Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with $n$ function values and $n$ full gradients in $d$ dimensions scales cubically in the joint state size, imposing an intractable $\mathcal{O}(n^3 d^3)$ computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most $m^2$ directional derivatives once a conditioning set of size $m$ is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples $n$ and $d$ from the dense matrix inversion. This reduces the per-target evaluation cost to $\mathcal{O}(dm^2 + m^6)$ time and $\mathcal{O}(dm^2 + m^4)$ memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to $d$, enabling highly scalable inference in high-dimensional spaces.

翻译：梯度观测可显著提升高斯过程（GP）代理模型性能，尤其在函数评估代价高昂的高维场景中。然而，对$n$个函数值和$n$个完整梯度（维度$d$）执行精确推断时，计算复杂度与联合状态规模呈三次方增长，造成难以承受的$\mathcal{O}(n^3 d^3)$计算瓶颈。我们提出TERA方法——一种基于目标特异性精确梯度缩减的高可扩展导数GP技术。理论证明：对于平稳核函数，与目标点-条件点连线方向正交的梯度分量与目标函数值条件独立；因此，当指定规模为$m$的条件集后，精确条件密度完全由至多$m^2$个方向导数刻画。通过将这些降维后的无维度条件量作为Vecchia近似中的局部因子，TERA有效解耦了$n$和$d$与稠密矩阵求逆的关联。这使得每个目标点的计算代价降至$\mathcal{O}(dm^2 + m^6)$时间复杂度和$\mathcal{O}(dm^2 + m^4)$内存复杂度，且底层导数GP模型在数学上保持不变。实验评估表明，TERA在达到最先进预测精度的同时，运算速度比标准导数GP快数个数量级。关键优势在于：计算时间和峰值GPU内存均随$d$增长保持基本恒定，从而在高维空间中实现高度可扩展的推断。