Gradient-Based Non-Linear Inverse Learning
This provides theoretical guarantees for gradient-based methods in nonlinear inverse problems, which is incremental but important for statistical learning applications.
The paper tackles nonlinear inverse learning problems under random design by analyzing gradient descent and stochastic gradient descent with constant step sizes, deriving minimax-optimal convergence rates under smoothness assumptions on the target function.
We study statistical inverse learning in the context of nonlinear inverse problems under random design. Specifically, we address a class of nonlinear problems by employing gradient descent (GD) and stochastic gradient descent (SGD) with mini-batching, both using constant step sizes. Our analysis derives convergence rates for both algorithms under classical a priori assumptions on the smoothness of the target function. These assumptions are expressed in terms of the integral operator associated with the tangent kernel, as well as through a bound on the effective dimension. Additionally, we establish stopping times that yield minimax-optimal convergence rates within the classical reproducing kernel Hilbert space (RKHS) framework. These results demonstrate the efficacy of GD and SGD in achieving optimal rates for nonlinear inverse problems in random design.