Graph topology invariant gradient and sampling complexity for decentralized and stochastic optimization

One fundamental problem in decentralized multi-agent optimization is the trade-off between gradient/sampling complexity and communication complexity. We propose new algorithms whose gradient and sampling complexities are graph topology invariant, while their communication complexities remain optimal. For convex smooth deterministic problems, we propose a primal dual sliding (PDS) algorithm that computes an $\epsilon$-solution with $O((\tilde{L}/\epsilon)^{1/2})$ gradient … Read more

Mirror-prox sliding methods for solving a class of monotone variational inequalities

In this paper we propose new algorithms for solving a class of structured monotone variational inequality (VI) problems over compact feasible sets. By identifying the gradient components existing in the operator of VI, we show that it is possible to skip computations of the gradients from time to time, while still maintaining the optimal iteration … Read more

Random Gradient Extrapolation for Distributed and Stochastic Optimization

In this paper, we consider a class of finite-sum convex optimization problems defined over a distributed multiagent network with $m$ agents connected to a central server. In particular, the objective function consists of the average of $m$ ($\ge 1$) smooth components associated with each network agent together with a strongly convex term. Our major contribution … Read more

Dynamic Stochastic Approximation for Multi-stage Stochastic Optimization

In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm, for solving these types of stochastic optimization problems. We show that DSA can achieve an optimal ${\cal O}(1/\epsilon^4)$ rate of convergence in terms … Read more

Communication-Efficient Algorithms for Decentralized and Stochastic Optimization

We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. We first propose a decentralized primal-dual … Read more

Accelerated gradient sliding for structured convex optimization

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show … Read more

Algorithms for stochastic optimization with expectation constraints

This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with an expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the expectation constraint on devision variables. We show that … Read more

Generalized Uniformly Optimal Methods for Nonlinear Programming

In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search step (gradient descent or Quasi-Newton iteration) into these uniformly optimal convex programming methods, and then enforce a monotone decreasing property of … Read more

An optimal randomized incremental gradient method

In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization … Read more

Fast Bundle-Level Type Methods for unconstrained and ball-constrained convex optimization

It has been shown in \cite{Lan13-1} that the accelerated prox-level (APL) method and its variant, the uniform smoothing level (USL) method, have optimal iteration complexity for solving black-box and structured convex programming problems without requiring the input of any smoothness information. However, these algorithms require the assumption on the boundedness of the feasible set and … Read more