Deeper Insights into Learning Performance of Stochastic Configuration Networks

Dianhui Wang; Xiufeng Yan

REVIEW 5 major objections 6 minor 28 references

Deeper Insights into Learning Performance of Stochastic Configuration Networks

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves an exact residual test for candidate basis functions in SCN-III, replacing a lower-bound score, and shows that the resulting RMPI-SCN converges faster and learns better on ten regression benchmarks.

desk verdict Correct and useful exact one-step selection condition for SCN-III, but the empirical evaluation confounds the new selector with other changes and misses the closest prior baselines. read the letter →

arxiv 2411.08544 v1 pith:PLFSIOSE submitted 2024-11-13 cs.AI

Xiufeng Yan , Dianhui Wang This is my paper · ORCID

classification cs.AI

keywords stochasticconfigurationnetworksrandomizedlearningincrementalMoore-Penroseinversesupervisorymechanismbasisfunctionselectionregression

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

Stochastic Configuration Networks are randomized neural networks that grow one hidden node at a time, accepting only candidate basis functions that meet a residual-shrinking test. This paper identifies a mismatch in the standard SCN-III test: its score $\xi$ measures how much the candidate would reduce the error if only its own output weight were updated, but SCN-III actually refits all output weights, so the score is only a lower bound on the candidate's true value. The paper derives a necessary-and-sufficient condition for a candidate to achieve $\|e_L\| \le \sqrt{r}\|e_{L-1}\|$ in SCN-III, based on a recursive Moore-Penrose inverse calculation, and uses it to build RMPI-SCN, a training scheme that selects the candidate with the smallest exact one-step residual. Simulations on ten regression datasets show RMPI-SCN reaches lower training and test error with fewer hidden nodes than SCN-III on most benchmarks.

What carries the argument

The load-bearing identity is the recursive Moore-Penrose update. For an existing hidden-layer matrix $H_{L-1}$ and a candidate output vector $h$, Greville's formulas give $H_L^\dagger$ in terms of $H_{L-1}^\dagger$, $d_L = H_{L-1}^\dagger h$, and $b_L$; substituting into the projection $Y - H_L H_L^\dagger Y$ collapses the new residual to $e_{L-1} - p_L\tau_L$, with $p_L = h - H_{L-1}H_{L-1}^\dagger h$ and $\tau_L = \langle Y,p_L\rangle/\|p_L\|^2$. This turns residual evaluation into a small inner-product computation and yields the necessary-and-sufficient inequalities that define the new supervisory mechanism. The same recursive update is reused inside RMPI-SCN to maintain $H_L^\dagger$ as nodes are added, so no full pseudoinverse of the growing output matrix is recomputed.

What would settle it

On a regression dataset with strongly overlapping basis functions, run RMPI-SCN and SCN-III to the same node budget and compare held-out RMSE across many seeds; if SCN-III is not worse on a majority of datasets, the claimed advantage of exact one-step selection is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the supervisory mechanism of SCN-III can be made exact. Theorem 2 states that for a candidate basis function $h$, the inequality $\|Y - [H_{L-1},h][H_{L-1},h]^\dagger Y\| \le \sqrt{r}\|e_{L-1}\|$ holds if and only if $p_L = h - H_{L-1}H_{L-1}^\dagger h$ is nonzero, $\langle e_{L-1},p_L\rangle^2/\|p_L\|^2 \ge (1-r)\|e_{L-1}\|^2$, and $\|\langle Y - e_{L-1},p_L\rangle\| \le \sqrt{\delta_L}$, where $\delta_L = \langle e_{L-1},p_L\rangle^2 - (1-r)\|p_L\|^2\|e_{L-1}\|^2$. Because $p_L$ and the scalar $\tau_L = \langle Y,p_L\rangle/\|p_L\|^2$ can be computed from the already-stored $H_{L-1}^\dagger$, the true one-step residual is available without forming the Moore-Penrose inverse of the augmented matrix. The authors fold this into RMPI-SCN, which also uses a size-dependent learning rate $r_L = r^{(1+1/L)^\alpha}$, and report that it consistently selects the most effective random basis function at each iteration, converges faster, and improves training and test error on most of ten benchmarks.

Load-bearing premise

The paper's results rest on the assumption that always adding the hidden node that reduces the current training error the most will also produce a better final model, an assumption that is supported empirically on most datasets but not proven.

