Title: Generalization Bounds for Kolmogorov-Arnold Networks (KANs) and Enhanced KANs with Lower Lipschitz Complexity
Venue: NeurIPS 2025
Accepted: yes
Keywords: Generalization bounds

Kolmogorov-Arnold Networks

Model complexity.
Abstract: Kolmogorov-Arnold Networks (KANs) have demonstrated remarkable expressive capacity and predictive power in symbolic learning. However, existing generalization errors of KANs primarily focus on approximation errors while neglecting estimation errors, leading to a suboptimal bias-variance trade-off and poor generalization performance. Meanwhile, the unclear generalization mechanism hinders the design of more effective KANs variants. As the authors of KANs highlighted, they ``would like to explore ways to restrict KANs' hypothesis space so that they can achieve good performance''. To address these challenges, we explore the generalization mechanism of KANs and design more effective KANs with lower model complexity and better generalization. We define \textit{Lipschitz complexity} as the first structural measure for deep functions represented by KANs and derive novel generalization bounds based on \textit{Lipschitz complexity}, establishing a theoretical foundation for understanding their generalization behavior. To reduce \textit{Lipschitz complexity} and boost the generalization mechanism of KANs, we propose Lipschitz-Enhanced KANs ($\textbf{LipKANs}$) by integrating the Lip layer and pioneering the $L_{1.5}$-regularized loss, contributing to tighter generalization bounds. Empirical experiments validate that the proposed LipKANs enhance the generalization mechanism of KANs when modeling complex distributions. We hope our theoretical bounds and LipKANs lay a foundation for the future development of KANs.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the generalization properties of Kolmogorov-Arnold Networks (KANs), a recently proposed architecture known for strong expressiveness in symbolic and scientific learning tasks. The authors address a critical gap in prior analyses, which largely focused on approximation error, by rigorously formulating generalization bounds for KANs through a novel complexity measure—Lipschitz complexity. Building on these theoretical insights, they introduce LipKANs, a variant that integrates Lipschitz-bounded layers and L1.5-regularization, leading to provably tighter generalization bounds and improved empirical performance on benchmark datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\nStrong Theoretical Foundation:\n\nThe paper makes a significant theoretical contribution by defining Lipschitz complexity as a structural complexity measure for deep KANs. The derived generalization bounds are carefully constructed, with a clear connection to covering numbers and tensor norms.\n\nThe generalization bounds are both novel and insightful, shedding light on how the architecture’s functional richness impacts its statistical behavior. The connection to known results for MLPs (e.g., Bartlett et al. 2017) is well-executed and positions the contribution in context.\n\nArchitectural Innovation – LipKANs:\n\nThe introduction of Lip layers and the shift from L1 to L1.5 regularization are elegant and well-motivated by the theory.\n\nThe reformulated architecture adheres closely to the theoretical framework, and the algorithmic instantiation (Algorithm 1) is clear, reproducible, and practically useful.\n\nClarity and Rigor:\n\nThe paper is written with excellent clarity, especially for a technically dense topic. Definitions, assumptions, and lemmas are well organized.\n\nThe proofs (particularly Theorems 4.4 and 4.5) are carefully structured and leverage standard but non-trivial tools in statistical learning theory (e.g., tensor covering bounds, sub-Gaussian concentration).\n\nEmpirical Validation:\n\nThe experiments, though not extensive, support the key claims. Notably, the improvements on more complex datasets (e.g., STL-10, CIFAR-100, MIMIC-III) convincingly demonstrate the generalization benefits of LipKANs.\n\nThe ablation study on L1.5 regularization is especially useful, highlighting the nontrivial gain over L1 or standard Lip layers alone.\n\nWeaknesses\nLimited Experimental Breadth:\n\nWhile the theoretical and algorithmic contributions are substantial, the empirical section could be strengthened. The paper evaluates a modest set of tasks and models; additional baselines (e.g., Transformer-based models, state-of-the-art MLPs, or regularization baselines like dropout) would provide a more complete picture of LipKANs' practical competitiveness.\n\nExperiments on regression tasks or out-of-distribution generalization could further validate the claims about improved generalization due to lower Lipschitz complexity.\n\nQUESTIONS:\nQuestions for the Authors\n\nRole of Activation Family Cardinality:\nWhile the bounds depend only logarithmically on the activation family size 𝐺, in practice how sensitive is model performance to the choice of \nG? Have you observed performance trade-offs when varying the number or type of basis functions?