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Lorenzo Rosasco

Lorenzo Rosasco appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Machine Learningmath.OCComputer VisionArtificial Intelligencemath.STStatistics Theorymath.FARoboticsSoundInformation Theorymath.DSmath.ITmath.NAmath.PRMathematical SoftwareNeural and Evolutionary ComputingNumerical Analysisphysics.ao-ph

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Published work

57 published item(s)

preprint2026arXiv

A New Formulation for Zeroth-Order Optimization of Adversarial EXEmples in Malware Detection

Machine learning malware detectors are vulnerable to adversarial EXEmples, i.e., carefully-crafted Windows programs tailored to evade detection. Unlike other adversarial problems, attacks in this context must be functionality-preserving, a constraint that is challenging to address. As a consequence, heuristic algorithms are typically used, which inject new content, either randomly-picked or harvested from legitimate programs. In this paper, we show how learning malware detectors can be cast within a zeroth-order optimization framework, which allows incorporating functionality-preserving manipulations. This permits the deployment of sound and efficient gradient-free optimization algorithms, which come with theoretical guarantees and allow for minimal hyper-parameters tuning. As a by-product, we propose and study ZEXE, a novel zeroth-order attack against Windows malware detection. Compared to state-of-the-art techniques, ZEXE provides improvement in the evasion rate, reducing to less than one third the size of the injected content.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Dynamic robotic cloth folding with efficient Koopman operator-based model predictive control

Robotic cloth folding is a challenging task, particularly when considering dynamic folding tasks, which aim at folding cloth by fast motions that leverage its dynamics. When subject to such fast motions, the complexity of cloth dynamics hinders both system identification and planning of folding trajectories, resulting in a difficult simulation-to-reality transfer when using physical models of cloth. Compared to the dexterity that humans exhibit when performing folding tasks, robotic approaches usually employ small garments with quite rigid dynamics, and are either too slow, or fast but imprecise, requiring several attempts to achieve a reasonably good fold. In this paper, we tackle these challenges by generating fast folding trajectories with a novel model predictive controller, integrating physics-based simulation of cloth dynamics and efficient, kernel-based Koopman operator regression. Koopman operator regression, an increasingly popular machine learning technique for nonlinear system identification, is used to obtain a linear model for the cloth being folded. Such a surrogate model, trained with data from a high-fidelity, physics-based cloth simulator, can then be employed within a suitable model predictive control algorithm, in place of the costly, nonlinear one, to efficiently generate folding trajectories to be executed by a robotic manipulator. Both in simulated and real-robot experiments, we show how the linearization supplied by the Koopman operator-based model can be employed to efficiently generate fast folding trajectories to unseen poses, without sacrificing folding accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching

Black-Box Variational Inference (BBVI) typically relies on Stochastic Gradient Descent (SGD) to optimize the Evidence Lower Bound (ELBO). However, the stochastic gradients in BBVI inherently exhibit unbounded variance, violating standard assumptions and instead satisfying the weaker Blum-Gladyshev (BG) condition, where variance grows quadratically with distance from the optimum. In this paper, we bridge the gap between stochastic optimization theory and the practical instances of BBVI. Focusing on the broad elliptic location-scale family of parameterized distributions, we offer two main contributions. First, we prove the existence of an ELBO solution, a foundational property usually assumed a priori in the literature. Second, we establish comprehensive convergence guarantees spanning finite-time and asymptotic regimes for Minibatch Projected SGD (PSGD) equipped with dynamic batching and preconditioning under the BG condition. Our theoretical framework demonstrates that dynamic batching combined with preconditioning systematically enables rigorous guarantees even in complex settings. We illustrate our theoretical findings with numerical results, highlighting the efficacy of our approach for modern inference tasks.

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preprint2022arXiv

Ada-BKB: Scalable Gaussian Process Optimization on Continuous Domains by Adaptive Discretization

Gaussian process optimization is a successful class of algorithms(e.g. GP-UCB) to optimize a black-box function through sequential evaluations. However, for functions with continuous domains, Gaussian process optimization has to rely on either a fixed discretization of the space, or the solution of a non-convex optimization subproblem at each evaluation. The first approach can negatively affect performance, while the second approach requires a heavy computational burden. A third option, only recently theoretically studied, is to adaptively discretize the function domain. Even though this approach avoids the extra non-convex optimization costs, the overall computational complexity is still prohibitive. An algorithm such as GP-UCB has a runtime of $O(T^4)$, where $T$ is the number of iterations. In this paper, we introduce Ada-BKB (Adaptive Budgeted Kernelized Bandit), a no-regret Gaussian process optimization algorithm for functions on continuous domains, that provably runs in $O(T^2 d_\text{eff}^2)$, where $d_\text{eff}$ is the effective dimension of the explored space, and which is typically much smaller than $T$. We corroborate our theoretical findings with experiments on synthetic non-convex functions and on the real-world problem of hyper-parameter optimization, confirming the good practical performances of the proposed approach.

