Gradient-Based Non-Linear Inverse Learning

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.

Statistical inverse learning problems with random observations

We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable solutions. We discuss recent results in spectral regularization methods and regularization by projection, exploring both approaches within the context of Hilbert scales and presenting new insights particularly in regularization by projection. Additionally, we overview recent advancements in regularization using convex penalties. Convergence rates are analyzed in terms of the sample size in a probabilistic sense, yielding minimax rates in both expectation and probability. To achieve these results, the structure of reproducing kernel Hilbert spaces is leveraged to establish minimax rates in the statistical learning setting. We detail the assumptions underpinning these key elements of our proofs. Finally, we demonstrate the application of these concepts to nonlinear inverse problems in pharmacokinetic/pharmacodynamic (PK/PD) models, where the task is to predict changes in drug concentrations in patients.