Blackbox Matrix Multiplication for GP Scaling

• BBMM is an algorithmic paradigm that reformulates Gaussian process inference as blackbox matrix-matrix multiplications using GPU acceleration, reducing computational barriers.
• It employs batched conjugate gradient solvers, pivoted-Cholesky preconditioning, and stochastic estimators to efficiently compute log-determinants and trace terms.
• Variants like AltBBMM further optimize performance for large-scale, low-noise settings, achieving significant speedups with minimal accuracy loss.

Blackbox Matrix-Matrix Multiplication (BBMM) is an algorithmic paradigm for scaling Gaussian process (GP) inference and learning by reformulating the core linear algebraic operations—matrix solves, log-determinants, and traces—as blackbox matrix-matrix multiplications. By exploiting batched conjugate gradient (mBCG) solves, pivoted-Cholesky preconditioning, and GPU-accelerated routines, BBMM reduces the cubic time and quadratic memory barriers of classic GP methods, enabling exact inference and learning in high-dimensional and large-$n$ settings. BBMM decouples GP inference from explicit matrix storage and leverages computational primitives extensible to structured, sparse, or approximate kernel methods, aligning exact Bayesian nonparametrics with modern GPU hardware and practical dataset sizes (Gardner et al., 2018, Sun et al., 2021).

1. Gaussian Process Inference: Computational Bottlenecks

In supervised learning with a zero-mean GP prior, training data $X \in \mathbb{R}^{n \times d}$ and $y \in \mathbb{R}^n$ define the covariance matrix $K = K(X,X;\theta)$, possibly regularized as $\widetilde{K} = K + \sigma^2 I$. Standard inference requires:

• Solving $\widetilde{K} \alpha = y$ for predictive means,
• Evaluating log-determinants $\log |\widetilde{K}|$ for marginal likelihood,
• Estimating trace terms $$\operatorname{tr}[\widetilde{K}^{-1} \frac{\partial \widetilde{K}}{\partial \theta}]$$ for hyperparameter gradients.

Direct approaches via Cholesky decomposition incur $O(n^3)$ time and $O(n^2)$ memory. These prohibitive scalings have historically constrained exact GPs to datasets with $X \in \mathbb{R}^{n \times d}$0 (Gardner et al., 2018). BBMM addresses these challenges by re-expressing inference as a sequence of blackbox matrix-matrix multiplications and stochastic estimators, vastly improving scalability.

2. BBMM Fundamentals: Algorithmic Structure

BBMM operates under the assumption of access to routines for

• $X \in \mathbb{R}^{n \times d}$1,
• $X \in \mathbb{R}^{n \times d}$2,

for arbitrary matrices $X \in \mathbb{R}^{n \times d}$3. The principal features include:

• Batched Conjugate Gradient (mBCG): Simultaneously solves $X \in \mathbb{R}^{n \times d}$4 for multiple right-hand sides $X \in \mathbb{R}^{n \times d}$5, stacking $X \in \mathbb{R}^{n \times d}$6 and probe vectors $X \in \mathbb{R}^{n \times d}$7.
• Pivoted-Cholesky Preconditioning: Builds a rank-$X \in \mathbb{R}^{n \times d}$8 preconditioner $X \in \mathbb{R}^{n \times d}$9, improving mBCG convergence.
• Stochastic Estimators: Estimates log-determinants and traces via Hutchinson's trace estimator and Stochastic Lanczos Quadrature (SLQ) utilizing the Krylov tridiagonalizations from mBCG.

The overall workflow replaces $y \in \mathbb{R}^n$0 Cholesky with $y \in \mathbb{R}^n$1-scaling matrix-matrix multiplications and converges in a small number of mBCG iterations for well-preconditioned systems (Gardner et al., 2018).

3. Practical Variants: AltBBMM for Large-Scale Low-Noise Settings

"AltBBMM" is a variant tailored for large-scale, low-noise learning tasks, particularly molecular energy prediction with the MOB-ML framework (Sun et al., 2021). It introduces several modifications to the standard BBMM scheme:

• Block Conjugate Gradient (BCG): Solves for all right-hand sides in one Krylov space, accelerating convergence due to richer subspace expansions.
• Symmetric Preconditioning: Transforms the system to $y \in \mathbb{R}^n$2, further enhancing numerical stability, especially at low regularization $y \in \mathbb{R}^n$3.
• Double-Precision Arithmetic: Avoids stagnation in low-noise regimes.
• Hyperparameter Tuning on Subsets: Optimizes kernel parameters on a small subset (e.g., 50 molecules), then applies them to the full dataset, eliminating expensive mBCG hyperloops.