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preprint2022arXiv

Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (linear) version of neural operators in the formalism of Gaussian processes. We then extend this analytic treatment to general deep neural operators using approximate methods from Bayesian deep learning. We extend previous results on neural operators by providing them with uncertainty quantification. As a result, our approach is able to identify cases, and provide structured uncertainty estimates, where the neural operator fails to predict well.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Efficient Hyperparameter Tuning for Large Scale Kernel Ridge Regression

Kernel methods provide a principled approach to nonparametric learning. While their basic implementations scale poorly to large problems, recent advances showed that approximate solvers can efficiently handle massive datasets. A shortcoming of these solutions is that hyperparameter tuning is not taken care of, and left for the user to perform. Hyperparameters are crucial in practice and the lack of automated tuning greatly hinders efficiency and usability. In this paper, we work to fill in this gap focusing on kernel ridge regression based on the Nyström approximation. After reviewing and contrasting a number of hyperparameter tuning strategies, we propose a complexity regularization criterion based on a data dependent penalty, and discuss its efficient optimization. Then, we proceed to a careful and extensive empirical evaluation highlighting strengths and weaknesses of the different tuning strategies. Our analysis shows the benefit of the proposed approach, that we hence incorporate in a library for large scale kernel methods to derive adaptively tuned solutions.

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preprint2022arXiv

Efficient Unsupervised Learning for Plankton Images

Monitoring plankton populations in situ is fundamental to preserve the aquatic ecosystem. Plankton microorganisms are in fact susceptible of minor environmental perturbations, that can reflect into consequent morphological and dynamical modifications. Nowadays, the availability of advanced automatic or semi-automatic acquisition systems has been allowing the production of an increasingly large amount of plankton image data. The adoption of machine learning algorithms to classify such data may be affected by the significant cost of manual annotation, due to both the huge quantity of acquired data and the numerosity of plankton species. To address these challenges, we propose an efficient unsupervised learning pipeline to provide accurate classification of plankton microorganisms. We build a set of image descriptors exploiting a two-step procedure. First, a Variational Autoencoder (VAE) is trained on features extracted by a pre-trained neural network. We then use the learnt latent space as image descriptor for clustering. We compare our method with state-of-the-art unsupervised approaches, where a set of pre-defined hand-crafted features is used for clustering of plankton images. The proposed pipeline outperforms the benchmark algorithms for all the plankton datasets included in our analysis, providing better image embedding properties.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Fast iterative regularization by reusing data

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is not smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to solve inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early-stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set, i.e. redundant information. More precisely we show that the linear systems can be used more than once along the iteration. Despite the simplicity of the idea, we show that this procedure brings surprising advantages in the numerical applications. We discuss various approaches to take advantage of redundant information, that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction through total variation. We confirm the efficiency of the proposed procedures, comparing the results with state-of-the-art methods.

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preprint2022arXiv

From Handheld to Unconstrained Object Detection: a Weakly-supervised On-line Learning Approach

Deep Learning (DL) based methods for object detection achieve remarkable performance at the cost of computationally expensive training and extensive data labeling. Robots embodiment can be exploited to mitigate this burden by acquiring automatically annotated training data via a natural interaction with a human showing the object of interest, handheld. However, learning solely from this data may introduce biases (the so-called domain shift), and prevents adaptation to novel tasks. While Weakly-supervised Learning (WSL) offers a well-established set of techniques to cope with these problems in general-purpose Computer Vision, its adoption in challenging robotic domains is still at a preliminary stage. In this work, we target the scenario of a robot trained in a teacher-learner setting to detect handheld objects. The aim is to improve detection performance in different settings by letting the robot explore the environment with a limited human labeling budget. We compare several techniques for WSL in detection pipelines to reduce model re-training costs without compromising accuracy, proposing solutions which target the considered robotic scenario. We show that the robot can improve adaptation to novel domains, either by interacting with a human teacher (Active Learning) or with an autonomous supervision (Semi-supervised Learning). We integrate our strategies into an on-line detection method, achieving efficient model update capabilities with few labels. We experimentally benchmark our method on challenging robotic object detection tasks under domain shift.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Iterative regularization for low complexity regularizers