\n\nInterpretability Impacts:\nOne often-cited feature of KANs is their symbolic interpretability. Do LipKANs (with additional Lipschitz layers and L1.5 regularization) preserve this interpretability, or does the added complexity obscure it?\n\nSuggestions for Improvement\n\nVisualization of Complexity Metrics:\nA plot showing how the Lipschitz complexity evolves during training across different variants (KAN, LipKAN, etc.) would offer intuitive insight into how the regularization actually impacts model complexity and generalization in practice.\n\nEmpirical Study of Covering Bounds:\nAlthough theoretical, it would be illuminating to empirically explore how covering numbers (or their proxies) behave in practice for different architectures. For example, measuring empirical Rademacher complexity or sharpness may help support the theoretical framework.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-05-31"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the generalization mechanisms of Kolmogorov-Arnold Networks (KANs), and, building on this understanding, introduces LipKANs that is more efficient owing to lower model complexity and yields better generalization. For the methodology, they derive a uniform high-probability generalization bound and introduce the concept off Lipschitz complexity accordingly. It is shown that the generalization bound with high probability for KANs is proportional to the Lipschitz complexity proposed. Lipschitz complexity is then regarded as a measure of structural complexity of a KAN model. LipKANs leverages this insights by encouraging the training process to be favor low-complexity models, replacing the conventional L1-regularized loss with the proposed Lipschitz complexity. Through extensive experiments, it demonstrates enhanced generalization performance compared to other variants of KAN and highlights the critical role of the Lipschitz through ablation study.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. Technically Solid Methodology:\nThe paper is technically rigorous, offering a comprehensive and logical flow that articulates the significance of generalization bounds in deep learning. The authors clearly identify the gap in understanding the generalization behavior of Kolmogorov-Arnold Networks (KANs), then systematically derive a high-probability generalization bound for KANs. The rationale for introducing Lipschitz complexity as a meaningful measure of model complexity is convincingly argued, connecting the theoretical analysis to practical implications.\n\n2. Well-Motivated Introduction of LipKANs:\nBuilding on the theoretical insights, the transition to proposing LipKANs is natural and well-justified. The authors make a strong case for why optimizing over Lipschitz complexity can lead to better generalization, and they effectively motivate the replacement of standard L1 regularization with their new complexity measure.\n\n3. Thorough and Consistent Experimental Validation:\nThe experimental section is comprehensive, covering various benchmarks and scenarios. The results consistently demonstrate the superior performance of LipKANs compared to baseline models, and the ablation studies effectively highlight the crucial role of Lipschitz complexity in improving generalization.\n\n4. Clarity in Theoretical Development:\nAlthough detailed proofs are relegated to the appendix, the main paper offers clear explanations of the core ideas and proof techniques, allowing readers to understand the theoretical underpinnings without being bogged down in technicalities. This clarity strengthens the paper’s overall accessibility.\n\n\nWeakness:\n1. Loose Generalization Bound:\nWhile the proposed generalization bound successfully captures key factors influencing KANs’ generalization, it does not exploit dependencies or relationships among different weight matrices. As a result, the bound can be quite loose. For instance, for two KAN models where each weight matrix has the same norm, the calculated complexity—and thus the bound—will be identical, regardless of the relative directions of the matrices. However, the actual Lipschitz constant, and thus the true generalization capability, can be significantly affected by these directional relationships. As a consequence, the proposed bound may fail to distinguish between models with genuinely different complexities, potentially affecting model selection.\n\n2. Insufficient Discussion of Bias-Variance Trade-Off:\nThe paper mentions the bias-variance trade-off, but does not clearly explain how it relates to their results or the broader context of estimation versus generalization. Clarifying this relationship could help readers understand how LipKANs balance model expressiveness and generalization, and why the trade-off is enhanced compared to the conventional method.\n\n3. (minor) It is better to move definition 4.2 before assumption 2 as the definition is mentioned in assumption.\n\nQUESTIONS:\n1. Is it common that assumption 2 holds?\n2. Is it possible that we take the relation among weights into consideration so as to refine Lipschitz complexity?\n3. The theory is based on sub-Gaussian distribution. Can it extend to other distribution?\n\nLIMITATIONS:\n1.The theory relies on sub-Gaussian distribution assumption.