Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning

• URL: http://arxiv.org/abs/2308.01358v1
• Date: Wed, 2 Aug 2023 18:02:00 GMT
• Title: Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning
• Authors: Constantin Philippenko and Aymeric Dieuleveut
• Abstract summary: We investigate the impact of compression on gradient algorithms for machine learning. We highlight differences in terms of convergence rates between several unbiased compression operators. We extend our results to the case of federated learning.
• Score: 9.31522898261934
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Learning general Gaussian mixtures with efficient score matching [16.06356123715737]
  We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components. We give an algorithm that draws $dmathrmpoly(k/varepsilon)$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler.
  arXiv  Detail & Related papers  (2024-04-29T17:30:36Z)
• Rate Analysis of Coupled Distributed Stochastic Approximation for Misspecified Optimization [0.552480439325792]
  We consider an $n$ agents distributed optimization problem with imperfect information characterized in a parametric sense. We propose a coupled distributed approximation algorithm, in which every agent updates the current beliefs of its unknown parameter. We quantitatively characterize the factors that affect the algorithm performance, and prove that the mean-squared error of the decision variable is bounded by $mathcalO(frac1nk)+mathcalOleft(frac1sqrtn (1-rho_w)right)frac1k1.5
  arXiv  Detail & Related papers  (2024-04-21T14:18:49Z)
• Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems [56.86067111855056]
  We consider clipped optimization problems with heavy-tailed noise with structured density. We show that it is possible to get faster rates of convergence than $mathcalO(K-(alpha - 1)/alpha)$, when the gradients have finite moments of order. We prove that the resulting estimates have negligible bias and controllable variance.
  arXiv  Detail & Related papers  (2023-11-07T17:39:17Z)
• Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization [49.58290066287418]
  We propose a novel method named Multi-block-probe Variance Reduced (MSVR) to alleviate the complexity of compositional problems. Our results improve upon prior ones in several aspects, including the order of sample complexities and dependence on strongity.
  arXiv  Detail & Related papers  (2022-07-18T12:03:26Z)
• Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization [11.017632675093628]
  Coordinate-type subgradient methods for addressing nonsmooth problems are relatively underexplored due to the set of properties of the Lipschitz-type assumption.
  arXiv  Detail & Related papers  (2022-06-30T02:17:11Z)
• Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization [116.89941263390769]
  We consider the smooth convex-concave bilinearly-coupled saddle-point problem, $min_mathbfxmax_mathbfyF(mathbfx) + H(mathbfx,mathbfy)$, where one has access to first-order oracles for $F$, $G$ as well as the bilinear coupling function $H$. We present a emphaccelerated gradient-extragradient (AG-EG) descent-ascent algorithm that combines extragrad
  arXiv  Detail & Related papers  (2022-06-17T06:10:20Z)
• An Improved Analysis of Gradient Tracking for Decentralized Machine Learning [34.144764431505486]
  We consider decentralized machine learning over a network where the training data is distributed across $n$ agents. The agent's common goal is to find a model that minimizes the average of all local loss functions. We improve the dependency on $p$ from $mathcalO(p-1)$ to $mathcalO(p-1)$ in the noiseless case.
  arXiv  Detail & Related papers  (2022-02-08T12:58:14Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Robustly Learning any Clusterable Mixture of Gaussians [55.41573600814391]
  We study the efficient learnability of high-dimensional Gaussian mixtures in the adversarial-robust setting. We provide an algorithm that learns the components of an $epsilon$-corrupted $k$-mixture within information theoretically near-optimal error proofs of $tildeO(epsilon)$. Our main technical contribution is a new robust identifiability proof clusters from a Gaussian mixture, which can be captured by the constant-degree Sum of Squares proof system.
  arXiv  Detail & Related papers  (2020-05-13T16:44:12Z)
• Outlier-Robust Clustering of Non-Spherical Mixtures [5.863264019032882]
  We give the first outlier-robust efficient algorithm for clustering a mixture of $k$ statistically separated d-dimensional Gaussians (k-GMMs) Our results extend to clustering mixtures of arbitrary affine transforms of the uniform distribution on the $d$-dimensional unit sphere.
  arXiv  Detail & Related papers  (2020-05-06T17:24:27Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.