Torino seminar series in Stochastics and Mathematical Statistics
TORINO SEMINAR SERIES IN
"Stochastics and Mathematical Statistics"
"Stochastics and Mathematical Statistics"
Department of Mathematics "Giuseppe Peano",
University of Torino
University of Torino
NOTE: All seminars are held in person at the Department of Mathematics in Via Carlo Alberto 10, Turin.
First semester 2024/2025
Thu 31 October, 2024. Aula C, Palazzo Campana, 14:30-15:30
Sandra Fortini (Bocconi)
Title: Large-width asymptotics for ReLU neural networks with alpha-Stable initializations
Abstract: There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized according to Gaussian distributions. In such a context, two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. We consider these problems for alpha-Stable NNs, namely NNs whose weights are initialized according to alpha-Stable distributions, i.e. distributions with heavy-tails. First, for alpha-Stable NNs , we show that if the NN's width goes to infinity then a rescaled NN converges weakly to an alpha-Stable stochastic process. As a difference with respect to the Gaussian setting, the choice of the activation function affects the scaling of the NN. Then, we study the large-width training dynamics of alpha-Stable ReLU-NNs, characterizing the infinitely wide dynamics in terms of a random kernel, referred to as the alpha-Stable NTK, and showing that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. The randomness of the alpha-Stable NTK is a further difference with respect to the Gaussian setting, that is: within the alpha-Stable setting, the randomness of the NN at initialization does not vanish in the large-width regime of the training.
Joint work with Stefano Favaro (Università di Torino) and Stefano Peluchetti (Sakana AI)
Thu 14 November, 2024. Sala Orsi, Palazzo Campana, 14:00-15:00
Marco Fuhrman (Università di Milano)
Title: Partially observed controlled Markov chains and optimal control of the Wonham filter.
Abstract: After a general introduction to the optimal control problem for a time-continuous Markov chain we will address the case of control with partial observation, where the control actions are chosen dynamically based upon the observation of an auxiliary process perturbed by a Brownian noise. Using techniques of optimal filtering the original problem is recast as a control problem with full observation for a process with values in the space of probabilities on the state space of the Markov chain (the controlled Wonham filter). The latter problem with be studied with various techniques, in particular by the dynamic programming equations.
This is joint work with Fulvia Confortola (Politecnico di Milano).Thu 5 December, 2024. Aula 5, Palazzo Campana, 14:30-15:30
Fabio Antonelli (Università degli Studi dell'Aquila)
Title: Differentiating a killed Brownian motion in a smooth domain
Abstract: In this work in progress, we try to extend the probabilistic representation formula for the derivative of the semigroup associated to a multidimensional killed diffusion process, that was obtained by one of the authors in the half space by employing the normally reflected process at the boundary. In particular, the authors showed that when the components are uncorrelated at the boundary, jumps in the derivative formula appear in the components unconcerned by the reflection. Here, we try to extend the formula in the case of a correlated Brownian motion killed at the boundary of a smooth multi-dimensional bounded domain. This makes the boundary curvature mixed with correlation appear in the jump-diffusion derivative process as well as in the obliquely reflected process. The main idea is to exploit the half space results, by constructing a modified Euler scheme for the reflected process, by mapping the boundary manifold to the boundary of the half plane and back and proving that, under appropriate conditions, tightness of the scheme’s derivative is ensured, so providing the limit formula.
Joint work with Arturo Kohatsu-Higa.
Organized by
Bruno Toaldo: bruno.toaldo@unito.it
Elena Issoglio: elena.issoglio@unito.it
Sponsors
Department of Mathematics "Giuseppe Peano", University of Torino