arXiv subjects

Nicolò Cesa-Bianchi

Publications and source records attributed to Nicolò Cesa-Bianchi.

At least 73 records · Page 4Linked to original sources

arXiv2020

Cooperative Online Learning: Keeping your Neighbors Updated

We study an asynchronous online learning setting with a network of agents. At each time step, some of the agents are activated, requested to make a prediction, and pay the corresponding loss. The loss function is then revealed to these agents and also to their neighbors in the network. Our results characterize how much knowing the network structure affects the regret as a function of the model of agent activations. When activations are stochastic, the optimal regret (up to constant factors) is shown to be of order $\sqrt{αT}$, where $T$ is the horizon and $α$ is the independence number of the network. We prove that the upper bound is achieved even when agents have no information about the network structure. When activations are adversarial the situation changes dramatically: if agents ignore the network structure, a $Ω(T)$ lower bound on the regret can be proven, showing that learning is impossible. However, when agents can choose to ignore some of their neighbors based on the knowledge of the network structure, we prove a $O(\sqrt{\overlineχ T})$ sublinear regret bound, where $\overlineχ \ge α$ is the clique-covering number of the network.

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arXiv2020

Correlation Clustering with Adaptive Similarity Queries

In correlation clustering, we are given $n$ objects together with a binary similarity score between each pair of them. The goal is to partition the objects into clusters so to minimise the disagreements with the scores. In this work we investigate correlation clustering as an active learning problem: each similarity score can be learned by making a query, and the goal is to minimise both the disagreements and the total number of queries. On the one hand, we describe simple active learning algorithms, which provably achieve an almost optimal trade-off while giving cluster recovery guarantees, and we test them on different datasets. On the other hand, we prove information-theoretical bounds on the number of queries necessary to guarantee a prescribed disagreement bound. These results give a rich characterization of the trade-off between queries and clustering error.

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arXiv2020

Bandit Regret Scaling with the Effective Loss Range

We study how the regret guarantees of nonstochastic multi-armed bandits can be improved, if the effective range of the losses in each round is small (e.g. the maximal difference between two losses in a given round). Despite a recent impossibility result, we show how this can be made possible under certain mild additional assumptions, such as availability of rough estimates of the losses, or advance knowledge of the loss of a single, possibly unspecified arm. Along the way, we develop a novel technique which might be of independent interest, to convert any multi-armed bandit algorithm with regret depending on the loss range, to an algorithm with regret depending only on the effective range, while avoiding predictably bad arms altogether.

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arXiv2019

Nonstochastic Multiarmed Bandits with Unrestricted Delays

We investigate multiarmed bandits with delayed feedback, where the delays need neither be identical nor bounded. We first prove that "delayed" Exp3 achieves the $O(\sqrt{(KT + D)\ln K} )$ regret bound conjectured by Cesa-Bianchi et al. [2019] in the case of variable, but bounded delays. Here, $K$ is the number of actions and $D$ is the total delay over $T$ rounds. We then introduce a new algorithm that lifts the requirement of bounded delays by using a wrapper that skips rounds with excessively large delays. The new algorithm maintains the same regret bound, but similar to its predecessor requires prior knowledge of $D$ and $T$. For this algorithm we then construct a novel doubling scheme that forgoes the prior knowledge requirement under the assumption that the delays are available at action time (rather than at loss observation time). This assumption is satisfied in a broad range of applications, including interaction with servers and service providers. The resulting oracle regret bound is of order $\min_β(|S_β|+β\ln K + (KT + D_β)/β)$, where $|S_β|$ is the number of observations with delay exceeding $β$, and $D_β$ is the total delay of observations with delay below $β$. The bound relaxes to $O (\sqrt{(KT + D)\ln K} )$, but we also provide examples where $D_β\ll D$ and the oracle bound has a polynomially better dependence on the problem parameters.

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arXiv2019

Dynamic Pricing with Finitely Many Unknown Valuations

Motivated by posted price auctions where buyers are grouped in an unknown number of latent types characterized by their private values for the good on sale, we investigate revenue maximization in stochastic dynamic pricing when the distribution of buyers' private values is supported on an unknown set of points in [0,1] of unknown cardinality $K$. This setting can be viewed as an instance of a stochastic $K$-armed bandit problem where the location of the arms (the $K$ unknown valuations) must be learned as well. In the distribution-free case, we prove that our setting is just as hard as $K$-armed stochastic bandits: no algorithm can achieve a regret significantly better than $\sqrt{KT}$, (where T is the time horizon); we present an efficient algorithm matching this lower bound up to logarithmic factors. In the distribution-dependent case, we show that for all $K>2$ our setting is strictly harder than $K$-armed stochastic bandits by proving that it is impossible to obtain regret bounds that grow logarithmically in time or slower. On the other hand, when a lower bound $γ>0$ on the smallest drop in the demand curve is known, we prove an upper bound on the regret of order $(1/Δ+(\log \log T)/γ^2)(K\log T)$. This is a significant improvement on previously known regret bounds for discontinuous demand curves, that are at best of order $(K^{12}/γ^8)\sqrt{T}$. When $K=2$ in the distribution-dependent case, the hardness of our setting reduces to that of a stochastic $2$-armed bandit: we prove that an upper bound of order $(\log T)/Δ$ (up to $\log\log$ factors) on the regret can be achieved with no information on the demand curve. Finally, we show a $O(\sqrt{T})$ upper bound on the regret for the setting in which the buyers' decisions are nonstochastic, and the regret is measured with respect to the best between two fixed valuations one of which is known to the seller.

