Warm-Starting Iterative Gaussian Processes for Faster Sequential Inference

The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization, adaptive and active sensing, and sequential detection and decision-making. By organizing these advances from a signal-processing perspective, we intend to equip practitioners with practical tools and a coherent roadmap for deploying sequential GP models in real-world systems.

Gaussian process (GP) inference in Python is dominated by libraries such as GPyTorch and GPflow, which are built on deep-learning frameworks and inherit their dispatch overhead and dependency footprint. We present LightGP, a dependency-free C++17 library for GP regression with Python bindings, supporting Apple Metal and NVIDIA CUDA backends alongside tuned CPU paths via Apple Accelerate and OpenBLAS. LightGP provides four inference paths -- exact Cholesky, matrix-free conjugate gradients, sparse variational free energy, and structured kernel interpolation with FFT -- covering problems from $N{=}100$ to $N{=}500{,}000$. On an Apple M4, LightGP CPU is 2.6--8.7$\times$ faster than GPyTorch CPU for exact GP and ${\sim}1.5\times$ faster for sparse GP at every scale tested. On an NVIDIA RTX~3060, LightGP CUDA is 2.3--6.7$\times$ faster than GPyTorch CUDA for exact GP up to $N{=}2{,}048$, with GPyTorch closing the gap at $N{=}4{,}096$. A fused matrix-free kernel-vector product on Metal achieves 32$\times$ over the explicit path at $N{=}20{,}000$ with $O(N)$ memory, and an FFT-accelerated SKI matvec via Accelerate vDSP runs in sub-millisecond time at $N{=}200{,}000$. LightGP compiles as a single static library with zero external dependencies and is installable via \texttt{pip install lightgp

Gaussian Processes (GPs) are widely used as surrogates for black-box functions in sequential decision-making problems such as Bayesian optimization (BO), level set estimation (LSE), and Bayesian active learning (BAL). GP performance critically depends on kernels, and standard kernels can lead to suboptimal decisions under misspecification. To address this, we introduce HACK GPs (Hedge Adaptive Cumulative Kernels), a method that views kernel selection as an online learning with expert advice problem. HACK treats each candidate kernel as a GP "expert" and updates a distribution over experts online using AdaHedge, based on a loss received as a proxy for their ability to fit the function and align with the task objective. We provide two variants of HACK: (i) Mixture of Gaussians (MoG) and (ii) categorical sampling. We establish general guarantees showing that, under a loss-gap condition, the weight concentrates on the best kernel and the resulting acquisition function is close to that of the best expert. Empirically, we observe robust performance across BO, LSE, and BAL compared to standard kernels such as Squared Exponential and Matern-5/2, as well as simple ensemble baselines.