Generalized Product of Experts for Automatic and Principled Fusion of Gaussian Process Predictions (1410.7827v2)

Abstract: In this work, we propose a generalized product of experts (gPoE) framework for combining the predictions of multiple probabilistic models. We identify four desirable properties that are important for scalability, expressiveness and robustness, when learning and inferring with a combination of multiple models. Through analysis and experiments, we show that gPoE of Gaussian processes (GP) have these qualities, while no other existing combination schemes satisfy all of them at the same time. The resulting GP-gPoE is highly scalable as individual GP experts can be independently learned in parallel; very expressive as the way experts are combined depends on the input rather than fixed; the combined prediction is still a valid probabilistic model with natural interpretation; and finally robust to unreliable predictions from individual experts.

• The paper introduces the gPoE framework which automatically adjusts each expert's influence based on reliability to maintain valid probabilistic predictions.
• It combines independent Gaussian process experts without joint training, preserving closed-form expressions for mean and covariance.
• Empirical evaluation on datasets like KIN40K and SARCOS shows improved SMSE and SNLP, demonstrating scalability and robust performance.

Generalized Product of Experts for Gaussian Processes

The paper "Generalized Product of Experts for Automatic and Principled Fusion of Gaussian Process Predictions" authored by Yanshuai Cao and David J. Fleet, addresses the challenges and provides solutions for combining predictions from multiple probabilistic models, specifically Gaussian processes. This paper introduces the Generalized Product of Experts (gPoE) framework, demonstrating its advantages in scalability, expressiveness, robustness, and maintaining probabilistic validity, which are essential properties in predictive model combination.

Core Concepts

The authors point out four desirable properties for effective model fusion: the ability to combine predictions without joint training or training auxiliary meta-models, an input-dependent combination mechanism, validity of the combined prediction as a probabilistic model, and automatic filtering of unreliable predictions. These properties are not collectively met by existing frameworks such as mixtures of experts, ensemble methods, or traditional product of experts (PoE). The gPoE, however, is designed to satisfy all these conditions when applied to Gaussian processes, making it a more suitable choice for complex model combinations.

Generalized Product of Experts (gPoE)

The gPoE framework extends the PoE by introducing a parameter, $\alpha_i(x)$, which adjusts each expert’s influence based on its reliability at a particular input $x$. This parameter provides the flexibility to down-weight or negate predictions from less reliable experts, remedying the typical issue in PoE where a single over-confident, yet inaccurate expert can dominate the combined model.

Mathematically, the gPoE maintains the Gaussian nature of combined predictions, offering closed analytical forms for mean and covariance calculations, akin to traditional Gaussian predictions but enhanced by reliability-adjusted weights. The reliability is assessed using the change in entropy from prior to posterior at each point, effectively capturing an expert’s knowledge gain about an input and determining its influence in the predictive model.

Empirical Evaluation

The paper reports robust empirical performance of gPoE against alternatives like bagging, mixtures of experts, and traditional PoE across datasets of varying dimensions and sizes (KIN40K, SARCOS, and UK apartment price dataset). The gPoE consistently outperformed other methods in terms of standardized mean square error (SMSE) and standardized negative log probability (SNLP). This implies that gPoE can maintain high prediction accuracy and uncertainty quantification even when combining relatively small and independently trained Gaussian process experts.

Implications and Future Directions

The introduction of gPoE provides significant practical implications, particularly in applications requiring large-scale probabilistic model fusion. It enables parallelization of model training due to independent expert learning, contributing to efficiency in computationally intensive environments. Moreover, the adaptability of expert contributions based on input points introduces a dynamic expressiveness vital for nuanced predictive tasks.

Future avenues for research could explore gPoE extensions to capture multimodal distributions, such as integrating mixtures of Gaussian processes with generalized product frameworks. Additionally, investigating other measures of reliability for Gaussian processes beyond entropy change could further enhance model fusion effectiveness and address potential mis-specification issues.

Conclusion

This work presents a sophisticated approach to combining Gaussian process predictions, addressing key limitations in existing fusion methodologies. The gPoE framework not only achieves desirable properties across scalability, expressiveness, robustness, and probabilistic validity but also sets a precedent for future innovation in predictive model combinations. The findings underscore the potential for gPoE to serve as a foundational framework in advancing probabilistic modeling and inference in various research and application domains.