We present skwdro, a Python library for training robust machine learning models. The library is based on distributionally robust optimization using Wasserstein distances, popular in optimal transport and machine learnings. The goal of the library is to make the training of robust models easier for a wide audience by proposing a wrapper for PyTorch modules, enabling model loss' robustification with minimal code changes. It comes along with scikit-learn compatible estimators for some popular objectives. The core of the implementation relies on an entropic smoothing of the original robust objective, in order to ensure maximal model flexibility. The library is available at https://github.com/iutzeler/skwdro and the documentation at https://skwdro.readthedocs.io

In this paper, we examine the time it takes for stochastic gradient descent (SGD) to reach the global minimum of a general, non-convex loss function. We approach this question through the lens of large deviations theory and randomly perturbed dynamical systems, and we provide a tight characterization of the associated hitting times of SGD with matching upper and lower bounds. Our analysis reveals that the global convergence time of SGD is dominated by the most "costly" set of obstacles that the algorithm may need to overcome in order to reach a global minimizer, coupling in this way the geometry of the underlying loss landscape with the statistics of the noise entering the process. Finally, motivated by applications to the training of deep neural networks, we provide a series of refinements and extensions of our analysis to, among others, loss functions with no spurious local minima or ones with bounded depths.

Estimating optimal transport maps between two distributions from respective samples is an important element for many machine learning methods. To do so, rather than extending discrete transport maps, it has been shown that estimating the Brenier potential of the transport problem and obtaining a transport map through its gradient is near minimax optimal for smooth problems. In this paper, we investigate the private estimation of such potentials and transport maps with respect to the distribution samples. We propose a differentially private transport map estimator with $L^2$ error at most $n^{-1} \vee n^{-\frac{2 \alpha}{2 \alpha - 2 + d}} \vee (n\epsilon)^{-\frac{2 \alpha}{2 \alpha + d}} $ up do polylog terms where $n$ is the sample size, $\epsilon$ is the desired level of privacy, $\alpha$ is the smoothness of the true transport map, and $d$ is the dimension of the feature space. We also provide a lower bound for the problem.

In this paper, we examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem's state space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method's step-size and energy levels determined by the problem's objective and the statistics of the noise. In particular, we show that, in the long run, (*a*) the problem's critical region is visited exponentially more often than any non-critical region; (*b*) the iterates of SGD are exponentially concentrated around the problem's minimum energy state (which does not always coincide with the global minimum of the objective); (*c*) all other connected components of critical points are visited with frequency that is exponentially proportional to their energy level; and, finally, (*d*) any component of local maximizers or saddle points is "dominated" by a component of local minimizers which is visited exponentially more often.

We consider stochastic optimization problems where the objective depends on some parameter, as commonly found in hyperparameter optimization for instance. We investigate the behavior of the derivatives of the iterates of Stochastic Gradient Descent (SGD) with respect to that parameter and show that they are driven by an inexact SGD recursion on a different objective function, perturbed by the convergence of the original SGD. This enables us to establish that the derivatives of SGD converge to the derivative of the solution mapping in terms of mean squared error whenever the objective is strongly convex. Specifically, we demonstrate that with constant step-sizes, these derivatives stabilize within a noise ball centered at the solution derivative, and that with vanishing step-sizes they exhibit $O(\log(k)^2 / k)$ convergence rates. Additionally, we prove exponential convergence in the interpolation regime. Our theoretical findings are illustrated by numerical experiments on synthetic tasks.

Wasserstein distributionally robust estimators have emerged as powerful models for prediction and decision-making under uncertainty. These estimators provide attractive generalization guarantees: the robust objective obtained from the training distribution is an exact upper bound on the true risk with high probability. However, existing guarantees either suffer from the curse of dimensionality, are restricted to specific settings, or lead to spurious error terms. In this paper, we show that these generalization guarantees actually hold on general classes of models, do not suffer from the curse of dimensionality, and can even cover distribution shifts at testing. We also prove that these results carry over to the newly-introduced regularized versions of Wasserstein distributionally robust problems.

In this paper, we provide a general framework for studying multi-agent online learning problems in the presence of delays and asynchronicities. Specifically, we propose and analyze a class of adaptive dual averaging schemes in which agents only need to accumulate gradient feedback received from the whole system, without requiring any between-agent coordination. In the single-agent case, the adaptivity of the proposed method allows us to extend a range of existing results to problems with potentially unbounded delays between playing an action and receiving the corresponding feedback. In the multi-agent case, the situation is significantly more complicated because agents may not have access to a global clock to use as a reference point; to overcome this, we focus on the information that is available for producing each prediction rather than the actual delay associated with each feedback. This allows us to derive adaptive learning strategies with optimal regret bounds, even in a fully decentralized, asynchronous environment. Finally, we also analyze an “optimistic” variant of the proposed algorithm which is capable of exploiting the predictability of problems with a slower variation and leads to improved regret bounds.

