Section: New Results

Decision-making Under Uncertainty

Reinforcement Learning

Nonparametric multiple change point estimation in highly dependent time series [7]

Given a heterogeneous time-series sample, the objective is to find points in time, called change points, where the probability distribution generating the data has changed. The data are assumed to have been generated by arbitrary unknown stationary ergodic distributions. No modelling, independence or mixing assumptions are made. A novel, computationally efficient, nonparametric method is proposed, and is shown to be asymptotically consistent in this general framework. The theoretical results are complemented with experimental evaluations.

Explore no more: Improved high-probability regret bounds for non-stochastic bandits [26]

This work addresses the problem of regret minimization in non-stochastic multi-armed bandit problems, focusing on performance guarantees that hold with high probability. Such results are rather scarce in the literature since proving them requires a large deal of technical effort and significant modifications to the standard, more intuitive algorithms that come only with guarantees that hold on expectation. One of these modifications is forcing the learner to sample arms from the uniform distribution at least Ω( T) times over T rounds, which can adversely affect performance if many of the arms are suboptimal. While it is widely conjectured that this property is essential for proving high-probability regret bounds, we show in this paper that it is possible to achieve such strong results without this undesirable exploration component. Our result relies on a simple and intuitive loss-estimation strategy called Implicit eXploration (IX) that allows a remarkably clean analysis. To demonstrate the flexibility of our technique, we derive several improved high-probability bounds for various extensions of the standard multi-armed bandit framework. Finally, we conduct a simple experiment that illustrates the robustness of our implicit exploration technique.

First-order regret bounds for combinatorial semi-bandits [27]

We consider the problem of online combinatorial optimization under semi-bandit feedback, where a learner has to repeatedly pick actions from a combinatorial decision set in order to minimize the total losses associated with its decisions. After making each decision, the learner observes the losses associated with its action, but not other losses. For this problem, there are several learning algorithms that guarantee that the learner's expected regret grows as O( T) with the number of rounds T. In this paper, we propose an algorithm that improves this scaling to O( L * T), where L * T is the total loss of the best action. Our algorithm is among the first to achieve such guarantees in a partial-feedback scheme, and the first one to do so in a combinatorial setting.

Random-Walk Perturbations for Online Combinatorial Optimization [4]

We study online combinatorial optimization problems where a learner is interested in minimizing its cumulative regret in the presence of switching costs. To solve such problems, we propose a version of the follow-the-perturbed-leader algorithm in which the cumulative losses are perturbed by independent symmetric random walks. In the general setting, our forecaster is shown to enjoy near-optimal guarantees on both quantities of interest, making it the best known efficient algorithm for the studied problem. In the special case of prediction with expert advice, we show that the forecaster achieves an expected regret of the optimal order O( n log N) where n is the time horizon and N is the number of experts, while guaranteeing that the predictions are switched at most O( n log N) times, in expectation.

Qualitative Multi-Armed Bandits: A Quantile-Based Approach [32]

We formalize and study the multi-armed bandit (MAB) problem in a generalized stochastic setting, in which rewards are not assumed to be numerical. Instead, rewards are measured on a qualitative scale that allows for comparison but invalidates arithmetic operations such as averaging. Correspondingly, instead of characterizing an arm in terms of the mean of the underlying distribution, we opt for using a quantile of that distribution as a representative value. We address the problem of quantile-based online learning both for the case of a finite (pure exploration) and infinite time horizon (cumulative regret minimization). For both cases, we propose suitable algorithms and analyze their properties. These properties are also illustrated by means of first experimental studies.

Predicting the outcomes of every process for which an asymptotically accurate stationary predictor exists is impossible [30]

The problem of prediction consists in forecasting the conditional distribution of the next outcome given the past. Assume that the source generating the data is such that there is a stationary predictor whose error converges to zero (in a certainsense). The question is whether there is a universal predictor for all such sources, that is, a predictor whose error goes to zero if any of the sources that have this property is chosen to generate the data. This question is answered in the negative, contrasting a number of previously established positive results concerning related but smaller sets of processes.

