AI 中文总结
本文针对不适定逆问题中的连续时间SGD，提出基于偏差原则的后验早停规则，证明其误差达到极小极大最优速率（至多差对数因子），为首个此类收敛速率结果。
AI 中文摘要
我们研究了不适定线性逆问题中随机梯度下降（SGD）的连续时间模型的早停问题。我们考虑一种基于偏差原则的后验停止规则，该规则在残差达到噪声水平时停止动力学过程。与确定性梯度流不同，随机动力学表现出持续的多乘性波动，其振幅与残差本身成比例。在初始误差满足源条件且经验协方差算子的谱满足多项式界的假设下，我们证明，以高概率，该随机停止时刻的误差在相应的源类上达到极小极大速率（至多相差一个对数因子）。因此，我们的结果确立了偏差原则作为连续时间SGD的自适应正则化策略，尽管随机采样引起持续波动，该策略在极小极大意义下最优（至多相差一个对数因子）。据我们所知，这是逆问题中连续时间随机梯度下降模型在后验停止规则下的首个收敛速率结果：现有分析（包括方差缩减变体的分析）仅界定了停止时间，但未量化该时刻达到的误差。
英文摘要
We study early stopping for a continuous-time model of stochastic gradient descent (SGD) in ill-posed linear inverse problems. We consider an a posteriori stopping rule based on the discrepancy principle, which stops the dynamics once the residual reaches the noise level. Unlike deterministic gradient flow, the stochastic dynamics exhibits persistent multiplicative fluctuations whose amplitude scales with the residual itself. Under a source condition on the initial error and a polynomial bound on the spectrum of the empirical covariance operator, we prove that, with high probability, the error at this random stopping time achieves the minimax rate over the corresponding source class, up to a logarithmic factor. Our results therefore establish the discrepancy principle as an adaptive regularization strategy for continuous-time SGD that is minimax-optimal up to a logarithmic factor, despite the persistent fluctuations induced by stochastic sampling. To our knowledge, this is the first convergence-rate result for a continuous-time model of stochastic gradient descent in inverse problems under an a posteriori stopping rule: existing analyses, including those of variance-reduced variants, bound the stopping time but do not quantify the error attained there.
