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A stochastic gradient method for finite-sum minimization subject to deterministic linear constraints is proposed and analyzed. The procedure presented adapts the projected gradient method on convex set to the use of both a stochastic…
In this paper we introduce a new gradient method which attains quadratic convergence in a certain sense. Applicable to infinite-dimensional unconstrained minimization problems posed in a Hilbert space $H$, the approach consists in finding…
Nonsmooth sparsity constrained optimization encompasses a broad spectrum of applications in machine learning. This problem is generally non-convex and NP-hard. Existing solutions to this problem exhibit several notable limitations,…
State-of-the-art methods for solving smooth optimization problems are nonlinear conjugate gradient, low memory BFGS, and Majorize-Minimize (MM) subspace algorithms. The MM subspace algorithm which has been introduced more recently has shown…
Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this paper we propose and investigate…
We present practical Levenberg-Marquardt variants of Gauss-Newton and natural gradient methods for solving non-convex optimization problems that arise in training deep neural networks involving enormous numbers of variables and huge data…
We extend the well-known BFGS quasi-Newton method and its memory-limited variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: the…
The conjugate gradient (CG) method, a standard and vital way of minimizing the energy of a variational state, is applied to solve several problems in Skyrmion physics. The single-Skyrmion profile optimizing the energy of a two-dimensional…
Quasi-Newton methods are ubiquitous in deterministic local search due to their efficiency and low computational cost. This class of methods uses the history of gradient evaluations to approximate second-order derivatives. However, only…
Update formulas for the Hessian approximations in quasi-Newton methods such as BFGS can be derived as analytical solutions to certain nearest-matrix problems. In this article, we propose a similar idea for deriving new limited memory…
We propose a new stochastic proximal quasi-Newton method for minimizing the sum of two convex functions in the particular context that one of the functions is the average of a large number of smooth functions and the other one is nonsmooth.…
In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to…
We introduce a proximal limited--memory quasi--Newton scheme for minimizing the sum of a continuously differentiable function and a proper, lower semicontinuous and prox-bounded, possibly nonsmooth, function. Both functions might be…
We analyze the convergence of a nonlocal gradient descent method for minimizing a class of high-dimensional non-convex functions, where a directional Gaussian smoothing (DGS) is proposed to define the nonlocal gradient (also referred to as…
The discrete nature of transmitted symbols poses challenges for achieving optimal detection in multiple-input multiple-output (MIMO) systems associated with a large number of antennas. Recently, the combination of two powerful machine…
This note provides a novel, simple analysis of the method of conjugate gradients for the minimization of convex quadratic functions. In contrast with standard arguments, our proof is entirely self-contained and does not rely on the…
We consider a class of difference-of-convex (DC) optimization problems where the objective function is the sum of a smooth function and a possible nonsmooth DC function. The application of proximal DC algorithms to address this problem…
Stochastic multi-objective optimization (SMOO) has recently emerged as a powerful framework for addressing machine learning problems with multiple objectives. The bias introduced by the nonlinearity of the subproblem solution mapping…
Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a…
Due to their high computational complexity, deep neural networks are still limited to powerful processing units. To promote a reduced model complexity by dint of low-bit fixed-point quantization, we propose a gradient-based optimization…