Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian~optimization~task.

翻译：高斯过程是一个强大的不确定性量化与序贯决策框架，但其应用受限于线性系统的求解需求。通常，该求解过程在数据集规模上具有三次复杂度，且对条件数敏感。我们探索将随机梯度算法作为近似求解这些线性系统的计算高效方法：针对后验采样开发低方差优化目标，并将其扩展至诱导点方法。反直觉的是，即便在未快速收敛至最优解的情况下，随机梯度下降仍能产生准确的预测。我们通过非收敛性隐式偏差的谱特征解释这一现象。研究表明，在数据覆盖充分的区域以及远离数据的区域，随机梯度下降产生的预测分布均接近真实后验。实验表明，随机梯度下降在足够大规模或病态回归任务中达到了最先进性能。在大规模贝叶斯优化任务中，其不确定性估计与显著更昂贵的基线方法性能相当。