Kernel-matrix multiplications are executed in 4096×4096 batches, with dynamic GPU scheduling. A noise "jitter" $y \in \mathbb{R}^n$4 is always added to prevent singularity. AltBBMM achieves a fourfold empirical speedup with minimal accuracy reduction (∼0.01–0.02 kcal/mol) in benchmark molecular regression (Sun et al., 2021).

4. Complexity Analysis

The computational and memory complexities of BBMM and AltBBMM are as follows:

Algorithm	Per-iteration Cost	Preconditioner Build	Overall Scaling	Comments
BBMM	$y \in \mathbb{R}^n$5 (block-size $y \in \mathbb{R}^n$6)	$y \in \mathbb{R}^n$7 (rank-$y \in \mathbb{R}^n$8)	$y \in \mathbb{R}^n$9	mBCG looped over hyperparam steps
AltBBMM	Fewer $K = K(X,X;\theta)$0 BCG iterations	Single $K = K(X,X;\theta)$1 build	$K = K(X,X;\theta)$2, 4× faster	Single block solve, fixed hyperparams

As $K = K(X,X;\theta)$3 and BCG iterations $K = K(X,X;\theta)$4, the overall time and memory scale quadratically (or better with structured kernels) (Gardner et al., 2018, Sun et al., 2021).

5. Empirical Performance and Applications

Extensive experiments in chemical physics demonstrate the scaling and accuracy of BBMM and AltBBMM. For MOB-ML molecular energy learning:

• BBMM and AltBBMM enable training on 6500 molecules ($K = K(X,X;\theta)$51 million pair energies), a $K = K(X,X;\theta)$6 expansion over prior limits.
• Mean absolute error (MAE) and wall-clock times:

Algorithm	QM7b-T MAE (kcal/mol)	GDB-13-T MAE/7HA (kcal/mol)	Time (hrs)
BBMM	0.185	0.490	26.52
AltBBMM	0.193	0.493	6.24

AltBBMM achieves nearly the same out-of-sample accuracy as BBMM with a fourfold reduction in training time (Sun et al., 2021). Both schemes preserve state-of-the-art efficiency in the low-data regime and extend it to the million-pair regime, outperforming previous learning methods on molecular energies.

6. Extensions and Generalizations

BBMM's reliance on blackbox matrix-matrix multiplication routines makes it extensible to structured kernel approximations (e.g., SKI/KISS-GP), sparse methods (e.g., SGPR), and scalable exact GPs. Implementations such as GPyTorch leverage batch tensor operations and GPU acceleration via PyTorch, yielding up to $K = K(X,X;\theta)$7 wall-clock speedups over CPU Cholesky for $K = K(X,X;\theta)$8, and strong gains for scalable approximations at $K = K(X,X;\theta)$9 (Gardner et al., 2018). Any kernel admitting fast $\widetilde{K} = K + \sigma^2 I$0 and $\widetilde{K} = K + \sigma^2 I$1 can integrate with the BBMM/mBCG pipeline without bespoke solvers or differentiation code.

7. Implications for Large-Scale Gaussian Processes

BBMM and AltBBMM compress the computational gap between exact GPs and their approximate or sparse variants. By reducing inference and learning cost from $\widetilde{K} = K + \sigma^2 I$2 to $\widetilde{K} = K + \sigma^2 I$3 or better, these schemes make exact Bayesian nonparametric learning feasible at scale, especially in domains demanding high-fidelity uncertainty quantification (e.g., molecular simulation, chemical physics). Unlike low-rank approximations that may deteriorate model calibration, BBMM-based methods retain the "gold-standard" predictive uncertainty characteristic of GPs, offering a practical route to trustworthy modeling as dataset sizes approach and exceed $\widetilde{K} = K + \sigma^2 I$4 (Sun et al., 2021).