Optimal Algorithms for Non-Smooth Distributed Optimization in Networks

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.Comment: 17 page

Reciprocity of Algorithms Solving Distributed Consensus-Based Optimization and Distributed Resource Allocation

This paper aims at proposing a procedure to derive distributed algorithms for distributed consensus-based optimization by using distributed algorithms for network resource allocation and vice versa over switching networks with/without synchronous protocol. It is shown that first-order gradient distributed consensus-based optimization algorithms can be used for finding an optimal solution of distributed resource allocation with synchronous protocol under weaker assumptions than those given in the literature for non-switching (static) networks. It is shown that first-order gradient distributed resource allocation algorithms can be utilized for finding an optimal solution of distributed consensus-based optimization. The results presented here can be applied to time-varying and random directed networks with or without synchronous protocol with arbitrary initialization. As a result, several algorithms can now be used to derive distributed algorithms for both consensus-based optimization and resource allocation, that can overcome limitations of the existing results. While the focus of this paper is on the first-order gradient algorithms, it is to be noted that the results also work with second-order gradient algorithms.Comment: 8 page

Optimal decentralized distributed algorithms for stochastic convex optimization

We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case we use special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in smooth convex case with deterministic first-order oracle. We propose and analyze three novel methods to handle stochastic convex optimization problems with affine constraints: SPDSTM, R-RRMA-AC-SA and SSTM_sc. All methods use stochastic dual oracle. SPDSTM is the stochastic primal-dual modification of STM and it is applied for the dual problem when the primal functional is strongly convex and Lipschitz continuous on some ball. R-RRMA-AC-SA is an accelerated stochastic method based on the restarts of RRMA-AC-SA and SSTM_sc is just stochastic STM for strongly convex problems. Both methods are applied to the dual problem when the primal functional is strongly convex, smooth and Lipschitz continuous on some ball and use stochastic dual first-order oracle. We develop convergence analysis for these methods for the unbiased and biased oracles respectively. Finally, we apply all aforementioned results and approaches to solve decentralized distributed optimization problem and discuss optimality of the obtained results in terms of communication rounds and number of oracle calls per node

Quantization Design for Distributed Optimization

We consider the problem of solving a distributed optimization problem using a distributed computing platform, where the communication in the network is limited: each node can only communicate with its neighbours and the channel has a limited data-rate. A common technique to address the latter limitation is to apply quantization to the exchanged information. We propose two distributed optimization algorithms with an iteratively refining quantization design based on the inexact proximal gradient method and its accelerated variant. We show that if the parameters of the quantizers, i.e. the number of bits and the initial quantization intervals, satisfy certain conditions, then the quantization error is bounded by a linearly decreasing function and the convergence of the distributed algorithms is guaranteed. Furthermore, we prove that after imposing the quantization scheme, the distributed algorithms still exhibit a linear convergence rate, and show complexity upper-bounds on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithms and the theoretical findings for solving a distributed optimal control problem