Differentially Private Random Block Coordinate Descent

Coordinate Descent (CD) methods have gained significant attention in machine learning due to their effectiveness in solving high-dimensional problems and their ability to decompose complex optimization tasks. However, classical CD methods were neither designed nor analyzed with data privacy in mind, a critical concern when handling sensitive information. This has led to the development of differentially private CD methods, such as DP-CD (Differentially Private Coordinate Descent) proposed by Mangold et al. (ICML 2022), yet a disparity remains between non-private CD and DP-CD methods. In our work, we propose a differentially private random block coordinate descent method that selects multiple coordinates with varying probabilities in each iteration using sketch matrices. Our algorithm generalizes both DP-CD and the classical DP-SGD (Differentially Private Stochastic Gradient Descent), while preserving the same utility guarantees. Furthermore, we demonstrate that better utility can be achieved through importance sampling, as our method takes advantage of the heterogeneity in coordinate-wise smoothness constants, leading to improved convergence rates.

MindFlayer: Efficient Asynchronous Parallel SGD in the Presence of Heterogeneous and Random Worker Compute Times

We study the problem of minimizing the expectation of smooth nonconvex functions with the help of several parallel workers whose role is to compute stochastic gradients. In particular, we focus on the challenging situation where the workers' compute times are arbitrarily heterogeneous and random. In the simpler regime characterized by arbitrarily heterogeneous but deterministic compute times, Tyurin and Richt\'arik (NeurIPS 2023) recently designed the first theoretically optimal asynchronous SGD method, called Rennala SGD, in terms of a novel complexity notion called time complexity. The starting point of our work is the observation that Rennala SGD can have arbitrarily bad performance in the presence of random compute times -- a setting it was not designed to handle. To advance our understanding of stochastic optimization in this challenging regime, we propose a new asynchronous SGD method, for which we coin the name MindFlayer SGD. Our theory and empirical results demonstrate the superiority of MindFlayer SGD over existing baselines, including Rennala SGD, in cases when the noise is heavy tailed.

LoCoDL: Communication-Efficient Distributed Learning with Local Training and Compression

In Distributed optimization and Learning, and even more in the modern framework of federated learning, communication, which is slow and costly, is critical. We introduce LoCoDL, a communication-efficient algorithm that leverages the two popular and effective techniques of Local training, which reduces the communication frequency, and Compression, in which short bitstreams are sent instead of full-dimensional vectors of floats. LoCoDL works with a large class of unbiased compressors that includes widely-used sparsification and quantization methods. LoCoDL provably benefits from local training and compression and enjoys a doubly-accelerated communication complexity, with respect to the condition number of the functions and the model dimension, in the general heterogenous regime with strongly convex functions. This is confirmed in practice, with LoCoDL outperforming existing algorithms.

GradSkip: Communication-Accelerated Local Gradient Methods with Better Computational Complexity

In this work, we study distributed optimization algorithms that reduce the high communication costs of synchronization by allowing clients to perform multiple local gradient steps in each communication round. Recently, Mishchenko et al. (2022) proposed a new type of local method, called ProxSkip, that enjoys an accelerated communication complexity without any data similarity condition. However, their method requires all clients to call local gradient oracles with the same frequency. Because of statistical heterogeneity, we argue that clients with well-conditioned local problems should compute their local gradients less frequently than clients with ill-conditioned local problems. Our first contribution is the extension of the original ProxSkip method to the setup where clients are allowed to perform a different number of local gradient steps in each communication round. We prove that our modified method, GradSkip, still converges linearly, has the same accelerated communication complexity, and the required frequency for local gradient computations is proportional to the local condition number. Next, we generalize our method by extending the randomness of probabilistic alternations to arbitrary unbiased compression operators and considering a generic proximable regularizer. This generalization, GradSkip+, recovers several related methods in the literature. Finally, we present an empirical study to confirm our theoretical claims.