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

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preprint2022arXiv

Mean Nyström Embeddings for Adaptive Compressive Learning

Compressive learning is an approach to efficient large scale learning based on sketching an entire dataset to a single mean embedding (the sketch), i.e. a vector of generalized moments. The learning task is then approximately solved as an inverse problem using an adapted parametric model. Previous works in this context have focused on sketches obtained by averaging random features, that while universal can be poorly adapted to the problem at hand. In this paper, we propose and study the idea of performing sketching based on data-dependent Nyström approximation. From a theoretical perspective we prove that the excess risk can be controlled under a geometric assumption relating the parametric model used to learn from the sketch and the covariance operator associated to the task at hand. Empirically, we show for k-means clustering and Gaussian modeling that for a fixed sketch size, Nyström sketches indeed outperform those built with random features.

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preprint2022arXiv

Multiclass learning with margin: exponential rates with no bias-variance trade-off

We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multiclass setting.

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preprint2022arXiv

Nyström Kernel Mean Embeddings

Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and illustrate our theoretical findings with numerical experiments.

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preprint2022arXiv

Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.

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preprint2022arXiv

Physics Informed Shallow Machine Learning for Wind Speed Prediction

The ability to predict wind is crucial for both energy production and weather forecasting. Mechanistic models that form the basis of traditional forecasting perform poorly near the ground. In this paper, we take an alternative data-driven approach based on supervised learning. We analyze a massive dataset of wind measured from anemometers located at 10 m height in 32 locations in two central and north west regions of Italy (Abruzzo and Liguria). We train supervised learning algorithms using the past history of wind to predict its value at a future time (horizon). Using data from a single location and time horizon we compare systematically several algorithms where we vary the input/output variables, the memory of the input and the linear vs non-linear learning model. We then compare performance of the best algorithms across all locations and forecasting horizons. We find that the optimal design as well as its performance vary with the location. We demonstrate that the presence of a reproducible diurnal cycle provides a rationale to understand this variation. We conclude with a systematic comparison with state of the art algorithms and show that, when the model is accurately designed, shallow algorithms are competitive with more complex deep architectures.

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preprint2022arXiv

Scaling Gaussian Process Optimization by Evaluating a Few Unique Candidates Multiple Times

Computing a Gaussian process (GP) posterior has a computational cost cubical in the number of historical points. A reformulation of the same GP posterior highlights that this complexity mainly depends on how many \emph{unique} historical points are considered. This can have important implication in active learning settings, where the set of historical points is constructed sequentially by the learner. We show that sequential black-box optimization based on GPs (GP-Opt) can be made efficient by sticking to a candidate solution for multiple evaluation steps and switch only when necessary. Limiting the number of switches also limits the number of unique points in the history of the GP. Thus, the efficient GP reformulation can be used to exactly and cheaply compute the posteriors required to run the GP-Opt algorithms. This approach is especially useful in real-world applications of GP-Opt with high switch costs (e.g. switching chemicals in wet labs, data/model loading in hyperparameter optimization). As examples of this meta-approach, we modify two well-established GP-Opt algorithms, GP-UCB and GP-EI, to switch candidates as infrequently as possible adapting rules from batched GP-Opt. These versions preserve all the theoretical no-regret guarantees while improving practical aspects of the algorithms such as runtime, memory complexity, and the ability of batching candidates and evaluating them in parallel.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Asymptotics of Ridge (less) Regression under General Source Condition

We analyze the prediction error of ridge regression in an asymptotic regime where the sample size and dimension go to infinity at a proportional rate. In particular, we consider the role played by the structure of the true regression parameter. We observe that the case of a general deterministic parameter can be reduced to the case of a random parameter from a structured prior. The latter assumption is a natural adaptation of classic smoothness assumptions in nonparametric regression, which are known as source conditions in the the context of regularization theory for inverse problems. Roughly speaking, we assume the large coefficients of the parameter are in correspondence to the principal components. In this setting a precise characterisation of the test error is obtained, depending on the inputs covariance and regression parameter structure. We illustrate this characterisation in a simplified setting to investigate the influence of the true parameter on optimal regularisation for overparameterized models. We show that interpolation (no regularisation) can be optimal even with bounded signal-to-noise ratio (SNR), provided that the parameter coefficients are larger on high-variance directions of the data, corresponding to a more regular function than posited by the regularization term. This contrasts with previous work considering ridge regression with isotropic prior, in which case interpolation is only optimal in the limit of infinite SNR.