Editorial extensions

If this is right

SCN-III implementations can replace the lower-bound score $\xi$ with the exact one-step residual, removing the need to search over $r$ in an inner loop and the need to compute $[H_{L-1},h]^\dagger$ for every candidate.
Because smaller $r$ values remain usable for longer, the algorithm can keep the uniform distribution's support tighter when appropriate and expand it more accurately when the exact criterion fails.
The adaptive schedule $r_L = r^{(1+1/L)^\alpha}$ preserves the geometric convergence guarantee, with $\limsup \|e_L\|/\|e_{L-1}\| \le r^\alpha$, while avoiding the practice of forcing $r$ close to one.
If the central claim is correct, switching to the exact residual test should reproduce the reported gains: lower training RMSE on nine of ten datasets, lower test RMSE on most, and fewer nodes (e.g., 268 vs 398 on DB4).

Reading between the lines

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

The greedy one-step residual is not obviously the best proxy for final generalization; the paper's own tables show RMPI-SCN trailing SCN-III on one training metric (DB5) and one test metric (DB10), so the advantage may be dataset-dependent.
The recursive pseudoinverse update is exact in exact arithmetic, but no numerical-stability analysis is given for hundreds of accumulated updates; a QR- or SVD-based refresh might be needed in very large networks.
The same residual identity could be carried into recurrent, 2D, and deep SCN variants, since Theorem 2 only treats single-hidden-layer feedforward networks; the authors themselves list these as future work.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is Theorem 2: a necessary and sufficient condition for ||Y - H_L H_L^† Y|| ≤ sqrt(r)||e_{L-1}||, expressed via the orthogonal component p_L = h - H_{L-1}H_{L-1}^†h. The derivation using Greville's recursion checks out as far as I followed it, and it yields a practical way to evaluate candidates without forming the new pseudoinverse each time. That is a real contribution, and it fixes a genuine weakness: the ξ-based criterion in SCN-III is only a lower bound for the one-step residual reduction, so maximizing ξ does not necessarily pick the best candidate. The new condition is exact for that one-step goal, which is a clean theoretical improvement.

The soft spots are all in the empirical section. Most importantly, RMPI-SCN changes three things at once: the selection rule, the r_L schedule, and the recursive update path. No ablation isolates the new selector, so the observed gains in training and test RMSE cannot be cleanly attributed to it. The theoretical result only says the exact selector minimizes the immediate residual among the candidate batch; it does not promise better final generalization. Second, the paper does not compare against the most relevant improved SCN baselines (refs [16] and [18]), which also revise the supervisory mechanism. Third, the abstract and introduction claim reduced computational complexity, but the tables show RMPI-SCN is slower than SCN-III on most datasets; the paper acknowledges this in the text, but it undercuts the scalability claim. There is also a notational inconsistency in the definition of e_{L-1} between Theorem 2 and the algorithm, and the sign in Eq. (57) looks wrong under one reading (it may be consistent with the flipped residual definition, but as written it will confuse). No significance tests are reported, and some differences on DB3, DB5, DB7, and DB9 are small enough that 100 runs without a test statistic do not settle them.

None of this kills the core contribution. The math is solid, the idea is clearly motivated, and the weaknesses are addressable: add an ablation, compare with the improved SCN variants, report significance, fix the notation, and release code or full hyperparameters. I would want those before trusting the empirical claims, but the theoretical result stands on its own.

This paper deserves a serious referee. It is exactly the kind of work that should go through peer review: a correct and checkable theoretical improvement with an evaluation that needs strengthening. I would take it with the expectation of major revisions.

Referee Report

5 major / 6 minor

Summary. This paper analyzes the supervisory mechanism of SCN-III, arguing that the standard lower-bound criterion ξ in Eq. (10) can select suboptimal random basis functions. It derives necessary and sufficient conditions (Theorem 2, Corollaries 2 and 3) for the one-step training residual to satisfy ||Y - H_L H_L^† Y|| ≤ √r ||e_{L-1}||, using Greville's recursive pseudoinverse and the projection residual p_L = h - H_{L-1} H_{L-1}^† h. The paper then proposes RMPI-SCN, which selects candidate nodes by the exact one-step residual, replaces SCN-III's inner r-search with a scheduled rate r_L = r^{(1+1/L)α}, and maintains H_L^† recursively. The empirical section compares RMPI-SCN with SCN-III, RVFL, and MLP-BP on ten regression datasets.