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arXiv2019

Distribution-Dependent Analysis of Gibbs-ERM Principle

Gibbs-ERM learning is a natural idealized model of learning with stochastic optimization algorithms (such as Stochastic Gradient Langevin Dynamics and ---to some extent--- Stochastic Gradient Descent), while it also arises in other contexts, including PAC-Bayesian theory, and sampling mechanisms. In this work we study the excess risk suffered by a Gibbs-ERM learner that uses non-convex, regularized empirical risk with the goal to understand the interplay between the data-generating distribution and learning in large hypothesis spaces. Our main results are distribution-dependent upper bounds on several notions of excess risk. We show that, in all cases, the distribution-dependent excess risk is essentially controlled by the effective dimension $\mathrm{tr}\left(\boldsymbol{H}^{\star} (\boldsymbol{H}^{\star} + λ\boldsymbol{I})^{-1}\right)$ of the problem, where $\boldsymbol{H}^{\star}$ is the Hessian matrix of the risk at a local minimum. This is a well-established notion of effective dimension appearing in several previous works, including the analyses of SGD and ridge regression, but ours is the first work that brings this dimension to the analysis of learning using Gibbs densities. The distribution-dependent view we advocate here improves upon earlier results of Raginsky et al. (2017), and can yield much tighter bounds depending on the interplay between the data-generating distribution and the loss function. The first part of our analysis focuses on the localized excess risk in the vicinity of a fixed local minimizer. This result is then extended to bounds on the global excess risk, by characterizing probabilities of local minima (and their complement) under Gibbs densities, a results which might be of independent interest.

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arXiv2019

Positive and Unlabeled Learning through Negative Selection and Imbalance-aware Classification

Motivated by applications in protein function prediction, we consider a challenging supervised classification setting in which positive labels are scarce and there are no explicit negative labels. The learning algorithm must thus select which unlabeled examples to use as negative training points, possibly ending up with an unbalanced learning problem. We address these issues by proposing an algorithm that combines active learning (for selecting negative examples) with imbalance-aware learning (for mitigating the label imbalance). In our experiments we observe that these two techniques operate synergistically, outperforming state-of-the-art methods on standard protein function prediction benchmarks.

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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.

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arXiv2017

Nonparametric Online Regression while Learning the Metric

We study algorithms for online nonparametric regression that learn the directions along which the regression function is smoother. Our algorithm learns the Mahalanobis metric based on the gradient outer product matrix $\boldsymbol{G}$ of the regression function (automatically adapting to the effective rank of this matrix), while simultaneously bounding the regret ---on the same data sequence--- in terms of the spectrum of $\boldsymbol{G}$. As a preliminary step in our analysis, we extend a nonparametric online learning algorithm by Hazan and Megiddo enabling it to compete against functions whose Lipschitzness is measured with respect to an arbitrary Mahalanobis metric.

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arXiv2017

Algorithmic Chaining and the Role of Partial Feedback in Online Nonparametric Learning

We investigate contextual online learning with nonparametric (Lipschitz) comparison classes under different assumptions on losses and feedback information. For full information feedback and Lipschitz losses, we design the first explicit algorithm achieving the minimax regret rate (up to log factors). In a partial feedback model motivated by second-price auctions, we obtain algorithms for Lipschitz and semi-Lipschitz losses with regret bounds improving on the known bounds for standard bandit feedback. Our analysis combines novel results for contextual second-price auctions with a novel algorithmic approach based on chaining. When the context space is Euclidean, our chaining approach is efficient and delivers an even better regret bound.

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arXiv2017

On the Troll-Trust Model for Edge Sign Prediction in Social Networks

In the problem of edge sign prediction, we are given a directed graph (representing a social network), and our task is to predict the binary labels of the edges (i.e., the positive or negative nature of the social relationships). Many successful heuristics for this problem are based on the troll-trust features, estimating at each node the fraction of outgoing and incoming positive/negative edges. We show that these heuristics can be understood, and rigorously analyzed, as approximators to the Bayes optimal classifier for a simple probabilistic model of the edge labels. We then show that the maximum likelihood estimator for this model approximately corresponds to the predictions of a Label Propagation algorithm run on a transformed version of the original social graph. Extensive experiments on a number of real-world datasets show that this algorithm is competitive against state-of-the-art classifiers in terms of both accuracy and scalability. Finally, we show that troll-trust features can also be used to derive online learning algorithms which have theoretical guarantees even when edges are adversarially labeled.

