Section: New Results

Decision-making Under Uncertainty

Reinforcement Learning

Selecting Near-Optimal Approximate State Representations in Reinforcement Learning [23]

We consider a reinforcement learning setting 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.

Online Stochastic Optimization under Correlated Bandit Feedback [15]

In this paper we consider the problem of online stochastic optimization of a locally smooth function under bandit feedback. We introduce the high-confidence tree (HCT) algorithm, a novel anytime X -armed bandit algorithm, and derive regret bounds matching the performance of state-of-the-art algorithms in terms of the dependency on number of steps and the near-optimality di-mension. The main advantage of HCT is that it handles the challenging case of correlated bandit feedback (reward), whereas existing meth-ods require rewards to be conditionally indepen-dent. HCT also improves on the state-of-the-art in terms of the memory requirement, as well as requiring a weaker smoothness assumption on the mean-reward function in comparison with the existing anytime algorithms. Finally, we discuss how HCT can be applied to the problem of policy search in reinforcement learning and we report preliminary empirical results.

Sparse Multi-task Reinforcement Learning [9]

In multi-task reinforcement learning (MTRL), the objective is to simultaneously learn multiple tasks and exploit their similarity to improve the performance w.r.t. single-task learning. In this paper we investigate the case when all the tasks can be accurately represented in a linear approximation space using the same small subset of the original (large) set of features. This is equivalent to assuming that the weight vectors of the task value functions are jointly sparse, i.e., the set of their non-zero components is small and it is shared across tasks. Building on existing results in multi-task regression, we develop two multi-task extensions of the fitted $Q$-iteration algorithm. While the first algorithm assumes that the tasks are jointly sparse in the given representation, the second one learns a transformation of the features in the attempt of finding a more sparse representation. For both algorithms we provide a sample complexity analysis and numerical simulations.

Multi-arm Bandit Theory

Spectral Bandits for Smooth Graph Functions with Applications in Recommender Systems [20]

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each recommended item is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens nodes evaluations.

Online combinatorial optimization with stochastic decision sets and adversarial losses [21]

Most work on sequential learning assumes a fixed set of actions that are available all the time. However, in practice, actions can consist of picking subsets of readings from sensors that may break from time to time, road segments that can be blocked or goods that are out of stock. In this paper we study learning algorithms that are able to deal with stochastic availability of such unreliable composite actions. We propose and analyze algorithms based on the Follow-The-Perturbed-Leader prediction method for several learning settings differing in the feedback provided to the learner. Our algorithms rely on a novel loss estimation technique that we call Counting Asleep Times. We deliver regret bounds for our algorithms for the previously studied full information and (semi-)bandit settings, as well as a natural middle point between the two that we call the restricted information setting. A special consequence of our results is a significant improvement of the best known performance guarantees achieved by an efficient algorithm for the sleeping bandit problem with stochastic availability. Finally, we evaluate our algorithms empirically and show their improvement over the known approaches.

Extreme bandits [10]

In many areas of medicine, security, and life sciences, we want to allocate limited resources to different sources in order to detect extreme values. In this paper, we study an efficient way to allocate these resources sequentially under limited feedback. While sequential design of experiments is well studied in bandit theory, the most commonly optimized property is the regret with respect to the maximum mean reward. However, in other problems such as network intrusion detection, we are interested in detecting the most extreme value output by the sources. Therefore, in our work we study extreme regret which measures the efficiency of an algorithm compared to the oracle policy selecting the source with the heaviest tail. We propose the ExtremeHunter algorithm, provide its analysis, and evaluate it empirically on synthetic and real-world experiments.

Efficient learning by implicit exploration in bandit problems with side observations [18]

We consider online learning problems under a a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.

Best-Arm Identification in Linear Bandits [29]

We study the best-arm identification problem in linear bandit, where the rewards of the arms depend linearly on an unknown parameter ${\theta }^{*}$ 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, we point out the connection to the $G$-optimality criterion used in optimal experimental design.

Exploiting easy data in online optimization [28]

We consider the problem of online optimization, where a learner chooses a decision from a given decision set and suffers some loss associated with the decision and the state of the environment. The learner's objective is to minimize its cumulative regret against the best fixed decision in hindsight. Over the past few decades numerous variants have been considered, with many algorithms designed to achieve sub-linear regret in the worst case. However, this level of robustness comes at a cost. Proposed algorithms are often over-conservative, failing to adapt to the actual complexity of the loss sequence which is often far from the worst case. In this paper we introduce a general algorithm that, provided with a "safe" learning algorithm and an opportunistic "benchmark", can effectively combine good worst-case guarantees with much improved performance on "easy" data. We derive general theoretical bounds on the regret of the proposed algorithm and discuss its implementation in a wide range of applications, notably in the problem of learning with shifting experts (a recent COLT open problem). Finally, we provide numerical simulations in the setting of prediction with expert advice with comparisons to the state of the art.

Spectral Bandits for Smooth Graph Functions [32]

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

Regret bounds for restless Markov bandits [5]

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 first represents the setting as an MDP which exhibits some special structural properties. In order to grasp this information we introduce the notion of $ϵ$-structured MDPs, which are a generalization of concepts like (approximate) state aggregation and MDP homomorphisms. We propose a general algorithm for learning $ϵ$-structured MDPs and show regret bounds that demonstrate that additional structural information enhances learning. Applied to the restless bandit setting, this algorithm achieves after any $T$ steps regret of order $\tilde{O}\left(T^{1/2}\right)$ with respect to the best policy that knows the distributions of all arms. We make no assumptions on the Markov chains underlying each arm except that they are irreducible. In addition, we show that index-based policies are necessarily suboptimal for the considered problem.

