Local Gaussian process approximation for large computer experiments (1303.0383v4)

Abstract: We provide a new approach to approximate emulation of large computer experiments. By focusing expressly on desirable properties of the predictive equations, we derive a family of local sequential design schemes that dynamically define the support of a Gaussian process predictor based on a local subset of the data. We further derive expressions for fast sequential updating of all needed quantities as the local designs are built-up iteratively. Then we show how independent application of our local design strategy across the elements of a vast predictive grid facilitates a trivially parallel implementation. The end result is a global predictor able to take advantage of modern multicore architectures, while at the same time allowing for a nonstationary modeling feature as a bonus. We demonstrate our method on two examples utilizing designs sized in the thousands, and tens of thousands of data points. Comparisons are made to the method of compactly supported covariances.

• The paper introduces a local GP approach that reduces computational complexity by focusing on local data subsets.
• It develops a sequential design strategy to optimize predictive accuracy using criteria like MSPE and ALC.
• The method supports efficient parallel computation and nonstationary modeling for scalable computer experiments.

Local Gaussian Process Approximation for Large Computer Experiments

In the paper titled "Local Gaussian Process Approximation for Large Computer Experiments," the authors, Robert B. Gramacy and Daniel W. Apley, present a novel approach to efficiently emulate large computer experiments using Gaussian processes (GPs). The emphasis is on overcoming the computational and stationarity challenges traditionally faced when employing GPs for large datasets.

Summary and Key Contributions

The paper introduces a method that adapts traditional GP modeling by focusing on local subsets of data. This local approach facilitates handling large datasets while maintaining computational feasibility. The authors present a local sequential design scheme that selects a subset of data dynamically, which supports a GP predictor, offering a principled manner to achieve efficient emulation. They also derive expressions for the fast sequential updating of models as these local designs iteratively build up. Importantly, the method supports parallel computation, thereby leveraging modern multicore architectures effectively.

Two empirical examples presented in the paper illustrate the viability of this method. The authors compare the performance of their approach against models using compactly supported covariance (CSC) functions. The local design strategy across various predictive grid elements allows for independent implementation, which results in a trivially parallel structure. The methodology also accommodates nonstationary modeling by allowing for variability in covariance structures, unlike traditional stationary assumptions in GP modeling.

Technical Insights

1. Localized Predictions and Computation: By focusing on predicting locally around a target input, the authors reduce the computational burden associated with inverting large covariance matrices. This method effectively utilizes the insight that distant data points contribute insignificantly to the local prediction.
2. Sequential Design and Active Learning: The paper extends active learning principles by crafting a local sequential design. This involves choosing new design points based on criteria such as mean-squared predictive error (MSPE) or active learning Cohn (ALC) heuristics, ensuring that new points optimize the prediction accuracy locally.
3. Nonstationary Modeling: The ability to handle nonstationary data through localized parameter inference is a significant leap forward. The method facilitates dynamic updating of covariance structures, which can account for varying spatial correlations within the data.
4. Parallelizability: The independence between local designs for different prediction points allows the entire process to be parallelized easily. This fits seamlessly with modern distributed computing environments, which is crucial given the size of the datasets considered.

Empirical Evaluation

The paper's empirical section demonstrates that the proposed local GP approximation scales better than traditional full GP models, especially as data size increases. The experiments highlight the method's efficiency, accuracy, and robustness in scenarios where full GP models would be computationally prohibitive. For example, the authors report computational times orders of magnitude faster than CSC methods while maintaining or improving predictive accuracy.

Implications and Future Directions

The research establishes a robust framework for addressing the challenges of scalability and nonstationarity in GP modeling for large-scale data. The combination of local modeling with parallel computing prepares the methodology for real-world applications in fields requiring high-dimensional data analysis, such as climate modeling and engineering simulations.

Future research directions could involve refining the selection criteria for local design points, exploring alternative local covariance structures, and integrating more sophisticated machine learning techniques that may further enhance the predictive power or reduce computation time. Additionally, further exploration into the nonstationary capabilities of the model could provide more insights into applications with inherently nonstationary data distributions.

In conclusion, the paper makes a valuable contribution to the field of computer experiment emulation by presenting a scalable, efficient, and theoretically grounded method for large-scale GP emulation, opening pathways for advanced statistical modeling in extensive data environments.