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preprint2021arXiv

Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

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preprint2020arXiv

A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings

We propose and analyze a novel theoretical and algorithmic framework for structured prediction. While so far the term has referred to discrete output spaces, here we consider more general settings, such as manifolds or spaces of probability measures. We define structured prediction as a problem where the output space lacks a vectorial structure. We identify and study a large class of loss functions that implicitly defines a suitable geometry on the problem. The latter is the key to develop an algorithmic framework amenable to a sharp statistical analysis and yielding efficient computations. When dealing with output spaces with infinite cardinality, a suitable implicit formulation of the estimator is shown to be crucial.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Decentralised Learning with Random Features and Distributed Gradient Descent

We investigate the generalisation performance of Distributed Gradient Descent with Implicit Regularisation and Random Features in the homogenous setting where a network of agents are given data sampled independently from the same unknown distribution. Along with reducing the memory footprint, Random Features are particularly convenient in this setting as they provide a common parameterisation across agents that allows to overcome previous difficulties in implementing Decentralised Kernel Regression. Under standard source and capacity assumptions, we establish high probability bounds on the predictive performance for each agent as a function of the step size, number of iterations, inverse spectral gap of the communication matrix and number of Random Features. By tuning these parameters, we obtain statistical rates that are minimax optimal with respect to the total number of samples in the network. The algorithm provides a linear improvement over single machine Gradient Descent in memory cost and, when agents hold enough data with respect to the network size and inverse spectral gap, a linear speed-up in computational runtime for any network topology. We present simulations that show how the number of Random Features, iterations and samples impact predictive performance.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Hyperbolic Manifold Regression

Geometric representation learning has recently shown great promise in several machine learning settings, ranging from relational learning to language processing and generative models. In this work, we consider the problem of performing manifold-valued regression onto an hyperbolic space as an intermediate component for a number of relevant machine learning applications. In particular, by formulating the problem of predicting nodes of a tree as a manifold regression task in the hyperbolic space, we propose a novel perspective on two challenging tasks: 1) hierarchical classification via label embeddings and 2) taxonomy extension of hyperbolic representations. To address the regression problem we consider previous methods as well as proposing two novel approaches that are computationally more advantageous: a parametric deep learning model that is informed by the geodesics of the target space and a non-parametric kernel-method for which we also prove excess risk bounds. Our experiments show that the strategy of leveraging the hyperbolic geometry is promising. In particular, in the taxonomy expansion setting, we find that the hyperbolic-based estimators significantly outperform methods performing regression in the ambient Euclidean space.

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preprint2020arXiv

Near-linear Time Gaussian Process Optimization with Adaptive Batching and Resparsification

Gaussian processes (GP) are one of the most successful frameworks to model uncertainty. However, GP optimization (e.g., GP-UCB) suffers from major scalability issues. Experimental time grows linearly with the number of evaluations, unless candidates are selected in batches (e.g., using GP-BUCB) and evaluated in parallel. Furthermore, computational cost is often prohibitive since algorithms such as GP-BUCB require a time at least quadratic in the number of dimensions and iterations to select each batch. In this paper, we introduce BBKB (Batch Budgeted Kernel Bandits), the first no-regret GP optimization algorithm that provably runs in near-linear time and selects candidates in batches. This is obtained with a new guarantee for the tracking of the posterior variances that allows BBKB to choose increasingly larger batches, improving over GP-BUCB. Moreover, we show that the same bound can be used to adaptively delay costly updates to the sparse GP approximation used by BBKB, achieving a near-constant per-step amortized cost. These findings are then confirmed in several experiments, where BBKB is much faster than state-of-the-art methods.

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preprint2020arXiv

Theory III: Dynamics and Generalization in Deep Networks

The key to generalization is controlling the complexity of the network. However, there is no obvious control of complexity -- such as an explicit regularization term -- in the training of deep networks for classification. We will show that a classical form of norm control -- but kind of hidden -- is present in deep networks trained with gradient descent techniques on exponential-type losses. In particular, gradient descent induces a dynamics of the normalized weights which converge for $t \to \infty$ to an equilibrium which corresponds to a minimum norm (or maximum margin) solution. For sufficiently large but finite $ρ$ -- and thus finite $t$ -- the dynamics converges to one of several margin maximizers, with the margin monotonically increasing towards a limit stationary point of the flow. In the usual case of stochastic gradient descent, most of the stationary points are likely to be convex minima corresponding to a constrained minimizer -- the network with normalized weights-- which corresponds to vanishing regularization. The solution has zero generalization gap, for fixed architecture, asymptotically for $N \to \infty$, where $N$ is the number of training examples. Our approach extends some of the original results of Srebro from linear networks to deep networks and provides a new perspective on the implicit bias of gradient descent. We believe that the elusive complexity control we describe is responsible for the puzzling empirical finding of good predictive performance by deep networks, despite overparametrization.

