Pierre Laforgue

Pierre Laforgue, Andrea Della Vecchia, Nicolò Cesa-Bianchi et al.

We introduce and analyze AdaTask, a multitask online learning algorithm that adapts to the unknown structure of the tasks. When the $N$ tasks are stochastically activated, we show that the regret of AdaTask is better, by a factor that can be as large as $\sqrt{N}$, than the regret achieved by running $N$ independent algorithms, one for each task. AdaTask can be seen as a comparator-adaptive version of Follow-the-Regularized-Leader with a Mahalanobis norm potential. Through a variational formulation of this potential, our analysis reveals how AdaTask jointly learns the tasks and their structure. Experiments supporting our findings are presented.

Pierre Laforgue, Stephan Clémençon, Patrice Bertail

Tournament procedures, recently introduced in Lugosi & Mendelson (2016), offer an appealing alternative, from a theoretical perspective at least, to the principle of Empirical Risk Minimization in machine learning. Statistical learning by Median-of-Means (MoM) basically consists in segmenting the training data into blocks of equal size and comparing the statistical performance of every pair of candidate decision rules on each data block: that with highest performance on the majority of the blocks is declared as the winner. In the context of nonparametric regression, functions having won all their duels have been shown to outperform empirical risk minimizers w.r.t. the mean squared error under minimal assumptions, while exhibiting robustness properties. It is the purpose of this paper to extend this approach in order to address other learning problems, in particular for which the performance criterion takes the form of an expectation over pairs of observations rather than over one single observation, as may be the case in pairwise ranking, clustering or metric learning. Precisely, it is proved here that the bounds achieved by MoM are essentially conserved when the blocks are built by means of independent sampling without replacement schemes instead of a simple segmentation. These results are next extended to situations where the risk is related to a pairwise loss function and its empirical counterpart is of the form of a $U$-statistic. Beyond theoretical results guaranteeing the performance of the learning/estimation methods proposed, some numerical experiments provide empirical evidence of their relevance in practice.

Pier Giuseppe Sessa, Pierre Laforgue, Nicolò Cesa-Bianchi et al.

Multitask learning is a powerful framework that enables one to simultaneously learn multiple related tasks by sharing information between them. Quantifying uncertainty in the estimated tasks is of pivotal importance for many downstream applications, such as online or active learning. In this work, we provide novel multitask confidence intervals in the challenging agnostic setting, i.e., when neither the similarity between tasks nor the tasks' features are available to the learner. The obtained intervals do not require i.i.d. data and can be directly applied to bound the regret in online learning. Through a refined analysis of the multitask information gain, we obtain new regret guarantees that, depending on a task similarity parameter, can significantly improve over treating tasks independently. We further propose a novel online learning algorithm that achieves such improved regret without knowing this parameter in advance, i.e., automatically adapting to task similarity. As a second key application of our results, we introduce a novel multitask active learning setup where several tasks must be simultaneously optimized, but only one of them can be queried for feedback by the learner at each round. For this problem, we design a no-regret algorithm that uses our confidence intervals to decide which task should be queried. Finally, we empirically validate our bounds and algorithms on synthetic and real-world (drug discovery) data.

Giulia Clerici, Pierre Laforgue, Nicolò Cesa-Bianchi

Nonstationary phenomena, such as satiation effects in recommendations, have mostly been modeled using bandits with finitely many arms. However, the richer action space provided by linear bandits is often preferred in practice. In this work, we introduce a novel nonstationary linear bandit model, where current rewards are influenced by the learner's past actions in a fixed-size window. Our model, which recovers stationary linear bandits as a special case, leverages two parameters: the window size $m \ge 0$, and an exponent $γ$ that captures the rotting ($γ< 0)$ or rising ($γ> 0$) nature of the phenomenon. When both $m$ and $γ$ are known, we propose and analyze a variant of OFUL which minimizes regret against cycling policies. By choosing the cycle length so as to trade-off approximation and estimation errors, we then prove a bound of order $\sqrt{d}\,(m+1)^{\frac{1}{2}+\max\{γ,0\}}\,T^{3/4}$ (ignoring log factors) on the regret against the optimal sequence of actions, where $T$ is the horizon and $d$ is the dimension of the linear action space. Through a bandit model selection approach, our results are extended to the case where $m$ and $γ$ are unknown. Finally, we complement our theoretical results with experiments against natural baselines.

Tamim El Ahmad, Pierre Laforgue, Florence d'Alché-Buc

Kernel methods are learning algorithms that enjoy solid theoretical foundations while suffering from important computational limitations. Sketching, which consists in looking for solutions among a subspace of reduced dimension, is a well studied approach to alleviate these computational burdens. However, statistically-accurate sketches, such as the Gaussian one, usually contain few null entries, such that their application to kernel methods and their non-sparse Gram matrices remains slow in practice. In this paper, we show that sparsified Gaussian (and Rademacher) sketches still produce theoretically-valid approximations while allowing for important time and space savings thanks to an efficient \emph{decomposition trick}. To support our method, we derive excess risk bounds for both single and multiple output kernel problems, with generic Lipschitz losses, hereby providing new guarantees for a wide range of applications, from robust regression to multiple quantile regression. Our theoretical results are complemented with experiments showing the empirical superiority of our approach over SOTA sketching methods.

