Stochastic Gradient Descent in Continuous Time
This provides a computationally efficient method for continuous-time optimization problems in fields like science, engineering, and finance, though it appears incremental as an extension of stochastic gradient descent to continuous time.
The paper tackles the problem of statistical learning for continuous-time models by introducing Stochastic Gradient Descent in Continuous Time (SGDCT), which performs online parameter updates via a stochastic differential equation and proves convergence to zero gradient of the objective function. As an example, it applies SGDCT with a deep neural network to price high-dimensional American options up to 100 dimensions.
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. SGDCT performs an online parameter update in continuous time, with the parameter updates $θ_t$ satisfying a stochastic differential equation. We prove that $\lim_{t \rightarrow \infty} \nabla \bar g(θ_t) = 0$ where $\bar g$ is a natural objective function for the estimation of the continuous-time dynamics. The convergence proof leverages ergodicity by using an appropriate Poisson equation to help describe the evolution of the parameters for large times. SGDCT can also be used to solve continuous-time optimization problems, such as American options. For certain continuous-time problems, SGDCT has some promising advantages compared to a traditional stochastic gradient descent algorithm. As an example application, SGDCT is combined with a deep neural network to price high-dimensional American options (up to 100 dimensions).