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On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators

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Pith's one-line read Spline-based Kolmogorov–Arnold Networks are claimed to attain the dimension-free minimax rate n^{-2r/(2r+1)} for regression targets whose univariate components have Sobolev smoothness r.

desk verdict The claimed log-free minimax rate for spline KAN sieves is not established—the upper bound rests on an invalid estimation-error step—but the lower bound and the non-identifiability discussion are decent. read the letter →

arxiv 2509.19830 v3 pith:TD7ZXGNE submitted 2025-09-24 cs.LG cs.AIstat.ML

Wei Liu , Eleni Chatzi , Zhilu Lai This is my paper
classification cs.LGcs.AIstat.ML MSC 62G0862G2041A15
keywords Kolmogorov-ArnoldnetworksnonparametricregressionB-splinessieveestimationminimaxoptimalitySobolevsmoothnessdimension-freerateidentifiability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to prove that Kolmogorov–Arnold Networks with univariate B-spline components, fit by least squares over a growing sieve of spline spaces, converge to the true regression function at the minimax-optimal rate O(n^{-2r/(2r+1)}) whenever the target admits an additive or hybrid additive–multiplicative KAN representation with components of Sobolev smoothness r. If true, this gives KANs a theoretical footing as structured, interpretable alternatives to black-box neural nets, with the rate free of the ambient dimension. The paper also derives a knot-selection rule k ~ n^{1/(2r+1)}, proves a matching lower bound over the KAN-representable class, and shows the univariate components are identifiable only up to constant shifts and permutations. The preprint's metadata abstract states the upper rate as O((log n / n)^{2r/(2r+1)}), attributing the log factor to the nonlinear sieve, while the main theorems state the log-free version.

What carries the argument

The engine is the spline-based KAN sieve estimator: each univariate component is a B-spline with k interior knots, so the whole network is a finite-dimensional function class F_n of dimension p_n ≍ k. The proof combines univariate spline approximation theory (which gives O(k^{-2r}) bias through Lipschitz composition) with a sieve least-squares empirical-process bound that gives O(p_n/n) estimation error from the metric entropy bound log N(ε,F_n,‖·‖∞) ≲ p_n log(1/ε). Balancing the two terms at k ≍ n^{1/(2r+1)} yields the rate.

What would settle it

Fit the spline KAN sieve for the univariate target h(x)=x^r with k ≍ n^{1/(2r+1)} knots and measure E||fhat - h||^2_{L2} over n = 10^4 to 10^6. If the squared error is not consistently bounded by C n^{-2r/(2r+1)} but instead tracks C (log n / n)^{2r/(2r+1)}, Theorem 1's no-log claim is false. A direct proof check: verify whether the inequality (39)–(41) can be derived without a log factor; standard chaining arguments for nonconvex sieves introduce one.

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Extended reading notes

Core claim

The central claim is that the spline-based KAN sieve least-squares estimator attains E||fhat - f0||^2_{L2([0,1]^d)} = O(n^{-2r/(2r+1)}) uniformly over a ball of KAN-representable functions with univariate Sobolev smoothness r, for both additive and hybrid multiplicative architectures. The proof splits the risk into a bias term from spline approximation, O(k^{-2r}), and a variance term from the p_n ~ k-dimensional empirical risk minimizer, O(p_n/n), then balances k ~ n^{1/(2r+1)}. Corollary 1 packages the upper bound with a univariate lower bound to assert minimax optimality over the KAN class; Corollary 2 turns the balance into a concrete knot-count rule.

Load-bearing premise

The proof assumes that the p_n-dimensional KAN sieve is well-behaved enough that its least-squares risk is O(p_n/n) with no log n factor; if the correct empirical-process bound is O(p_n log n / n), the log-free rate collapses to the (log n / n)^{2r/(2r+1)} stated in the abstract.

Editorial extensions

If this is right

  • KAN sieve estimators are minimax optimal over the additive and hybrid KAN classes at the univariate Sobolev rate, so dimension does not enter the exponent.
  • Practitioners get a concrete knot budget: k_n ≍ n^{1/(2r+1)}.
  • Hybrid multiplicative nodes retain the same rate up to a constant overhead that grows with dimension, so expressiveness need not degrade convergence order.
  • Fitted KANs can be consistent even when their internal univariate components are not; interpretation requires centering constraints or permutation fixes.
  • The guarantees apply to the empirical risk minimizer, independent of optimizer noise; stochastic or gradient-based training dynamics are outside the paper's scope.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the log-free estimation-error step fails, the true upper rate is likely (log n / n)^{2r/(2r+1)}; then the minimax claim holds only up to log, which matters for theory but is often invisible in practice.
  • The dimension-free exponent is not a free lunch: it rests on the KAN structure assumption; on a full Sobolev class over [0,1]^d the usual n^{-2r/(2r+d)} curse still applies.
  • The simulations' 'steeper than theoretical' slopes are consistent with finite-sample constants hiding a log factor, so they do not discriminate between the two rate statements.
  • One testable byproduct: for univariate targets in the class, the spline sieve should achieve exactly n^{-2r/(2r+1)} squared error; measuring this directly would settle whether the no-log bound holds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper analyzes least-squares regression over Kolmogorov-Arnold networks whose univariate components are B-splines. It claims that for target functions admitting an additive or multiplicative KAN representation with univariate components in W^r([0,1]), the spline-based KAN sieve estimator achieves squared L2 risk O(n^{-2r/(2r+1)}), that this rate is minimax optimal over the KAN class, that k_n ≍ n^{1/(2r+1)} is the optimal knot scaling, and that the univariate components can be estimated at the same rate. The lower bound is obtained by embedding the univariate Sobolev class into the KAN class. The upper bound is decomposed into an approximation error step and an estimation error step. The paper also contains simulation evidence and a discussion of non-identifiability of the KAN representation.

