Convergence Rates of Distributed Gradient Methods Under Random Quantization: A Stochastic Approximation Approach

Doan, Thinh T.; Maguluri, Siva Theja; Romberg, Justin

doi:10.1109/tac.2020.3031018

Cited by 44 publications

(45 citation statements)

References 36 publications

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“…algorithms; [15] employed biased but contractive compressors to design a decentralized SGD algorithm; [16] and [17], [18] utilized unbiased compressors to respectively design decentralized gradient descent and primal-dual algorithms; [19] and [20] made use of the standard uniform quantizer to respectively design decentralized subgradient methods and alternating direction method of multipliers appraoches; [21], [22] and [23] respectively adopted the unbiased random quantization and the adaptive quantization to design decentralized projected subgradient algorithms; [24] and [25]- [28] exploited the standard uniform quantizer with dynamic quantization level to respectively design decentralized subgradient and primal-dual algorithms; and [29] applied the standard uniform quantizer with a fixed quantization level to design a decentralized gradient descent algorithm. The compressors mentioned above can be unified into three general classes.…”

Section: A Related Work and Motivationmentioning

confidence: 99%

Communication Compression for Distributed Nonconvex Optimization

Yi¹,

Zhang²,

Yang³

et al. 2022

Preprint

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This paper considers decentralized nonconvex optimization with the cost functions being distributed over agents. Noting that information compression is a key tool to reduce the heavy communication load for decentralized algorithms as agents iteratively communicate with neighbors, we propose three decentralized primal-dual algorithms with compressed communication. The first two algorithms are applicable to a general class of compressors with bounded relative compression error and the third algorithm is suitable for two general classes of compressors with bounded absolute compression error.We show that the proposed decentralized algorithms with compressed communication have comparable convergence properties as state-of-the-art algorithms without communication compression. Specifically, we show that they can find first-order stationary points with sublinear convergence rate O(1/T ) when each local cost function is smooth, where T is the total number of iterations, and find global optima with linear convergence rate under an additional condition that the global cost function satisfies the Polyak-Łojasiewicz condition. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

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Section: A Related Work and Motivationmentioning

confidence: 99%

Communication Compression for Distributed Nonconvex Optimization

Yi¹,

Zhang²,

Yang³

et al. 2022

Preprint

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“…In this scheme, the agents update locally their state variables by averaging the state variables received from their neighbors, and then follow the subgradient descent direction. More recently, additional convergence results were presented in [53], where random (dithered) quantization is applied, along with a weighting scheme to give more or less importance to the analog local state and the quantized averaged state of the neighbors. In [54], the randomized quantizers proposed in [41] are considered for a distributed gradient descent implementation using consensus with compressed iterates and an update rule similar to the one adopted in [53].…”

Section: A Distributed Quantizationmentioning

confidence: 99%

“…, N . To this end, we exploit the block decomposition of matrix V in (53). Regarding si , since v 1k = π k , from (160) we have that:…”

Section: Appendix B Proof Of Lemmamentioning

confidence: 99%

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Distributed Adaptive Learning Under Communication Constraints

Carpentiero¹,

Matta²,

Sayed³

2021

Preprint

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This work examines adaptive distributed learning strategies designed to operate under communication constraints. We consider a network of agents that must solve an online optimization problem from continual observation of streaming data. To this end, the agents implement a distributed cooperative strategy where each agent is allowed to perform local exchange of information with its neighbors. In order to cope with communication constraints, the exchanged information must be unavoidably compressed to some extent. We propose a distributed diffusion strategy nicknamed as ACTC (Adapt-Compress-Then-Combine), which relies on the following main steps: i) an adaptation step where each agent performs an individual stochastic-gradient update with constant step-size; ii) a compression step that leverages a recently introduced class of stochastic compression operators; and iii) a combination step where each agent combines the compressed updates received from its neighbors. The distinguishing elements of novelty of this work are as follows. First, we focus on adaptive strategies, where constant (as opposed to diminishing) step-sizes are critical to infuse the agents with the ability of responding in real time to nonstationary variations in the observed model. Second, our study considers the general class of directed (i.e., non-symmetric) and left-stochastic combination policies, which allow us to enhance the role played by the network topology in the learning performance. Third, in contrast with related works that assume strong convexity for all individual agents' cost functions, we require only a global strong convexity at a network level. For global strong convexity, it suffices that a single agent has a strongly-convex cost, while the remaining agents might feature even a non-convex cost. Fourth, we focus on a diffusion (as opposed to consensus) strategy, which will be shown to entail some gains in terms of learning performance. Under the demanding setting of global strong convexity and compressed information, we are able to prove that the iterates of the ACTC strategy fluctuate around the right global optimizer (with a mean-square-deviation in the order of the step-size), achieving remarkable savings in terms of bits exchanged between neighboring agents. Illustrative examples are provided to validate the theoretical results, and to compare the ACTC strategy against up-to-date stochastic-gradient solutions with compressed data, highlighting the benefits of the proposed solution.

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“…Other Settings. We also want to mention some related literature in game theory [39,40,41,42,43], two-time-scale stochastic approximation [44,45,46,47,48,49,50,51,52,53], reinforcement learning [54,55,56,57,58], two-time-scale optimization [59,60], and decentralized optimization [61,62,63,64,65,66,67]. These works study different variants of two-time-scale methods mostly for solving a single optimization problem, and often aim to find global optimality (or fixed points) using different structure of the underlying problems (e.g., Markov structure in stochastic games and reinforcement learning or strong monotonicity in stochastic approximation).…”

Section: Related Workmentioning

confidence: 99%

Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems

Doan¹

2021

Preprint

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There are much recent interests in solving noncovnex min-max optimization problems due to its broad applications in many areas including machine learning, networked resource allocations, and distributed optimization. Perhaps, the most popular first-order method in solving min-max optimization is the so-called simultaneous (or single-loop) gradient descent-ascent algorithm due to its simplicity in implementation. However, theoretical guarantees on the convergence of this algorithm is very sparse since it can diverge even in a simple bilinear problem.In this paper, our focus is to characterize the finite-time performance (or convergence rates) of the continuous-time variant of simultaneous gradient descent-ascent algorithm. In particular, we derive the rates of convergence of this method under a number of different conditions on the underlying objective function, namely, two-sided Polyak-Lojasiewicz (P L), one-sided P L, nonconvex-strongly concave, and strongly convex-nonconcave conditions. Our convergence results improve the ones in prior works under the same conditions of objective functions. The key idea in our analysis is to use the classic singular perturbation theory and coupling Lyapunov functions to address the time-scale difference and interactions between the gradient descent and ascent dynamics. Our results on the behavior of continuous-time algorithm may be used to enhance the convergence properties of its discrete-time counterpart.

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Convergence Rates of Distributed Gradient Methods Under Random Quantization: A Stochastic Approximation Approach

Cited by 44 publications

References 36 publications

Communication Compression for Distributed Nonconvex Optimization

Communication Compression for Distributed Nonconvex Optimization

Distributed Adaptive Learning Under Communication Constraints

Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems

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