Distributed Adaptive Learning Under Communication Constraints

TL;DR

This work addresses distributed adaptive learning over networks where communication is bandwidth-constrained. It introduces ACTC, an Adapt-Compress-Then-Combine diffusion strategy that operates with constant step-sizes on directed, left-stochastic graphs and uses stochastic quantizers to compress exchanged information. The authors establish mean-square stability with a steady-state MSD of $O(\mu)$ in the small-step regime, and show two distinct transient phases: a fast network-coordination phase and a slower centralized-convergence phase. They analyze how network topology, quantization quality, and gradient noise interact, revealing that compression speeds learning at the cost of a controlled increase in steady-state error, and demonstrate favorable comparisons against CHOCO-SGD and DUAL-SGD in illustrative scenarios. The findings highlight a practical trade-off between bit-rate savings and learning speed, with diffusion-based cooperation mitigating identifiability issues and enabling scalable, real-time distributed inference.

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. 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. We propose a diffusion strategy nicknamed as ACTC (Adapt-Compress-Then-Combine), which relies on the following 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 this work are as follows. First, we focus on adaptive strategies, where constant (as opposed to diminishing) step-sizes are critical to respond in real time to nonstationary variations. Second, we consider the general class of directed graphs and left-stochastic combination policies, which allow us to enhance the interplay between topology and learning. Third, in contrast with related works that assume strong convexity for all individual agents' cost functions, we require strong convexity only at a network level, a condition satisfied even if a single agent has a strongly-convex cost and the remaining agents have non-convex costs. Fourth, we focus on a diffusion (as opposed to consensus) strategy. Under the demanding setting of compressed information, we establish that the ACTC iterates fluctuate around the desired optimizer, achieving remarkable savings in terms of bits exchanged between neighboring agents.

Figures (9)

• Figure 1: Sketch of the randomized quantizer described in Sec. \ref{['sec:AlistarhQuant']}, for the case where the bit-rate is $r=2$.
• Figure 2: Left plot. ACTC network mean-square-deviation in \ref{['eq:netMSD']} as a function of the iteration $i$, for different values of the bit-rate $r$. We considered the distributed regression problem in Sec. \ref{['sec:experiments']}, with dimensionality $M=50$, Gaussian regressors $\bm{u}_{k,i}$ with diagonal matrices and variances drawn as independent realizations from a uniform distribution in $(1,4)$, and Gaussian disturbances $\bm{v}_{k,i}$ with variances drawn as independent realizations from a uniform distribution in $(0.25,1)$. The ACTC algorithm is run with stability parameter $\zeta = 0.25$, and with equal step-sizes $\mu_k=\mu=4 \times10^{-3}$. All agents use the randomized quantizer in Sec. \ref{['sec:AlistarhQuant']} with bit-rate $r$. The mean-square-deviations are estimated by means of $10^2$ Monte Carlo runs. Right plot. Network topology used in the experiments, on top of which we build a Metropolis combination matrix to run the ACTC algorithm. All nodes have a self-loop, not shown in the figure.
• Figure 3: Difference between the network mean-square-deviation of the ACTC strategy and the mean-square-deviation of the uncompressed strategies. The label "uncompressed ACTC" stems for the ACTC strategy in \ref{['ACTC']} with $\bm{\mathcal{Q}}(x)=x$, whereas "ATC" stems for the classical ATC in Sayed. The dashed curve is obtained by depicting a curve proportional to $(2^r - 1)^{-2}$, with proportionality constant set so as to match the second point of the uncompressed ACTC curve. The relevant parameters of the ACTC strategy and of the distributed regression problem are set as in Fig. \ref{['fig:1']}, but for the dimensionality $M=10$.
• Figure 4: ACTC mean-square-deviation of the individual agents achieved using $r=2$ bits, under the same setting of Fig. \ref{['fig:1']}. The inset plot zooms in on the faster initial transient needed by the agents to reach a coordinated evolution.
• Figure 5: Network mean-square-deviation of the ACTC strategy, for different $(\mu,\zeta)$ pairs guaranteeing the same value of the product $\mu\,\zeta$ (and, hence, the same value of convergence rate $\rho_{\rm{cen}}$). The simulation setting is the same as in Fig. \ref{['fig:1']}. The inset plot highlights the impact of $\zeta$ on the faster initial transient needed by the agents to reach a coordinated evolution.

Theorems & Definitions (14)

• Definition 1: Average Energy Operator
• Lemma 1: Gradient Noise Energy Transfer
• Lemma 2: Quantization Error Energy Transfer
• Lemma 3: Quantized Iterates Energy Transfer
• Theorem 1: Recursion on Quantized Iterates
• Theorem 2: Mean-Square Stability
• Theorem 3: ACTC Learning Behavior
• Definition 2: Energy Vector Operator
• Definition 3: Norm Matrix Operator
• Lemma 4: Characterization of $\bm{\mathcal{G}}_{i-1}$