In this paper, we study the minimax optimization problem in the smooth and strongly convex-strongly concave setting when we have access to noisy estimates of gradients. In particular, we first analyze the stochastic Gradient Descent Ascent (GDA) method with constant stepsize, and show that it converges to a neighborhood of the solution of the minimax problem. We further provide tight bounds on the convergence rate and the size of this neighborhood. Next, we propose a multistage variant of stochastic GDA (M-GDA) that runs in multiple stages with a particular learning rate decay schedule and converges to the exact solution of the minimax problem. We show M-GDA achieves the lower bounds in terms of noise dependence without any assumptions on the knowledge of noise characteristics. We also show that M-GDA obtains a linear decay rate with respect to the error's dependence on the initial error, although the dependence on condition number is suboptimal. In order to improve this dependence, we apply the multistage machinery to the stochastic Optimistic Gradient Descent Ascent (OGDA) algorithm and propose the M-OGDA algorithm which also achieves the optimal linear decay rate with respect to the initial error. To the best of our knowledge, this method is the first to simultaneously achieve the best dependence on noise characteristic as well as the initial error and condition number.

We study the problem of learning halfspaces with Massart noise in the distribution-specific PAC model. We give the first computationally efficient algorithm for this problem with respect to a broad family of distributions, including log-concave distributions. This resolves an open question posed in a number of prior works. Our approach is extremely simple: We identify a smooth {\em non-convex} surrogate loss with the property that any approximate stationary point of this loss defines a halfspace that is close to the target halfspace. Given this structural result, we can use SGD to solve the underlying learning problem.

In this paper, we study the problem of sampling from distributions of the form p(x) \propto e^{-\beta f(x)} for some function f whose values and gradients we can query. This mode of access to f is natural in the scenarios in which such problems arise, for instance sampling from posteriors in parametric Bayesian models. Classical results show that a natural random walk, Langevin diffusion, mixes rapidly when f is convex. Unfortunately, even in simple examples, the applications listed above will entail working with functions f that are nonconvex -- for which sampling from p may in general require an exponential number of queries. In this paper, we study one aspect of nonconvexity relevant for modern machine learning applications: existence of invariances (symmetries) in the function f, as a result of which the distribution p will have manifolds of points with equal probability. We give a recipe for proving mixing time bounds of Langevin dynamics in order to sample from manifolds of local optima of the function f in settings where the distribution is well-concentrated around them. We specialize our arguments to classic matrix factorization-like Bayesian inference problems where we get noisy measurements A(XX^T), X \in R^{d \times k} of a low-rank matrix, i.e. f(X) = \|A(XX^T) - b\|^2_2, X \in R^{d \times k}, and \beta the inverse of the variance of the noise. Such functions f are invariant under orthogonal transformations, and include problems like matrix factorization, sensing, completion. Beyond sampling, Langevin dynamics is a popular toy model for studying stochastic gradient descent. Along these lines, we believe that our work is an important first step towards understanding how SGD behaves when there is a high degree of symmetry in the space of parameters the produce the same output.

Meta-learning can successfully acquire useful inductive biases from data, especially when a large number of meta-tasks are available. Yet, its generalization properties to unseen tasks are poorly understood. Particularly if the number of meta-tasks is small, this raises concerns for potential overfitting. We provide a theoretical analysis using the PAC-Bayesian framework and derive novel generalization bounds for meta-learning with unbounded loss functions and Bayesian base learners. Using these bounds, we develop a class of PAC-optimal meta-learning algorithms with performance guarantees and a principled meta-regularization. When instantiating our PAC-optimal hyper-posterior (PACOH) with Gaussian processes as base learners, the resulting approach consistently outperforms several popular meta-learning methods, both in terms of predictive accuracy and the quality of its uncertainty estimates.

This problem arises in many cases in machine learning, most notably in large-scale (mini-batch) Bayesian inference (Welling and Teh, 2011, Ahn et al., 2012) and nonconvex stochastic optimization (Raginsky et al., 2017). For the setting of Bayesian inference, one is interested in sampling from a posterior probability measure where U corresponds to the sum of the log-likelihood and the log-prior. For the nonconvex optimization, U(·) is the nonconvex cost function to be minimized. For large values ofβ, a sample from the target measure (1) is an approximate minimizer of the potential U (Raginsky et al., 2017). Consequently, nonasymptotic error bounds for the schemes, which are designed to sample from (1), can be used to obtain guarantees for Bayesian inference or nonconvex optimization. Sampling from a measure of the form (1) is also central in statistical physics (Binder et al., 1993), most notably in molecular dynamics Haile (1992).

