arXiv subjects

Gergely Neu

Publications and source records attributed to Gergely Neu.

At least 55 records · Page 3Linked to original sources

arXiv2020

Fast Rates for Online Prediction with Abstention

In the setting of sequential prediction of individual $\{0, 1\}$-sequences with expert advice, we show that by allowing the learner to abstain from the prediction by paying a cost marginally smaller than $\frac 12$ (say, $0.49$), it is possible to achieve expected regret bounds that are independent of the time horizon $T$. We exactly characterize the dependence on the abstention cost $c$ and the number of experts $N$ by providing matching upper and lower bounds of order $\frac{\log N}{1-2c}$, which is to be contrasted with the best possible rate of $\sqrt{T\log N}$ that is available without the option to abstain. We also discuss various extensions of our model, including a setting where the sequence of abstention costs can change arbitrarily over time, where we show regret bounds interpolating between the slow and the fast rates mentioned above, under some natural assumptions on the sequence of abstention costs.

cs.LG↗

arXiv2020

Faster saddle-point optimization for solving large-scale Markov decision processes

We consider the problem of computing optimal policies in average-reward Markov decision processes. This classical problem can be formulated as a linear program directly amenable to saddle-point optimization methods, albeit with a number of variables that is linear in the number of states. To address this issue, recent work has considered a linearly relaxed version of the resulting saddle-point problem. Our work aims at achieving a better understanding of this relaxed optimization problem by characterizing the conditions necessary for convergence to the optimal policy, and designing an optimization algorithm enjoying fast convergence rates that are independent of the size of the state space. Notably, our characterization points out some potential issues with previous work.

math.OC↗

arXiv2019

Adaptive Temporal-Difference Learning for Policy Evaluation with Per-State Uncertainty Estimates

We consider the core reinforcement-learning problem of on-policy value function approximation from a batch of trajectory data, and focus on various issues of Temporal Difference (TD) learning and Monte Carlo (MC) policy evaluation. The two methods are known to achieve complementary bias-variance trade-off properties, with TD tending to achieve lower variance but potentially higher bias. In this paper, we argue that the larger bias of TD can be a result of the amplification of local approximation errors. We address this by proposing an algorithm that adaptively switches between TD and MC in each state, thus mitigating the propagation of errors. Our method is based on learned confidence intervals that detect biases of TD estimates. We demonstrate in a variety of policy evaluation tasks that this simple adaptive algorithm performs competitively with the best approach in hindsight, suggesting that learned confidence intervals are a powerful technique for adapting policy evaluation to use TD or MC returns in a data-driven way.

cs.LG↗

arXiv2019

Beating SGD Saturation with Tail-Averaging and Minibatching

While stochastic gradient descent (SGD) is one of the major workhorses in machine learning, the learning properties of many practically used variants are poorly understood. In this paper, we consider least squares learning in a nonparametric setting and contribute to filling this gap by focusing on the effect and interplay of multiple passes, mini-batching and averaging, and in particular tail averaging. Our results show how these different variants of SGD can be combined to achieve optimal learning errors, hence providing practical insights. In particular, we show for the first time in the literature that tail averaging allows faster convergence rates than uniform averaging in the nonparametric setting. Finally, we show that a combination of tail-averaging and minibatching allows more aggressive step-size choices than using any one of said components.

stat.ML↗

arXiv2019

Bandit Principal Component Analysis

We consider a partial-feedback variant of the well-studied online PCA problem where a learner attempts to predict a sequence of $d$-dimensional vectors in terms of a quadratic loss, while only having limited feedback about the environment's choices. We focus on a natural notion of bandit feedback where the learner only observes the loss associated with its own prediction. Based on the classical observation that this decision-making problem can be lifted to the space of density matrices, we propose an algorithm that is shown to achieve a regret of $O(d^{3/2}\sqrt{T})$ after $T$ rounds in the worst case. We also prove data-dependent bounds that improve on the basic result when the loss matrices of the environment have bounded rank or the loss of the best action is bounded. One version of our algorithm runs in $O(d)$ time per trial which massively improves over every previously known online PCA method. We complement these results by a lower bound of $Ω(d\sqrt{T})$.

