Revisiting Generalization Power of a DNN in Terms of Symbolic Interactions

Junpeng Zhang; Lei Cheng; Qihan Ren; Quanshi Zhang

REVIEW 4 major objections 5 minor 14 references

Revisiting Generalization Power of a DNN in Terms of Symbolic Interactions

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The generalization power of a deep network can be summarized by the distribution of its symbolic interactions: generalizable ones decay with order, non-generalizable ones form a spindle.

desk verdict A compact parametric summary of interaction-order distributions with a practical disentangling objective, but the central decay/spindle-to-generalization dichotomy is never directly tested—the components are assumed, then fit to the same data. read the letter →

arxiv 2502.10162 v1 pith:NSQDS4NL submitted 2025-02-14 cs.LG cs.AIcs.CLcs.CV

Lei Cheng , Junpeng Zhang , Qihan Ren , Quanshi Zhang This is my paper · ORCID

classification cs.LGcs.AIcs.CLcs.CV

keywords symbolicinteractionsAND-ORgeneralizationpoweroverfittinginteractionorderinterpretablemachinelearningdeeptheory

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

The paper argues that a deep network's generalization behavior can be understood not from high-dimensional features but from the distribution of symbolic interactions—AND/OR inference patterns that provably reproduce the network on all masked versions of an input. Its central hypothesis is that generalizable interactions decay in strength as their order (number of variables involved) grows, while non-generalizable interactions follow a spindle shape: weak at very low and very high orders, strongest at medium orders. If true, the generalization power of a trained network is summarized by two fitted curves, and overfitting can be detected as the re-emergence of the spindle component. The authors fit both shapes simultaneously and report close matches with interaction distributions extracted from real networks.

What carries the argument

The load-bearing object is the AND-OR interaction decomposition: for any input and network output, there is a logical model of AND interactions (activated when all variables in a set are present) and OR interactions (activated when any variable is present) whose sum equals the network's outputs on all $2^n$ masked samples. Interaction order $m=|S|$ is the complexity measure, and the aggregated positive/negative strengths $A^{(m),+}$, $A^{(m),-}$ form the per-order distribution. The spindle component is modeled by a scaled binomial $A^{(m),+}_{\text{spindle}}(\alpha,\beta)=\beta\,\Gamma(\alpha n+1)/(\Gamma(m+1)\Gamma(\alpha n-m+1))$, and the decay component by applying the noise matrix $M(\delta)$ from Theorem 2.3 to penalize high-order interactions; the disentanglement fits the sum of the two parametric curves to the observed distribution.

What would settle it

One concrete check would settle it: compute the Jaccard similarity of Definition 2.2 separately for interactions in the fitted decay component and in the fitted spindle component across train/test splits. If spindle-component interactions generalize as well as decay-component ones, or if decay-component interactions fail to generalize, the central dichotomy is false.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Hypothesis 1: generalizable interactions encoded by a DNN usually follow a decay-shaped distribution over interaction order, while non-generalizable interactions usually follow a spindle-shaped distribution. The paper supports this with two-stage training dynamics—random-initialization noise interactions (spindle-shaped) are removed during normal learning, and spindle-shaped interactions re-emerge during overfitting—and with noise-injection experiments in which added parameter or input noise generates new spindle-shaped interactions. It then proposes parametric forms: a generalized binomial (gamma-function) curve for the spindle component and a noise-penalized interaction model for the decay component, and disentangles the two by fitting their sum to the observed per-order interaction strengths. The authors report that the fitted theory matches real interaction distributions across VGG, AlexNet, ResNet, and BERT models on several datasets.

Load-bearing premise

The paper assumes that interactions created by injecting parameter or input noise are exactly the non-generalizable interactions, and therefore that the spindle-shaped component recovered by the fit is non-generalizable; this identification is never checked directly with the Jaccard-similarity definition of generalization.

Editorial extensions

If this is right

Generalization of an entire DNN reduces to the generalization of its symbolic interactions, so network-level generalization can be summarized by the interaction-order distribution rather than by features.
Overfitting is characterized by re-learning spindle-shaped interactions; tracking the spindle component during training can locate the onset of the overfitting phase.
Adding non-generalizable representations (parameter or input noise) increases the spindle component while leaving the decay component roughly unchanged, so the decomposition separates the two sources.
The parametric forms predict the dynamics of interaction strengths in real DNNs, meaning the fitted curves can serve as a compact descriptor of a network's generalization state.