Improved Regret Bounds for Undiscounted Continuous Reinforcement Learning [22]

We consider the problem of undiscounted reinforcement learning in continuous state space. Regret bounds in this setting usually hold under various assumptions on the structure of the reward andtransition function. Under the assumption that the rewards andtransition probabilities are Lipschitz, for 1-dimensional state space a regret bound of O˜(T34) after any T steps has been given by.Here we improve upon this result by using non-parametric kernel density estimation for estimating the transition probability distributions,and obtain regret bounds that depend on the smoothness of the transition probability distributions.In particular, under the assumption that the transition probability functions are smoothly differentiable, the regret bound is shown to be O˜(T23) asymptotically for reinforcement learning in 1-dimensional state space. Finally, we also derive improved regret bounds for higher dimensional state space.

Maximum Entropy Semi-Supervised Inverse Reinforcement Learning [9]

A popular approach to apprenticeship learning (AL) is to formulate itas an inverse reinforcement learning (IRL) problem. The MaxEnt-IRL algorithm successfully integrates the maximum entropy principleinto IRL and unlike its predecessors, it resolves theambiguity arising from the fact that a possibly large number of policies couldmatch the expert's behavior. In this paper, we study an AL setting in which inaddition to the expert's trajectories,a number of unsupervised trajectories is available. We introduce MESSI,a novel algorithm that combines MaxEnt-IRLwith principles coming from semi-supervised learning. In particular, MESSIintegrates the unsupervised data intothe MaxEnt-IRL framework using a pairwise penalty on trajectories. Empiricalresults in a highway driving and grid-world problems indicate that MESSI is able to take advantage of the unsupervised trajectories and improve the performance ofMaxEnt-IRL.

Direct Policy Iteration with Demonstrations [12]

We consider the problem of learning the optimal policy of an unknown Markov decision process (MDP) when expert demonstrations are available along with interaction samples. We build on classification-based policy iteration to perform a seamless integration of interaction and expert data, thus obtaining an algorithm which can benefit from both sources of information at the same time. Furthermore , we provide a full theoretical analysis of the performance across iterations providing insights on how the algorithm works. Finally, we report an empirical evaluation of the algorithm and a comparison with the state-of-the-art algorithms.

Approximate Modified Policy Iteration and its Application to the Game of Tetris [8]

Modified policy iteration (MPI) is a dynamic programming (DP) algorithm that contains the two celebrated policy and value iteration methods. Despite its generality, MPI has not been thoroughly studied, especially its approximation form which is used when the state and/or action spaces are large or infinite. In this paper, we propose three implementations of approximate MPI (AMPI) that are extensions of the well-known approximate DP algorithms: fitted-value iteration, fitted-Q iteration, and classification-based policy iteration. We provide error propagation analysis that unify those for approximate policy and value iteration. We develop the finite-sample analysis of these algorithms, which highlights the influence of their parameters. In the classification-based version of the algorithm (CBMPI), the analysis shows that MPI's main parameter controls the balance between the estimation error of the classifier and the overall value function approximation. We illustrate and evaluate the behavior of these new algorithms in the Mountain Car and Tetris problems. Remarkably, in Tetris, CBMPI outperforms the existing DP approaches by a large margin, and competes with the current state-of-the-art methods while using fewer samples.

Multi-arm Bandit Theory

Simple regret for infinitely many armed bandits [11]

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits , the rate of the simple regret will depend on a parameter β characterizing the distribution of the near-optimal arms. We prove that depending on β, our algorithm is minimax optimal either up to a multiplicative constant or up to a log(n) factor. We also provide extensions to several important cases: when β is unknown, in a natural setting where the near-optimal arms have a small variance , and in the case of unknown time horizon.

Black-box optimization of noisy functions with unknown smoothness [20]

We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of lnn away from the error of the best known optimization algorithms using the knowledge of the smoothness.

Cheap Bandits [21]

We consider stochastic sequential learning problems where the learner can observe the average reward of several actions. Such a setting is interesting in many applications involving monitoring and surveillance, where the set of the actions to observe represent some (geographical) area. The importance of this setting is that in these applications , it is actually cheaper to observe average reward of a group of actions rather than the reward of a single action. We show that when the reward is smooth over a given graph representing the neighboring actions, we can maximize the cumulative reward of learning while minimizing the sensing cost. In this paper we propose CheapUCB, an algorithm that matches the regret guarantees of the known algorithms for this setting and at the same time guarantees a linear cost again over them. As a by-product of our analysis , we establish a ⌦(p dT) lower bound on the cumulative regret of spectral bandits for a class of graphs with effective dimension d.