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arXiv2016

Active Learning for Online Recognition of Human Activities from Streaming Videos

Recognising human activities from streaming videos poses unique challenges to learning algorithms: predictive models need to be scalable, incrementally trainable, and must remain bounded in size even when the data stream is arbitrarily long. Furthermore, as parameter tuning is problematic in a streaming setting, suitable approaches should be parameterless, and make no assumptions on what class labels may occur in the stream. We present here an approach to the recognition of human actions from streaming data which meets all these requirements by: (1) incrementally learning a model which adaptively covers the feature space with simple local classifiers; (2) employing an active learning strategy to reduce annotation requests; (3) achieving promising accuracy within a fixed model size. Extensive experiments on standard benchmarks show that our approach is competitive with state-of-the-art non-incremental methods, and outperforms the existing active incremental baselines.

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arXiv2015

The ABACOC Algorithm: a Novel Approach for Nonparametric Classification of Data Streams

Stream mining poses unique challenges to machine learning: predictive models are required to be scalable, incrementally trainable, must remain bounded in size (even when the data stream is arbitrarily long), and be nonparametric in order to achieve high accuracy even in complex and dynamic environments. Moreover, the learning system must be parameterless ---traditional tuning methods are problematic in streaming settings--- and avoid requiring prior knowledge of the number of distinct class labels occurring in the stream. In this paper, we introduce a new algorithmic approach for nonparametric learning in data streams. Our approach addresses all above mentioned challenges by learning a model that covers the input space using simple local classifiers. The distribution of these classifiers dynamically adapts to the local (unknown) complexity of the classification problem, thus achieving a good balance between model complexity and predictive accuracy. We design four variants of our approach of increasing adaptivity. By means of an extensive empirical evaluation against standard nonparametric baselines, we show state-of-the-art results in terms of accuracy versus model size. For the variant that imposes a strict bound on the model size, we show better performance against all other methods measured at the same model size value. Our empirical analysis is complemented by a theoretical performance guarantee which does not rely on any stochastic assumption on the source generating the stream.

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arXiv2015

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetildeΘ(α^{1/2} T^{1/2})$ minimax regret, where $α$ is the independence number of the underlying graph; the second class induces problems with $\widetildeΘ(δ^{1/3}T^{2/3})$ minimax regret, where $δ$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time.

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arXiv2014

On the Complexity of Learning with Kernels

A well-recognized limitation of kernel learning is the requirement to handle a kernel matrix, whose size is quadratic in the number of training examples. Many methods have been proposed to reduce this computational cost, mostly by using a subset of the kernel matrix entries, or some form of low-rank matrix approximation, or a random projection method. In this paper, we study lower bounds on the error attainable by such methods as a function of the number of entries observed in the kernel matrix or the rank of an approximate kernel matrix. We show that there are kernel learning problems where no such method will lead to non-trivial computational savings. Our results also quantify how the problem difficulty depends on parameters such as the nature of the loss function, the regularization parameter, the norm of the desired predictor, and the kernel matrix rank. Our results also suggest cases where more efficient kernel learning might be possible.

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arXiv2014

Nonstochastic Multi-Armed Bandits with Graph-Structured Feedback

We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide several algorithms addressing different variants of our setting, and provide tight regret bounds depending on combinatorial properties of the information feedback structure.

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arXiv2014

A Generalized Online Mirror Descent with Applications to Classification and Regression

Online learning algorithms are fast, memory-efficient, easy to implement, and applicable to many prediction problems, including classification, regression, and ranking. Several online algorithms were proposed in the past few decades, some based on additive updates, like the Perceptron, and some on multiplicative updates, like Winnow. A unifying perspective on the design and the analysis of online algorithms is provided by online mirror descent, a general prediction strategy from which most first-order algorithms can be obtained as special cases. We generalize online mirror descent to time-varying regularizers with generic updates. Unlike standard mirror descent, our more general formulation also captures second order algorithms, algorithms for composite losses and algorithms for adaptive filtering. Moreover, we recover, and sometimes improve, known regret bounds as special cases of our analysis using specific regularizers. Finally, we show the power of our approach by deriving a new second order algorithm with a regret bound invariant with respect to arbitrary rescalings of individual features.

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arXiv2013

A Gang of Bandits

Multi-armed bandit problems are receiving a great deal of attention because they adequately formalize the exploration-exploitation trade-offs arising in several industrially relevant applications, such as online advertisement and, more generally, recommendation systems. In many cases, however, these applications have a strong social component, whose integration in the bandit algorithm could lead to a dramatic performance increase. For instance, we may want to serve content to a group of users by taking advantage of an underlying network of social relationships among them. In this paper, we introduce novel algorithmic approaches to the solution of such networked bandit problems. More specifically, we design and analyze a global strategy which allocates a bandit algorithm to each network node (user) and allows it to "share" signals (contexts and payoffs) with the neghboring nodes. We then derive two more scalable variants of this strategy based on different ways of clustering the graph nodes. We experimentally compare the algorithm and its variants to state-of-the-art methods for contextual bandits that do not use the relational information. Our experiments, carried out on synthetic and real-world datasets, show a marked increase in prediction performance obtained by exploiting the network structure.

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