Significance. The theoretical core is sound and useful: Theorem 2 reduces candidate evaluation to vector operations involving p_L and gives a clean orthogonal-projection characterization of the one-step residual reduction; the proof does not depend on the data and introduces no tunable constants other than r. If validated in isolation, the new selection rule would be a meaningful improvement over the lower-bound-based selection in SCN-III. The main weakness is that the empirical claims are not isolated: RMPI-SCN changes the selector, the r-schedule, and the update path simultaneously, and no ablation, statistical significance testing, or comparison with the closest improved-SCN baselines is provided.

major comments (5)

[Section III.C, Eqs. (47), (55)-(57), (58)-(60)] RMPI-SCN differs from SCN-III in at least three coupled design changes: the candidate score (the exact one-step residual ξ_j instead of SCN-III's lower-bound ξ), the scheduled learning rate r_L that eliminates the inner r-search, and the recursive Greville update of H^†. The paper attributes the better results in Table III to the new 'accurate' supervisory mechanism, but no experiment isolates this rule from the other two modifications. Add at least two ablations on the same candidate draws: exact-residual selection with SCN-III's r-search, and SCN-III's ξ selection with the r_L schedule. Without these, the observed gains cannot be attributed to the new evaluation mechanism.
[Section IV, text around Figs. 4 and 8] The paper states that RMPI-SCN's faster convergence is 'not an experiment coincidence but a mathematical consequence' because it selects the candidate minimizing the current residual. This overreaches: minimizing the immediate one-step residual among the current candidates gives no logical guarantee about the final training or test RMSE, since the greedy choice affects future candidate distributions and the residual path. The reported exceptions in Table III (e.g., DB5 training RMSE, where RMPI-SCN is slightly worse than SCN-III, and DB10 test RMSE, where RVFL is best) are acknowledged but not explained. Either provide a theoretical argument for the greedy trajectory or replace the causal claim by the directly supported statement that RMPI-SCN minimizes the one-step residual.
[Table III and Section IV] The 100-run comparisons are reported only as means and standard deviations; no statistical significance tests are given. For several datasets the differences are small relative to the reported variation (e.g., DB3 test RMSE: 3.9136 ± 0.0243 vs. 3.9635 ± 0.0171; DB5; DB7; DB9), so the claim that RMPI-SCN 'outperforms' SCN-III is not established. Report paired tests or bootstrap confidence intervals over the 100 repetitions, preferably using identical candidate draws for both methods.
[Section IV, Table III and abstract] The paper claims reduced computational complexity and enhanced scalability, but the reported wall-clock times are not consistent with that claim: RMPI-SCN is slower than SCN-III on DB4 (43.031 s vs. 24.373 s), DB6 (0.160 s vs. 0.130 s), DB9 (0.381 s vs. 0.221 s), and DB10 (0.334 s vs. 0.198 s). Since RMPI-SCN also uses fewer nodes on some datasets, a fair complexity comparison would require operation counts or runtime to reach a fixed residual threshold. Without such an analysis, the efficiency claim is not supported by the experiments.
[Section IV, comparisons] The experimental evaluation does not include the closest prior-art improvements to SCN's supervisory mechanism, in particular Refs. [16], [17], [18], and [25]. These methods directly address the same lower-bound limitation that motivates RMPI-SCN. The paper should compare against these methods on the same benchmarks, or explicitly justify their exclusion; without such comparisons, the contribution relative to the state of the art is not established.

minor comments (6)

