arXiv subjects

Francesco Cosentino

Publications and source records attributed to Francesco Cosentino.

4 recordsLinked to original sources

arXiv2023

Grid-Free Computation of Probabilistic Safety with Malliavin Calculus

This work concerns continuous-time, continuous-space stochastic dynamical systems described by stochastic differential equations (SDE). It presents a new approach to compute probabilistic safety regions, namely sets of initial conditions of the SDE associated to trajectories that are safe with a probability larger than a given threshold. The approach introduces a functional that is minimised at the border of the probabilistic safety region, then solves an optimisation problem using techniques from Malliavin Calculus, which computes such region. Unlike existing results in the literature, the new approach allows one to compute probabilistic safety regions without gridding the state space of the SDE.

math.PR↗

arXiv2021

Markov Chain Approximations to Stochastic Differential Equations by Recombination on Lattice Trees

We revisit the classical problem of approximating a stochastic differential equation by a discrete-time and discrete-space Markov chain. Our construction iterates Caratheodory's theorem over time to match the moments of the increments locally. This allows to construct a Markov chain with a sparse transition matrix where the number of attainable states grows at most polynomially as time increases. Moreover, the MC evolves on a tree whose nodes lie on a "universal lattice" in the sense that an arbitrary number of different SDEs can be approximated on the same tree. The construction is not tailored to specific models, we discuss both the case of uni-variate and multi-variate case SDEs, and provide an implementation and numerical experiments.

math.PR↗

arXiv2020

A Randomized Algorithm to Reduce the Support of Discrete Measures

Given a discrete probability measure supported on $N$ atoms and a set of $n$ real-valued functions, there exists a probability measure that is supported on a subset of $n+1$ of the original $N$ atoms and has the same mean when integrated against each of the $n$ functions. If $ N \gg n$ this results in a huge reduction of complexity. We give a simple geometric characterization of barycenters via negative cones and derive a randomized algorithm that computes this new measure by "greedy geometric sampling". We then study its properties, and benchmark it on synthetic and real-world data to show that it can be very beneficial in the $N\gg n$ regime. A Python implementation is available at \url{https://github.com/FraCose/Recombination_Random_Algos}.

cs.LG↗

arXiv2020

Carathéodory Sampling for Stochastic Gradient Descent

Many problems require to optimize empirical risk functions over large data sets. Gradient descent methods that calculate the full gradient in every descent step do not scale to such datasets. Various flavours of Stochastic Gradient Descent (SGD) replace the expensive summation that computes the full gradient by approximating it with a small sum over a randomly selected subsample of the data set that in turn suffers from a high variance. We present a different approach that is inspired by classical results of Tchakaloff and Carathéodory about measure reduction. These results allow to replace an empirical measure with another, carefully constructed probability measure that has a much smaller support, but can preserve certain statistics such as the expected gradient. To turn this into scalable algorithms we firstly, adaptively select the descent steps where the measure reduction is carried out; secondly, we combine this with Block Coordinate Descent so that measure reduction can be done very cheaply. This makes the resulting methods scalable to high-dimensional spaces. Finally, we provide an experimental validation and comparison.

cs.LG↗