Adding Heavy-Ball Momentum to Distributed Constrained Optimization for Energy Resource Management Subject to Time-Delays and Quantization

This paper proposes a distributed resource allocation algorithm in which each agent communicates only the sign of its local cost gradient, a single bit per directed link per iteration. The algorithm is based on Laplacian-gradient dynamics and preserves the equality (demand) constraint at every iterate, guaranteeing anytime feasibility irrespective of termination time. First, we provide a complete Lyapunov-based convergence analysis for the fixed-step-size regime: the iterates are shown to enter, in at most O(1/\varepsilon^2) iterations, a neighborhood of the optimizer whose radius scales as O(τ) in the decision variable and O(τ^2) in the cost residual, where $τ$ is the step-size. Crucially, the step-size bound is a time-invariant network constant (depending only on the graph Laplacian spectrum and the cost curvature bounds), so no adaptive tuning is required. Second, we characterize the invariant set under B-connected, weight-balanced digraphs, strictly weaker than the all-time connectivity and doubly-stochastic weights assumed in most prior work, and show that any odd, sign-preserving nonlinearity yields the same invariant-set and descent structure, with the single-bit protocol being the extreme communication-minimizing instance of a broader nonlinear class. Third, we address local box constraints via smooth penalty relaxation and quantify explicitly how the penalty weight interacts with the step-size bound and the optimality gap. Simulations on economic dispatch (1000 generators) and CPU scheduling (500 servers) demonstrate that the single-bit protocol achieves 5-60 times reduction in total communication cost relative to multi-bit baselines, at the price of a tunable and explicitly bounded optimality gap. A diminishing step-size schedule is shown empirically to eliminate this gap; its formal convergence analysis is identified as an open problem.

We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.

Byzantine-robust distributed optimization relies on robust aggregation rules to mitigate the influence of malicious Byzantine workers. Despite the proliferation of such rules, a unified convergence analysis framework that accommodates general data heterogeneity is lacking. In this work, we provide a thorough convergence theory of Byzantine-robust distributed stochastic gradient descent (SGD), analyzing variants both with and without local momentum. We establish the convergence rates for nonconvex smooth objectives and those satisfying the Polyak-Lojasiewicz condition under a general data heterogeneity assumption. Our analysis reveals that while stochasticity and data heterogeneity introduce unavoidable error floors, local momentum provably reduces the error component induced by stochasticity. Furthermore, we derive matching lower bounds to demonstrate that the upper bounds obtained in our analysis are tight and characterize the fundamental limits of Byzantine resilience under stochasticity and data heterogeneity. Empirical results support our theoretical findings.