Javier Pena — Page 2

arXiv2012

Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision

A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite precision. In this paper we describe a finite-precision version of this algorithm. Our main result shows that this version works within the same time bounds and requires a precision which, on the average, amounts to a polynomial amount of bits in the mantissa of the intervening floating-point numbers.

math.NA↗

arXiv2011

A Smoothing Stochastic Gradient Method for Composite Optimization

We consider the unconstrained optimization problem whose objective function is composed of a smooth and a non-smooth conponents where the smooth component is the expectation a random function. This type of problem arises in some interesting applications in machine learning. We propose a stochastic gradient descent algorithm for this class of optimization problem. When the non-smooth component has a particular structure, we propose another stochastic gradient descent algorithm by incorporating a smoothing method into our first algorithm. The proofs of the convergence rates of these two algorithms are given and we show the numerical performance of our algorithm by applying them to regularized linear regression problems with different sets of synthetic data.

math.OC↗