We present a novel stochastic variational Gaussian process ($\mathcal{GP}$) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for $\mathcal{GP}$s (CVTGP) is defined in terms of the $\mathcal{GP}$ prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent $\mathcal{GP}$ coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to $\mathcal{O}(M)$, enjoys numerical stability, and maintains $\mathcal{O}(M^3)$ time- and $\mathcal{O}(M^2)$ space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic $\mathcal{GP}$ inference methods.

翻译：我们提出了一种新颖的随机变分高斯过程($\mathcal{GP}$)推断方法，该方法基于对一组可学习的带权伪输入-输出点（核心集）的后验。所提出的基于核心集的$\mathcal{GP}$温和变分族（CVTGP）并非采用自由形式的变分族，而是依据$\mathcal{GP}$先验和数据似然来定义，从而容纳了模型归纳偏差。我们通过对潜在$\mathcal{GP}$核心集变量上的建议后验进行边缘化，推导出CVTGP的对数边缘似然的变分下界，并证明其适用于随机优化。通过利用基于核心集的温和后验（该后验进一步提供了数据的稀疏且可解释的表示），CVTGP将可学习参数规模缩减至$\mathcal{O}(M)$，具有数值稳定性，并维持了$\mathcal{O}(M^3)$的时间复杂度和$\mathcal{O}(M^2)$的空间复杂度。在具有高斯观测噪声的模拟和真实回归问题上的结果验证了，相较于其他随机$\mathcal{GP}$推断方法，CVTGP能提供更好的证据下界估计和预测均方根误差。