Section: New Results

Decision-making Under Uncertainty

Reinforcement Learning

A Fitted-Q Algorithm for Budgeted MDPs, [26]

We address the problem of budgeted reinforcement learning, in continuous state-space, using a batch of transitions. To this extend, we introduce a novel algorithm called Budgeted Fitted-Q (BFTQ). Benchmarks show that BFTQ performs as well as a regular Fitted-Q algorithm in a continuous 2-D world but also allows one to choose the right amount of budget that fits to a given task without the need of engineering the rewards. We believe that the general principles used to design BFTQ can be applied to extend others classical reinforcement learning algorithms for budgeted oriented applications.

Safe transfer learning for dialogue applications, [27]

In this paper, we formulate the hypothesis that the first dialogues with a new user should be handle in a very conservative way, for two reasons : avoid user dropout; gather more successful dialogues to speedup the learning of the asymptotic strategy. To this extend, we propose to transfer a safe strategy to initiate the first dialogues.

Variance-Aware Regret Bounds for Undiscounted Reinforcement Learning in MDPs, [17]

The problem of reinforcement learning in an unknown and discrete Markov Decision Process (MDP) under the average-reward criterion is considered, when the learner interacts with the system in a single stream of observations, starting from an initial state without any reset. We revisit the minimax lower bound for that problem by making appear the local variance of the bias function in place of the diameter of the MDP. Furthermore, we provide a novel analysis of the KL-Ucrl algorithm establishing a high-probability regret bound scaling as O S s,a V s,a T for this algorithm for ergodic MDPs, where S denotes the number of states and where V s,a is the variance of the bias function with respect to the next-state distribution following action a in state s. The resulting bound improves upon the best previously known regret bound O(DS $\sqrt{}$ AT) for that algorithm, where A and D respectively denote the maximum number of actions (per state) and the diameter of MDP. We finally compare the leading terms of the two bounds in some benchmark MDPs indicating that the derived bound can provide an order of magnitude improvement in some cases. Our analysis leverages novel variations of the transportation lemma combined with Kullback-Leibler concentration inequalities, that we believe to be of independent interest.

Efficient Bias-Span-Constrained Exploration-Exploitation in Reinforcement Learning, [29]

We introduce SCAL, an algorithm designed to perform efficient exploration-exploitation in any unknown weakly-communicating Markov decision process (MDP) for which an upper bound $c$ on the span of the optimal bias function is known. For an MDP with $S$ states, $A$ actions and $\Gamma \le S$ possible next states, we prove a regret bound of $\tilde{O}\left(c\sqrt{\Gamma SAT}\right)$, which significantly improves over existing algorithms (e.g., UCRL and PSRL), whose regret scales linearly with the MDP diameter $D$. In fact, the optimal bias span is finite and often much smaller than $D$ (e.g., $D=\infty$ in non-communicating MDPs). A similar result was originally derived by Bartlett and Tewari (2009) for REGAL.C, for which no tractable algorithm is available. In this paper, we relax the optimization problem at the core of REGAL.C, we carefully analyze its properties, and we provide the first computationally efficient algorithm to solve it. Finally, we report numerical simulations supporting our theoretical findings and showing how SCAL significantly outperforms UCRL in MDPs with large diameter and small span.

Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes, [28]

While designing the state space of an MDP, it is common to include states that are transient or not reachable by any policy (e.g., in mountain car, the product space of speed and position contains configurations that are not physically reachable). This leads to defining weakly-communicating or multi-chain MDPs. In this paper, we introduce TUCRL, the first algorithm able to perform efficient exploration-exploitation in any finite Markov Decision Process (MDP) without requiring any form of prior knowledge. In particular, for any MDP with $S^{𝙲}$ communicating states, $A$ actions and ${\Gamma }^{𝙲}\le S^{𝙲}$ possible communicating next states, we derive a $\tilde{O}\left(D^{𝙲}\sqrt{{\Gamma }^{𝙲}{S}^{𝙲}AT}\right)$ regret bound, where $D^{𝙲}$ is the diameter (i.e., the longest shortest path) of the communicating part of the MDP. This is in contrast with optimistic algorithms (e.g., UCRL, Optimistic PSRL) that suffer linear regret in weakly-communicating MDPs, as well as posterior sampling or regularized algorithms (e.g., REGAL), which require prior knowledge on the bias span of the optimal policy to bias the exploration to achieve sub-linear regret. We also prove that in weakly-communicating MDPs, no algorithm can ever achieve a logarithmic growth of the regret without first suffering a linear regret for a number of steps that is exponential in the parameters of the MDP. Finally, we report numerical simulations supporting our theoretical findings and showing how TUCRL overcomes the limitations of the state-of-the-art.

