arXiv subjects

Alessandro Rudi

Publications and source records attributed to Alessandro Rudi.

At least 73 records · Page 4Linked to original sources

arXiv2019

Affine Invariant Covariance Estimation for Heavy-Tailed Distributions

In this work we provide an estimator for the covariance matrix of a heavy-tailed multivariate distributionWe prove that the proposed estimator $\widehat{\mathbf{S}}$ admits an \textit{affine-invariant} bound of the form \[(1-\varepsilon) \mathbf{S} \preccurlyeq \widehat{\mathbf{S}} \preccurlyeq (1+\varepsilon) \mathbf{S}\]in high probability, where $\mathbf{S}$ is the unknown covariance matrix, and $\preccurlyeq$ is the positive semidefinite order on symmetric matrices. The result only requires the existence of fourth-order moments, and allows for $\varepsilon = O(\sqrt{κ^4 d\log(d/δ)/n})$ where $κ^4$ is a measure of kurtosis of the distribution, $d$ is the dimensionality of the space, $n$ is the sample size, and $1-δ$ is the desired confidence level. More generally, we can allow for regularization with level $λ$, then $d$ gets replaced with the degrees of freedom number. Denoting $\text{cond}(\mathbf{S})$ the condition number of $\mathbf{S}$, the computational cost of the novel estimator is $O(d^2 n + d^3\log(\text{cond}(\mathbf{S})))$, which is comparable to the cost of the sample covariance estimator in the statistically interesing regime $n \ge d$. We consider applications of our estimator to eigenvalue estimation with relative error, and to ridge regression with heavy-tailed random design.

math.ST↗

arXiv2019

Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling

In this paper, we propose and study a Nyström based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nyström sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings, which are also illustrated by numerical results, are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning.

stat.ML↗

arXiv2019

Beyond Least-Squares: Fast Rates for Regularized Empirical Risk Minimization through Self-Concordance

We consider learning methods based on the regularization of a convex empirical risk by a squared Hilbertian norm, a setting that includes linear predictors and non-linear predictors through positive-definite kernels. In order to go beyond the generic analysis leading to convergence rates of the excess risk as $O(1/\sqrt{n})$ from $n$ observations, we assume that the individual losses are self-concordant, that is, their third-order derivatives are bounded by their second-order derivatives. This setting includes least-squares, as well as all generalized linear models such as logistic and softmax regression. For this class of losses, we provide a bias-variance decomposition and show that the assumptions commonly made in least-squares regression, such as the source and capacity conditions, can be adapted to obtain fast non-asymptotic rates of convergence by improving the bias terms, the variance terms or both.

cs.LG↗

arXiv2019

Localized Structured Prediction

Key to structured prediction is exploiting the problem structure to simplify the learning process. A major challenge arises when data exhibit a local structure (e.g., are made by "parts") that can be leveraged to better approximate the relation between (parts of) the input and (parts of) the output. Recent literature on signal processing, and in particular computer vision, has shown that capturing these aspects is indeed essential to achieve state-of-the-art performance. While such algorithms are typically derived on a case-by-case basis, in this work we propose the first theoretical framework to deal with part-based data from a general perspective. We derive a novel approach to deal with these problems and study its generalization properties within the setting of statistical learning theory. Our analysis is novel in that it explicitly quantifies the benefits of leveraging the part-based structure of the problem with respect to the learning rates of the proposed estimator.

stat.ML↗

arXiv2019

Efficient online learning with kernels for adversarial large scale problems

We are interested in a framework of online learning with kernels for low-dimensional but large-scale and potentially adversarial datasets. We study the computational and theoretical performance of online variations of kernel Ridge regression. Despite its simplicity, the algorithm we study is the first to achieve the optimal regret for a wide range of kernels with a per-round complexity of order $n^α$ with $α< 2$. The algorithm we consider is based on approximating the kernel with the linear span of basis functions. Our contributions is two-fold: 1) For the Gaussian kernel, we propose to build the basis beforehand (independently of the data) through Taylor expansion. For $d$-dimensional inputs, we provide a (close to) optimal regret of order $O((\log n)^{d+1})$ with per-round time complexity and space complexity $O((\log n)^{2d})$. This makes the algorithm a suitable choice as soon as $n \gg e^d$ which is likely to happen in a scenario with small dimensional and large-scale dataset; 2) For general kernels with low effective dimension, the basis functions are updated sequentially in a data-adaptive fashion by sampling Nystr{ö}m points. In this case, our algorithm improves the computational trade-off known for online kernel regression.

stat.ML↗

arXiv2019

A General Theory for Structured Prediction with Smooth Convex Surrogates

In this work we provide a theoretical framework for structured prediction that generalizes the existing theory of surrogate methods for binary and multiclass classification based on estimating conditional probabilities with smooth convex surrogates (e.g. logistic regression). The theory relies on a natural characterization of structural properties of the task loss and allows to derive statistical guarantees for many widely used methods in the context of multilabeling, ranking, ordinal regression and graph matching. In particular, we characterize the smooth convex surrogates compatible with a given task loss in terms of a suitable Bregman divergence composed with a link function. This allows to derive tight bounds for the calibration function and to obtain novel results on existing surrogate frameworks for structured prediction such as conditional random fields and quadratic surrogates.

