arXiv subjects

Shivam Singh Dhama

Publications and source records attributed to Shivam Singh Dhama.

4 recordsLinked to original sources

arXiv2026

Quantitative Fluctuation Analysis for Continuous-Time Stochastic Gradient Descent via Malliavin Calculus

In this paper, we establish a Quantitative Central Limit Theorem (QCLT) for the Stochastic Gradient Descent in Continuous Time (SGDCT) algorithm, whose parameter updates are governed by a stochastic differential equation. We derive an explicit rate at which the SGDCT iterates converge, in the Wasserstein metric, to a critical point of the objective function. This rate is driven primarily by the magnitude of the learning rate: for a fixed convexity constant of the objective function, smaller learning rates lead to slower convergence. Our approach relies on tools from Malliavin calculus. In particular, we apply a second-order Poincaré inequality and obtain explicit bounds by estimating the first- and second-order Malliavin derivatives separately. Controlling the second-order derivative requires several delicate calculations and a careful sequence of decompositions in order to achieve sharp estimates. We complement the theoretical results with several numerical experiments that illustrate the predicted convergence behavior.

math.PR↗

arXiv2026

Uniform-in-time Gaussian fluctuations for multiscale nonlinear stochastic systems via Malliavin Calculus

We establish a uniform-in-time quantitative central limit theorem (QCLT) for a nonlinear slow-fast stochastic system. We identify significant weaker sufficient conditions that enable us to obtain time-independent bounds for the Wasserstein distance between the fluctuation process and a centered Gaussian random variable. To prove our main result, we utilize tools from Malliavin calculus, specifically the second-order Poincaré inequality. In this context, applying the Poincaré inequality requires demonstrating uniform bounds over time for both the first- and second-order Malliavin derivatives.

math.PR↗

arXiv2025

Uniform-in-time bounds for a stochastic hybrid system with fast periodic sampling and small white-noise

We study the asymptotic behavior, uniform-in-time, of a non-linear dynamical system under the combined effects of fast periodic sampling with period $δ$ and small white noise of size $\varepsilon,\thinspace 0<\varepsilon,δ\ll 1$. The dynamics depend on both the current and recent measurements of the state, and as such it is not Markovian. Our main results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation (ODE) uniformly in time as $\varepsilon,δ\searrow 0.$ Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the remainder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided.

math.PR↗

arXiv2024

Asymptotic analysis of dynamical systems driven by Poisson random measures with periodic sampling

In this article, we study the dynamics of a nonlinear system governed by an ordinary differential equation under the combined influence of fast periodic sampling with period $δ$ and small jump noise of size $\varepsilon, 0< \varepsilon,δ\ll 1.$ The noise is a combination of Brownian motion and Poisson random measure. The instantaneous rate of change of the state depends not only on its current value but on the most recent measurement of the state, as the state is measured at certain discrete-time instants. As $\varepsilon,δ\searrow 0,$ the stochastic process of interest converges, in a suitable sense, to the dynamics of the deterministic equation. Next, the study of rescaled fluctuations of the stochastic process around its mean is found to vary depending on the relative rates of convergence of small parameters $\varepsilon, δ$ in different asymptotic regimes. We show that the rescaled process converges, in a strong (path-wise) sense, to an effective process having an extra drift term capturing both the sampling and noise effect. Consequently, we obtain a first-order perturbation expansion of the stochastic process of interest, in terms of the effective process along with error bounds on the remainder.

math.PR↗