Upper Confidence Reinforcement Learning exploiting state-action equivalence, [53]

Stochastic Variance-Reduced Policy Gradient, [34]

In this paper, we propose a novel reinforcement-learning algorithm consisting in a stochastic variance-reduced version of policy gradient for solving Markov Decision Processes (MDPs). Stochastic variance-reduced gradient (SVRG) methods have proven to be very successful in supervised learning. However, their adaptation to policy gradient is not straightforward and needs to account for I) a non-concave objective function; II) approximations in the full gradient computation; and III) a non-stationary sampling process. The result is SVRPG, a stochastic variance-reduced policy gradient algorithm that leverages on importance weights to preserve the unbiasedness of the gradient estimate. Under standard assumptions on the MDP, we provide convergence guarantees for SVRPG with a convergence rate that is linear under increasing batch sizes. Finally, we suggest practical variants of SVRPG, and we empirically evaluate them on continuous MDPs.

Importance Weighted Transfer of Samples in Reinforcement Learning, [38]

We consider the transfer of experience samples (i.e., tuples < s, a, s', r >) in reinforcement learning (RL), collected from a set of source tasks to improve the learning process in a given target task. Most of the related approaches focus on selecting the most relevant source samples for solving the target task, but then all the transferred samples are used without considering anymore the discrepancies between the task models. In this paper, we propose a model-based technique that automatically estimates the relevance (importance weight) of each source sample for solving the target task. In the proposed approach, all the samples are transferred and used by a batch RL algorithm to solve the target task, but their contribution to the learning process is proportional to their importance weight. By extending the results for importance weighting provided in supervised learning literature, we develop a finite-sample analysis of the proposed batch RL algorithm. Furthermore, we empirically compare the proposed algorithm to state-of-the-art approaches, showing that it achieves better learning performance and is very robust to negative transfer, even when some source tasks are significantly different from the target task.

Training Dialogue Systems With Human Advice, [20]

One major drawback of Reinforcement Learning (RL) Spoken Dialogue Systems is that they inherit from the general exploration requirements of RL which makes them hard to deploy from an industry perspective. On the other hand, industrial systems rely on human expertise and hand written rules so as to avoid irrelevant behavior to happen and maintain acceptable experience from the user point of view. In this paper, we attempt to bridge the gap between those two worlds by providing an easy way to incorporate all kinds of human expertise in the training phase of a Reinforcement Learning Dialogue System. Our approach, based on the TAMER framework, enables safe and efficient policy learning by combining the traditional Reinforcement Learning reward signal with an additional reward, encoding expert advice. Experimental results show that our method leads to substantial improvements over more traditional Reinforcement Learning methods.

Deep reinforcement learning

FiLM: Visual Reasoning with a General Conditioning Layer, [35]

We introduce a general-purpose conditioning method for neural networks called FiLM: Feature-wise Linear Modulation. FiLM layers influence neural network computation via a simple , feature-wise affine transformation based on conditioning information. We show that FiLM layers are highly effective for visual reasoning - answering image-related questions which require a multi-step, high-level process - a task which has proven difficult for standard deep learning methods that do not explicitly model reasoning. Specifically, we show on visual reasoning tasks that FiLM layers 1) halve state-of-the-art error for the CLEVR benchmark, 2) modulate features in a coherent manner, 3) are robust to ablations and architectural modifications, and 4) generalize well to challenging, new data from few examples or even zero-shot.