cs.LG↗

arXiv2018

Potential and Pitfalls of Multi-Armed Bandits for Decentralized Spatial Reuse in WLANs

Spatial Reuse (SR) has recently gained attention to maximize the performance of IEEE 802.11 Wireless Local Area Networks (WLANs). Decentralized mechanisms are expected to be key in the development of SR solutions for next-generation WLANs, since many deployments are characterized by being uncoordinated by nature. However, the potential of decentralized mechanisms is limited by the significant lack of knowledge with respect to the overall wireless environment. To shed some light on this subject, we show the main considerations and possibilities of applying online learning to address the SR problem in uncoordinated WLANs. In particular, we provide a solution based on Multi-Armed Bandits (MABs) whereby independent WLANs dynamically adjust their frequency channel, transmit power and sensitivity threshold. To that purpose, we provide two different strategies, which refer to selfish and environment-aware learning. While the former stands for pure individual behavior, the second one considers the performance experienced by surrounding networks, thus taking into account the impact of individual actions on the environment. Through these two strategies we delve into practical issues of applying MABs in wireless networks, such as convergence guarantees or adversarial effects. Our simulation results illustrate the potential of the proposed solutions for enabling SR in future WLANs. We show that substantial improvements on network performance can be achieved regarding throughput and fairness.

cs.NI↗

arXiv2018

Collaborative Spatial Reuse in Wireless Networks via Selfish Multi-Armed Bandits

Next-generation wireless deployments are characterized by being dense and uncoordinated, which often leads to inefficient use of resources and poor performance. To solve this, we envision the utilization of completely decentralized mechanisms to enable Spatial Reuse (SR). In particular, we focus on dynamic channel selection and Transmission Power Control (TPC). We rely on Reinforcement Learning (RL), and more specifically on Multi-Armed Bandits (MABs), to allow networks to learn their best configuration. In this work, we study the exploration-exploitation trade-off by means of the $\varepsilon$-greedy, EXP3, UCB and Thompson sampling action-selection, and compare their performance. In addition, we study the implications of selecting actions simultaneously in an adversarial setting (i.e., concurrently), and compare it with a sequential approach. Our results show that optimal proportional fairness can be achieved, even when no information about neighboring networks is available to the learners and Wireless Networks (WNs) operate selfishly. However, there is high temporal variability in the throughput experienced by the individual networks, specially for $\varepsilon$-greedy and EXP3. These strategies, contrary to UCB and Thompson sampling, base their operation on the absolute experienced reward, rather than on its distribution. We identify the cause of this variability to be the adversarial setting of our setup in which the set of most played actions provide intermittent good/poor performance depending on the neighboring decisions. We also show that learning sequentially, even if using a selfish strategy, contributes to minimize this variability. The sequential approach is therefore shown to effectively deal with the challenges posed by the adversarial settings that are typically found in decentralized WNs.

cs.NI↗

arXiv2018

Online Influence Maximization with Local Observations

We consider an online influence maximization problem in which a decision maker selects a node among a large number of possibilities and places a piece of information at the node. The node transmits the information to some others that are in the same connected component in a random graph. The goal of the decision maker is to reach as many nodes as possible, with the added complication that feedback is only available about the degree of the selected node. Our main result shows that such local observations can be sufficient for maximizing global influence in two broadly studied families of random graph models: stochastic block models and Chung--Lu models. With this insight, we propose a bandit algorithm that aims at maximizing local (and thus global) influence, and provide its theoretical analysis in both the subcritical and supercritical regimes of both considered models. Notably, our performance guarantees show no explicit dependence on the total number of nodes in the network, making our approach well-suited for large-scale applications.

