We consider estimators obtained by iterates of the conjugate gradient (CG) algorithm applied to the normal equation of prototypical statistical inverse problems. Stopping the CG algorithm early induces regularisation, and optimal convergence rates of prediction and reconstruction error are established in wide generality for an ideal oracle stopping time. Based on this insight, a fully data-driven early stopping rule $\tau$ is constructed, which also attains optimal rates, provided the error in estimating the noise level is not dominant. The error analysis of CG under statistical noise is subtle due to its nonlinear dependence on the observations. We provide an explicit error decomposition and identify two terms in the prediction error, which share important properties of classical bias and variance terms. Together with a continuous interpolation between CG iterates, this paves the way for a comprehensive error analysis of early stopping. In particular, a general oracle-type inequality is proved for the prediction error at $\tau$. For bounding the reconstruction error, a more refined probabilistic analysis, based on concentration of self-normalised Gaussian processes, is developed. The methodology also provides some new insights into early stopping for CG in deterministic inverse problems. A numerical study for standard examples shows good results in practice for early stopping at $\tau$.

翻译：本文研究了通过共轭梯度（CG）算法迭代求解典型统计逆问题正规方程所获得的估计量。提前终止CG算法会引入正则化效应，我们针对理想的神谕停止时间，在广泛条件下建立了预测误差与重构误差的最优收敛速率。基于这一认识，我们构建了完全数据驱动的早停规则$\tau$，该规则在噪声水平估计误差不占主导地位时同样能达到最优速率。由于CG算法对观测数据的非线性依赖特性，其在统计噪声下的误差分析具有复杂性。我们提供了显式的误差分解，并识别出预测误差中的两项——这两项具有经典偏差项与方差项的重要特性。结合CG迭代间的连续插值技术，这为早停机制的全面误差分析奠定了基础。特别地，我们证明了在$\tau$时刻预测误差的广义神谕型不等式。为界定重构误差，我们基于自归一化高斯过程的集中性开展了更精细的概率分析。该方法也为确定性逆问题中CG早停机制提供了新的理论视角。针对标准算例的数值研究表明，采用$\tau$进行早停在实践中能获得良好效果。