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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In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes µ, the strategy is stable both in terms of mean-square error and average bit rate: by reducing µ, it is possible to keep the estimation errors small (on the order of µ) without increasing indefinitely the bit rate as µ → 0. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.

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In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus optimization as special case, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study a quantized differential based approach where the communicated estimates among agents are quantized. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes µ, the strategy is stable in the meansquare error sense. The analysis also reveals the influence of the gradient and quantization noises on the performance.