Truthful Learning Mechanisms for Multi–Slot Sponsored Search Auctions with Externalities [5]

Sponsored Search Auctions (SSAs) constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and, at the same time, to guarantee both the advertisers and the auctioneer a non–negative utility. Nonetheless, in sponsored search auctions, the Click–Through–Rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard truthful mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs at the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to the mechanism that can exploit the true CTRs. Previous work showed that, when dominant–strategy truthfulness is adopted, in single–slot auctions the problem can be solved using suitable exploration–exploitation mechanisms able to achieve a cumulative regret (on the auctioneer's revenue) of order O(T(2/3)), where T is the number of times the auction is repeated. It is also known that, when truthfulness in expectation is adopted, a cumulative regret (over the social welfare) of order O(T(1/2)) can be obtained. In this paper, we extend the results available in the literature to the more realistic case of multi–slot auctions. In this case, a model of the user is needed to characterize how the CTR of an ad changes as its position in the allocation changes. In particular, we adopt the cascade model, one of the most popular models for sponsored search auctions, and we prove a number of novel upper bounds and lower bounds on both auctioneer’s revenue loss and social welfare w.r.t. to the Vickrey–Clarke–Groves (VCG) auction. Furthermore, we report numerical simulations investigating the accuracy of the bounds in predicting the dependency of the regret on the auction parameters.

A Relative Exponential Weighing Algorithm for Adversarial Utility-based Dueling Bandits [37]

We study the K-armed dueling bandit problem which is a variation of the classical Multi-Armed Bandit (MAB) problem in which the learner receives only relative feedback about the selected pairs of arms. We propose a new algorithm called Relative Exponential-weight algorithm for Exploration and Exploitation (REX3) to handle the adversarial utility-based formulation of this problem. This algorithm is a non-trivial extension of the Exponential-weight algorithm for Exploration and Exploitation (EXP3) algorithm. We prove a finite time expected regret upper bound of order O(sqrt(K ln(K)T)) for this algorithm and a general lower bound of order omega(sqrt(KT)). At the end, we provide experimental results using real data from information retrieval applications.

Simultaneous Optimistic Optimization on the Noiseless BBOB Testbed [15]

We experiment the SOO (Simultaneous Optimistic Optimization) global optimizer on the BBOB testbed. We report results for both the unconstrained-budget setting and the expensive setting, as well as a comparison with the DiRect algorithm to which SOO is mostly related. Overall, SOO is shown to perform rather poorly in the highest dimensions while agreeably exhibiting interesting performance for the most difficult functions, which is to be attributed to its global nature and to the fact that its design was guided by the goal of obtaining theoretically provable performance. The greedy exploration-exploitation sampling strategy underlying SOO design is also shown to be a viable alternative for the expensive setting which gives rooms for further improvements in this direction.

Recommendation systems

Bandits and Recommender Systems [23]

This paper addresses the on-line recommendation problem facing new users and new items; we assume that no information is available neither about users, nor about the items. The only source of information is a set of ratings given by users to some items. By on-line, we mean that the set of users, and the set of items, and the set of ratings is evolving along time and that at any moment, the recommendation system has to select items to recommend based on the currently available information, that is basically the sequence of past events. We also mean that each user comes with her preferences which may evolve along short and longer scales of time; so we have to continuously update their preferences. When the set of ratings is the only available source of information , the traditional approach is matrix factorization. In a decision making under uncertainty setting, actions should be selected to balance exploration with exploitation; this is best modeled as a bandit problem. Matrix factors provide a latent representation of users and items. These representations may then be used as contextual information by the bandit algorithm to select items. This last point is exactly the originality of this paper: the combination of matrix factorization and bandit algorithms to solve the on-line recommendation problem. Our work is driven by considering the recommendation problem as a feedback controlled loop. This leads to interactions between the representation learning, and the recommendation policy.

