用梯度下降解非线性逆问题，证明能达最优收敛速度。
Gradient-Based Non-Linear Inverse Learning
- 用固定步长的梯度下降和随机梯度下降求解非线性逆问题。
- 在经典假设下，证明了算法收敛速度达到极小极大最优。
- 适用于研究随机设计下非线性逆问题的理论工作者。
我们在随机设计背景下研究非线性逆问题的统计逆学习。具体地，采用固定步长的梯度下降（GD）和带小批量的随机梯度下降（SGD）来处理一类非线性问题。分析在目标函数光滑性经典假设下，推导出两类算法的收敛速率。这些假设通过与切线核相关的积分算子平滑性以及有效维数有界来表达。此外，我们建立了停止时间，使收敛速率在经典的再生核希尔伯特空间（RKHS）框架内达到极小极大最优。结果表明，GD和SGD在随机设计下能实现非线性逆问题的最优收敛速率。
原文摘要 · Abstract (English)
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.
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