cs.LG↗

arXiv2018

Iterate averaging as regularization for stochastic gradient descent

We propose and analyze a variant of the classic Polyak-Ruppert averaging scheme, broadly used in stochastic gradient methods. Rather than a uniform average of the iterates, we consider a weighted average, with weights decaying in a geometric fashion. In the context of linear least squares regression, we show that this averaging scheme has a the same regularizing effect, and indeed is asymptotically equivalent, to ridge regression. In particular, we derive finite-sample bounds for the proposed approach that match the best known results for regularized stochastic gradient methods.

cs.LG↗

arXiv2018

Wireless Optimisation via Convex Bandits: Unlicensed LTE/WiFi Coexistence

Bandit Convex Optimisation (BCO) is a powerful framework for sequential decision-making in non-stationary and partially observable environments. In a BCO problem, a decision-maker sequentially picks actions to minimize the cumulative cost associated with these decisions, all while receiving partial feedback about the state of the environment. This formulation is a very natural fit for wireless-network optimisation problems and has great application potential since: i) instead of assuming full observability of the network state, it only requires the metric to optimise as input, and ii) it provides strong performance guarantees while making only minimal assumptions about the network dynamics. Despite these advantages, BCO has not yet been explored in the context of wireless-network optimisation. In this paper, we make the first steps to demonstrate the potential of BCO techniques by formulating an unlicensed LTE/WiFi fair coexistence use case in the framework, and providing experimental results in a simulated environment. On the algorithmic front, we propose a simple and natural sequential multi-point BCO algorithm amenable to wireless networking optimisation, and provide its theoretical analysis. We expect the contributions of this paper to pave the way to further research on the application of online convex methods in the bandit setting.

cs.NI↗

arXiv2017

Boltzmann Exploration Done Right

Boltzmann exploration is a classic strategy for sequential decision-making under uncertainty, and is one of the most standard tools in Reinforcement Learning (RL). Despite its widespread use, there is virtually no theoretical understanding about the limitations or the actual benefits of this exploration scheme. Does it drive exploration in a meaningful way? Is it prone to misidentifying the optimal actions or spending too much time exploring the suboptimal ones? What is the right tuning for the learning rate? In this paper, we address several of these questions in the classic setup of stochastic multi-armed bandits. One of our main results is showing that the Boltzmann exploration strategy with any monotone learning-rate sequence will induce suboptimal behavior. As a remedy, we offer a simple non-monotone schedule that guarantees near-optimal performance, albeit only when given prior access to key problem parameters that are typically not available in practical situations (like the time horizon $T$ and the suboptimality gap $Δ$). More importantly, we propose a novel variant that uses different learning rates for different arms, and achieves a distribution-dependent regret bound of order $\frac{K\log^2 T}Δ$ and a distribution-independent bound of order $\sqrt{KT}\log K$ without requiring such prior knowledge. To demonstrate the flexibility of our technique, we also propose a variant that guarantees the same performance bounds even if the rewards are heavy-tailed.

cs.LG↗

arXiv2017

On the Hardness of Inventory Management with Censored Demand Data

We consider a repeated newsvendor problem where the inventory manager has no prior information about the demand, and can access only censored/sales data. In analogy to multi-armed bandit problems, the manager needs to simultaneously "explore" and "exploit" with her inventory decisions, in order to minimize the cumulative cost. We make no probabilistic assumptions---importantly, independence or time stationarity---regarding the mechanism that creates the demand sequence. Our goal is to shed light on the hardness of the problem, and to develop policies that perform well with respect to the regret criterion, that is, the difference between the cumulative cost of a policy and that of the best fixed action/static inventory decision in hindsight, uniformly over all feasible demand sequences. We show that a simple randomized policy, termed the Exponentially Weighted Forecaster, combined with a carefully designed cost estimator, achieves optimal scaling of the expected regret (up to logarithmic factors) with respect to all three key primitives: the number of time periods, the number of inventory decisions available, and the demand support. Through this result, we derive an important insight: the benefit from "information stalking" as well as the cost of censoring are both negligible in this dynamic learning problem, at least with respect to the regret criterion. Furthermore, we modify the proposed policy in order to perform well in terms of the tracking regret, that is, using as benchmark the best sequence of inventory decisions that switches a limited number of times. Numerical experiments suggest that the proposed approach outperforms existing ones (that are tailored to, or facilitated by, time stationarity) on nonstationary demand models. Finally, we extend the proposed approach and its analysis to a "combinatorial" version of the repeated newsvendor problem.

