An Analysis Tool for Push-Sum Based Distributed Optimization

This paper establishes the explicit absolute probability sequence for the push-sum algorithm, and based on which, constructs quadratic Lyapunov functions for push-sum based distributed optimization algorithms. As illustrative examples, the proposed novel analysis tool can establish optimal convergence rates for the subgradient-push and stochastic gradient-push, two important algorithms for distributed convex optimization over directed graphs. Specifically, the paper proves that the subgradient-push algorithm with a constant stepsize for finite $T$ steps converges at a rate of $O(1/\sqrt{T})$ for general convex functions, and the stochastic gradient-push algorithm with a time $t$ dependent diminishing stepsize converges at a rate of $O(1/t)$ for strongly convex functions over time-varying directed graphs. Both rates are respectively the same as the state-of-the-art rates of their single-agent counterparts and thus optimal.