Spectral Thompson Sampling [19]

Thompson Sampling (TS) has surged a lot of interest due to its good empirical performance, in particular in the computational advertising. Though successful, the tools for its performance analysis appeared only recently. In this paper, we describe and analyze SpectralTS algorithm for a bandit problem, where the payoffs of the choices are smooth given an underlying graph. In this setting, each choice is a node of a graph and the expected payoffs of the neighboring nodes are assumed to be similar. Although the setting has application both in recommender systems and advertising, the traditional algorithms would scale poorly with the number of choices. For that purpose we consider an effective dimension d, which is small in real-world graphs. We deliver the analysis showing that the regret of SpectralTS scales as $d{\left(T\ln N\right)}^{1/2}$ with high probability, where T is the time horizon and N is the number of choices. Since a d sqrt(T ln N) regret is comparable to the known results, SpectralTS offers a computationally more efficient alternative. We also show that our algorithm is competitive on both synthetic and real-world data.

Recommendation systems

User Engagement as Evaluation: a Ranking or a Regression Problem? [39]

In this paper, we describe the winning approach used on the RecSys Challenge 2014 which focuses on employing user en-gagement as evaluation of recommendations. On one hand, we regard the challenge as a ranking problem and apply the LambdaMART algorithm, which is a listwise model special-ized in a Learning To Rank approach. On the other hand, after noticing some specific characteristics of this challenge, we also consider it as a regression problem and use pointwise regression models such as Random Forests. We compare how these different methods can be modified or combined to improve the accuracy and robustness of our model and we draw the advantages or disadvantages of each approach.

Improving offline evaluation of contextual bandit algorithms via bootstrapping techniques [22]

In many recommendation applications such as news recommendation, the items that can be recommended come and go at a very fast pace. This is a challenge for recommender systems (RS) to face this setting. Online learning algorithms seem to be the most straight forward solution. The contextual bandit framework was introduced for that very purpose. In general the evaluation of a RS is a critical issue. Live evaluation is often avoided due to the potential loss of revenue, hence the need for offline evaluation methods. Two options are available. Model based meth- ods are biased by nature and are thus difficult to trust when used alone. Data driven methods are therefore what we consider here. Evaluat- ing online learning algorithms with past data is not simple but some methods exist in the litera- ture. Nonetheless their accuracy is not satisfac- tory mainly due to their mechanism of data re- jection that only allow the exploitation of a small fraction of the data. We precisely address this issue in this paper. After highlighting the limita- tions of the previous methods, we present a new method, based on bootstrapping techniques. This new method comes with two important improve- ments: it is much more accurate and it provides a measure of quality of its estimation. The latter is a highly desirable property in order to minimize the risks entailed by putting online a RS for the first time. We provide both theoretical and ex- perimental proofs of its superiority compared to state-of-the-art methods, as well as an analysis of the convergence of the measure of quality.

Bandits Warm-up Cold Recommender Systems [35]

We address the cold start problem in recommendation systems assuming no contextual information is available neither about users, nor items. We consider the case in which we only have access to a set of ratings of items by users. Most of the existing works consider a batch setting, and use cross-validation to tune parameters. The classical method consists in minimizing the root mean square error over a training subset of the ratings which provides a factorization of the matrix of ratings, interpreted as a latent representation of items and users. Our contribution in this paper is 5-fold. First, we explicit the issues raised by this kind of batch setting for users or items with very few ratings. Then, we propose an online setting closer to the actual use of recommender systems; this setting is inspired by the bandit framework. The proposed methodology can be used to turn any recommender system dataset (such as Netflix, MovieLens,...) into a sequential dataset. Then, we explicit a strong and insightful link between contextual bandit algorithms and matrix factorization; this leads us to a new algorithm that tackles the exploration/exploitation dilemma associated to the cold start problem in a strikingly new perspective. Finally, experimental evidence confirm that our algorithm is effective in dealing with the cold start problem on publicly available datasets. Overall, the goal of this paper is to bridge the gap between recommender systems based on matrix factorizations and those based on contextual bandits.

Nonparametric statistics of time series

Uniform hypothesis testing for finite-valued stationary processes [6]

Given a discrete-valued sample $X_1,\cdots ,X_n$ we wish to decide whether it was generated by a distribution belonging to a family $H_0$, or it was generated by a distribution belonging to a family $H_1$. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each $ϵ$ there exist a sample size $n$ such that probability of error is upper-bounded by $ϵ$ for samples longer than $n$. We find some necessary and some sufficient conditions on $H_0$ and $H_1$ under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance.

Asymptotically consistent estimation of the number of change points in highly dependent time series [17]

The problem of change point estimation is considered in a general framework where the data are generated by arbitrary unknown stationary ergodic process distributions. This means that the data may have long-range dependencies of an arbitrary form. In this context the consistent estimation of the number of change points is provably impossible. A formulation is proposed which overcomes this obstacle: it is possible to find the correct number of change points at the expense of introducing the additional constraint that the correct number of process distributions that generate the data is provided. This additional parameter has a natural interpretation in many real-world applications. It turns out that in this formulation change point estimation can be reduced to time series clustering. Based on this reduction, an algorithm is proposed that finds the number of change points and locates the changes. This algorithm is shown to be asymptotically consistent. The theoretical results are complemented with empirical evaluations.