Deep Gaussian Processes (1211.0358v2)

Abstract: In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples.

• The paper introduces a novel deep GP framework that layers multiple Gaussian Processes using variational inference to capture intricate data representations.
• The model employs inducing points to significantly reduce computational complexity, enhancing scalability for both sparse and dense datasets.
• Empirical results on synthetic and real-world datasets validate the framework’s ability to uncover latent structures and improve performance in tasks like motion capture and digit classification.

Insights into Deep Gaussian Processes

The paper "Deep Gaussian Processes" by Andreas C. Damianou and Neil D. Lawrence introduces an innovative extension of Gaussian Process (GP) models in the framework of deep learning. Unlike traditional shallow GPs, the proposed Deep Gaussian Processes (DGPs) consist of multiple layers of GPs, enabling the model to capture more intricate representations and mappings in the data. This paper provides a substantial contribution to the field of probabilistic modeling by leveraging recent advances in variational inference to deal with the intractabilities associated with deep hierarchical models.

Overview

The central idea of the paper is to model data as the output of a multivariate GP where the inputs to this GP are determined by another GP. This recursive structure can be extended to multiple layers, effectively making DGPs deep belief networks based on GP mappings. The inference in these models is handled through an approximate Bayesian learning framework using variational inference, which provides a strict lower bound on the marginal likelihood of the model. This lower bound is utilized for model selection - determining the optimal number of layers and nodes per layer.

Methodology and Contributions

1. Bayesian Inference via Variational Approximations: The authors adopt a fully Bayesian treatment for the DGP model. They exploit recent advancements in variational inference to marginalize the latent variables in the GP hierarchy. This approach builds upon and extends previous work by Damianou et al. (2011), allowing multiple GPs to be stacked with variational approximations. The resulting variational lower bound on the marginal likelihood of the model is critical for rigorous model selection and provides a benchmark for assessing the model's applicability.
2. Scalability and Flexibility: By introducing inducing points for augmenting the probability space of the GP priors, the complexity of inference is reduced from typical O(N^2) to O(NM), where M is the number of inducing points, making the method scalable. The model can handle both sparse and dense data effectively and allows automatic relevance determination (ARD), which helps in discovering the appropriate latent structure automatically.
3. Hierarchical Structure: The DGP constructed by the authors allows several layers of hierarchical GP mappings. Each node in the hierarchy can act both as an input for the layer below and an output for the layer above, with the observed outputs positioned at the leaves of the hierarchy. This hierarchical approach is capable of learning both local and abstract features at varying depths of the model.

Experimental Results

The paper provides a rigorous experimental validation of the proposed DGP model with both synthetic and real-world data.

1. Toy Data: The authors create a toy dataset by sampling from a three-level stack of GPs and assess the ability of the DGP to recover the original latent structure. Comparisons with stacked Isomap and PCA show that the DGP model not only discovers the correct dimensionality for each hidden layer but also recovers latent signals closer to the actual ones generated.
2. Human Motion Modeling: On a motion capture dataset from the CMU MOCAP database, the DGP model is evaluated in an unsupervised learning setting. It successfully captures the shared subspace between two interacting subjects, which is demonstrated through the learned ARD weights and latent space projections.
3. Handwritten Digit Classification: The DGP is tested on a subsampled version of the USPS handwritten digit dataset. The model's performance improves with increasing depth, validating the advantage of deeper hierarchies. The ability of the DGP to learn abstract features at higher layers is showcased through nearest neighbor classification in the latent spaces.

Implications and Future Directions

The ability of DGPs to handle small datasets effectively is of particular interest, given the common necessity for large datasets in traditional deep learning approaches. The Bayesian treatment ensures a well-regularized model, even with scarce data, providing robustness and flexibility across different tasks.

Theoretical Implications:

• The approach opens up avenues for further exploration in the probabilistic treatment of deep learning models, providing a principled way to perform model selection and capacity control.

Practical Implications:

• DGPs can enhance various applications, such as multi-task learning, where shared representations across tasks can be effectively modeled. They are also potentially powerful in handling nonstationary data or data involving structural breaks.

Future Work:

• The extension of DGP methodologies to very large datasets remains an open problem. Incorporating stochastic variational inference techniques could be a feasible solution to scale DGPs further, thereby broadening their applicability.
• Additional investigation into combining DGPs with other deep learning methods for unsupervised pre-training or guiding deep models could further enhance performance.

Conclusion

The paper by Damianou and Lawrence makes significant strides in extending Gaussian Processes to deep hierarchical models. With the strategic use of variational inference, the proposed DGP model demonstrates a robust and flexible approach to capturing complex data structures. The methodological rigor and empirical validation provided create a firm foundation for future research and development in this domain.