SkyGP-Fast: Real-Time GP Learning

• SkyGP-Fast is a streaming, online Gaussian process framework that uses lightweight expert ensembles to achieve low-latency updates via efficient rank-one Cholesky modifications.
• It adaptively allocates data to experts using a kernel-induced distance metric, ensuring robust statistical calibration and effective handling of nonstationary data.
• The framework excels in real-time, safety-critical applications like robotics and adaptive control by maintaining scalability and rapid inference in rapidly evolving environments.

SkyGP-Fast is an algorithmic framework for streaming, online Gaussian process (GP) learning, prioritized for computational efficiency in real-time and safety-critical applications. As part of the Streaming Kernel-induced Progressively generated Gaussian Process expert framework ("SkyGP"), SkyGP-Fast addresses the scalability constraints of online GP inference for large and rapidly evolving datasets by distributing the computational load across a dynamically maintained ensemble of lightweight GP "experts". The distinguishing feature of SkyGP-Fast is its avoidance of expensive data replacement and recomputation steps; instead, it guarantees low-latency updates via rank-one Cholesky modifications, ensuring time complexity per update never exceeds $\mathcal{O}(N^2)$ per expert. This enables application to domains including closed-loop control and robot learning where strict real-time guarantees are required (Yang et al., 5 Aug 2025).

1. Framework Architecture and Update Principles

The core architecture of SkyGP-Fast is a streaming, distributed ensemble of independent GP experts, each associated with a representative "center" in the feature space, and each processing a bounded window of local data. As new datapoints $(x^k, y^k)$ arrive in stream:

• An incremental update rule maintains the center $c_i^k$ of each expert (with $k$ the time index):

$c_i^k = \frac{(k-1) \, c_i^{k-1} + x^k}{k}$

• For each datapoint, a kernel-induced distance $d_i^k(c_i^k, x^k) = 1/\kappa(c_i^k, x^k)$ governs the assignment of data to experts, where $\kappa(\cdot, \cdot)$ is the kernel function.
• Assignment of points is adaptively managed using a dynamic, windowed allocation algorithm. This adaptive window size $W$ is computed based on recent data pairs and kernel similarity metrics.

Upon reaching capacity (maximal number of datapoints per expert), SkyGP-Fast appends a new expert, in contrast to the SkyGP-Dense variant, which triggers a costly replacement update and full Cholesky recomputation ($\mathcal{O}(N^3)$). Each update in SkyGP-Fast therefore is computationally bounded by a single rank-one Cholesky modification per expert.

2. Data Allocation, Expert Generation, and Aggregation

SkyGP-Fast implements a hierarchical data allocation protocol based on the kernel-induced distance metric to manage which expert ingests new data. The detailed data allocation pseudocode, termed "Expert Localization with an Adaptive Window" (Algorithm 1 in (Yang et al., 5 Aug 2025)), leverages previous pairwise data and similarity information to determine the optimal target expert or to instantiate a new expert when all existing ones are saturated.

Aggregated predictions from multiple experts are computed using ensemble aggregation methods:

• The ensemble mean is given by

$$\tilde{\mu}(x) = \sum_{i \in N} \omega_i(x) \mu_i(x)$$

• The variance is aggregated analogously, with weights $\omega_i(x)$ set according to the chosen committee mechanism (mixture-of-experts (MoE), product-of-experts (PoE), or robust Bayesian committee machine (rBCM)).

This ensemble mechanism ensures both statistical calibration and robustness in the presence of streaming, nonstationary data.

3. Efficient Online Inference and Computational Analysis

SkyGP-Fast achieves a marked reduction in computational complexity by constraining each inference step to rank-one Cholesky factor updates (costing $\mathcal{O}(N^2)$), rather than requiring recomputation of Cholesky factors across entire data buffers as in competing approaches. In contrast, the SkyGP-Dense variant supports selective data replacement driven by an event-triggered update law, which, while maximally utilizing local data, incurs $\mathcal{O}(N^3)$ recomputation costs.

Performance evaluations across regression benchmarks (e.g., SARCOS, PUMA, and ELECTRIC datasets) demonstrate that SkyGP-Fast achieves:

• Update latencies of $\sim0.04$ seconds and prediction times as low as $0.16$ seconds per expert on the SARCOS dataset.
• When compared to baselines (e.g., ISSGP), SkyGP-Fast offers significantly lower latencies; ISSGP required up to $18$ seconds for prediction and $7$ seconds for updates, making it impractical for real-time scenarios.
• Although in some cases the SMSE and MSLL metrics for certain baselines were competitive, SkyGP-Fast's update and prediction speed is decisive in streaming and safety-critical contexts.

In closed-loop control tasks for nonlinear dynamical systems and Euler–Lagrange systems, SkyGP-Fast provided sufficiently fast predictions for the learning-based controller to deliver safe and accurate trajectory tracking, validated via Monte Carlo experiments with lower tracking error distributions.

4. Application Domains: Safety-Critical Control and Streaming Decision Problems

SkyGP-Fast is specifically suited to dynamical systems where computational frugality and real-time response are critical, such as:

• Autonomous underwater vehicles and aerial platforms requiring continuous online model updates for robust trajectory planning.
• Healthcare robotics, which must adapt to ongoing interaction data streams while sustaining safe operation.
• Broad classes of adaptive control systems facing nonstationary environments or abrupt regime changes ("jump data").

SkyGP-Fast's capability to maintain statistical performance while updating rapidly is especially relevant in applications where safety and reactivity to environmental changes are paramount.

5. Limitations and Prospects for Future Development

Several limitations are acknowledged in the current formulation of SkyGP-Fast:

• The center update mechanism is an incremental mean, which may not accurately reflect complex or multi-modal data distributions (such as ring-shaped data manifolds).
• The reliance on a dynamic list or table for expert management may result in suboptimal adaptation to sudden, large shifts in the data distribution, potentially slowing response in the face of "jump data".
• Partitioning strategies remain largely local; future improvements could involve more sophisticated nonparametric clustering or adaptive expert allocation strategies for multimodal or high-variance data.

Prospective future research includes:

• Enhanced center update rules to handle intricate or non-convex distributions.
• Mechanisms for dynamic expert merging, splitting, or recycling, particularly for abrupt or transient data regime transitions.
• Exploring alternative predictive aggregation schemes and adaptive memory-management to further optimize efficiency in resource-constrained environments.

6. Summary Table: Features and Trade-Offs

Variant	Update Complexity	Data Replacement	Suitability
SkyGP-Fast	$\mathcal{O}(N^2)$	No	Real-time, streaming control
SkyGP-Dense	$\mathcal{O}(N^3)$	Yes	Maximum accuracy, static

SkyGP-Fast thus occupies a design space oriented toward real-time deployment, emphasizing update speed and bounded computation, with some sacrifice in terms of data utilization efficiency relative to methods that optimize local accuracy via exhaustive data replacement.

7. Significance in the Context of Online Learning and Control

The introduction of SkyGP-Fast presents a scalable, computationally tractable paradigm for Gaussian process inference in the streaming setting. By structuring GP inference around dynamically allocated, computationally efficient experts, SkyGP-Fast enables online learning and control with guarantees on latency while inheriting calibration properties of GPs. Its ensemble aggregation structure facilitates robust handling of nonstationary data. While limitations exist in managing abrupt or highly complex data regime shifts, the proposed framework forms a foundation upon which more intricate adaptive strategies may be developed to further extend the method's applicability and performance (Yang et al., 5 Aug 2025).