cs.LG↗

arXiv2019

Learning with SGD and Random Features

Sketching and stochastic gradient methods are arguably the most common techniques to derive efficient large scale learning algorithms. In this paper, we investigate their application in the context of nonparametric statistical learning. More precisely, we study the estimator defined by stochastic gradient with mini batches and random features. The latter can be seen as form of nonlinear sketching and used to define approximate kernel methods. The considered estimator is not explicitly penalized/constrained and regularization is implicit. Indeed, our study highlights how different parameters, such as number of features, iterations, step-size and mini-batch size control the learning properties of the solutions. We do this by deriving optimal finite sample bounds, under standard assumptions. The obtained results are corroborated and illustrated by numerical experiments.

stat.ML↗

arXiv2019

On Fast Leverage Score Sampling and Optimal Learning

Leverage score sampling provides an appealing way to perform approximate computations for large matrices. Indeed, it allows to derive faithful approximations with a complexity adapted to the problem at hand. Yet, performing leverage scores sampling is a challenge in its own right requiring further approximations. In this paper, we study the problem of leverage score sampling for positive definite matrices defined by a kernel. Our contribution is twofold. First we provide a novel algorithm for leverage score sampling and second, we exploit the proposed method in statistical learning by deriving a novel solver for kernel ridge regression. Our main technical contribution is showing that the proposed algorithms are currently the most efficient and accurate for these problems.

stat.ML↗

arXiv2018

Approximating the Quadratic Transportation Metric in Near-Linear Time

Computing the quadratic transportation metric (also called the $2$-Wasserstein distance or root mean square distance) between two point clouds, or, more generally, two discrete distributions, is a fundamental problem in machine learning, statistics, computer graphics, and theoretical computer science. A long line of work has culminated in a sophisticated geometric algorithm due to Agarwal and Sharathkumar in 2014, which runs in time $\tilde{O}(n^{3/2})$, where $n$ is the number of points. However, obtaining faster algorithms has proven difficult since the $2$-Wasserstein distance is known to have poor sketching and embedding properties, which limits the effectiveness of geometric approaches. In this paper, we give an extremely simple deterministic algorithm with $\tilde{O}(n)$ runtime by using a completely different approach based on entropic regularization, approximate Sinkhorn scaling, and low-rank approximations of Gaussian kernel matrices. We give explicit dependence of our algorithm on the dimension and precision of the approximation.

cs.DS↗

arXiv2018

Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes

We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for low-dimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinite-dimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with non-linear kernel methods and on a classical benchmark with a linear model.

cs.LG↗

arXiv2018

Exponential convergence of testing error for stochastic gradient methods

We consider binary classification problems with positive definite kernels and square loss, and study the convergence rates of stochastic gradient methods. We show that while the excess testing loss (squared loss) converges slowly to zero as the number of observations (and thus iterations) goes to infinity, the testing error (classification error) converges exponentially fast if low-noise conditions are assumed.

cs.LG↗

arXiv2018

Sharp Analysis of Learning with Discrete Losses

The problem of devising learning strategies for discrete losses (e.g., multilabeling, ranking) is currently addressed with methods and theoretical analyses ad-hoc for each loss. In this paper we study a least-squares framework to systematically design learning algorithms for discrete losses, with quantitative characterizations in terms of statistical and computational complexity. In particular we improve existing results by providing explicit dependence on the number of labels for a wide class of losses and faster learning rates in conditions of low-noise. Theoretical results are complemented with experiments on real datasets, showing the effectiveness of the proposed general approach.

cs.LG↗

arXiv2018

Manifold Structured Prediction

Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Promising experimental results on both simulated and real data complete our study.

stat.ML↗

arXiv2018

Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance

Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn distance, proving that it enjoys the same smoothness as its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula allows us to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.

stat.ML↗

arXiv2018

NYTRO: When Subsampling Meets Early Stopping

Early stopping is a well known approach to reduce the time complexity for performing training and model selection of large scale learning machines. On the other hand, memory/space (rather than time) complexity is the main constraint in many applications, and randomized subsampling techniques have been proposed to tackle this issue. In this paper we ask whether early stopping and subsampling ideas can be combined in a fruitful way. We consider the question in a least squares regression setting and propose a form of randomized iterative regularization based on early stopping and subsampling. In this context, we analyze the statistical and computational properties of the proposed method. Theoretical results are complemented and validated by a thorough experimental analysis.

stat.ML↗

arXiv2018

FALKON: An Optimal Large Scale Kernel Method

Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal statistical accuracy is achieved requiring essentially $O(n)$ memory and $O(n\sqrt{n})$ time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures.

stat.ML↗

arXiv2017

Consistent Multitask Learning with Nonlinear Output Relations

Key to multitask learning is exploiting relationships between different tasks to improve prediction performance. If the relations are linear, regularization approaches can be used successfully. However, in practice assuming the tasks to be linearly related might be restrictive, and allowing for nonlinear structures is a challenge. In this paper, we tackle this issue by casting the problem within the framework of structured prediction. Our main contribution is a novel algorithm for learning multiple tasks which are related by a system of nonlinear equations that their joint outputs need to satisfy. We show that the algorithm is consistent and can be efficiently implemented. Experimental results show the potential of the proposed method.

cs.LG↗

arXiv2017

A Consistent Regularization Approach for Structured Prediction

We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design learning algorithms using a surrogate loss approach and regularization techniques. We prove universal consistency and finite sample bounds characterizing the generalization properties of the proposed methods. Experimental results are provided to demonstrate the practical usefulness of the proposed approach.

cs.LG↗