Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection (2407.18707v1)

Abstract: Infinitely wide or deep neural networks (NNs) with independent and identically distributed (i.i.d.) parameters have been shown to be equivalent to Gaussian processes. Because of the favorable properties of Gaussian processes, this equivalence is commonly employed to analyze neural networks and has led to various breakthroughs over the years. However, neural networks and Gaussian processes are equivalent only in the limit; in the finite case there are currently no methods available to approximate a trained neural network with a Gaussian model with bounds on the approximation error. In this work, we present an algorithmic framework to approximate a neural network of finite width and depth, and with not necessarily i.i.d. parameters, with a mixture of Gaussian processes with error bounds on the approximation error. In particular, we consider the Wasserstein distance to quantify the closeness between probabilistic models and, by relying on tools from optimal transport and Gaussian processes, we iteratively approximate the output distribution of each layer of the neural network as a mixture of Gaussian processes. Crucially, for any NN and $\epsilon >0$ our approach is able to return a mixture of Gaussian processes that is $\epsilon$-close to the NN at a finite set of input points. Furthermore, we rely on the differentiability of the resulting error bound to show how our approach can be employed to tune the parameters of a NN to mimic the functional behavior of a given Gaussian process, e.g., for prior selection in the context of Bayesian inference. We empirically investigate the effectiveness of our results on both regression and classification problems with various neural network architectures. Our experiments highlight how our results can represent an important step towards understanding neural network predictions and formally quantifying their uncertainty.

• The paper presents an iterative algorithm that approximates each NN layer with a Gaussian mixture, achieving provable error bounds.
• It utilizes Wasserstein distance to compress and propagate mixture models through layers, ensuring convergence with increased components.
• Empirical results validate that even small GMMs provide accurate approximations, while the approach refines prior selection for Bayesian inference.

Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection

The paper "Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection" by Steven Adams, Andrea Patanè, Morteza Lahijanian, and Luca Laurenti introduces a novel approach to approximate finite neural networks (NNs) with mixtures of Gaussian processes (GMMs), with formal guarantees on the approximation error. The research addresses the challenge that while infinitely wide or deep NNs are known to converge to Gaussian processes (GPs) in the limit, no methods so far provided guaranteed approximations for finite NNs.

Contributions

The principal contributions of the paper can be summarized as follows:

1. Algorithmic Framework: The authors present an iterative algorithmic framework to approximate the output distribution of each layer of a finite neural network with a GMM. This approach relies on the Wasserstein distance to quantify the proximity between distributions, combining techniques from optimal transport (OT) theory and Gaussian processes.
2. Error Bounds: The framework includes error bounds on the approximation, ensuring that the GMM approximations are $\epsilon$-close to the original neural network outputs for any epsilon greater than zero.
3. Empirical Validation: Experiments on various regression and classification tasks with different neural network architectures empirically validate the proposed method. The empirical results demonstrate that even a relatively small number of Gaussian components can suffice for accurate approximations.
4. Prior Selection: Additionally, the framework allows the tuning of neural network parameters to mimic desired GP behavior, significantly impacting Bayesian inference. This prior selection mechanism enhances posterior performance and addresses the critical issue of encoding functional prior knowledge into NNs.

Methodology

The methodology is built on representing neural network layers iteratively as mixtures of Gaussian processes. The key steps include:

• Initialization: The first step involves computing the output distribution of the first layer using the input and layer parameters, which collectively form a GMM.
• Signature Operation: Each layer's output distribution is approximated by a discrete distribution (called a signature), simplifying the propagation through layers by enabling exact integration.
• Compression: To manage computational complexity, a compression step reduces the size of the GMM, maintaining error bounds on the distance between the original and compressed models.
• Iteration: These steps are iteratively applied layer-by-layer through the neural network until the final output distribution is obtained.

Theoretical Insights

The theoretical analysis includes:

• Wasserstein Distance Bounds: The authors derive bounds on the Wasserstein distance for each approximation step, effectively constraining the propagation of errors through the network layers.
• Convergence Guarantees: By increasing the number of components in the GMM and the support of the signature distributions, the approximation error can be made arbitrarily small.
• Closed-Form Solutions: The paper presents closed-form solutions for the Wasserstein distance in several key steps, notably for GMM compression and signature approximation.

Experimental Results

Empirical evaluations using datasets such as MNIST, CIFAR-10, and various UCI datasets demonstrate the practical utility of the framework. Notably, small GMMs already provide accurate approximations, underscoring the efficiency of the proposed method. For instance, it was shown that even a GMM with 10 components suffices for high accuracy in approximating several neural network architectures.

Implications and Future Directions

The implications of this research are manifold:

• Theoretical Advancement: The work provides a significant theoretical foundation for approximating finite neural networks with probabilistic guarantees, filling a notable gap in the literature.
• Practical Applications: In practical terms, the framework enhances the interpretability and uncertainty quantification of neural network predictions by approximating outputs with GMs.
• Prior Selection: The ability to encode functional priors presents a refined approach to Bayesian neural networks, potentially improving the performance of machine learning models in various domains.

Future research may explore extending this approximation technique to other types of neural network structures, such as recurrent neural networks, or applying the framework to robustness verification and adversarial training scenarios.

In conclusion, the paper presents a comprehensive and theoretically sound framework to approximate finite NNs with GMMs, accompanied by formal error bounds and practical applications in uncertainty quantification and prior selection. The detailed derivation and empirical support indicate a significant advancement in the intersection of neural networks and probabilistic modeling.