cs.LG↗

arXiv2017

Algorithmic stability and hypothesis complexity

We introduce a notion of algorithmic stability of learning algorithms---that we term \emph{argument stability}---that captures stability of the hypothesis output by the learning algorithm in the normed space of functions from which hypotheses are selected. The main result of the paper bounds the generalization error of any learning algorithm in terms of its argument stability. The bounds are based on martingale inequalities in the Banach space to which the hypotheses belong. We apply the general bounds to bound the performance of some learning algorithms based on empirical risk minimization and stochastic gradient descent.

stat.ML↗

arXiv2017

Fast rates for online learning in Linearly Solvable Markov Decision Processes

We study the problem of online learning in a class of Markov decision processes known as linearly solvable MDPs. In the stationary version of this problem, a learner interacts with its environment by directly controlling the state transitions, attempting to balance a fixed state-dependent cost and a certain smooth cost penalizing extreme control inputs. In the current paper, we consider an online setting where the state costs may change arbitrarily between consecutive rounds, and the learner only observes the costs at the end of each respective round. We are interested in constructing algorithms for the learner that guarantee small regret against the best stationary control policy chosen in full knowledge of the cost sequence. Our main result is showing that the smoothness of the control cost enables the simple algorithm of following the leader to achieve a regret of order $\log^2 T$ after $T$ rounds, vastly improving on the best known regret bound of order $T^{3/4}$ for this setting.

cs.LG↗

arXiv2017

A unified view of entropy-regularized Markov decision processes

We propose a general framework for entropy-regularized average-reward reinforcement learning in Markov decision processes (MDPs). Our approach is based on extending the linear-programming formulation of policy optimization in MDPs to accommodate convex regularization functions. Our key result is showing that using the conditional entropy of the joint state-action distributions as regularization yields a dual optimization problem closely resembling the Bellman optimality equations. This result enables us to formalize a number of state-of-the-art entropy-regularized reinforcement learning algorithms as approximate variants of Mirror Descent or Dual Averaging, and thus to argue about the convergence properties of these methods. In particular, we show that the exact version of the TRPO algorithm of Schulman et al. (2015) actually converges to the optimal policy, while the entropy-regularized policy gradient methods of Mnih et al. (2016) may fail to converge to a fixed point. Finally, we illustrate empirically the effects of using various regularization techniques on learning performance in a simple reinforcement learning setup.

cs.LG↗

arXiv2016

Importance weighting without importance weights: An efficient algorithm for combinatorial semi-bandits

We propose a sample-efficient alternative for importance weighting for situations where one only has sample access to the probability distribution that generates the observations. Our new method, called Geometric Resampling (GR), is described and analyzed in the context of online combinatorial optimization under semi-bandit feedback, where a learner sequentially selects its actions from a combinatorial decision set so as to minimize its cumulative loss. In particular, we show that the well-known Follow-the-Perturbed-Leader (FPL) prediction method coupled with Geometric Resampling yields the first computationally efficient reduction from offline to online optimization in this setting. We provide a thorough theoretical analysis for the resulting algorithm, showing that its performance is on par with previous, inefficient solutions. Our main contribution is showing that, despite the relatively large variance induced by the GR procedure, our performance guarantees hold with high probability rather than only in expectation. As a side result, we also improve the best known regret bounds for FPL in online combinatorial optimization with full feedback, closing the perceived performance gap between FPL and exponential weights in this setting.

cs.LG↗

arXiv2015

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

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 $Ω(\sqrt{T})$ 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.

cs.LG↗

arXiv2015

First-order regret bounds for combinatorial semi-bandits

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 $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this paper, we propose an algorithm that improves this scaling to $\widetilde{O}(\sqrt{L_T^*})$, 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.

cs.LG↗