Accuracy and fairness trade-offs in machine learning: A stochastic multi-objective approach

In the application of machine learning to real life decision-making systems, e.g., credit scoring and criminal justice, the prediction outcomes might discriminate against people with sensitive attributes, leading to unfairness. The commonly used strategy in fair machine learning is to include fairness as a constraint or a penalization term in the minimization of the prediction … Read more

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 … Read more

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 … Read more

Trust-region methods for the derivative-free optimization of nonsmooth black-box functions

In this paper we study the minimization of a nonsmooth black-box type function, without assuming any access to derivatives or generalized derivatives and without any knowledge about the analytical origin of the function nonsmoothness. Directional methods have been derived for such problems but to our knowledge no model-based method like a trust-region one has yet … Read more

Complexity of gradient descent for multiobjective optimization

A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper we analyze the rate of convergence of gradient descent for smooth unconstrained multiobjective … Read more

A decoupled first/second-order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds

In order to be provably convergent towards a second-order stationary point, optimization methods applied to nonconvex problems must necessarily exploit both first and second-order information. However, as revealed by recent complexity analyzes of some of these methods, the overall effort to reach second-order points is significantly larger when compared to the one of approaching first-order … Read more

Complexity and global rates of trust-region methods based on probabilistic models

Trust-region algorithms have been proved to globally converge with probability one when the accuracy of the trust-region models is imposed with a certain probability conditioning on the iteration history. In this paper, we study their complexity, providing global rates and worst case complexity bounds on the number of iterations (with overwhelmingly high probability), for both … Read more

Direct search based on probabilistic feasible descent for bound and linearly constrained problems

Direct search is a methodology for derivative-free optimization whose iterations are characterized by evaluating the objective function using a set of polling directions. In deterministic direct search applied to smooth objectives, these directions must somehow conform to the geometry of the feasible region and typically consist of positive generators of approximate tangent cones (which then … Read more

A second-order globally convergent direct-search method and its worst-case complexity

Direct-search algorithms form one of the main classes of algorithms for smooth unconstrained derivative-free optimization, due to their simplicity and their well-established convergence results. They proceed by iteratively looking for improvement along some vectors or directions. In the presence of smoothness, first-order global convergence comes from the ability of the vectors to approximate the steepest … Read more

Trust-region methods without using derivatives: Worst case complexity and the non-smooth case

Trust-region methods are a broad class of methods for continuous optimization that found application in a variety of problems and contexts. In particular, they have been studied and applied for problems without using derivatives. The analysis of trust-region derivative-free methods has focused on global convergence, and they have been proved to generate a sequence of … Read more