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We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone…
In 1986, Dixon and McKee developed a discrete fractional Gr\"{o}nwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Gr\"{o}nwall inequality. However, this…
Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders…
In this note, under a weak monotonicity and a weak coercivity, we address strong well-posedness of McKean-Vlasov stochastic differential equations (SDEs) driven by L\'{e}vy jump processes, where the coefficients are Lipschitz continuous…
We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation…
We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a…
Concentration inequalities form an essential toolkit in the study of high dimensional (HD) statistical methods. Most of the relevant statistics literature in this regard is based on sub-Gaussian or sub-exponential tail assumptions. In this…
In this paper, we derive a characterization theorem for the path-independent property of the density of the Girsanov transformation for {\it degenerated} stochastic differential equations (SDEs), extending the characterization theorem of…
We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to…
We develop a functional extension of an extremal principle by Schneider (Monatsh. Math., 1967) by introducing generalized outer linearizations of convex functions. Given a coercive convex function on $\mathbb{R}^n$, a generalized outer…
Stochastic Gradient Descent (SGD) based methods have been widely used for training large-scale machine learning models that also generalize well in practice. Several explanations have been offered for this generalization performance, a…
This paper focuses on the invariant measure of McKean-Vlasov (MV) stochastic differential equations (SDEs) with common noise (wCN) whose coefficients depend on both the state and the measure. Using the existence of the unique solution of…
We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random…
We consider stochastic differential equations on $\mathbb R^d$ with coefficients depending on the path and distribution for the whole history. Under a local integrability condition on the time-spatial singular drift, the well-posedness and…
In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential…
This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for…
Stochastic gradient methods (SGMs) are predominant approaches for solving stochastic optimization. On smooth nonconvex problems, a few acceleration techniques have been applied to improve the convergence rate of SGMs. However, little…
A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding…
Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for McKean-Vlasov type SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient…
We study trade-offs between convergence rate and robustness to gradient errors in the context of first-order methods. Our focus is on generalized momentum methods (GMMs)--a broad class that includes Nesterov's accelerated gradient,…