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preprint2019arXiv

Monte Carlo wavelets: a randomized approach to frame discretization

In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by spectral calculus. Then, we propose a stochastic discretization based on Monte Carlo estimates of integral operators. Using concentration of measure results, we establish the convergence of such a discretization and derive convergence rates under natural regularity assumptions.

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preprint2016arXiv

A first-order stochastic primal-dual algorithm with correction step

We investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I. Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty, 2011} for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models.

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preprint2016arXiv

Generalization Properties and Implicit Regularization for Multiple Passes SGM

We study the generalization properties of stochastic gradient methods for learning with convex loss functions and linearly parameterized functions. We show that, in the absence of penalizations or constraints, the stability and approximation properties of the algorithm can be controlled by tuning either the step-size or the number of passes over the data. In this view, these parameters can be seen to control a form of implicit regularization. Numerical results complement the theoretical findings.

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preprint2016arXiv

Less is More: Nyström Computational Regularization

We study Nyström type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling and high probability estimates are considered. In particular, we prove that these approaches can achieve optimal learning bounds, provided the subsampling level is suitably chosen. These results suggest a simple incremental variant of Nyström Kernel Regularized Least Squares, where the subsampling level implements a form of computational regularization, in the sense that it controls at the same time regularization and computations. Extensive experimental analysis shows that the considered approach achieves state of the art performances on benchmark large scale datasets.

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preprint2015arXiv

A Stochastic forward-backward splitting method for solving monotone inclusions in Hilbert spaces

We propose and analyze the convergence of a novel stochastic forward-backward splitting algorithm for solving monotone inclusions given by the sum of a maximal monotone operator and a single-valued maximal monotone cocoercive operator. This latter framework has a number of interesting special cases, including variational inequalities and convex minimization problems, while stochastic approaches are practically relevant to account for perturbations in the data. The algorithm we propose is a stochastic extension of the classical deterministic forward-backward method, and is obtained considering the composition of the resolvent of the maximal monotone operator with a forward step based on a stochastic estimate of the single-valued operator. Our study provides a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. The approach we consider allows to avoid averaging, a feature critical when considering methods based on sparsity, and, for minimization problems, it allows to obtain convergence rates matching those obtained by stochastic extensions of so called accelerated methods. Stochastic quasi Fejer's sequences are a key technical tool to prove almost sure convergence.

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preprint2015arXiv

A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions

We propose an inertial forward-backward splitting algorithm to compute the zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal-dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.

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preprint2015arXiv

Convex Learning of Multiple Tasks and their Structure

Reducing the amount of human supervision is a key problem in machine learning and a natural approach is that of exploiting the relations (structure) among different tasks. This is the idea at the core of multi-task learning. In this context a fundamental question is how to incorporate the tasks structure in the learning problem.We tackle this question by studying a general computational framework that allows to encode a-priori knowledge of the tasks structure in the form of a convex penalty; in this setting a variety of previously proposed methods can be recovered as special cases, including linear and non-linear approaches. Within this framework, we show that tasks and their structure can be efficiently learned considering a convex optimization problem that can be approached by means of block coordinate methods such as alternating minimization and for which we prove convergence to the global minimum.

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preprint2015arXiv

Deep Convolutional Networks are Hierarchical Kernel Machines

In i-theory a typical layer of a hierarchical architecture consists of HW modules pooling the dot products of the inputs to the layer with the transformations of a few templates under a group. Such layers include as special cases the convolutional layers of Deep Convolutional Networks (DCNs) as well as the non-convolutional layers (when the group contains only the identity). Rectifying nonlinearities -- which are used by present-day DCNs -- are one of the several nonlinearities admitted by i-theory for the HW module. We discuss here the equivalence between group averages of linear combinations of rectifying nonlinearities and an associated kernel. This property implies that present-day DCNs can be exactly equivalent to a hierarchy of kernel machines with pooling and non-pooling layers. Finally, we describe a conjecture for theoretically understanding hierarchies of such modules. A main consequence of the conjecture is that hierarchies of trained HW modules minimize memory requirements while computing a selective and invariant representation.

