We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model parameters by maximising our lower bound retains many of the sparse variational approach benefits while reducing the bias introduced into parameter learning. The basis of our bound is a more careful analysis of the log-determinant term appearing in the log marginal likelihood, as well as using the method of conjugate gradients to derive tight lower bounds on the term involving a quadratic form. Our approach is a step forward in unifying methods relying on lower bound maximisation (e.g. variational methods) and iterative approaches based on conjugate gradients for training Gaussian processes. In experiments, we show improved predictive performance with our model for a comparable amount of training time compared to other conjugate gradient based approaches.

本文提出一种无需完全核矩阵的矩阵分解即可计算的高斯过程回归模型的对数边际似然的下界。我们通过最大化我们的下界来学习模型参数的近似最大似然方法保留了许多稀疏变分方法的优点，同时减少了参数学习中引入的偏差。我们的方法通过对出现在对数边际似然中的对数行列式项进行更仔细的分析，以及使用共轭梯度法导出涉及二次形式的项的紧凑下界，从而在统一依赖下界最大化的方法和基于共轭梯度的迭代方法的训练高斯过程方面迈出了一步。实验结果表明，相对于其他基于共轭梯度的方法，在相当的训练时间内，我们的模型具有更好的预测性能。

共轭梯度在高斯过程回归对数边缘似然度上的更紧确界