Distributed Adaptive Learning Under Communication Constraints

Abstract

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.

Quantization for decentralized learning under subspace constraints

“…As explained in [30], deterministic quantizers can lead to severe estimation biases in inference problems. To overcome this issue, randomized quantizers Q(•) are commonly used to compensate for the bias (on average, over time) [12], [29]- [33]. This section is devoted to describing the general class of randomized quantizers considered throughout the study.…”

“…For any deterministic input x ∈ R L with L representing a generic vector length, the randomized quantizer Q(•) is characterized in terms of a probability P[Q(x) = y] for any y belonging to the set of output levels of the quantizer. We consider randomized quantizers Q(•) satisfying the following general property, which as explained in the sequel, relaxes the condition on the mean-square error from [5], [12], [29]- [32].…”

“…Many existing works focus on studying decentralized learning approaches in the presence of randomized quantizers that satisfy the unbiasedness condition (4) and the variance bound (5) with the absolute noise term σ 2 q = 0 [5], [12], [29]- [32]. In contrast, the analysis in the current work is general and does not require σ 2 q to be zero.…”

“…Motivated by the approaches proposed in [12], [29]- [33], we equip strategy (1) with a quantization mechanism by proposing the following decentralized multitask learning approach -see Fig. 3:…”

“…The analysis shows that, by properly designing the quantization operators, the iterates generated by the quantized decentralized adaptive implementation lead to small estimation errors on the order of µ (as it happens in the ideal case without quantization), while concurrently guaranteeing a bounded average bit rate as µ → 0. While there exist several useful works in the literature that study decentralized learning approaches in the presence of differential quantization [12], [29]- [34], these works investigate standard consensus or single-task optimization, and do not consider the subspace constrained formulation (3) or multitask variants. Moreover, with some exceptions that consider deterministic optimization [33], [34], the analyses conducted in these works assume that some quantities (e.g., the norm or some components of the vector to be quantized) are represented with very high precision (e.g., machine precision) and the associated quantization error is neglected.…”