Related papers: Second order linear differential equations with an…
We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion.…
This paper deals with the filtering problem for a class of discrete time stochastic volatility models in which the disturbances have rational probability density functions. This includes the Cauchy distributions and Student t-distributions…
We study the convergence properties of the original and away-step Frank-Wolfe algorithms for linearly constrained stochastic optimization assuming the availability of unbiased objective function gradient estimates. The objective function is…
Optimization in Deep Learning is mainly dominated by first-order methods which are built around the central concept of backpropagation. Second-order optimization methods, which take into account the second-order derivatives are far less…
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…
This thesis studies high-dimensional, continuous-valued pairwise Markov Random Fields. We are particularly interested in approximating pairwise densities whose logarithm belongs to a Sobolev space. For this problem we propose the method of…
The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this…
We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the…
Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient…
Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of…
In this study, we present a novel algorithm for determining directionality in 2D distributions of discrete data. We compare a reference dataset with a known direction to a measured dataset with an unknown direction by the Frobenius norm of…
This work focuses on the numerical approximations of random periodic solutions of stochastic differential equations (SDEs). Under non-globally Lipschitz conditions, we prove the existence and uniqueness of random periodic solutions for the…
We study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau-Lifshitz-Bloch (LLB) equation on a bounded domain in $\mathbb R^d$ for $d=1,2$. Our main results are…
We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in…
The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…
Models of stochastic processes are widely used in almost all fields of science. Theory validation, parameter estimation, and prediction all require model calibration and statistical inference using data. However, data are almost always…
We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on…
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in…
We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random…
Zeroth-order optimization aims to minimize an objective function using only function evaluations, and is therefore fundamental in black-box optimization, hyperparameter tuning, bandit learning, and adversarial machine learning. While…