This article describes the implementation and use of PHOTON, a high-level Python API designed to simplify and accelerate the process of machine learning model development. It enables designing both basic and advanced machine learning pipeline architectures and automatizes the repetitive training, optimization and evaluation workflow. PHOTON offers easy access to established machine learning toolboxes as well as the possibility to integrate custom algorithms and solutions for any part of the model construction and evaluation process. By adding a layer of abstraction incorporating current best practices it offers an easy-to-use, flexible approach to implementing fast, reproducible, and unbiased machine learning solutions.

Parkinson's disease is a neurodegenerative disorder characterized by the presence of different motor impairments. Information from speech, handwriting, and gait signals have been considered to evaluate the neurological state of the patients. On the other hand, user models based on Gaussian mixture models - universal background models (GMM-UBM) and i-vectors are considered the state-of-the-art in biometric applications like speaker verification because they are able to model specific speaker traits. This study introduces the use of GMM-UBM and i-vectors to evaluate the neurological state of Parkinson's patients using information from speech, handwriting, and gait. The results show the importance of different feature sets from each type of signal in the assessment of the neurological state of the patients.

Background and objectives: Parkinson's disease is a neurological disorder that affects the motor system producing lack of coordination, resting tremor, and rigidity. Impairments in handwriting are among the main symptoms of the disease. Handwriting analysis can help in supporting the diagnosis and in monitoring the progress of the disease. This paper aims to evaluate the importance of different groups of features to model handwriting deficits that appear due to Parkinson's disease; and how those features are able to discriminate between Parkinson's disease patients and healthy subjects. Methods: Features based on kinematic, geometrical and non-linear dynamics analyses were evaluated to classify Parkinson's disease and healthy subjects. Classifiers based on K-nearest neighbors, support vector machines, and random forest were considered. Results: Accuracies of up to $93.1\%$ were obtained in the classification of patients and healthy control subjects. A relevance analysis of the features indicated that those related to speed, acceleration, and pressure are the most discriminant. The automatic classification of patients in different stages of the disease shows $\kappa$ indexes between $0.36$ and $0.44$. Accuracies of up to $83.3\%$ were obtained in a different dataset used only for validation purposes. Conclusions: The results confirmed the negative impact of aging in the classification process when we considered different groups of healthy subjects. In addition, the results reported with the separate validation set comprise a step towards the development of automated tools to support the diagnosis process in clinical practice.

We demonstrate that the Conditional Entropy Bottleneck (CEB) can improve model robustness. CEB is an easy strategy to implement and works in tandem with data augmentation procedures. We report results of a large scale adversarial robustness study on CIFAR-10, as well as the ImageNet-C Common Corruptions Benchmark, ImageNet-A, and PGD attacks.

Decentralized methods to solve finite-sum minimization problems are important in many signal processing and machine learning tasks where the data is distributed over a network of nodes and raw data sharing is not permitted due to privacy and/or resource constraints. In this article, we review decentralized stochastic first-order methods and provide a unified algorithmic framework that combines variance-reduction with gradient tracking to achieve both robust performance and fast convergence. We provide explicit theoretical guarantees of the corresponding methods when the objective functions are smooth and strongly-convex, and show their applicability to non-convex problems via numerical experiments. Throughout the article, we provide intuitive illustrations of the main technical ideas by casting appropriate tradeoffs and comparisons among the methods of interest and by highlighting applications to decentralized training of machine learning models. I. INTRODUCTION In multi-agent networks and large-scale machine learning, when data is available at different devices with limited communication, it is often desirable to seek scalable learning methods that do not require bringing, storing, and processing data at one single location. In this article, we describe decentralized, stochastic first-order methods, which are particularly favorable to such ad-hoc and resource-constrained settings. Specifically, we describe a unified algorithmic framework for combining different variance reduction methods with gradient tracking in order to significantly improve upon the performance of the standard decentralized stochastic gradient descent (DSGD). However, this improvement comes at a price of losing the simplicity of DSGD and we study the added communication, computation, and storage requirements with the help of precise technical statements. For the ease of accessibility, we restrict the theoretical arguments to smooth and strongly-convex objectives, while the applicability to non-convex problems is shown with the help of numerical experiments.