A Proximal Zeroth-Order Algorithm for Nonconvex Nonsmooth Problems

Abstract

In this paper, we focus on solving an important class of nonconvex optimization problems which includes many problems for example signal processing over a networked multi-agent system and distributed learning over networks. Motivated by many applications in which the local objective function is the sum of smooth but possibly nonconvex part, and non-smooth but convex part subject to a linear equality constraint, this paper proposes a proximal zeroth-order primal dual algorithm (PZO-PDA) that accounts for the information structure of the problem. This algorithm only utilize the zeroth-order information (i.e., the functional values) of smooth functions, yet the flexibility is achieved for applications that only noisy information of the objective function is accessible, where classical methods cannot be applied. We prove convergence and rate of convergence for PZO-PDA. Numerical experiments are provided to validate the theoretical results.

Distributed Nash equilibrium learning: A second‐order proximal algorithm

“…Let's postpone the proof of Proposition and give priority to discuss how Proposition 1 indicates the convergence rate of each algorithm. Inequality (25) asserts that error norm ||e so i,k || is bounded by a linear and quadratic term of the iterate difference norm ||x i i (k + 1) − x i i (k)||, so the approximation error vanishes as the sequence {x i i (k)} converges. Notice that x i (k) converges by Algorithm 1, so after a number of iterations, the term ||x i i (k + 1) − x i i (k)|| → 0, and l 2 ||x i i (k + 1) − x i i (k)|| becomes smaller than 2M, this implies that the error norm ||e so i,k || eventually becomes proportional to the quadratic term…”

“…The operator splitting method is soon extended to deal with aggregative game with time‐varying communication graph 23 . Other works like dynamic NE seeking strategy for multiagent networked game with disturbance rejection is considered by Romano and Pavel, 7 distributed computation algorithm for ‐NE is investigated by Parise et al 24 Notice that there are some works on nonconvex problems, 25,26 but we only focus on convex games in this article.…”

Zeroth-Order Algorithms for Stochastic Distributed Nonconvex Optimization

“…Various ZO optimization methods have been proposed, e.g., ZO (stochastic) gradient descent algorithms [20]- [31], ZO stochastic coordinate descent algorithms [32], ZO (stochastic) variance reduction algorithms [24], [25], [29], [30], [33]- [45], ZO (stochastic) proximal algorithms [33], [41], [46], [47], ZO Frank-Wolfe algorithms [24], [43], [45], [48], ZO mirror descent algorithms [18], [39], [49], ZO adaptive momentum methods [47], [50], ZO methods of multipliers [34], [35], [51], [52], ZO stochastic path-integrated differential estimator [37], [42], [52]. Convergence properties of these algorithms have been analyzed in detail.…”

“…Assumptions 3 and 4 are standard in stochastic optimization with ZO information feedback, e.g., [22], [24], [32]- [35], [39], [40], [47], [48], [60]. Assumption 5 is slightly weaker than the assumption that each ∇f i is bounded, which is normally used in the literature studying finite-sum ZO optimization, e.g., [18], [29], [30], [34]- [36], [41], [48], [51]- [53], [57], [58],…”

“…Here, we assume that u i,k and ξ i,k , ∀i ∈ [n], k ≥ 1 are mutually independent, which is commonly assumed in stochastic optimization, e.g., [18], [22], [24], [32]- [35], [39], [40], [47], [48], [50], [54], [55], [60]. Let L k denote the σ-algebra generated by the independent random variables…”