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Posterior Contraction Rates for sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Variational Bayesian deep learning · 0 citations (OpenAlex)

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Proves near-minimax posterior contraction rates (up to log factor) for sparse fixed-depth Bayesian KANs with spike-and-slab priors over anisotropic and compositional Besov spaces, with adaptation via a hyperprior on a model-size parameter.

Problem

Posterior contraction and adaptivity guarantees for Bayesian Kolmogorov-Arnold networks (spline-edge architectures) over anisotropic Besov spaces, where prior theory has been developed mainly for ReLU MLPs.

Approach

Sparse fixed-depth KANs with B-spline edges achieve approximation via width, grid range/size and sparsity rather than depth or finer grids; combined with spike-and-slab priors and a single model-size hyperprior, this yields near-minimax contraction rates governed by intrinsic anisotropic smoothness and layerwise smoothness for compositional structure.

Key results

• Sparse Bayesian KANs with spike-and-slab priors contract at rate n^{-s~/(2s~+1)} (log n)^{1/2}, near-minimax over anisotropic Besov spaces.
• A hyperprior on a single model-size parameter N yields adaptive contraction at the same rate without knowing the smoothness vector.
• Extension to compositional anisotropic Besov spaces gives rates governed by intrinsic layerwise smoothness s~* rather than ambient dimension d.

Surprising findings

• KAN depth can be held fixed (L0 = 3 + 2 ceil(log2 d)) while approximation power comes from width, grid range/size and sparsity, unlike deep ReLU theory.

Limitations

• Analysis restricted to fixed-knot equally spaced B-spline grids; adaptive knot placement not treated.
• Rates hold up to a (log n)^{1/2} factor relative to minimax.
• Purely theoretical; no computational or empirical validation; posterior computation not addressed.

Experimental setup

Metrics: posterior contraction rate

Baselines: sparse MLP Bayesian neural networks (ReLU)

Builds on: Kolmogorov-Arnold networks spike-and-slab sparse Bayesian neural network contraction theory

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