Juliette Achddou, Nicolò Cesa-Bianchi, Pierre Laforgue

We study multitask online learning in a setting where agents can only exchange information with their neighbors on an arbitrary communication network. We introduce $\texttt{MT-CO}_2\texttt{OL}$, a decentralized algorithm for this setting whose regret depends on the interplay between the task similarities and the network structure. Our analysis shows that the regret of $\texttt{MT-CO}_2\texttt{OL}$ is never worse (up to constants) than the bound obtained when agents do not share information. On the other hand, our bounds significantly improve when neighboring agents operate on similar tasks. In addition, we prove that our algorithm can be made differentially private with a negligible impact on the regret. Finally, we provide experimental support for our theory.

Tamim El Ahmad, Luc Brogat-Motte, Pierre Laforgue et al.

Leveraging the kernel trick in both the input and output spaces, surrogate kernel methods are a flexible and theoretically grounded solution to structured output prediction. If they provide state-of-the-art performance on complex data sets of moderate size (e.g., in chemoinformatics), these approaches however fail to scale. We propose to equip surrogate kernel methods with sketching-based approximations, applied to both the input and output feature maps. We prove excess risk bounds on the original structured prediction problem, showing how to attain close-to-optimal rates with a reduced sketch size that depends on the eigendecay of the input/output covariance operators. From a computational perspective, we show that the two approximations have distinct but complementary impacts: sketching the input kernel mostly reduces training time, while sketching the output kernel decreases the inference time. Empirically, our approach is shown to scale, achieving state-of-the-art performance on benchmark data sets where non-sketched methods are intractable.

Tamim El Ahmad, Junjie Yang, Pierre Laforgue et al.

By leveraging the kernel trick in the output space, kernel-induced losses provide a principled way to define structured output prediction tasks for a wide variety of output modalities. In particular, they have been successfully used in the context of surrogate non-parametric regression, where the kernel trick is typically exploited in the input space as well. However, when inputs are images or texts, more expressive models such as deep neural networks seem more suited than non-parametric methods. In this work, we tackle the question of how to train neural networks to solve structured output prediction tasks, while still benefiting from the versatility and relevance of kernel-induced losses. We design a novel family of deep neural architectures, whose last layer predicts in a data-dependent finite-dimensional subspace of the infinite-dimensional output feature space deriving from the kernel-induced loss. This subspace is chosen as the span of the eigenfunctions of a randomly-approximated version of the empirical kernel covariance operator. Interestingly, this approach unlocks the use of gradient descent algorithms (and consequently of any neural architecture) for structured prediction. Experiments on synthetic tasks as well as real-world supervised graph prediction problems show the relevance of our method.

Pierre Laforgue, Giulia Clerici, Nicolò Cesa-Bianchi et al.

Motivated by the fact that humans like some level of unpredictability or novelty, and might therefore get quickly bored when interacting with a stationary policy, we introduce a novel non-stationary bandit problem, where the expected reward of an arm is fully determined by the time elapsed since the arm last took part in a switch of actions. Our model generalizes previous notions of delay-dependent rewards, and also relaxes most assumptions on the reward function. This enables the modeling of phenomena such as progressive satiation and periodic behaviours. Building upon the Combinatorial Semi-Bandits (CSB) framework, we design an algorithm and prove a bound on its regret with respect to the optimal non-stationary policy (which is NP-hard to compute). Similarly to previous works, our regret analysis is based on defining and solving an appropriate trade-off between approximation and estimation. Preliminary experiments confirm the superiority of our algorithm over both the oracle greedy approach and a vanilla CSB solver.

Stephan Clémençon, Pierre Laforgue, Robin Vogel

In practice, and especially when training deep neural networks, visual recognition rules are often learned based on various sources of information. On the other hand, the recent deployment of facial recognition systems with uneven performances on different population segments has highlighted the representativeness issues induced by a naive aggregation of the datasets. In this paper, we show how biasing models can remedy these problems. Based on the (approximate) knowledge of the biasing mechanisms at work, our approach consists in reweighting the observations, so as to form a nearly debiased estimator of the target distribution. One key condition is that the supports of the biased distributions must partly overlap, and cover the support of the target distribution. In order to meet this requirement in practice, we propose to use a low dimensional image representation, shared across the image databases. Finally, we provide numerical experiments highlighting the relevance of our approach.

Nicolò Cesa-Bianchi, Pierre Laforgue, Andrea Paudice et al.