[Algorithm 1, line 10] The pseudocode sets r* = r^{(1+1/L)γ}, but the hyperparameter introduced in Section III.B and used in Corollary 3 is α; please align the notation.
[Abstract and Section III.A] The abstract says the method works 'without requiring the computation of the Moore-Penrose inverse of the output matrix', but Eqs. (58)-(60) recursively maintain H_L^†. Please rephrase to 'without recomputing H_L^† from scratch for each candidate'.
[Section IV, dataset list] The sentence 'The datasets DB3-DB10 are benchmark datasets: CCPP, Superconduct, Delta Ail, Stock, Concrete, and CCPP' repeats CCPP and does not match Table II; the list should be corrected to match DB3-DB10.
[Algorithm 1, line 17] The pseudocode line 'eL = HLH†LT − T' contains a typo; it should read 'eL = HLH†LY − Y'.
[Section III.C, Eqs. (56)-(57)] The definition of ξ_j in Eqs. (56)-(57) is not explicitly derived from the minimization of ||Y - H_L H_L^† Y|| in Eq. (52); a short derivation or explanatory sentence would aid reproducibility.
[Theorem 2, Eq. (29)] Condition Eq. (29) is redundant: because p_L is orthogonal to range(H_{L-1}) and Y - e_{L-1} belongs to range(H_{L-1}), we have ⟨Y - e_{L-1}, p_L⟩ = 0, so Eq. (29) reduces to the nonnegativity already implied by Eq. (28). This does not affect the correctness of the theorem, but the statement can be simplified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorem is derived from Greville's recursive Moore-Penrose formula and least-squares definitions, not from fitted values or self-citation chains.

full rationale

The paper's load-bearing theoretical result is Theorem 2, with Corollaries 2 and 3. The proof starts from Greville's external recursive Moore-Penrose identity, defines p_L = h - H_{L-1} H_{L-1}^\dagger h, and derives the exact post-addition residual Y - H_L H_L^\dagger Y = e_{L-1} - p_L \tau_L. The necessary-and-sufficient condition for ||e_L|| <= sqrt(r)||e_{L-1}|| follows by analyzing when the quadratic form ||e_{L-1} - \tau p_L||^2 - r||e_{L-1}||^2 is non-positive. This is a self-contained derivation that does not fit any parameter to the benchmark data. The only tuned hyperparameter, alpha, appears in the schedule r_L = r^{(1+1/L)^\alpha} and is not an output of the theorem. The statement that RMPI-SCN selects the most effective random candidate is a definitional property of the selection rule: the algorithm computes the exact one-step residual for each candidate and chooses the minimum. That is a design choice, not a circular prediction. Citations to Wang and Li for Theorem 1 provide background on the original SCN framework and are not load-bearing for the new recursive evaluation; no step reduces a predicted quantity to a fitted input, and no imported uniqueness theorem is used to force the main result. Therefore no significant circularity is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central derivation relies on standard linear algebra (Greville's inverse) plus the SCN density assumption. The algorithm introduces one new hyperparameter alpha and inherits the existing SCN parameters r, lambda, T_max, L_max. No new physical entities are postulated.

free parameters (5)

alpha = not specified (selected via cross-validation)
Controls the growth rate of r_L = r^{(1+1/L)alpha} in Corollary 3 and Algorithm 1; the paper says a higher alpha relaxes the error-reduction constraint faster.
r = not specified (selected via cross-validation)
Base learning-rate parameter in the inequality ||e_L|| <= sqrt(r_L)||e_{L-1}||; initialized in Algorithm 1.
lambda = set from a sequence Upsilon = {lambda_1,...,lambda_K} (values not given)
Support scale for the uniform distribution used to draw random weights and biases; expanded when no candidate satisfies the inequality.
T_max = not specified
Number of candidate basis functions generated per iteration.
L_max = not specified
Maximum number of hidden nodes allowed.

assumptions (4)

standard math Greville's recursive formula for the Moore-Penrose inverse (Eq. 31-33)
Used in Theorem 2 to compute the residual after adding a column.
domain assumption span(Gamma) is dense in L2(K) and the basis functions are bounded in L2(K)
Inherited from Theorem 1 of Wang and Li [14] to guarantee existence of suitable random basis functions.
domain assumption The candidate component p_L = h - H_{L-1} H_{L-1}^† h is nonzero
Required in Theorem 2 and in the algorithm; the paper does not discuss the p_L = 0 case.
domain assumption Training data are normalized and the logistic sigmoid activation is used
Stated in Section IV for the experiments; the theory only requires bounded activation and compact domain.

