CMAP
Abstract:In this paper, we present a novel analysis of FedAvg with constant step size, relying on the Markov property of the underlying process. We demonstrate that the global iterates of the algorithm converge to a stationary distribution and analyze its resulting bias and variance relative to the problem's solution. We provide a first-order expansion of the bias in both homogeneous and heterogeneous settings. Interestingly, this bias decomposes into two distinct components: one that depends solely on stochastic gradient noise and another on client heterogeneity. Finally, we introduce a new algorithm based on the Richardson-Romberg extrapolation technique to mitigate this bias.
Abstract:Distributed approaches have many computational benefits, but they are vulnerable to attacks from a subset of devices transmitting incorrect information. This paper investigates Byzantine-resilient algorithms in a decentralized setting, where devices communicate directly with one another. We investigate the notion of breakdown point, and show an upper bound on the number of adversaries that decentralized algorithms can tolerate. We introduce $\mathrm{CG}^+$, an algorithm at the intersection of $\mathrm{ClippedGossip}$ and $\mathrm{NNA}$, two popular approaches for robust decentralized learning. $\mathrm{CG}^+$ meets our upper bound, and thus obtains optimal robustness guarantees, whereas neither of the existing two does. We provide experimental evidence for this gap by presenting an attack tailored to sparse graphs which breaks $\mathrm{NNA}$ but against which $\mathrm{CG}^+$ is robust.
Abstract:Distributed approaches have many computational benefits, but they are vulnerable to attacks from a subset of devices transmitting incorrect information. This paper investigates Byzantine-resilient algorithms in a decentralized setting, where devices communicate directly with one another. We leverage the so-called dual approach to design a general robust decentralized optimization method. We provide both global and local clipping rules in the special case of average consensus, with tight convergence guarantees. These clipping rules are practical, and yield results that finely characterize the impact of Byzantine nodes, highlighting for instance a qualitative difference in convergence between global and local clipping thresholds. Lastly, we demonstrate that they can serve as a basis for designing efficient attacks.
Abstract:Constant (naive) imputation is still widely used in practice as this is a first easy-to-use technique to deal with missing data. Yet, this simple method could be expected to induce a large bias for prediction purposes, as the imputed input may strongly differ from the true underlying data. However, recent works suggest that this bias is low in the context of high-dimensional linear predictors when data is supposed to be missing completely at random (MCAR). This paper completes the picture for linear predictors by confirming the intuition that the bias is negligible and that surprisingly naive imputation also remains relevant in very low dimension.To this aim, we consider a unique underlying random features model, which offers a rigorous framework for studying predictive performances, whilst the dimension of the observed features varies.Building on these theoretical results, we establish finite-sample bounds on stochastic gradient (SGD) predictors applied to zero-imputed data, a strategy particularly well suited for large-scale learning.If the MCAR assumption appears to be strong, we show that similar favorable behaviors occur for more complex missing data scenarios.
Abstract:The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.
Abstract:Compression schemes have been extensively used in Federated Learning (FL) to reduce the communication cost of distributed learning. While most approaches rely on a bounded variance assumption of the noise produced by the compressor, this paper investigates the use of compression and aggregation schemes that produce a specific error distribution, e.g., Gaussian or Laplace, on the aggregated data. We present and analyze different aggregation schemes based on layered quantizers achieving exact error distribution. We provide different methods to leverage the proposed compression schemes to obtain compression-for-free in differential privacy applications. Our general compression methods can recover and improve standard FL schemes with Gaussian perturbations such as Langevin dynamics and randomized smoothing.
Abstract:The energy landscape of high-dimensional non-convex optimization problems is crucial to understanding the effectiveness of modern deep neural network architectures. Recent works have experimentally shown that two different solutions found after two runs of a stochastic training are often connected by very simple continuous paths (e.g., linear) modulo a permutation of the weights. In this paper, we provide a framework theoretically explaining this empirical observation. Based on convergence rates in Wasserstein distance of empirical measures, we show that, with high probability, two wide enough two-layer neural networks trained with stochastic gradient descent are linearly connected. Additionally, we express upper and lower bounds on the width of each layer of two deep neural networks with independent neuron weights to be linearly connected. Finally, we empirically demonstrate the validity of our approach by showing how the dimension of the support of the weight distribution of neurons, which dictates Wasserstein convergence rates is correlated with linear mode connectivity.
Abstract:In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance $\mathfrak{C}_{\mathrm{ania}}$ of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with $\mathrm{Tr}(\mathfrak{C}_{\mathrm{ania}} H^{-1})/K$ (where $H$ is the Hessian of the optimization problem and $K$ the number of iterations) generalizing the rate for the vanilla LSR case where it is $\sigma^2 \mathrm{Tr}(H H^{-1}) / K = \sigma^2 d / K$ (Bach and Moulines, 2013). Then, we analyze the dependency of $\mathfrak{C}_{\mathrm{ania}}$ on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks.
Abstract:Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates -- a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.
Abstract:Stochastic approximation (SA) is a classical algorithm that has had since the early days a huge impact on signal processing, and nowadays on machine learning, due to the necessity to deal with a large amount of data observed with uncertainties. An exemplar special case of SA pertains to the popular stochastic (sub)gradient algorithm which is the working horse behind many important applications. A lesser-known fact is that the SA scheme also extends to non-stochastic-gradient algorithms such as compressed stochastic gradient, stochastic expectation-maximization, and a number of reinforcement learning algorithms. The aim of this article is to overview and introduce the non-stochastic-gradient perspectives of SA to the signal processing and machine learning audiences through presenting a design guideline of SA algorithms backed by theories. Our central theme is to propose a general framework that unifies existing theories of SA, including its non-asymptotic and asymptotic convergence results, and demonstrate their applications on popular non-stochastic-gradient algorithms. We build our analysis framework based on classes of Lyapunov functions that satisfy a variety of mild conditions. We draw connections between non-stochastic-gradient algorithms and scenarios when the Lyapunov function is smooth, convex, or strongly convex. Using the said framework, we illustrate the convergence properties of the non-stochastic-gradient algorithms using concrete examples. Extensions to the emerging variance reduction techniques for improved sample complexity will also be discussed.