Adjusting Model Size in Continual Gaussian Processes: How Big is Big Enough?

TL;DR

This work tackles capacity selection in continual Gaussian Process regression by introducing Vegas Inducing Point Selection (VIPS), an online method that grows the inducing set adaptively using a single pre-set hyperparameter. By deriving an online LML upper bound and accompanying approximation guarantees, VIPS stops growing when capacity suffices to retain near-optimal predictive performance. Empirical results on streaming UCI benchmarks and magnetic-field robotics show VIPS closely matches full-batch GP accuracy while using far fewer inducing points and without dataset-specific tuning, outperforming competing adaptive methods. The approach offers a principled, resource-efficient framework for online Bayesian nonparametric learning with potential implications for adaptive neural networks.

Abstract

Many machine learning models require setting a parameter that controls their size before training, e.g. number of neurons in DNNs, or inducing points in GPs. Increasing capacity typically improves performance until all the information from the dataset is captured. After this point, computational cost keeps increasing, without improved performance. This leads to the question "How big is big enough?" We investigate this problem for Gaussian processes (single-layer neural networks) in continual learning. Here, data becomes available incrementally, and the final dataset size will therefore not be known before training, preventing the use of heuristics for setting a fixed model size. We develop a method to automatically adjust model size while maintaining near-optimal performance. Our experimental procedure follows the constraint that any hyperparameters must be set without seeing dataset properties, and we show that our method performs well across diverse datasets without the need to adjust its hyperparameter, showing it requires less tuning than others.

Figures (13)

• Figure 1: Three continual learning scenarios with different capacity requirements. Top: Three consecutive batches for 1) a growing input space, 2) i.i.d. samples from a uniform distribution, and 3) narrow-range samples with occasional outliers. Bottom: Number of inducing points selected using the VIPS algorithm at each batch. When selecting model size with our method, we observed: 1) a linear increase, 2) after initial training, we see a halt in growth, and 3) a low model size until it encounters outliers.
• Figure 2: (a) Performance comparison of fixed memory approaches (blue curves with $M = 10, 20, 30$ inducing points) and VIPS, with $M$ (shown at the top) inducing points at each batch. (b) (log) Time taken to train the online GP model on the "naval" dataset divided into 20 batches with fixed size (oracle: $M=100$, heuristic: $M=1000$) and VIPS, our adaptive method.
• Figure 3: Number of datasets where each method achieves minimal model size at different outlier allowances. A "win" is assigned to the dataset with the smallest model which satisfies the (a) RMSE and (b) NLPD thresholds of 10%. Counts represent absolute wins. Higher counts indicate better method robustness across datasets.
• Figure 4: A small robot is used to perform sequential estimation of magnetic field anomalies. The strength of the magnetic field is given by 10 $\mu \mathrm{T}$to 90 $\mu \mathrm{T}$. (a) shows the final estimate of the magnitude field learned sequentially through different paths. (b) show the outcome of learning a single path continuously with the black dots representing the chosen inducing points.
• Figure 5: Plot of the three datasets considered in Section \ref{['appendix:datatypes']}.

Theorems & Definitions (2)

• proof
• proof