GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration (1809.11165v6)

Abstract: Despite advances in scalable models, the inference tools used for Gaussian processes (GPs) have yet to fully capitalize on developments in computing hardware. We present an efficient and general approach to GP inference based on Blackbox Matrix-Matrix multiplication (BBMM). BBMM inference uses a modified batched version of the conjugate gradients algorithm to derive all terms for training and inference in a single call. BBMM reduces the asymptotic complexity of exact GP inference from $O(n^3)$ to $O(n^2)$. Adapting this algorithm to scalable approximations and complex GP models simply requires a routine for efficient matrix-matrix multiplication with the kernel and its derivative. In addition, BBMM uses a specialized preconditioner to substantially speed up convergence. In experiments we show that BBMM effectively uses GPU hardware to dramatically accelerate both exact GP inference and scalable approximations. Additionally, we provide GPyTorch, a software platform for scalable GP inference via BBMM, built on PyTorch.

• The paper introduces the Blackbox Matrix-Matrix Multiplication (BBMM) method that reduces GP inference complexity from O(n³) to O(n²) using batched conjugate gradients.
• It proposes a pivoted Cholesky preconditioner that significantly accelerates convergence and reduces the number of iterations in GP computations.
• The GPyTorch platform enables scalable, GPU-accelerated GP inference, achieving up to 20× speedups for exact models on both moderate and large datasets.

Overview of "GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration"

The paper "GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration" by Jacob R. Gardner et al. addresses the inefficiencies in inference tools commonly used for Gaussian Processes (GPs). The authors introduce the Blackbox Matrix-Matrix Multiplication (BBMM) inference method, which leverages modern GPU hardware to significantly accelerate both exact GP inference and scalable approximations.

Key Contributions

1. BBMM Inference Method: The core contribution is the BBMM inference method that reduces the asymptotic complexity of exact GP inference from $n^3$ to $n^2$. This method utilizes a batched version of the conjugate gradients algorithm (mBCG) for all terms necessary for GP training and inference.
2. Preconditioner for Speedup: The paper proposes a specialized preconditioner based on the pivoted Cholesky decomposition to enhance the convergence speed of the mBCG algorithm.
3. GPyTorch Software Platform: The authors provide GPyTorch, a software platform built on PyTorch, enabling scalable GP inference via the BBMM method.

Detailed Contributions and Results

BBMM Inference:

• The BBMM inference method computes the GP marginal log likelihood and its derivatives using matrix-matrix multiplications, which are more hardware-efficient compared to traditional Cholesky decomposition.
• This approach enables significant reductions in both time and space complexity. Specifically, it reduces the computational time from $O(n^3)$ to $O(n^2)$ and operates effectively within the memory constraints of modern GPUs.

Batched Conjugate Gradients Algorithm (mBCG):

• The mBCG algorithm iterates using multiple right-hand sides simultaneously, allowing efficient utilization of GPU hardware.
• BBMM computes required GP inference terms, including the linear solve $K^{-1}y$, the log determinant $\log|K|$, and the trace term $$\mathrm{Tr}(K^{-1} \frac{\partial K}{\partial \theta})$$, in parallel, unlike existing methods that are inherently sequential.

Preconditioning with Pivoted Cholesky Decomposition:

• The pivoted Cholesky decomposition preconditioner is efficient, theoretically sound, and tested empirically.
• It significantly accelerates the convergence of conjugate gradients, with notable reductions in the number of iterations required.
• For univariate RBF kernels, the paper derives bounds showing exponential improvement in the condition number with the rank of the pivoted Cholesky decomposition.

Performance and Comparative Analysis:

• Empirical results show substantial speedups in inference times. For instance, on datasets with up to 3000 data points, exact GPs with BBMM are up to 20 times faster than Cholesky-based approaches.
• For larger datasets up to 500,000 data points, BBMM achieves up to 15 times and 4 times speedups for SKI and SGPR frameworks, respectively.
• The results also demonstrate that BBMM achieves comparable or better test errors compared to traditional methods.

Implications and Future Work

Practical Implications:

• The BBMM method represents a significant improvement in computational efficiency for GP inference tasks, making it feasible to work with larger datasets and more complex models.
• It allows for rapid prototyping and experimentation with new GP models due to its blackbox nature, requiring only a kernel matrix multiplication routine.

Theoretical Implications:

• The paper's theoretical contributions provide a solid foundation for understanding the convergence behavior of preconditioned BBMM algorithms.
• Future research could extend these results to more complex kernel functions and multidimensional settings.

Speculation on Future Developments:

• Future work may explore integration with variational inference methods for non-Gaussian likelihoods.
• Advancements in GPU hardware are likely to further enhance the performance benefits of the BBMM approach.

Conclusion

The paper "GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration" presents a highly efficient approach to GP inference that leverages modern GPU capabilities. By reducing the complexity of exact GP inference and introducing an effective preconditioning strategy, the BBMM method significantly accelerates both exact and approximate GP models. The GPyTorch software platform facilitates scalable GP inference and positions BBMM as a robust tool for both the research community and practical applications.