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Cite this review

Pith. "Pith review of Deeper Insights into Learning Performance of Stochastic Configuration Networks." pith.science (2026). https://pith.science/paper/PLFSIOSE

@misc{pith2026241108544,
  author       = {Pith},
  title        = {Pith review of: Deeper Insights into Learning Performance of Stochastic Configuration Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLFSIOSE}},
  note         = {Machine review of arXiv:2411.08544}
}

read the original abstract

Stochastic Configuration Networks (SCNs) are a class of randomized neural networks that integrate randomized algorithms within an incremental learning framework. A defining feature of SCNs is the supervisory mechanism, which adaptively adjusts the distribution to generate effective random basis functions, thereby enabling error-free learning. In this paper, we present a comprehensive analysis of the impact of the supervisory mechanism on the learning performance of SCNs. Our findings reveal that the current SCN framework evaluates the effectiveness of each random basis function in reducing residual errors using a lower bound on its error reduction potential, which constrains SCNs' overall learning efficiency. Specifically, SCNs may fail to consistently select the most effective random candidate as the new basis function during each training iteration. To overcome this problem, we propose a novel method for evaluating the hidden layer's output matrix, supported by a new supervisory mechanism that accurately assesses the error reduction potential of random basis functions without requiring the computation of the Moore-Penrose inverse of the output matrix. This approach enhances the selection of basis functions, reducing computational complexity and improving the overall scalability and learning capabilities of SCNs. We introduce a Recursive Moore-Penrose Inverse-SCN (RMPI-SCN) training scheme based on the new supervisory mechanism and demonstrate its effectiveness through simulations over some benchmark datasets. Experiments show that RMPI-SCN outperforms the conventional SCN in terms of learning capability, underscoring its potential to advance the SCN framework for large-scale data modeling applications.

Figures

Figures reproduced from arXiv: 2411.08544 by the authors.

**Figure 1.** Training diagram of SCN The key distinction between SCN-I and SCN-III lies in the method used to update the output weights after the addition of the L-th basis function gL. SCN-III employs: β = [β ∗ 1 , . . . , β∗ L] T = arg min β ∥Y − X L i=1 βihi∥ = H † L Y, (6) where HL = [HL−1, hL] is the augmented hidden layer output matrix, and H † L denotes the Moore-Penrose inverse of HL. This approach allows for a comprehen… view at source ↗

**Figure 2.** The curve of rL = r (1+1/L) α for α = [0.1, 0.3, 0.5, 0.7, 0.9]. hidden layer output matrix be HL−1 = [h1, . . . , hL−1] and hi = σ(Xwi + bi), 1 ≤ i ≤ L − 1, and H † L−1 is the MoorePenrose inverse of HL−1. The current training error is defined as eL−1 = Y − HL−1H † L−1 Y = [eL−1,1, . . . , eL−1,m], where eL−1,q = Yq −HL−1H † L−1 Yq and Yq = [y1,q, y2,q, . . . , yN,q] T , 1 ≤ q ≤ m. Let h be the output of a candida… view at source ↗

**Figure 4.** Training errors of RMPI-SCN and SCN-III on DB1 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: The histogram of random input weights of RMPI-SCN and SCN-III [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: The target function of DB1 and its 1st and 2nd order derivatives in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: The approximation of RVFL where the support of the uniform [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Training error reduction curve of IRVFL, RMPI-SCN and SCN-III on [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Training error reduction curve of IRVFL, RMPI-SCN and SCN-III on [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [16]

X. Zhu, X. Feng, W. Wang, X. Jia, and R. He, ”A further study on the inequality constraints in stochastic configuration networks,” Information Sciences, vol. 487, pp. 77-83, 2019

work page 2019

[2] [18]

J. Nan, W. Dai, and D. Wang, ”Stochastic configuration networks with improved supervisory mechanism,” Information Sciences, p. 120885, 2024

work page 2024

[3] [17]

K. Li, C. Yang, W. Wang, and J. Qiao, ”An improved stochastic con- figuration network for concentration prediction in wastewater treatment process,” Information Sciences, vol. 622, pp. 148-160, 2023

work page 2023

[4] [25]

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work page 2024

[5] [1]

Cybenko, ”Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals and Systems, vol

G. Cybenko, ”Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals and Systems, vol. 2, no. 4, pp. 303-314, 1989

work page 1989

[6] [2]

Hornik, M

K. Hornik, M. Stinchcombe, and H. White, ”Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, no. 5, pp. 359-366, 1989

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[7] [3]

Park and I

J. Park and I. W. Sandberg, ”Universal approximation using radial-basis- function networks,” Neural Computation, vol. 3, no. 2, pp. 246-257, 1991

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