Collaborative Filtering as a Multi-Armed Bandit [35]

Recommender Systems (RS) aim at suggesting to users one or several items in which they might have interest. Following the feedback they receive from the user, these systems have to adapt their model in order to improve future recommendations. The repetition of these steps defines the RS as a sequential process. This sequential aspect raises an exploration-exploitation dilemma, which is surprisingly rarely taken into account for RS without contextual information. In this paper we present an explore-exploit collaborative filtering RS, based on Matrix Factor-ization and Bandits algorithms. Using experiments on artificial and real datasets, we show the importance and practicability of using sequential approaches to perform recommendation. We also study the impact of the model update on both the quality and the computation time of the recommendation procedure.

AUC Optimisation and Collaborative Filtering [39]

In recommendation systems, one is interested in the ranking of the predicted items as opposed to other losses such as the mean squared error. Although a variety of ways to evaluate rankings exist in the literature, here we focus on the Area Under the ROC Curve (AUC) as it widely used and has a strong theoretical underpinning. In practical recommendation, only items at the top of the ranked list are presented to the users. With this in mind, we propose a class of objective functions over matrix factorisations which primarily represent a smooth surrogate for the real AUC, and in a special case we show how to prioritise the top of the list. The objectives are differentiable and optimised through a carefully designed stochastic gradient-descent-based algorithm which scales linearly with the size of the data. In the special case of square loss we show how to improve computational complexity by leveraging previously computed measures. To understand theoretically the underlying matrix factorisation approaches we study both the consistency of the loss functions with respect to AUC, and generalisation using Rademacher theory. The resulting generalisation analysis gives strong motivation for the optimisation under study. Finally, we provide computation results as to the efficacy of the proposed method using synthetic and real data.

Collaborative Filtering with Localised Ranking [16]

In recommendation systems, one is interested in the ranking of the predicted items as opposed to other losses such as the mean squared error. Although a variety of ways to evaluate rankings exist in the literature, here we focus on the Area Under the ROC Curve (AUC) as it widely used and has a strong theoretical underpinning. In practical recommendation, only items at the top of the ranked list are presented to the users. With this in mind we propose a class of objective functions which primarily represent a smooth surrogate for the real AUC, and in a special case we show how to prioritise the top of the list. This loss is differentiable and is optimised through a carefully designed stochastic gradient-descent-based algorithm which scales linearly with the size of the data. We mitigate sample bias present in the data by sampling observations according to a certain power-law based distribution. In addition, we provide computation results as to the efficacy of the proposed method using synthetic and real data.

Collaborative Filtering with Stacked Denoising AutoEncoders and Sparse Inputs [36]

Neural networks have not been widely studied in Collaborative Filtering. For instance, no paper using neural networks was published during the Net-flix Prize apart from Salakhutdinov et al's work on Restricted Boltzmann Machine (RBM) [14]. While deep learning has tremendous success in image and speech recognition, sparse inputs received less attention and remains a challenging problem for neural networks. Nonetheless, sparse inputs are critical for collaborative filtering. In this paper, we introduce a neural network architecture which computes a non-linear matrix factorization from sparse rating inputs. We show experimentally on the movieLens and jester dataset that our method performs as well as the best collaborative filtering algorithms. We provide an implementation of the algorithm as a reusable plugin for Torch [4], a popular neural network framework.

Nonparametric statistics of time series

The Replacement Bootstrap for Dependent Data [31]

Applications that deal with time-series data often require evaluating complex statistics for which each time series is essentially one data point. When only a few time series are available, bootstrap methods are used to generate additional samples that can be used to evaluate empirically the statistic of interest. In this work a novel bootstrap method is proposed, which is shown to have some asymptotic consistency guarantees under the only assumption that the time series are stationary and ergodic. This contrasts previously available results that impose mixing or finite-memory assumptions on the data. Empirical evaluation on simulated and real data, using a practically relevant and complex extrema statistic is provided.

