Deep Gaussian Processes for Regression using Approximate Expectation Propagation (1602.04133v1)

Abstract: Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.

• The paper presents a novel Bayesian scheme applying deep Gaussian processes to regression through approximate expectation propagation for efficient inference.
• It extends Probabilistic Backpropagation to accommodate non-linear mappings, ensuring scalable, accurate, and well-calibrated uncertainty estimates.
• Extensive experiments show that the proposed method outperforms traditional GPs and state-of-the-art Bayesian neural networks on multiple real-world datasets.

An Analysis of Deep Gaussian Processes for Regression Using Approximate Expectation Propagation

The discussed paper presents an innovative approach to leveraging Deep Gaussian Processes (DGPs) in regression tasks. Unlike traditional shallow Gaussian Processes (GPs), DGPs provide a hierarchical, multi-layer structure that enhances flexibility and predictive capabilities by resembling infinitely wide neural networks with multiple hidden layers. This architecture effectively uses GPs to parameterize each layer, thereby maintaining nonparametric properties and producing well-calibrated uncertainty estimates.

Core Contributions and Methodological Advances

The paper introduces a novel Bayesian learning scheme to apply DGPs to medium and large-scale regression problems. The center of this approach is the use of an approximate Expectation Propagation (EP) process combined with a stochastic variant denoted as SEP to efficiently handle the complexity of DGPs. The authors extend the Probabilistic Backpropagation (PBP) algorithm to accommodate non-linear mappings within this structure, ensuring computational efficiency and scalability. This approach allows for an adaptable and near-exact moment matching technique to estimate the posterior distributions in a DGP, essential for effective inference in such complex models.

Performance Evaluation and Results

The authors conducted extensive experiments across eleven real-world datasets to evaluate their proposed method. The results consistently showed superior performance of DGPs using the new framework over traditional GP regression models. The DGPs also outperformed state-of-the-art approaches for Bayesian neural networks (BNNs), not only providing enhanced accuracy but also delivering more reliable estimates of uncertainty in predictions. This performance is attributed to the structural depth of DGPs and their ability to model non-Gaussian output distributions, which are often encountered in real-world data scenarios.

Implications and Future Directions

The paper's findings suggest significant implications for both the theoretical understanding and practical deployment of probabilistic models in machine learning. The step toward scalable Bayesian deep learning through DGPs opens avenues in applications demanding robust predictive models with uncertainty quantification, such as climate modeling, financial forecasting, and more.

Despite these advancements, challenges remain, especially concerning classification tasks where the improvement margins over shallow GPs are minimal. Future research could explore addressing these challenges through more refined initialization strategies or adopting non-diagonal Gaussian approximations in multiclass classification contexts.

Conclusion

This paper is a valuable contribution to the field of machine learning, particularly within Bayesian regression frameworks. By innovating on the inference methods for DGPs and demonstrating their practical applicability through comprehensive empirical evaluation, it sets a benchmark for future research exploring the depths of probabilistic modeling in neural architectures. The work uniquely combines elements from GPs and neural networks, advocating for further exploration of hybrid models that capitalize on the strengths of both worlds.