Reading between the lines

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

A practical extension the authors do not draw: the fitted spindle weight could serve as an early overfitting alarm that does not require a validation set, since it can be computed from training data alone.
If the dichotomy holds distributionally rather than per-interaction, one testable corollary is that penalizing the spindle component during training should improve test accuracy without hurting training accuracy much; the paper does not run this intervention.
The spindle shape may be a signature of memorization of hard or noisy samples; connecting it to per-sample memorization scores would be a natural next test.
Because the paper's identification of spindle with non-generalizable rests on noise injection, a direct per-order Jaccard check between the fitted spindle component and Definition 2.2 would strengthen the theory beyond the current verification.

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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

Colleague,

The headline: this paper has a genuinely useful idea and a load-bearing gap between claim and evidence. The new element is the additive disentangling objective in Eq. (9), which summarizes a DNN's interaction strength over orders as a sum of a binomial-like spindle component and a M(δ)-penalized decay component. That gives a low-dimensional, tractable description of how interaction strength varies with order. The experimental sweep across VGG, ResNet, AlexNet, BERT, and several datasets is real work, and the figures do show the two shapes emerging during training dynamics and under noise injection.

What does not hold up is the claim that the two fitted components correspond to generalizable and non-generalizable interactions. The paper never measures those sets directly. Definition 2.2 defines generalization via Jaccard similarity between train and test interaction distributions, but the paper never computes that similarity for the decay and spindle components. Instead, it identifies decay with the end of the normal learning phase and spindle with newly emerged interactions under overfitting or injected noise. That identification is an assumption, not a test. Noise-induced interactions may be transient or atypical rather than representative of non-generalizable interactions generally. The paper's own conclusion concedes the components are not strictly aligned with generalization.

The verification problem is structural. Eq. (9) fits α, β, δ, γ to the very A(m) curves whose residuals are then reported as small. That is fitting, not prediction. Figures 3 and 7 have the same character. A four-parameter sum of a binomial and a penalized interaction term will flexibly match many shaped curves, so a small residual does not validate the semantic labeling. The spindle derivation also leans on an ad hoc effective count αn, and the decay formula is an opaque matrix from prior work, so neither component is independently pinned down.

To be fair, the paper is not careless. It states its assumptions plainly and builds on prior findings that low-order interactions generalize better. The two-stage dynamics plots in Figure 2 offer some independent evidence that the shapes appear as training moves from normal to overfitting. The problem is that observing a shape is not enough to establish the dichotomy.

My recommendation: send it to a serious referee, but require a direct test of Hypothesis 1. The authors need to extract decay and spindle components and then measure Sim(Dtrain, Dtest) separately for each component, or otherwise show that the components differ in measured generalization behavior. Without that, the paper is a modeling contribution with an unsupported interpretation. The disentangling objective is worth building on.

For a reading group, I'd say maybe—it's a good case study in fitting versus prediction. I would not cite it in the next 12 months until the dichotomy is tested directly.

Referee Report

4 major / 5 minor

Summary. The paper proposes that the generalization power of a DNN can be understood through symbolic interactions: generalizable interactions follow a decay-shaped distribution over interaction order, while non-generalizable interactions follow a spindle-shaped distribution. The authors derive parametric forms for both distributions, propose an optimization problem (Eq. 9) to disentangle the two components from a real DNN, and report experiments on image and text models showing that the fitted sum matches measured interaction-strength curves. The paper also claims that injecting Gaussian parameter noise or FGSM perturbations creates non-generalizable interactions that follow the spindle-shaped distribution.

Significance. If the central dichotomy were established, the paper would offer a novel and potentially useful lens for diagnosing overfitting: separating a network's interaction distribution into a generalizable decay component and a non-generalizable spindle component. The paper's universal-matching theorem (Theorem 2.1) is a genuine theoretical asset, and the spindle-vs-decay visualization across training timepoints is suggestive. However, the load-bearing identification of the two fitted components with the paper's own definition of generalization (Definition 2.2) is never directly tested, and the quantitative evidence rests on in-sample curve fitting. The paper itself concedes in the Conclusion that not every decay-component interaction is generalizable and not every spindle-component interaction is non-generalizable, which substantially weakens the claimed dichotomy.

major comments (4)