Significance. If the main theorems were established, the result would be significant: it would show that spline-based KANs achieve a dimension-free minimax rate for a structured compositional class, in contrast to the usual n^{-2r/(2r+d)} rate for general Sobolev classes. The lower bound and the non-identifiability discussion are useful, and the paper provides reproducible code and simulations that directly check the predicted knot scaling. However, the central upper-bound proof contains a load-bearing step that is not justified, and the claims in the body contradict the paper's own metadata abstract. The claimed log-free minimax optimality is therefore not established as written.

major comments (4)
  1. [Appendix A.1, Step 2, Eqs. (39)-(41)] The proof asserts that the metric entropy bound log N(ε, F_n, ||·||_∞) ≤ C p_n log(1/ε) implies E||f̂_n - f*_n||² = O(p_n/n). This does not follow from the cited 'standard empirical-process argument' for a nonconvex nonlinear sieve. For classes satisfying only this entropy bound, the standard least-squares bound is O(p_n log n / n); the log factor is material because balancing k^{-2r} + k log n / n gives k ≍ (n/log n)^{1/(2r+1)} and rate (log n/n)^{2r/(2r+1)}, not n^{-2r/(2r+1)}. Indeed, the metadata abstract states exactly this log-rate and says the lower bound matches 'up to a logarithmic factor,' while Theorem 1 and Corollary 1 claim the log-free rate. A valid proof of O(p_n/n) for this nonconvex class requires additional structure, such as VC-type or localized Rademacher conditions, which is not supplied.
  2. [Appendix A.1, Eq. (35); Theorem 2 proof, Eqs. (47)-(50)] The approximation step is missing the approximation error of the outer spline g_{q,k_n}. In Theorem 1, with f_{0,q,k_n} = g_{q,k_n}(Σ_j ψ_{qj,k_n}(x_j)), the inequality ||f_{0,q,k_n} - f_{0,q}|| ≤ L_q ||Σ_j(ψ_{qj,k_n}-ψ_{qj})|| omits the term involving g_{q,k_n} - g_q. One cannot conclude the displayed bound from L2 spline approximation of g_q alone; controlling g_{q,k_n}(S_{k_n}) - g_q(S_{k_n}) requires a sup-norm or range/density condition. The same issue appears in the Theorem 2 proof, where the products also require sup-norm control. This affects the approximation error term in both theorems.
  3. [Corollary 2, Eq. (70)] The corollary claims E||ψ̂_{qj} - ψ_{qj}||²_{L2} = O(n^{-2r/(2r+1)}) for each univariate spline unit. This is not established by the preceding proofs. Proposition 1 shows that, under the stated centering conditions, the representation is identifiable only up to permutation and constant shifts only after additional normalizations, and Remark 2 explicitly says consistency of the fit does not imply consistency of the components. The proof of Corollary 2 simply asserts that components inherit the overall rate, without addressing the non-identifiability. This claim should either be proved under explicit identifiability constraints or removed/weakened.
  4. [Corollary 1, Eq. (21) and Eq. (60)] The upper bound in Corollary 1 is justified by saying F_KAN^r is contained in the union of the additive subclass of Theorem 1 and the hybrid subclass of Theorem 2. But F_KAN^r as defined in Eq. (21) allows each node T_q to be either additive or multiplicative, including mixed architectures. Theorem 1 handles all-additive targets and Theorem 2 handles all-multiplicative targets; neither proof covers a target with both additive and multiplicative nodes. Thus the upper bound over the full class is not established as stated. The lower bound via the univariate subclass is fine.
minor comments (4)
  1. [Abstract] The body abstract and Theorem 1 state the log-free rate O(n^{-2r/(2r+1)}), while the arXiv metadata abstract states O((log n/n)^{2r/(2r+1)}) and describes the lower bound as matching only up to a logarithmic factor. These are materially different claims; the paper should state one consistent rate.
  2. [References] The lower-bound proof cites 'Tsybakov, 2009' but the bibliography lists Tsybakov (2008). Please harmonize.
  3. [Notation] The proof uses p_n and k_n interchangeably; while p_n ≍ k_n is stated, the notation would be clearer if the dimension of F_n were tracked consistently throughout the appendix.
  4. [Figure 1] The simulation figure reports empirical slopes but no confidence intervals or repeated-trial variability; adding error bars or multiple seeds would strengthen the empirical claim, though this is not central to the theoretical result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central KAN rate is derived from external spline approximation and classical minimax lower bounds; the flagged component-convergence claim is a proof gap, not a circular reduction.