We introduce and analyze MT-OMD, a multitask generalization of Online Mirror Descent (OMD) which operates by sharing updates between tasks. We prove that the regret of MT-OMD is of order $\sqrt{1 + σ^2(N-1)}\sqrt{T}$, where $σ^2$ is the task variance according to the geometry induced by the regularizer, $N$ is the number of tasks, and $T$ is the time horizon. Whenever tasks are similar, that is $σ^2 \le 1$, our method improves upon the $\sqrt{NT}$ bound obtained by running independent OMDs on each task. We further provide a matching lower bound, and show that our multitask extensions of Online Gradient Descent and Exponentiated Gradient, two major instances of OMD, enjoy closed-form updates, making them easy to use in practice. Finally, we present experiments which support our theoretical findings.

Guillaume Staerman, Pierre Laforgue, Pavlo Mozharovskyi et al.

Issued from Optimal Transport, the Wasserstein distance has gained importance in Machine Learning due to its appealing geometrical properties and the increasing availability of efficient approximations. In this work, we consider the problem of estimating the Wasserstein distance between two probability distributions when observations are polluted by outliers. To that end, we investigate how to leverage Medians of Means (MoM) estimators to robustify the estimation of Wasserstein distance. Exploiting the dual Kantorovitch formulation of Wasserstein distance, we introduce and discuss novel MoM-based robust estimators whose consistency is studied under a data contamination model and for which convergence rates are provided. These MoM estimators enable to make Wasserstein Generative Adversarial Network (WGAN) robust to outliers, as witnessed by an empirical study on two benchmarks CIFAR10 and Fashion MNIST. Eventually, we discuss how to combine MoM with the entropy-regularized approximation of the Wasserstein distance and propose a simple MoM-based re-weighting scheme that could be used in conjunction with the Sinkhorn algorithm.

Pierre Laforgue, Guillaume Staerman, Stephan Clémençon

In contrast to the empirical mean, the Median-of-Means (MoM) is an estimator of the mean $θ$ of a square integrable r.v. $Z$, around which accurate nonasymptotic confidence bounds can be built, even when $Z$ does not exhibit a sub-Gaussian tail behavior. Thanks to the high confidence it achieves on heavy-tailed data, MoM has found various applications in machine learning, where it is used to design training procedures that are not sensitive to atypical observations. More recently, a new line of work is now trying to characterize and leverage MoM's ability to deal with corrupted data. In this context, the present work proposes a general study of MoM's concentration properties under the contamination regime, that provides a clear understanding of the impact of the outlier proportion and the number of blocks chosen. The analysis is extended to (multisample) $U$-statistics, i.e. averages over tuples of observations, that raise additional challenges due to the dependence induced. Finally, we show that the latter bounds can be used in a straightforward fashion to derive generalization guarantees for pairwise learning in a contaminated setting, and propose an algorithm to compute provably reliable decision functions.

Pierre Laforgue, Alex Lambert, Luc Brogat-Motte et al.

Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous works have focused on the square norm loss function, completely neglecting robustness issues that may arise in such surrogate problems. To overcome this limitation, this paper develops a duality approach that allows to solve OVK machines for a wide range of loss functions. The infinite dimensional Lagrange multipliers are handled through a Double Representer Theorem, and algorithms for $ε$-insensitive losses and the Huber loss are thoroughly detailed. Robustness benefits are emphasized by a theoretical stability analysis, as well as empirical improvements on structured data applications.

Stephan Clémençon, Pierre Laforgue

With the deluge of digitized information in the Big Data era, massive datasets are becoming increasingly available for learning predictive models. However, in many practical situations, the poor control of the data acquisition processes may naturally jeopardize the outputs of machine learning algorithms, and selection bias issues are now the subject of much attention in the literature. The present article investigates how to extend Empirical Risk Minimization, the principal paradigm in statistical learning, when training observations are generated from biased models, i.e., from distributions that are different from that in the test/prediction stage, and absolutely continuous with respect to the latter. Precisely, we show how to build a "nearly debiased" training statistical population from biased samples and the related biasing functions, following in the footsteps of the approach originally proposed in Vardi (1985). Furthermore, we study from a nonasymptotic perspective the performance of minimizers of an empirical version of the risk computed from the statistical population thus created. Remarkably, the learning rate achieved by this procedure is of the same order as that attained in absence of selection bias. Beyond the theoretical guarantees, we also present experimental results supporting the relevance of the algorithmic approach promoted in this paper.

Pierre Laforgue, Stephan Clémençon, Florence d'Alché-Buc

This paper investigates a novel algorithmic approach to data representation based on kernel methods. Assuming that the observations lie in a Hilbert space X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the expected reconstruction error. Beyond a first extension of the autoencoding scheme to possibly infinite dimensional Hilbert spaces, KAE further allows to autoencode any kind of data by choosing X to be itself a RKHS. A theoretical analysis of the model is carried out, providing a generalization bound, and shedding light on its connection with Kernel Principal Component Analysis. The proposed algorithms are then detailed at length: they crucially rely on the form taken by the minimizers, revealed by a dedicated Representer Theorem. Finally, numerical experiments on both simulated data and real labeled graphs (molecules) provide empirical evidence of the KAE performances.