arXiv subjects

Luis Nunes Vicente

Publications and source records attributed to Luis Nunes Vicente.

24 records · Page 2Linked to original sources

arXiv2021

The Sharpe predictor for fairness in machine learning

In machine learning (ML) applications, unfair predictions may discriminate against a minority group. Most existing approaches for fair machine learning (FML) treat fairness as a constraint or a penalization term in the optimization of a ML model, which does not lead to the discovery of the complete landscape of the trade-offs among learning accuracy and fairness metrics, and does not integrate fairness in a meaningful way. Recently, we have introduced a new paradigm for FML based on Stochastic Multi-Objective Optimization (SMOO), where accuracy and fairness metrics stand as conflicting objectives to be optimized simultaneously. The entire trade-offs range is defined as the Pareto front of the SMOO problem, which can then be efficiently computed using stochastic-gradient type algorithms. SMOO also allows defining and computing new meaningful predictors for FML, a novel one being the Sharpe predictor that we introduce and explore in this paper, and which gives the highest ratio of accuracy-to-unfairness. Inspired from SMOO in finance, the Sharpe predictor for FML provides the highest prediction return (accuracy) per unit of prediction risk (unfairness).

cs.LG↗

arXiv2021

The stochastic multi-gradient algorithm for multi-objective optimization and its application to supervised machine learning

Optimization of conflicting functions is of paramount importance in decision making, and real world applications frequently involve data that is uncertain or unknown, resulting in multi-objective optimization (MOO) problems of stochastic type. We study the stochastic multi-gradient (SMG) method, seen as an extension of the classical stochastic gradient method for single-objective optimization. At each iteration of the SMG method, a stochastic multi-gradient direction is calculated by solving a quadratic subproblem, and it is shown that this direction is biased even when all individual gradient estimators are unbiased. We establish rates to compute a point in the Pareto front, of order similar to what is known for stochastic gradient in both convex and strongly convex cases. The analysis handles the bias in the multi-gradient and the unknown a priori weights of the limiting Pareto point. The SMG method is framed into a Pareto-front type algorithm for the computation of the entire Pareto front. The Pareto-front SMG algorithm is capable of robustly determining Pareto fronts for a number of synthetic test problems. One can apply it to any stochastic MOO problem arising from supervised machine learning, and we report results for logistic binary classification where multiple objectives correspond to distinct-sources data groups.

math.NA↗

arXiv2019

Modeling Hessian-vector products in nonlinear optimization: New Hessian-free methods

In this paper, we suggest two ways of calculating interpolation models for unconstrained smooth nonlinear optimization when Hessian-vector products are available. The main idea is to interpolate the objective function using a quadratic on a set of points around the current one and concurrently using the curvature information from products of the Hessian times appropriate vectors, possibly defined by the interpolating points. These enriched interpolating conditions form then an affine space of model Hessians or model Newton directions, from which a particular one can be computed once an equilibrium or least secant principle is defined. A first approach consists of recovering the Hessian matrix satisfying the enriched interpolating conditions, from which then a Newton direction model can be computed. In a second approach we pose the recovery problem directly in the Newton direction. These techniques can lead to a significant reduction in the overall number of Hessian-vector products when compared to the inexact or truncated Newton method, although simple implementations may pay a cost in linear algebra or number of function evaluations.

math.NA↗

arXiv2013

Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse structure in the Hessian matrix. To be able to exploit Hessian sparsity, existing optimization approaches require the knowledge of the sparsity structure. The goal of this paper is to develop and analyze a method where the sparse models are constructed automatically. The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements. Such a recovery is achieved by minimizing the l1-norm of a vector subject to the measurements constraints. We suggest an approach for building sparse quadratic polynomial interpolation models by minimizing the l1-norm of the entries of the model Hessian subject to the interpolation conditions. We show that this procedure recovers accurate models when the function Hessian is sparse, using relatively few randomly selected sample points. Motivated by this result, we developed a practical interpolation-based trust-region method using deterministic sample sets and minimum l1-norm quadratic models. Our computational results show that the new approach exhibits a promising numerical performance both in the general case and in the sparse one.

math.OC↗

arXiv2013

Convergence of trust-region methods based on probabilistic models

In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Secondly, we use random models of higher quality than those produced by usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability greater than or equal to 1/2. This is in contrast with stochastic gradient approaches, where the model is assumed to be "correct" only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Hence we show that a standard optimization framework can be used in cases when models are random and may or may not provide good approximations, as long as "good" models are more likely than "bad" models. Our results are based on the use of properties of martingales. Our motivation comes from using random sample sets and interpolation models in derivative-free optimization. However, our framework is general and can be applied with any source of uncertainty in the model. We discuss various applications for our methods in the paper.

math.OC↗

arXiv2013

On partial sparse recovery

We consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the $\ell_1$-norm of the part of the solution vector which is known to be sparse. Such a problem is closely related to a classical problem in Compressed Sensing where the $\ell_1$-norm of the whole solution vector is minimized. We introduce analogues of restricted isometry and null space properties for the recovery of partially sparse vectors and show that these new properties are implied by their original counterparts. We show also how to extend recovery under noisy measurements to the partially sparse case.

cs.IT↗