Imitation and Inverse Reinforcement Learning

Inverse Reinforcement Learning in Relational Domains [24]

In this work, we introduce the first approach to the Inverse Reinforcement Learning (IRL) problem in relational domains. IRL has been used to recover a more compact representation of the expert policy leading to better generalization performances among different contexts. On the other hand, rela-tional learning allows representing problems with a varying number of objects (potentially infinite), thus provides more generalizable representations of problems and skills. We show how these different formalisms allow one to create a new IRL algorithm for relational domains that can recover with great efficiency rewards from expert data that have strong generalization and transfer properties. We evaluate our algorithm in representative tasks and study the impact of diverse experimental conditions such as : the number of demonstrations, knowledge about the dynamics, transfer among varying dimensions of a problem, and changing dynamics.

Imitation Learning Applied to Embodied Conversational Agents [29]

Embodied Conversational Agents (ECAs) are emerging as a key component to allow human interact with machines. Applications are numerous and ECAs can reduce the aversion to interact with a machine by providing user-friendly interfaces. Yet, ECAs are still unable to produce social signals appropriately during their interaction with humans, which tends to make the interaction less instinctive. Especially, very little attention has been paid to the use of laughter in human-avatar interactions despite the crucial role played by laughter in human-human interaction. In this paper, methods for predicting when and how to laugh during an interaction for an ECA are proposed. Different Imitation Learning (also known as Apprenticeship Learning) algorithms are used in this purpose and a regularized classification algorithm is shown to produce good behavior on real data.

Stochastic Games

Optimism in Active Learning [3]

Active learning is the problem of interactively constructing the training set used in classification in order to reduce its size. It would ideally successively add the instance-label pair that decreases the classification error most. However, the effect of the addition of a pair is not known in advance. It can still be estimated with the pairs already in the training set. The online minimization of the classification error involves a tradeoff between exploration and exploitation. This is a common problem in machine learning for which multiarmed bandit, using the approach of Optimism int the Face of Uncertainty, has proven very efficient these last years. This paper introduces three algorithms for the active learning problem in classification using Optimism in the Face of Uncertainty. Experiments lead on built-in problems and real world datasets demonstrate that they compare positively to state-of-the-art methods.

Bayesian Credible Intervals for Online and Active Learning of Classification Trees [13]

Classification trees have been extensively studied for decades. In the online learning scenario, a whole class of algorithms for decision trees has been introduced, called incremental decision trees. In the case where subtrees may not be discarded, an incremental decision tree can be seen as a sequential decision process, consisting in deciding to extend the existing tree or not. This problem involves an trade-off between exploration and exploitation, which is addressed in recent work with the use of Hoeffding's bounds. This paper proposes to use Bayesian Credible Intervals instead, in order to get the most out of the knowledge of the output's distribution's shape. It also studies the case of Active Learning in such a tree following the Optimism in the Face of Uncertainty paradigm. Two novel algorithms are introduced for the online and active learning problems. Evaluations on real-world datasets show that these algorithms compare positively to state-of-the-art.

Optimism in Active Learning with Gaussian Processes [14]

In the context of Active Learning for classification, the classification error depends on the joint distribution of samples and their labels which is initially unknown. The minimization of this error requires estimating this distribution. Online estimation of this distribution involves a trade-off between exploration and exploitation. This is a common problem in machine learning for which multi-armed bandit theory, building upon Optimism in the Face of Uncertainty, has been proven very efficient these last years. We introduce two novel algorithms that use Optimism in the Face of Uncertainty along with Gaussian Processes for the Active Learning problem. The evaluation lead on real world datasets shows that these new algorithms compare positively to state-of-the-art methods.

Approximate Dynamic Programming for Two-Player Zero-Sum Markov Games [28]

This paper provides an analysis of error propagation in Approximate Dynamic Programming applied to zero-sum two-player Stochastic Games. We provide a novel and unified error propagation analysis in L p-norm of three well-known algorithms adapted to Stochastic Games (namely Approximate Value Iteration, Approximate Policy Iteration and Approximate Generalized Policy Iteratio,n). We show that we can achieve a stationary policy which is 2γ+ (1–γ) 2-optimal, where is the value function approximation error and is the approximate greedy operator error. In addition , we provide a practical algorithm (AGPI-Q) to solve infinite horizon γ-discounted two-player zero-sum Stochastic Games in a batch setting. It is an extension of the Fitted-Q algorithm (which solves Markov Decisions Processes from data) and can be non-parametric. Finally, we demonstrate experimentally the performance of AGPI-Q on a simultaneous two-player game, namely Alesia.