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preprint2015arXiv

Enabling Depth-driven Visual Attention on the iCub Humanoid Robot: Instructions for Use and New Perspectives

The importance of depth perception in the interactions that humans have within their nearby space is a well established fact. Consequently, it is also well known that the possibility of exploiting good stereo information would ease and, in many cases, enable, a large variety of attentional and interactive behaviors on humanoid robotic platforms. However, the difficulty of computing real-time and robust binocular disparity maps from moving stereo cameras often prevents from relying on this kind of cue to visually guide robots' attention and actions in real-world scenarios. The contribution of this paper is two-fold: first, we show that the Efficient Large-scale Stereo Matching algorithm (ELAS) by A. Geiger et al. 2010 for computation of the disparity map is well suited to be used on a humanoid robotic platform as the iCub robot; second, we show how, provided with a fast and reliable stereo system, implementing relatively challenging visual behaviors in natural settings can require much less effort. As a case of study we consider the common situation where the robot is asked to focus the attention on one object close in the scene, showing how a simple but effective disparity-based segmentation solves the problem in this case. Indeed this example paves the way to a variety of other similar applications.

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preprint2015arXiv

Holographic Embeddings of Knowledge Graphs

Learning embeddings of entities and relations is an efficient and versatile method to perform machine learning on relational data such as knowledge graphs. In this work, we propose holographic embeddings (HolE) to learn compositional vector space representations of entire knowledge graphs. The proposed method is related to holographic models of associative memory in that it employs circular correlation to create compositional representations. By using correlation as the compositional operator HolE can capture rich interactions but simultaneously remains efficient to compute, easy to train, and scalable to very large datasets. In extensive experiments we show that holographic embeddings are able to outperform state-of-the-art methods for link prediction in knowledge graphs and relational learning benchmark datasets.

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preprint2015arXiv

Iterative Regularization for Learning with Convex Loss Functions

We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint or penalization is considered, and generalization is achieved by (early) stopping an empirical iteration. We consider a nonparametric setting, in the framework of reproducing kernel Hilbert spaces, and prove finite sample bounds on the excess risk under general regularity conditions. Our study provides a new class of efficient regularized learning algorithms and gives insights on the interplay between statistics and optimization in machine learning.

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preprint2015arXiv

Learning Multiple Visual Tasks while Discovering their Structure

Multi-task learning is a natural approach for computer vision applications that require the simultaneous solution of several distinct but related problems, e.g. object detection, classification, tracking of multiple agents, or denoising, to name a few. The key idea is that exploring task relatedness (structure) can lead to improved performances. In this paper, we propose and study a novel sparse, non-parametric approach exploiting the theory of Reproducing Kernel Hilbert Spaces for vector-valued functions. We develop a suitable regularization framework which can be formulated as a convex optimization problem, and is provably solvable using an alternating minimization approach. Empirical tests show that the proposed method compares favorably to state of the art techniques and further allows to recover interpretable structures, a problem of interest in its own right.

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preprint2015arXiv

Learning with incremental iterative regularization

Within a statistical learning setting, we propose and study an iterative regularization algorithm for least squares defined by an incremental gradient method. In particular, we show that, if all other parameters are fixed a priori, the number of passes over the data (epochs) acts as a regularization parameter, and prove strong universal consistency, i.e. almost sure convergence of the risk, as well as sharp finite sample bounds for the iterates. Our results are a step towards understanding the effect of multiple epochs in stochastic gradient techniques in machine learning and rely on integrating statistical and optimization results.

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preprint2015arXiv

Modified Fejér sequences and applications

In this note, we propose and study the notion of modified Fejér sequences. Within a Hilbert space setting, we show that it provides a unifying framework to prove convergence rates for objective function values of several optimization algorithms. In particular, our results apply to forward-backward splitting algorithm, incremental subgradient proximal algorithm, and the Douglas-Rachford splitting method including and generalizing known results.

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preprint2015arXiv

On Invariance and Selectivity in Representation Learning

We discuss data representation which can be learned automatically from data, are invariant to transformations, and at the same time selective, in the sense that two points have the same representation only if they are one the transformation of the other. The mathematical results here sharpen some of the key claims of i-theory -- a recent theory of feedforward processing in sensory cortex.

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preprint2015arXiv

Real-world Object Recognition with Off-the-shelf Deep Conv Nets: How Many Objects can iCub Learn?

The ability to visually recognize objects is a fundamental skill for robotics systems. Indeed, a large variety of tasks involving manipulation, navigation or interaction with other agents, deeply depends on the accurate understanding of the visual scene. Yet, at the time being, robots are lacking good visual perceptual systems, which often become the main bottleneck preventing the use of autonomous agents for real-world applications. Lately in computer vision, systems that learn suitable visual representations and based on multi-layer deep convolutional networks are showing remarkable performance in tasks such as large-scale visual recognition and image retrieval. To this regard, it is natural to ask whether such remarkable performance would generalize also to the robotic setting. In this paper we investigate such possibility, while taking further steps in developing a computational vision system to be embedded on a robotic platform, the iCub humanoid robot. In particular, we release a new dataset ({\sc iCubWorld28}) that we use as a benchmark to address the question: {\it how many objects can iCub recognize?} Our study is developed in a learning framework which reflects the typical visual experience of a humanoid robot like the iCub. Experiments shed interesting insights on the strength and weaknesses of current computer vision approaches applied in real robotic settings.