In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm’s rate of convergence and the local geometry induced by the method’s underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this exponent determines both the optimal step-size policy of the algorithm and the optimal rates attained, explaining in this way the differences observed for some popular Bregman functions (Euclidean projection, negative entropy, fractional power, etc.).

In this paper, we propose a theoretically supported sequential strategy for training a large-scale Recommender System (RS) over implicit feedback, mainly in the form of clicks. The proposed approach consists in minimizing pairwise ranking loss over blocks of consecutive items constituted by a sequence of non-clicked items followed by a clicked one for each user. We present two variants of this strategy where model parameters are updated using either the momentum method or a gradient-based approach. To prevent updating the parameters for an abnormally high number of clicks over some targeted items (mainly due to bots), we introduce an upper and a lower threshold on the number of updates for each user. These thresholds are estimated over the distribution of the number of blocks in the training set. They affect the decision of RS by shifting the distribution of items that are shown to the users. Furthermore, we provide a convergence analysis of both algorithms and demonstrate their practical efficiency over six large-scale collections with respect to various ranking measures and computational time.

Owing to their stability and convergence speed, extragradient methods have become a staple for solving large-scale saddle-point problems in machine learning. The basic premise of these algorithms is the use of an extrapolation step before performing an update; thanks to this exploration step, extra-gradient methods overcome many of the non-convergence issues that plague gradient descent/ascent schemes. On the other hand, as we show in this paper, running vanilla extragradient with stochastic gradients may jeopardize its convergence, even in simple bilinear models. To overcome this failure, we investigate a double stepsize extragradient algorithm where the exploration step evolves at a more aggressive time-scale compared to the update step. We show that this modification allows the method to converge even with stochastic gradients, and we derive sharp convergence rates under an error bound condition.

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal O(1/t) convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a $O(1/t)$ ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a $O(1/t)$ local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.

Distributed learning aims at computing high-quality models by training over scattered data. This covers a diversity of scenarios, including computer clusters or mobile agents. One of the main challenges is then to deal with heterogeneous machines and unreliable communications. In this setting, we propose and analyze a flexible asynchronous optimization algorithm for solving nonsmooth learning problems. Unlike most existing methods, our algorithm is adjustable to various levels of communication costs, machines computational powers, and data distribution evenness. We prove that the algorithm converges linearly with a fixed learning rate that does not depend on communication delays nor on the number of machines. Although long delays in communication may slow down performance, no delay can break convergence.

We derive distributed algorithms for projecting the local values of the agents of a computing network on the simplex or on the$\ell _1$ball. These algorithms are based on the distributed alternating direction method of multipliers to solve a convex optimization problem of the form$\min _x \sum _n f_n(x)$, where each function$f_n$is local to node$n$and has an easy-to-compute proximity operator.

We address the problem of multi-class classification in the case where the number of classes is very large. We propose a double sampling strategy on top of a multi-class to binary reduction strategy, which transforms the original multi-class problem into a binary classification problem over pairs of examples. The aim of the sampling strategy is to overcome the curse of long-tailed class distributions exhibited in majority of large-scale multi-class classification problems and to reduce the number of pairs of examples in the expanded data. We show that this strategy does not alter the consistency of the empirical risk minimization principle defined over the double sample reduction. Experiments are carried out on DMOZ and Wikipedia collections with 10,000 to 100,000 classes where we show the efficiency of the proposed approach in terms of training and prediction time, memory consumption, and predictive performance with respect to state-of-the-art approaches.

In this letter, we derive distributed synchronous and asynchronous algorithms for computing quantiles of the agents' local values. These algorithms are based on the formulation of a suitable problem, explicitly solvable by the alternating direction method of multipliers (ADMM), and recent randomized optimization methods.

We introduce an asynchronous distributed stochastic gradient algorithm for matrix factorization based collaborative filtering. The main idea of this approach is to distribute the user-rating matrix across different machines, each having access only to a part of the information, and to asynchronously propagate the updates of the stochastic gradient optimization across the network. Each time a machine receives a parameter vector, it averages its current parameter vector with the received one, and continues its iterations from this new point. Additionally, we introduce a similarity based regularization that constrains the user and item factors to be close to the average factors of their similar users and items found on subparts of the distributed user-rating matrix. We analyze the impact of the regularization terms on MovieLens (100K, 1M, 10M) and NetFlix datasets and show that it leads to a more efficient matrix factorization in terms of Root Mean Square Error (RMSE) and Mean Absolute Error (MAE), and that the asynchronous distributed approach significantly improves in convergence time as compared to an equivalent synchronous distributed approach.

The distributed estimation of the average value of the sensors initial measures is one of the most popular issues in the Wireless Sensor Networks (WSN) area. In WSNs, broadcasting data seems natural to exchange information quickly because of the broadcast nature of the Wireless channel. Nevertheless, although broadcast-based algorithms converge faster than pairwise algorithms, the obtained consensus is not necessarily the true average. By the means of additional side-information exchange, we propose a broadcast-based algorithm converging rapidly to the true average. The convergence of this new algorithm is established and its convergence speed is exhibited. We remark that the proposed algorithm outperforms the existing ones.