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We develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear…
This paper presents a study of the effectiveness of Neural Network (NN) techniques for deconvolution inverse problems relevant for applications in Quantum Field Theory, but also in more general contexts. We consider NN's asymptotic limits,…
Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems…
Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies…
This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs…
The majority of machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, given samples provided in a streaming fashion, we define a general stochastic Newton algorithm and its…
This paper introduces a new derivative parsing algorithm for recognition of parsing expression grammars. Derivative parsing is shown to have a polynomial worst-case time bound, an improvement on the exponential bound of the recursive…
We introduce a cutting-plane framework for nonconvex quadratic programs (QPs) that progressively tightens convex relaxations. Our approach leverages the doubly nonnegative (DNN) relaxation to compute strong lower bounds and generate…
We study differentiable strongly quasiconvex functions for providing new properties for algorithmic and monotonicity purposes. Furthemore, we provide insights into the decreasing behaviour of strongly quasiconvex functions, applying this…
In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming…
In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the…
Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a…
We solve the analysis sparse coding problem considering a combination of convex and non-convex sparsity promoting penalties. The multi-penalty formulation results in an iterative algorithm involving proximal-averaging. We then unfold the…
Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups.…
In this paper we propose a variant of the random coordinate descent method for solving linearly constrained convex optimization problems with composite objective functions. If the smooth part of the objective function has Lipschitz…
Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…
We present a quasi-Newton method for unconstrained stochastic optimization. Most existing literature on this topic assumes a setting of stochastic optimization in which a finite sum of component functions is a reasonable approximation of an…
We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice ${\bf Z}^n$. We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the…
A weighted regression procedure is proposed for regression type problems where the innovations are heavy-tailed. This method approximates the least absolute regression method in large samples, and the main advantage will be if the sample is…
In this paper, we study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we…