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preprint2015arXiv

Stochastic inertial primal-dual algorithms

We propose and study a novel stochastic inertial primal-dual approach to solve composite optimization problems. These latter problems arise naturally when learning with penalized regularization schemes. Our analysis provide convergence results in a general setting, that allows to analyze in a unified framework a variety of special cases of interest. Key in our analysis is considering the framework of splitting algorithm for solving a monotone inclusions in suitable product spaces and for a specific choice of preconditioning operators.

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preprint2014arXiv

A Deep Representation for Invariance And Music Classification

Representations in the auditory cortex might be based on mechanisms similar to the visual ventral stream; modules for building invariance to transformations and multiple layers for compositionality and selectivity. In this paper we propose the use of such computational modules for extracting invariant and discriminative audio representations. Building on a theory of invariance in hierarchical architectures, we propose a novel, mid-level representation for acoustical signals, using the empirical distributions of projections on a set of templates and their transformations. Under the assumption that, by construction, this dictionary of templates is composed from similar classes, and samples the orbit of variance-inducing signal transformations (such as shift and scale), the resulting signature is theoretically guaranteed to be unique, invariant to transformations and stable to deformations. Modules of projection and pooling can then constitute layers of deep networks, for learning composite representations. We present the main theoretical and computational aspects of a framework for unsupervised learning of invariant audio representations, empirically evaluated on music genre classification.

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preprint2014arXiv

Convergence of Stochastic Proximal Gradient Algorithm

We prove novel convergence results for a stochastic proximal gradient algorithm suitable for solving a large class of convex optimization problems, where a convex objective function is given by the sum of a smooth and a possibly non-smooth component. We consider the iterates convergence and derive $O(1/n)$ non asymptotic bounds in expectation in the strongly convex case, as well as almost sure convergence results under weaker assumptions. Our approach allows to avoid averaging and weaken boundedness assumptions which are often considered in theoretical studies and might not be satisfied in practice.

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preprint2014arXiv

Learning An Invariant Speech Representation

Recognition of speech, and in particular the ability to generalize and learn from small sets of labelled examples like humans do, depends on an appropriate representation of the acoustic input. We formulate the problem of finding robust speech features for supervised learning with small sample complexity as a problem of learning representations of the signal that are maximally invariant to intraclass transformations and deformations. We propose an extension of a theory for unsupervised learning of invariant visual representations to the auditory domain and empirically evaluate its validity for voiced speech sound classification. Our version of the theory requires the memory-based, unsupervised storage of acoustic templates -- such as specific phones or words -- together with all the transformations of each that normally occur. A quasi-invariant representation for a speech segment can be obtained by projecting it to each template orbit, i.e., the set of transformed signals, and computing the associated one-dimensional empirical probability distributions. The computations can be performed by modules of filtering and pooling, and extended to hierarchical architectures. In this paper, we apply a single-layer, multicomponent representation for phonemes and demonstrate improved accuracy and decreased sample complexity for vowel classification compared to standard spectral, cepstral and perceptual features.

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preprint2014arXiv

Learning Sets with Separating Kernels

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.

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preprint2014arXiv

On the Sample Complexity of Subspace Learning

A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from samples. In this paper we introduce a general formulation of this problem and derive novel learning error estimates. Our results rely on natural assumptions on the spectral properties of the covariance operator associated to the data distribu- tion, and hold for a wide class of metrics between subspaces. As special cases, we discuss sharp error estimates for the reconstruction properties of PCA and spectral support estimation. Key to our analysis is an operator theoretic approach that has broad applicability to spectral learning methods.