Feature-wise transformations, [13]

Deep Reinforcement Learning and the Deadly Triad, [55]

We know from reinforcement learning theory that temporal difference learning can fail in certain cases. Sutton and Barto (2018) identify a deadly triad of function approximation, bootstrapping, and off-policy learning. When these three properties are combined, learning can diverge with the value estimates becoming unbounded. However, several algorithms successfully combine these three properties, which indicates that there is at least a partial gap in our understanding. In this work, we investigate the impact of the deadly triad in practice, in the context of a family of popular deep reinforcement learning models - deep Q-networks trained with experience replay - analyzing how the components of this system play a role in the emergence of the deadly triad, and in the agent's performance

Multi-armed Bandit Theory

Corrupt Bandits for Preserving Local Privacy, [30]

We study a variant of the stochastic multi-armed bandit (MAB) problem in which the rewards are corrupted. In this framework, motivated by privacy preservation in online recommender systems, the goal is to maximize the sum of the (unobserved) rewards, based on the observation of transformation of these rewards through a stochastic corruption process with known parameters. We provide a lower bound on the expected regret of any bandit algorithm in this corrupted setting. We devise a frequentist algorithm, KLUCB-CF, and a Bayesian algorithm, TS-CF and give upper bounds on their regret. We also provide the appropriate corruption parameters to guarantee a desired level of local privacy and analyze how this impacts the regret. Finally, we present some experimental results that confirm our analysis.

A simple parameter-free and adaptive approach to optimization under a minimal local smoothness assumption, [21]

We study the problem of optimizing a function under a budgeted number of evaluations. We only assume that the function is locally smooth around one of its global optima. The difficulty of optimization is measured in terms of 1) the amount of noise b of the function evaluation and 2) the local smoothness, d, of the function. A smaller d results in smaller optimization error. We come with a new, simple, and parameter-free approach. First, for all values of b and d, this approach recovers at least the state-of-the-art regret guarantees. Second, our approach additionally obtains these results while being agnostic to the values of both b and d. This leads to the first algorithm that naturally adapts to an unknown range of noise b and leads to significant improvements in a moderate and low-noise regime. Third, our approach also obtains a remarkable improvement over the state-of-the-art SOO algorithm when the noise is very low which includes the case of optimization under deterministic feedback ($b=0$). There, under our minimal local smoothness assumption, this improvement is of exponential magnitude and holds for a class of functions that covers the vast majority of functions that practitioners optimize ($d=0$). We show that our algorithmic improvement is also borne out in the numerical experiments, where we empirically show faster convergence on common benchmark functions.

Best of both worlds: Stochastic & adversarial best-arm identification, [18]

We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.

Optimistic optimization of a Brownian, [31]

We address the problem of optimizing a Brownian motion. We consider a (random) realization $W$ of a Brownian motion with input space in $[0,1]$. Given $W$, our goal is to return an $ϵ$-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order $log2$ (1/$ϵ$). This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive-each query depends on previous values-and is an instance of the optimism-in-the-face-of-uncertainty principle.

Rotting bandits are no harder than stochastic ones, [37]

In bandits, arms' distributions are stationary. This is often violated in practice, where rewards change over time. In applications as recommendation systems, online advertising, and crowdsourcing, the changes may be triggered by the pulls, so that the arms' rewards change as a function of the number of pulls. In this paper, we consider the specific case of non-parametric rotting bandits, where the expected reward of an arm may decrease every time it is pulled. We introduce the filtering on expanding window average (FEWA) algorithm that at each round constructs moving averages of increasing windows to identify arms that are more likely to return high rewards when pulled once more. We prove that, without any knowledge on the decreasing behavior of the arms, FEWA achieves similar anytime problem-dependent, $\widetilde{𝒪}(log\left(KT\right)),$ and problem-independent, $\widetilde{𝒪}\left(\sqrt{KT}\right)$, regret bounds of near-optimal stochastic algorithms as UCB1 of Auer et al. (2002a). This result substantially improves the prior result of Levine et al. (2017) which needed knowledge of the horizon and decaying parameters to achieve problem-independent bound of only $\widetilde{𝒪}\left(K^{1/3}{T}^{2/3}\right)$. Finally, we report simulations confirming the theoretical improvements of FEWA.

