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KernelMatmul: Scaling Gaussian Processes to Large Time Series
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Introduces KernelMatmul, a GPU CUDA kernel for on-the-fly, cutoff-sparsified kernel matrix multiplication that lets conjugate-gradient GP inference scale linearly in the number of time series samples.
Problem
Exact Gaussian process regression on large time series is limited by cubic time and quadratic memory costs of kernel matrix inference.
Approach
Combine conjugate-gradient GP inference with on-the-fly kernel matrix computation in a custom CUDA kernel and a cutoff-based sparsity approximation, with the cutoff derived from an RBF mass-based error bound (Theorem 1), giving linear time and memory in N.
Key results
- KernelMatmul is up to 4x faster than baseline methods on low-to-medium RHS configurations and achieves ~9x over KeOps dense where cuBLAS runs out of memory.
- For cutoff error epsilon <= 1e-5 sparse solutions are indistinguishable from dense solutions in residual and test MSE.
- KernelMatmul achieved significantly lower test MASE than SKI and VNNGP on all three Monash datasets (London Smart Meters, Solar, Traffic).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use KernelMatmul for kernel matrix multiplication instead of KeOps | Kernel matrix multiplication · Tested configurations | runtime | Helped |
| Use KernelMatmul instead of baseline methods | Kernel matrix multiplication · Low to medium RHS columns | speedup: up to 4 times faster | Helped |
| Use KernelMatmul instead of KeOps dense | Kernel matrix multiplication · Dense, N=2^17, high RHS columns | speedup: 9-fold improvement | Helped |
| Use KernelMatmul with cutoff-based sparsity | Gaussian process forecasting · London Smart Meters | MASE | Helped |
| Use KernelMatmul with cutoff-based sparsity | Gaussian process forecasting · Solar | MASE | Helped |
| Use KernelMatmul with cutoff-based sparsity | Gaussian process forecasting · Traffic | MASE | Helped |
| Set sparsity approximation ϵ to 10^-5 or lower | KernelMatmul · Synthetic spectral-kernel time series | test MSE: ϵ ≤ 10−5 | No clear effect |
| Increase ϵ beyond the range where the sparse-solution residual exceeds the GPyTorch tolerance | KernelMatmul · Synthetic spectral-kernel time series | test MSE | Hurt |
| Use KernelMatmul instead of SKI | Gaussian process forecasting · London Smart Meters, Solar, and Traffic | MASE | Helped |
| Use KernelMatmul instead of VNNGP | Gaussian process forecasting · London Smart Meters, Solar, and Traffic | MASE | Helped |
Surprising findings
- On-the-fly kernel computation outperforms cuBLAS dense matmul on large dense matrices with few RHS columns because the workload is memory-bound.
- SKI is slightly faster per trial than KernelMatmul on the tested kernel size, though it scales worse.
Limitations
- Sparsity approximation error grows with lengthscale and for larger epsilon; spectral kernel cutoff is only approximately controlled by the RBF-derived formula.
- Preconditioning in GPyTorch does not account for the introduced sparsity.
- Deep GPs and compactly supported kernels are not covered by the approach.
Experimental setup
Datasets: London Smart Meters Solar Traffic Monash Forecasting Repository
Benchmarks: kernel matrix multiplication benchmark (N up to 2^17) Monash London Smart Meters Monash Solar Monash Traffic
Metrics: wallclock time residual test MSE MASE seconds per trial
Baselines: naive dense matmul (cuBLAS) cuSPARSE BSR KeOps SKI VNNGP
Builds on: GPyTorch conjugate gradient GP inference (Gardner et al. 2018) KeOps structured kernel interpolation VNNGP
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