This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, and non-smooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used to treat several application examples including Markov decision process, Lasso algorithms, Pegasos algorithms, support vector machine classification, and learning. Some numerical demonstrations are also provided.
We study one-dimensional stochastic differential equations of form $dX_t = σ(X_t)dY_t$, where $Y$ is a suitable Hölder continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $σ$ for which we assume very mild conditions. In particular, we allow $σ$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
We establish the existence and uniqueness for a one-dimensional stochastic differential equation driven by a Brownian motion and a pure jump {\levy} process. It is shown that under fairly general conditions on the coefficients, pathwise uniqueness holds based on the methods of weak uniqueness and local time technique.
Solutions of martingale problems for Lévy-type operators and stochastic differential equations driven by Lévy processes with discontinuous coefficients
Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is a Markov processes $\{X_t, Y_t, φ_t\}_{t \geq 0}$, where the first fluid $X_t$ is driven by the Markov chain $φ_t$, and the second fluid $Y_t$ is driven by $φ_t$ as well as by $X_t$. That paper derived a closed-form expression for the joint stationary distribution, given in terms of operators acting on measures, which does not lend itself easily to numerical computations. Here, we construct a discontinuous Galerkin method for approximating this stationary distribution, and illustrate the methodology using an on-off ban…
In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and Hölder continuous random processes $X$ and $Y$. We discuss a notion of sufficient variability for the process $X$ which ensures that the paths of the composite process $t \mapsto f(X_t)$ are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann-Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new es…
Non-asymptotic convergence analysis of the stochastic gradient Hamiltonian Monte Carlo algorithm with discontinuous stochastic gradient with applications to training of ReLU neural networks
School of Mathematics, Renmin University of China, Beijing, China（中国人民大学数学学院）；Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore（南洋理工大学数学科学学院）；Financial Technology Thrust, Society Hub, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China（香港科技大学（广州）金融科技中心）
A variety of real-world applications are modeled via hyperbolic conservation laws. To account for uncertainties or insufficient measurements, random coefficients may be incorporated. These random fields may depend discontinuously on the state space, e.g., to represent permeability in a heterogeneous or fractured medium. We introduce a suitable admissibility criterion for the resulting stochastic discontinuous-flux conservation law and prove its well-posedness. Therefore, we ensure the pathwise existence and uniqueness of the corresponding deterministic setting and present a novel proof for the…
A new approach in stochastic optimization via the use of stochastic gradient Langevin dynamics (SGLD) algorithms, which is a variant of stochastic gradient decent (SGD) methods, allows us to efficiently approximate global minimizers of possibly complicated, high-dimensional landscapes. With this in mind, we extend here the non-asymptotic analysis of SGLD to the case of discontinuous stochastic gradients. We are thus able to provide theoretical guarantees for the algorithm's convergence in (standard) Wasserstein distances for both convex and non-convex objective functions. We also provide expli…
Motivated by studying stochastic systems with non-Gaussian Lévy noise, spectral properties for a type of linear cocycles are considered. These linear cocycles have countable jump discontinuities in time. A multiplicative ergodic theorem is proved for such linear cocycles. Then, the result is illustrated for two linear stochastic systems with general Lévy motions.
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
Motivated by an equilibrium problem, we establish the existence of a solution for a family of Markovian backward stochastic differential equations with quadratic nonlinearity and discontinuity in $Z$. Using unique continuation and backward uniqueness, we show that the set of discontinuity has measure zero. In a continuous-time stochastic model of an endowment economy, we prove the existence of an incomplete Radner equilibrium with nondegenerate endogenous volatility.
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equation…
This paper deals with generalized backward doubly stochastic differential equations driven by a Lévy process (GBDSDEL, in short). Under left or right continuous and linear growth conditions, we prove the existence of minimal (resp. maximal) solutions.
Langevin dynamics based algorithm e-TH$\varepsilon$O POULA for stochastic optimization problems with discontinuous stochastic gradient
UNIST（蔚山科学技术院）；Nanyang Technological University（南洋理工大学）；The University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（阿兰·图灵研究所）；National Technical University of Athens（雅典国家技术大学）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in energy markets. We prove existence and uniqueness of strong solutions. In addition we study the strong convergence order of the Euler-Maruyama scheme and recover the optimal rate $1/2$.
We consider a Markov chain $(x_n)$ whose kernel is indexed by a scaling parameter $γ>0$, refered to as the step size. The aim is to analyze the behavior of the Markov chain in the doubly asymptotic regime where $n\to\infty$ then $γ\to 0$. First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putti…
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly modeled by random coefficients. For flows in heterogeneous\textbackslash fractured\textbackslash porous media, the coefficients are modeled as discontinuous random fields, where the interfaces along the stochastic discontinuities represent transitions in the media. More precisely, the random coefficient is given by the sum of a (continuous) Gaussian random field a…
We present a new Aumann-like integral for a Hölder multifunction with respect to a Hölder signal, based on the Young integral of a particular set of Hölder selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.