\n2. Although the paper presents strong empirical results, it does not deeply discuss potential discrepancies between theoretical bounds and observed performance.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper points out that existing research on KANs generalization error mainly focuses on approximation error, but ignores estimation error, resulting in a poor bias-variance trade-off and poor generalization performance. Therefore, this paper introduces Lipschitz complexity as a novel structural complexity measure for Kolmogorov-Arnold Networks (KANs), derives generalization bounds based on this measure. Based on Lipschitz complexity, this paper proposes LipKANs, which enhanced architecture with Lipschitz layers and L1.5 regularization to improve generalization.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n- The paper provides a comprehensive theoretical foundation, including formal definitions of Lipschitz complexity (Definition 4.3), covering number bounds (Theorem 4.4), and generalization bounds (Theorem 4.5). Proofs are detailed in the appendix.\n- Experiments across vision (MNIST, CIFAR), text (AG News), and multimodal (MIMIC-III) tasks demonstrate consistent improvements in generalization for LipKANs over baselines (Table 1). Ablation studies (Figure 1) disentangle the contributions of Lipschitz layers vs. L1.5 regularization.\n- The paper is well-organized, with clear sections for reformulation (Sec 4.1), theory (Sec 4.3), and architecture (Sec 5). Figures and tables effectively summarize results.\n- The core theoretical contribution—defining Lipschitz complexity for deep compositional functions—is novel and distinct from existing measures (e.g., VC dimension, Rademacher complexity).\n\nWeaknesses\n\n- While experiments cover multiple domains, the gains on simpler datasets (e.g., MNIST, CIFAR-10, AVMNIST) are marginal (<1% in Table 1), suggesting limited utility for low-complexity tasks.\n- The cost of Lipschitz layers and L1.5 regularization is not analyzed, leaving practical efficiency unclear.\n\nQUESTIONS:\n- From the ablation study results in Figure 1, the main performance improvement comes from the L_{1.5}-regularization. Does this mean that it is not important to keep low Lipschitz complexity using Lip layer in practical applications?\n- This paper starts with the lack of KAN's generalization, but only verifies the performance on several small-scaled datasets. Discussing the transfer learning performance of KAN (for example, evaluating the performance of a model trained on the MNIST dataset on CIFAR-10) should better verify the generalization ability of the model.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper establishes generalization bounds for Kolmogorov-Arnold Networks (KANs), and proposes a new architecture for improving generalization, called LipKANs. Bounds for KANs, unlike MLPs, are complexity independent, and a mathematical measure, ‘Lipschitz complexity’, is introduced to represent the structural complexity of functions represented by KANs. Then, an architectural innovation is introduced where Lip layers are inserted in the KAN with explicit L1.5 regularization to bound network complexity. Finally, the proposed framework is evaluated on multiple benchmarks for vision, text, and multimodal data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1.\tThis paper explicitly addresses what is called the ‘approximation-estimation’ trade-off for KANs. This is important because naively reducing parameter count does not improve generalization in KANs proportionately, unlike MLPs. By representing the capacity of KANs in terms of properties of their activations, specifically in terms of the ‘Lipschitz complexity’, it no longer has a dependence on the number of parameters resulting in tighter bounds. \n2.\tWhile the paper provides theoretical arguments, this is also converted into a relatively simple, practical implementation for improving generalization in KANs.\n\nWeaknesses:\n1.\tThe results of the ablation study make me question the value of the Lip layers in improving generalization. Improvements seem to be mainly driven by the L1.5 regularization. I would want to see more experiments showing that Lip layers reliably contribute to generalization for MNIST and other datasets to be convinced that the proposed practical architecture genuinely improve generalization in line with theoretical results.\n\nQUESTIONS:\n1.\tIs the F1/accuracy score shown in Table 1 on a held-out test set? If you haven’t, can you specifically mention this somewhere? Also, an explicit quantification of the generalization gap would be useful.\n2.\tCould you provide confidence intervals for the output metrics shown in Table 1?\n3.\tFrom the ablation study, it seems that the L1.5 generalization is the biggest driver of improvements in generalization, and the Lip layers only help for MNIST. I wonder if this effect is just result of optimization/training splits? Can you show that LipKANs consistently show an improvement for MNIST? This for me is the biggest weakness in the paper, and any additional experiments to improve confidence in this would be useful.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-23"}