# Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem

arXiv: https://arxiv.org/abs/1710.04273 · HTML: https://mlsanity.com/paper/1710.04273/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2017.

## Built on by (most cited first, 10 of 10)

- [Stochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis](https://mlsanity.com/paper/2106.02588/) (2021)
- [Parameter Symmetry and Noise Equilibrium of Stochastic Gradient Descent](https://mlsanity.com/paper/2402.07193/) (2024)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Noise Balance and Stationary Distribution of Stochastic Gradient Descent](https://mlsanity.com/paper/2308.06671/) (2023)
- [Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations](https://mlsanity.com/paper/2202.06637/) (2022)
- [A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations](https://mlsanity.com/paper/2207.04496/) (2022)
- [Two-Timescale Stochastic Approximation for Bilevel Optimisation Problems in Continuous-Time Models](https://mlsanity.com/paper/2206.06995/) (2022)
- [Neuronal diversity can improve machine learning for physics and beyond](https://mlsanity.com/paper/2204.04348/) (2022)
- [Convergence of the generalization error for deep gradient flow methods for PDEs](https://mlsanity.com/paper/2512.25017/) (2025)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)

## Abstract (arXiv)

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. The parameter updates occur in continuous time and satisfy a stochastic differential equation. This paper analyzes the asymptotic convergence rate of the SGDCT algorithm by proving a central limit theorem (CLT) for strongly convex objective functions and, under slightly stronger conditions, for non-convex objective functions as well. An Lp convergence rate is also proven for the algorithm in the strongly convex case. The mathematical analysis lies at the intersection of stochastic analysis and statistical learning.