected regret of HOO is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of growth of the regret is independent of the dimension of the space. We also prove the minimax optimality of our algorithm when the dissimilarity is a metric. Our basic strategy has quadratic computational complexity as a function of the number of time steps and does not rely on the doubling trick. We also introduce a modified strategy, which relies on the doubling trick but runs in linearithmic time. Both results are improvements with respect to previous approaches.

cs.LG↗