子群秩1格用于实用高维黑盒积分近似
Subgroup Rank-1 Lattice for Practical High-dimensional Black-box Integral Approximation
- Centre for Frontier AI Research (CFAR)（前沿人工智能研究中心）
- Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出子群秩1格，利用FFT将高维黑盒积分近似复杂度降至O(n log m)，并证明收敛性，实验优于现有方法。
AI中文摘要：
估计黑盒高维函数的积分，从期望和核均值嵌入到自注意力中的softmax核，是机器学习中的基本子程序。秩1格规则适用于此设置：它们仅在固定点集处查询被积函数，且不需要梯度。然而，当这n个点作为特征映射的设计矩阵X∈R^{n×d}时，对于逐元素非线性Ψ，计算Ψ(X)^⊤v或Ψ(X)w对于任何标准拟蒙特卡洛点集都需要O(nd)的时间和内存。我们研究了子群秩1格，其Korobov生成元(1,t,…,t^{d-1})使用具有固定乘法阶m的标量t。将F_n^×分割为⟨t⟩的陪集，将两个映射简化为由FFT评估的短循环相关，对于任意Ψ在O(n log m)时间和O(n)内存内给出精确结果，而无需形成X。由于固定m超出了经典分量逐分量理论，我们直接证明收敛性：通过与分圆多项式Φ_m的结式，Korobov空间中的平方最坏情况误差对于素数m≥d+1以O(n^{-(α-1)/(m-1)})衰减，且该阈值是精确的。利用n在Q(ζ_m)中的分割，对m-1个可容许生成元进行平均，将常数改善Θ(m-1)倍。实验上，子群格在54个合成核估计设置中的49个以及九个真实数据集上的所有45个softmax注意力设置中击败了高斯和正交随机特征以及乱序Sobol'和Halton点，并在2.3毫秒内构建了d=2048，n≈4.1×10^7的样本集。
英文摘要：
Estimating integrals of black-box, high-dimensional functions, from expectations and kernel mean embeddings to the softmax kernel in self-attention, is a basic subroutine in machine learning. Rank-1 lattice rules suit this setting: they query the integrand only at a fixed point set and need no gradients. When the $n$ points serve as a design matrix $X\in\mathbb{R}^{n\times d}$ for a feature map, however, computing $Ψ(X)^\top v$ or $Ψ(X)w$ for an elementwise nonlinearity $Ψ$ costs $O(nd)$ time and memory for any standard quasi-Monte Carlo point set. We study subgroup rank-1 lattices, whose Korobov generator $(1,t,\dots,t^{d-1})$ uses a scalar $t$ of fixed multiplicative order $m$. Splitting $\mathbb{F}_n^\times$ into cosets of $\langle t\rangle$ reduces both maps to short cyclic correlations evaluated by FFT, giving exact results for arbitrary $Ψ$ in $O(n\log m)$ time and $O(n)$ memory, without forming $X$. Since fixing $m$ falls outside classical component-by-component theory, we prove convergence directly: via resultants with the cyclotomic polynomial $Φ_m$, the squared worst-case error in the Korobov space decays as $O(n^{-(α-1)/(m-1)})$ for prime $m\ge d+1$, and this threshold is exact. Using the splitting of $n$ in $\mathbb{Q}(ζ_m)$, averaging over the $m-1$ admissible generators improves the constant by a factor $Θ(m-1)$. Empirically, the subgroup lattice beats Gaussian and orthogonal random features and scrambled Sobol' and Halton points in 49 of 54 synthetic kernel-estimation settings and all 45 softmax-attention settings on nine real datasets, and builds a sample set with $d=2048$, $n\approx4.1\times10^7$ in 2.3 ms.