full rationale

The paper's load-bearing derivation (Theorem 1 and 2, Corollary 1) is not circular. Step 1 bounds the sieve approximation error by classical univariate B-spline approximation (eq. 33-37) and Step 2 invokes a sieve least-squares estimation bound O(k_n/n) from external empirical-process references (eq. 39-41); the final rate balances k_n^{-2r} against k_n/n exactly as in classical sieve theory (eq. 42-43). The minimax lower bound is imported from standard univariate Sobolev regression over the embedded subclass W_r^(1) (eq. 62-66). These anchors are external, parameter-free, and do not depend on fitting constants to the paper's own simulations. No fitted parameter is renamed as a prediction, and no load-bearing premise is justified by a self-citation. Two concerns are real but are not circularity: (i) the assertion that metric entropy O(p_n log(1/eps)) yields E||\hat f_n - f_n^*||^2 = O(p_n/n) is under-supported, since the standard empirical-process route generally produces a log n factor unless additional uniformity/bracketing conditions are supplied; this is a correctness/rigor gap, and it is consistent with the metadata abstract's log-rate versus the body's log-free Theorem 1. (ii) Corollary 2's per-component bound E||\hat\psi_{qj} - \psi_{qj}||^2 = O(n^{-2r/(2r+1)}) is not entailed by the preceding results: Proposition 1 (eq. 19) establishes non-identifiability under centering alone, so the empirical risk minimizer over the full sieve need not converge to the specific true components. That is an unsupported inference effectively assuming the identification structure, but it is not load-bearing for the main function-level rate and does not reduce the theorem to its inputs. Because the central claim does not become equivalent to a fitted quantity, a self-citation chain, or a definitional identity, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central proof rests on the KAN representation assumption, classical spline approximation, and two unproved technical transfers: the O(p_n/n) empirical-process bound for a nonlinear composition sieve, and the L2 composition error transfer. No parameters are fitted to data, and no new entities are postulated.

assumptions (4)
  • domain assumption All univariate components g_q and psi_qj lie in W^r([0,1]) with r > 1/2, are uniformly bounded by M, and each g_q is Lipschitz.
    Section 3 states these assumptions after equation (13); they define the class F_KAN_r over which the rate is claimed.
  • standard math Classical spline approximation gives L2 error O(k^{-r}) for each univariate W^r component.
    Invoked in Appendix A.1 Step 1, equation (33), as standard spline theory.
  • ad hoc to paper For the nonlinear KAN sieve, the sieve least-squares estimation error satisfies E||f_hat - f*_n||^2 = O(p_n/n), with p_n approximately k_n.
    Asserted in Appendix A.1 Step 2 from a metric-entropy bound log N(eps) <= C p_n log(1/eps). This is the load-bearing estimation step, and it is not proven for a nonconvex composition class.
  • ad hoc to paper L2 spline errors for psi and g can be propagated through the KAN composition to give overall approximation error O(k^{-2r}) without a sup-norm or range/density condition.
    Used in Appendix A.1 Step 1, equations (34)-(37). The paper passes from ||g - g_k||_{L2([0,1])} to ||g_k(S_k) - g(S)||_{L2([0,1]^d)} without controlling the distribution of S_k(X) or using sup-norm approximation.

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Pith. "Pith review of On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators." pith.science (2026). https://pith.science/paper/TD7ZXGNE

@misc{pith2026250919830,
  author       = {Pith},
  title        = {Pith review of: On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TD7ZXGNE}},
  note         = {Machine review of arXiv:2509.19830}
}
abstract

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions by composing univariate transformations through additive or multiplicative aggregation. We establish convergence guarantees for KANs whose univariate components are B-splines. The least-squares estimator over the KAN spline sieve attains the rate $O((\log n / n)^{2r/(2r+1)})$, uniformly over a ball of regression functions admitting a KAN representation with univariate components of Sobolev smoothness $r$; a matching lower bound of order $n^{-2r/(2r+1)}$ shows this is minimax optimal up to the logarithmic factor, which we trace to the nonlinearity of the sieve rather than to the architecture. The rate is free of the ambient dimension $d$; this dimension-free exponent reflects the assumed KAN structure of the target, not an escape from the minimax rate $n^{-2r/(2r+d)}$ on Sobolev classes over $[0,1]^d$. We derive a knot-selection rule, show that penalized selection over a dyadic knot grid attains the rate adaptively in the unknown smoothness, and show that univariate components are not identifiable under centering alone, so consistency of the fit does not imply consistency of the components. On targets of exactly known smoothness the fitted risk exponent is at least as steep as the bound in every configuration, and the predicted knot scaling and $k^{-r}$ approximation decay are checked directly.

Figures

Figures reproduced from arXiv: 2509.19830 by the authors.

Figure 1
Figure 1. Convergence rates of additive KAN, hybrid KAN, and MLP estimators on two synthetic [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Forward citations

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