Walid Hachem

Pascal Bianchi, Gersende Fort, Walid Hachem

In this paper, a distributed stochastic approximation algorithm is studied. Applications of such algorithms include decentralized estimation, optimization, control or computing. The algorithm consists in two steps: a local step, where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size, and a gossip step, where a node computes a local weighted average between its estimates and those of its neighbors. Convergence of the estimates toward a consensus is established under weak assumptions. The approach relies on two main ingredients: the existence of a Lyapunov function for the mean field in the agreement subspace, and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace. A second order analysis of the algorithm is also performed under the form of a Central Limit Theorem. The Polyak-averaged version of the algorithm is also considered.

Pascal Bianchi, Walid Hachem, Sholom Schechtman

In non-smooth stochastic optimization, we establish the non-convergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold $M$ where the function $f$ has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of $f$ is lower-bounded. We require two conditions on $f$. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a reinforced version of the projection formula of Bolte \emph{et.al.} for Whitney stratifiable functions, and which is of independent interest. The second assumption, termed the angle condition, allows to control the distance of the iterates to $M$. When $f$ is weakly convex, our assumptions are generic. Consequently, generically in the class of definable weakly convex functions, the SGD converges to a local minimizer.

Pascal Bianchi, Walid Hachem, Adil Salim

A new stochastic primal--dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions/operators that enter the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d. realizations. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators. An example of convex optimization under stochastic linear constraints is considered.

Adil Salim, Pascal Bianchi, Walid Hachem

The Douglas Rachford algorithm is an algorithm that converges to a minimizer of a sum of two convex functions. The algorithm consists in fixed point iterations involving computations of the proximity operators of the two functions separately. The paper investigates a stochastic version of the algorithm where both functions are random and the step size is constant. We establish that the iterates of the algorithm stay close to the set of solution with high probability when the step size is small enough. Application to structured regularization is considered.

Adil Salim, Pascal Bianchi, Walid Hachem

A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, there exist quite affordable methods to implement the proximity operator (backward step) in the special case where the graph is a simple path without loops. In this paper, an algorithm, referred to as "Snake", is proposed to solve such regularized problems over general graphs, by taking benefit of these fast methods. The algorithm consists in properly selecting random simple paths in the graph and performing the proximal gradient algorithm over these simple paths. This algorithm is an instance of a new general stochastic proximal gradient algorithm, whose convergence is proven. Applications to trend filtering and graph inpainting are provided among others. Numerical experiments are conducted over large graphs.

Pascal Bianchi, Walid Hachem, Franck Iutzeler

Based on the idea of randomized coordinate descent of $α$-averaged operators, a randomized primal-dual optimization algorithm is introduced, where a random subset of coordinates is updated at each iteration. The algorithm builds upon a variant of a recent (deterministic) algorithm proposed by Vũ and Condat that includes the well known ADMM as a particular case. The obtained algorithm is used to solve asynchronously a distributed optimization problem. A network of agents, each having a separate cost function containing a differentiable term, seek to find a consensus on the minimum of the aggregate objective. The method yields an algorithm where at each iteration, a random subset of agents wake up, update their local estimates, exchange some data with their neighbors, and go idle. Numerical results demonstrate the attractive performance of the method. The general approach can be naturally adapted to other situations where coordinate descent convex optimization algorithms are used with a random choice of the coordinates.