Adaptive black-box optimization got easier: HCT only needs local smoothness, [41]

Hierarchical bandits is an approach for global optimization of extremely irregular functions. This paper provides new elements regarding POO, an adaptive meta-algorithm that does not require the knowledge of local smoothness of the target function. We first highlight the fact that the subroutine algorithm used in POO should have a small regret under the assumption of local smoothness with respect to the chosen partitioning, which is unknown if it is satisfied by the standard subroutine HOO. In this work, we establish such regret guarantee for HCT, which is another hierarchical optimistic optimization algorithm that needs to know the smoothness. This confirms the validity of POO. We show that POO can be used with HCT as a subroutine with a regret upper bound that matches the one of best-known algorithms using the knowledge of smoothness up to a $\sqrt{}$ log n factor.

Boundary Crossing Probabilities for General Exponential Families, [16]

Multi-Player Bandits Revisited, [22]

Multi-player Multi-Armed Bandits (MAB) have been extensively studied in the literature, motivated by applications to Cognitive Radio systems. Driven by such applications as well, we motivate the introduction of several levels of feedback for multi-player MAB algorithms. Most existing work assume that sensing information is available to the algorithm. Under this assumption, we improve the state-of-the-art lower bound for the regret of any decentralized algorithms and introduce two algorithms, RandTopM and MCTopM, that are shown to empirically outperform existing algorithms. Moreover, we provide strong theoretical guarantees for these algorithms, including a notion of asymptotic optimality in terms of the number of selections of bad arms. We then introduce a promising heuristic, called Selfish, that can operate without sensing information, which is crucial for emerging applications to Internet of Things networks. We investigate the empirical performance of this algorithm and provide some first theoretical elements for the understanding of its behavior.

Pure Exploration in Infinitely-Armed Bandit Models with Fixed-Confidence, [19]

We consider the problem of near-optimal arm identification in the fixed confidence setting of the infinitely armed bandit problem when nothing is known about the arm reservoir distribution. We (1) introduce a PAC-like framework within which to derive and cast results; (2) derive a sample complexity lower bound for near-optimal arm identification; (3) propose an algorithm that identifies a nearly-optimal arm with high probability and derive an upper bound on its sample complexity which is within a log factor of our lower bound; and (4) discuss whether our ${log}^2(1/\delta )$ dependence is inescapable for “two-phase” (select arms first, identify the best later) algorithms in the infinite setting. This work permits the application of bandit models to a broader class of problems where fewer assumptions hold.

Aggregation of Multi-Armed Bandits Learning Algorithms for Opportunistic Spectrum Access, [23]

Multi-armed bandit algorithms have been recently studied and evaluated for Cognitive Radio (CR), especially in the context of Opportunistic Spectrum Access (OSA). Several solutions have been explored based on various models, but it is hard to exactly predict which could be the best for real-world conditions at every instants. Hence, expert aggregation algorithms can be useful to select on the run the best algorithm for a specific situation. Aggregation algorithms, such as Exp4 dating back from 2002, have never been used for OSA learning, and we show that it appears empirically sub-efficient when applied to simple stochastic problems. In this article, we present an improved variant, called Aggregator. For synthetic OSA problems modeled as Multi-Armed Bandit (MAB) problems, simulation results are presented to demonstrate its empirical efficiency. We combine classical algorithms, such as Thompson sampling, Upper-Confidence Bounds algorithms (UCB and variants), and Bayesian or Kullback-Leibler UCB. Our algorithm offers good performance compared to state-of-the-art algorithms (Exp4, CORRAL or LearnExp), and appears as a robust approach to select on the run the best algorithm for any stochastic MAB problem, being more realistic to real-world radio settings than any tuning-based approach.

What Doubling Tricks Can and Can't Do for Multi-Armed Bandits, [47]

An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the ”Doubling Trick”. In the context of adversarial or stochastic multi-armed bandits, the performance of an algorithm is measured by its regret, and we study two families of sequences of growing horizons (geometric and exponential) to generalize previously known results that certain doubling tricks can be used to conserve certain regret bounds. In a broad setting, we prove that a geometric doubling trick can be used to conserve (minimax) bounds in $R\_T=O\left(\sqrt{T}\right)$ but cannot conserve (distribution-dependent) bounds in $R\_T=O(logT)$. We give insights as to why exponential doubling tricks may be better, as they conserve bounds in $R\_T=O(logT)$, and are close to conserving bounds in $R\_T=O\left(\sqrt{T}\right)$.

Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals, [50]

This paper presents new deviation inequalities that are valid uniformly in time under adaptive sampling in a multi-armed bandit model. The deviations are measured using the Kullback-Leibler divergence in a given one-dimensional exponential family, and may take into account several arms at a time. They are obtained by constructing for each arm a mixture martingale based on a hierarchical prior, and by multiplying those martingales. Our deviation inequalities allow us to analyze stopping rules based on generalized likelihood ratios for a large class of sequential identification problems, and to construct tight confidence intervals for some functions of the means of the arms.

Stochastic Games

Actor-Critic Fictitious Play in Simultaneous Move Multistage Games, [36]

Fictitious play is a game theoretic iterative procedure meant to learn an equilibrium in normal form games. However, this algorithm requires that each player has full knowledge of other players' strategies. Using an architecture inspired by actor-critic algorithms, we build a stochastic approximation of the fictitious play process. This procedure is on-line, decentralized (an agent has no information of others' strategies and rewards) and applies to multistage games (a generalization of normal form games). In addition, we prove convergence of our method towards a Nash equilibrium in both the cases of zero-sum two-player multistage games and cooperative multistage games. We also provide empirical evidence of the soundness of our approach on the game of Alesia with and without function approximation.

Sequential Test for the Lowest Mean: From Thompson to Murphy Sampling, [39]

Learning the minimum/maximum mean among a finite set of distributions is a fundamental sub-task in planning, game tree search and reinforcement learning. We formalize this learning task as the problem of sequentially testing how the minimum mean among a finite set of distributions compares to a given threshold. We develop refined non-asymptotic lower bounds, which show that optimality mandates very different sampling behavior for a low vs high true minimum. We show that Thompson Sampling and the intuitive Lower Confidence Bounds policy each nail only one of these cases. We develop a novel approach that we call Murphy Sampling. Even though it entertains exclusively low true minima, we prove that MS is optimal for both possibilities. We then design advanced self-normalized deviation inequalities, fueling more aggressive stopping rules. We complement our theoretical guarantees by experiments showing that MS works best in practice.

Online Kernel and Graph-Based Methods

Improved large-scale graph learning through ridge spectral sparsification, [25]

The representation and learning benefits of methods based on graph Laplacians, such as Laplacian smoothing or harmonic function solution for semi-supervised learning (SSL), are empirically and theoretically well supported. Nonetheless , the exact versions of these methods scale poorly with the number of nodes n of the graph. In this paper, we combine a spectral sparsification routine with Laplacian learning. Given a graph $G$ as input, our algorithm computes a sparsifier in a distributed way in $O(nlog3(n\left)\right)$ time, $O(mlog3(n\left)\right)$ work and $O(nlog(n\left)\right)$ memory, using only $log\left(n\right)$ rounds of communication. Furthermore , motivated by the regularization often employed in learning algorithms, we show that constructing sparsifiers that preserve the spectrum of the Laplacian only up to the regularization level may drastically reduce the size of the final graph. By constructing a spectrally-similar graph, we are able to bound the error induced by the sparsification for a variety of downstream tasks (e.g., SSL). We empirically validate the theoretical guarantees on Amazon co-purchase graph and compare to the state-of-the-art heuristics.

DPPy: Sampling Determinantal Point Processes with Python, [49]

Determinantal point processes (DPPs) are specific probability distributions over clouds of points that are used as models and computational tools across physics, probability, statistics, and more recently machine learning. Sampling from DPPs is a challenge and therefore we present DPPy, a Python toolbox that gathers known exact and approximate sampling algorithms. The project is hosted on GitHub and equipped with an extensive documentation. This documentation takes the form of a short survey of DPPs and relates each mathematical property with DPPy objects.

Streaming kernel regression with provably adaptive mean, variance, and regularization, [14]

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bersntein-like concentration bounds. The later is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.