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preprint2014arXiv

Unsupervised Learning of Invariant Representations in Hierarchical Architectures

The present phase of Machine Learning is characterized by supervised learning algorithms relying on large sets of labeled examples ($n \to \infty$). The next phase is likely to focus on algorithms capable of learning from very few labeled examples ($n \to 1$), like humans seem able to do. We propose an approach to this problem and describe the underlying theory, based on the unsupervised, automatic learning of a ``good'' representation for supervised learning, characterized by small sample complexity ($n$). We consider the case of visual object recognition though the theory applies to other domains. The starting point is the conjecture, proved in specific cases, that image representations which are invariant to translations, scaling and other transformations can considerably reduce the sample complexity of learning. We prove that an invariant and unique (discriminative) signature can be computed for each image patch, $I$, in terms of empirical distributions of the dot-products between $I$ and a set of templates stored during unsupervised learning. A module performing filtering and pooling, like the simple and complex cells described by Hubel and Wiesel, can compute such estimates. Hierarchical architectures consisting of this basic Hubel-Wiesel moduli inherit its properties of invariance, stability, and discriminability while capturing the compositional organization of the visual world in terms of wholes and parts. The theory extends existing deep learning convolutional architectures for image and speech recognition. It also suggests that the main computational goal of the ventral stream of visual cortex is to provide a hierarchical representation of new objects/images which is invariant to transformations, stable, and discriminative for recognition---and that this representation may be continuously learned in an unsupervised way during development and visual experience.

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preprint2013arXiv

GURLS: a Least Squares Library for Supervised Learning

We present GURLS, a least squares, modular, easy-to-extend software library for efficient supervised learning. GURLS is targeted to machine learning practitioners, as well as non-specialists. It offers a number state-of-the-art training strategies for medium and large-scale learning, and routines for efficient model selection. The library is particularly well suited for multi-output problems (multi-category/multi-label). GURLS is currently available in two independent implementations: Matlab and C++. It takes advantage of the favorable properties of regularized least squares algorithm to exploit advanced tools in linear algebra. Routines to handle computations with very large matrices by means of memory-mapped storage and distributed task execution are available. The package is distributed under the BSD licence and is available for download at https://github.com/CBCL/GURLS.

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preprint2013arXiv

iCub World: Friendly Robots Help Building Good Vision Data-Sets

In this paper we present and start analyzing the iCub World data-set, an object recognition data-set, we acquired using a Human-Robot Interaction (HRI) scheme and the iCub humanoid robot platform. Our set up allows for rapid acquisition and annotation of data with corresponding ground truth. While more constrained in its scopes -- the iCub world is essentially a robotics research lab -- we demonstrate how the proposed data-set poses challenges to current recognition systems. The iCubWorld data-set is publicly available. The data-set can be downloaded from: http://www.iit.it/en/projects/data-sets.html.

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preprint2013arXiv

Learning Manifolds with K-Means and K-Flats

We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for k-means in two separate directions. First, we provide new results for k-means reconstruction on manifolds and, secondly, we prove reconstruction bounds for higher-order approximation (k-flats), for which no known results were previously available. While the results for k-means are novel, some of the technical tools are well-established in the literature. In the case of k-flats, both the results and the mathematical tools are new.

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preprint2013arXiv

q-ary Compressive Sensing

We introduce q-ary compressive sensing, an extension of 1-bit compressive sensing. We propose a novel sensing mechanism and a corresponding recovery procedure. The recovery properties of the proposed approach are analyzed both theoretically and empirically. Results in 1-bit compressive sensing are recovered as a special case. Our theoretical results suggest a tradeoff between the quantization parameter q, and the number of measurements m in the control of the error of the resulting recovery algorithm, as well its robustness to noise.

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preprint2012arXiv

Kernels for Vector-Valued Functions: a Review

Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a Bayesian/generative perspective they are the key in the context of Gaussian processes, where the kernel function is also known as the covariance function. Traditionally, kernel methods have been used in supervised learning problem with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partly by frameworks like multitask learning. In this paper, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods.

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preprint2012arXiv

Learning Probability Measures with respect to Optimal Transport Metrics

We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of the empirical law of large numbers, which, unlike existing bounds, are applicable to a wide class of measures.

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preprint2012arXiv

Multiclass Learning with Simplex Coding

In this paper we discuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework, a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this setting it is possible to derive the first provably consistent regularized method with training/tuning complexity which is independent to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper.

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preprint2012arXiv

Nonparametric sparsity and regularization

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm corresponds to a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance. An extensive empirical analysis shows that the proposed method performs favorably with respect to the state-of-the-art methods.

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preprint2012arXiv

Proximal methods for the latent group lasso penalty

We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.

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preprint2011arXiv

Online Learning, Stability, and Stochastic Gradient Descent

In batch learning, stability together with existence and uniqueness of the solution corresponds to well-posedness of Empirical Risk Minimization (ERM) methods; recently, it was proved that CV_loo stability is necessary and sufficient for generalization and consistency of ERM. In this note, we introduce CV_on stability, which plays a similar note in online learning. We show that stochastic gradient descent (SDG) with the usual hypotheses is CVon stable and we then discuss the implications of CV_on stability for convergence of SGD.

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