Fixed-Point Automatic Differentiation of Forward–Backward Splitting Algorithms for Partly Smooth Functions

A large class of non-smooth practical optimization problems can be written as minimization of a sum of smooth and partly smooth functions. We consider such structured problems which also depend on a parameter vector and study the problem of differentiating its solution mapping with respect to the parameter which has far reaching applications in sensitivity … Read more

Optimized convergence of stochastic gradient descent by weighted averaging

Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the arithmetic mean of all iterates converges considerably slower to the optimal solution than the iterates themselves. And also in the … Read more

Using Taylor-Approximated Gradients to Improve the Frank-Wolfe Method for Empirical Risk Minimization

\(\) The Frank-Wolfe method has become increasingly useful in statistical and machine learning applications, due to the structure-inducing properties of the iterates, and especially in settings where linear minimization over the feasible set is more computationally efficient than projection. In the setting of Empirical Risk Minimization — one of the fundamental optimization problems in statistical … Read more

Accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient

In this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are … Read more

A Proximal Gradient Method for Multi-objective Optimization Problems Using Bregman Functions

In this paper, a globally convergent proximal gradient method is developed for convex multi-objective optimization problems using Bregman distance. The proposed method is free from any kind of a priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a descent direction. This … Read more

Convergence Results for Primal-Dual Algorithms in the Presence of Adjoint Mismatch

Most optimization problems arising in imaging science involve high-dimensional linear operators and their adjoints. In the implementations of these operators, approximations may be introduced for various practical considerations (e.g., memory limitation, computational cost, convergence speed), leading to an adjoint mismatch. This occurs for the X-ray tomographic inverse problems found in Computed Tomography (CT), where the … Read more

A family of accelerated inexact augmented Lagrangian methods with applications to image restoration

In this paper, we focus on a class of convex optimization problems subject to equality or inequality constraints and have developed an Accelerated Inexact Augmented Lagrangian Method (AI-ALM). Different relative error criteria are designed to solve the subproblem of AI-ALM inexactly, and the popular used relaxation step is exploited to accelerate the convergence. By a … Read more

An Asynchronous Proximal Bundle Method

We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle for computing function values and subgradients. The algorithm allows for an arbitrary number of master problem processes computing … Read more

New Penalized Stochastic Gradient Methods for Linearly Constrained Strongly Convex Optimization

For minimizing a strongly convex objective function subject to linear inequality constraints, we consider a penalty approach that allows one to utilize stochastic methods for problems with a large number of constraints and/or objective function terms. We provide upper bounds on the distance between the solutions to the original constrained problem and the penalty reformulations, … Read more

A nested primal–dual FISTA-like scheme for composite convex optimization problems

We propose a nested primal–dual algorithm with extrapolation on the primal variable suited for minimizing the sum of two convex functions, one of which is continuously differentiable. The proposed algorithm can be interpreted as an inexact inertial forward–backward algorithm equipped with a prefixed number of inner primal–dual iterations for the proximal evaluation and a “warm–start” … Read more