DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networks (2407.21176v1)

Abstract: The need for scalable and expressive models in machine learning is paramount, particularly in applications requiring both structural depth and flexibility. Traditional deep learning methods, such as multilayer perceptrons (MLP), offer depth but lack ability to integrate structural characteristics of deep learning architectures with non-parametric flexibility of kernel methods. To address this, deep kernel learning (DKL) was introduced, where inputs to a base kernel are transformed using a deep learning architecture. These kernels can replace standard kernels, allowing both expressive power and scalability. The advent of Kolmogorov-Arnold Networks (KAN) has generated considerable attention and discussion among researchers in scientific domain. In this paper, we introduce a scalable deep kernel using KAN (DKL-KAN) as an effective alternative to DKL using MLP (DKL-MLP). Our approach involves simultaneously optimizing these kernel attributes using marginal likelihood within a Gaussian process framework. We analyze two variants of DKL-KAN for a fair comparison with DKL-MLP: one with same number of neurons and layers as DKL-MLP, and another with approximately same number of trainable parameters. To handle large datasets, we use kernel interpolation for scalable structured Gaussian processes (KISS-GP) for low-dimensional inputs and KISS-GP with product kernels for high-dimensional inputs. The efficacy of DKL-KAN is evaluated in terms of computational training time and test prediction accuracy across a wide range of applications. Additionally, the effectiveness of DKL-KAN is also examined in modeling discontinuities and accurately estimating prediction uncertainty. The results indicate that DKL-KAN outperforms DKL-MLP on datasets with a low number of observations. Conversely, DKL-MLP exhibits better scalability and higher test prediction accuracy on datasets with large number of observations.

• The paper presents DKL-KAN to enhance deep kernel learning by leveraging learnable Kolmogorov-Arnold Networks for improved expressiveness and reliable uncertainty quantification.
• It demonstrates the integration of scalable methods like KISS-GP to efficiently handle datasets of varying sizes, outperforming traditional DKL-MLP in limited-data scenarios.
• The study reveals trade-offs between model expressiveness and scalability, suggesting opportunities for refining KAN architectures to better accommodate large datasets.

Overview of "DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networks"

The paper "DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networks" by Shrenik Zinage, Sudeepta Mondal, and Soumalya Sarkar, explores the integration of Kolmogorov-Arnold Networks (KAN) into Deep Kernel Learning (DKL) frameworks. The paper presents DKL-KAN as an alternative to traditional DKL approaches using multilayer perceptrons (MLP), focusing on improving expressiveness and scalability while maintaining robust uncertainty quantification inherent to Gaussian Processes (GP).

Introduction

The motivation for this work stems from the inherent limitations of both standard GPs and conventional deep neural networks (DNN). While GPs excel in uncertainty quantification, they lack scalability and struggle with high-dimensional, highly structured data. On the other hand, DNNs, despite their expressiveness and representation learning capabilities, often lack reliable uncertainty estimates and pose challenges in Bayesian inference. DKL addresses these issues by leveraging neural networks to transform inputs into a feature space conducive for GPs. However, conventional DKL typically uses MLPs, which are prone to overfitting, computational inefficiency, and interpretability issues.

Methodology

Kolmogorov-Arnold Networks (KAN): KANs leverage the Kolmogorov-Arnold representation theorem, which asserts that any multivariate continuous function can be expressed as a finite sum of univariate functions. Exploiting this theorem allows KANs to use learnable activation functions, typically splines or B-splines, which adapt to the data more effectively than fixed activation functions used in MLPs. These characteristics make KANs promising candidates for enhancing the DKL framework.

Gaussian Processes and Deep Kernels: The paper extensively uses GP models with RBF kernels and incorporates KANs to transform the input space before feeding into the GP. Two configurations of DKL-KAN are evaluated:

1. DKL-KAN1: Matches the neuron and layer count of DKL-MLP.
2. DKL-KAN2: Has a similar number of trainable parameters as DKL-MLP.

Both versions are compared against traditional DKL-MLP and standard GPs.

Scalability Improvements: The authors employ Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP) and its variant with scalable kernel interpolation of product kernels (SKIP) for efficient handling of large datasets. These methods significantly reduce computational costs associated with GPs.

Experimental Results

The proposed DKL-KAN methods are evaluated on various UCI regression datasets of differing sizes and complexities. Key observations include:

• Low Observation Datasets: DKL-KAN1 outperforms DKL-MLP, suggesting that KAN's expressiveness and adaptability are beneficial in scenarios with limited data.
• Large Observation Datasets: DKL-MLP demonstrates better scalability and test prediction accuracy than DKL-KAN variants, indicating the current version of KANs struggles with larger datasets.
• Handling Discontinuities: DKL-KAN maintains higher epistemic uncertainty in regions lacking training data, unlike DKL-MLP which tends to be overconfident outside the training regions. This implies a more faithful uncertainty representation.

Numerical and Computational Observations

Performance metrics such as Root Mean Squared Error (RMSE) and computational training runtime are provided:

• Numerical Performance: The RMSE values indicate that DKL-KAN1 generally performs well, especially in datasets with fewer observations. However, for larger datasets like Protein and KEGG, DKL-MLP achieves better accuracy.
• Computational Efficiency: The average training times for DKL-KANs are comparable to those of other models when efficient KANs are used. Nevertheless, DKL-KAN's complexity poses a greater computational burden as the dataset size increases.

Implications and Future Work

This research offers substantial theoretical and practical insights:

• Theoretical Innovations: The introduction of KANs into the DKL framework exemplifies an innovative approach to enhancing model expressiveness and reliability in uncertainty estimation.
• Practical Applications: DKL-KAN’s ability to accurately estimate uncertainties can lead to more reliable deployment in real-world applications, particularly where the integrity of uncertainty quantification is crucial.

Future Developments: There is potential for further research to address the scalability and computational challenges observed with KANs in this paper. Optimizations in KAN architectures or hybrid models that blend the strengths of KANs with other scalable neural networks could offer promising directions.

In conclusion, the paper demonstrates that DKL-KAN introduces meaningful enhancements to the DKL framework but also highlights areas for improvement in terms of scalability and efficiency. The balanced combination of GPs and adaptable neural network structures like KANs could pave the way for more robust machine learning models in diverse applications.