[§2.2, §2.2.2, Definition 2.2] Hypothesis 1 is a claim about which interactions generalize, but the paper never connects its two fitted components to the Jaccard-similarity definition of generalization given in Definition 2.2. In §2.2.1 the decay-shaped interactions are identified with the interactions remaining after normal training, and in §2.2.2 the spindle-shaped interactions are identified with interactions newly emerged after parameter or input perturbation. Neither identification is checked by measuring Sim(Dtrain, Dtest) on the extracted decay and spindle components. The spindle-shaped distribution could be a property of perturbation-induced transient interactions rather than of non-generalizable interactions in general, so the central dichotomy is unsupported.
[§2.3.3, Eq. (9), §3] The 'predictions' reported in Figures 3, 4, and 5 are fitting residuals. Equation (9) minimizes the squared error between A_theory and the measured A(m), with α, β, γ, δ free, so a small residual only shows that a four-parameter family of curves can approximate the measured distribution; it does not independently confirm the decay/spindle decomposition. The paper does not report error bars, the number of random seeds, or any out-of-sample validation, so the evidence for the two-component model is currently in-sample and underdetermined.
[§2.3.1, Eq. (6)] The derivation of the spindle-shaped distribution assumes that non-generalizable interactions in a randomly initialized DNN follow a binomial form and then introduces an ad hoc effective variable count αn, with α treated as a free fitting parameter. No derivation is given for why a trained DNN's non-generalizable interactions should follow the same binomial form with a scaled n. The claim that 'not all input variables interact' motivates α, but the value of α is never independently measured or constrained, so the spindle model is a fitted family of curves rather than a falsifiable prediction.
[§4 (Conclusion)] The manuscript's own conclusion states that 'We cannot ensure that every detail interaction in a decay-shaped distribution is generalizable, or ensure every interaction in a spindle-shaped distribution is non-generalizable.' This concession directly weakens Hypothesis 1 as stated in §2.2. If the decay and spindle components are only trends and can each contain both generalizable and non-generalizable interactions, then the disentanglement method of Eq. (9) does not separate generalizable from non-generalizable interactions, and the paper's central claim needs to be reformulated and retested.

minor comments (5)

[§2.2.2] Typo: 'to to obtain' should be 'to obtain'.
[§2.2.2] The claim that injecting Gaussian parameter noise or FGSM perturbations creates 'non-generalizable representations' is stated as a supported finding, but Appendix E only provides a brief heuristic discussion and no direct measurement of non-generalizability.
[§3] Figures 3 and 4 report 'well matched' curves without quantitative goodness-of-fit metrics, making it difficult to compare across models, noise levels, or the fitted parameters.
[Appendix H] Theorem 2.3 is stated with a matrix M(δ) whose formula is deferred to Appendix H; the appendix gives the formula but no derivation. The paper would benefit from a self-contained derivation or a clear pointer to the prior work for the key decay property of M(δ).
[Throughout] Several figures and captions contain typos, e.g., 'resdual' in Figures 4 and 9, and 'Define2.2' in Appendix G. A careful proofreading pass is needed.

Circularity Check

4 steps flagged · score 6.0 of 10

Hypothesis 1 is tested by redefining "non-generalizable" as noise-induced or overfitting-emerged interactions, and the "theoretical" matches in Figures 3, 4, and 7 are least-squares fits of the same quantities; the central dichotomy is therefore only partially supported.

self definitional [Section 2.2.2, first paragraph ('verifying the spindle-shaped distribution of non-generalizable interactions')]
"In this subsection, we revised a well-trained DNN by injecting non-generalizable representations into it to to obtain a DNN with non-generalizable features. In this way, we could examine whether interactions newly emerged in the revised DNN followed a spindle-shaped distribution."

Definition 2.2 defines the generalization of interactions by the Jaccard similarity Sim(Dtrain, Dtest), and the paper's Hypothesis 1 is a claim about generalizable versus non-generalizable interactions under that definition. The verification, however, never computes Sim(Dtrain, Dtest) for the extracted components; instead, "non-generalizable representations" are produced by adding Gaussian parameter noise or FGSM input perturbations, and the interactions that newly emerge are then reported as the non-generalizable ones. The spindle-shaped claim is therefore tested on an operationally defined class of noise-induced interactions, not on the class defined by the paper's own Definition 2.2.
fitted input called prediction [Section 3, 'Fitness to the distribution of non-generalizable interactions' (Figure 3; Eq. (6))]
"We further used the derived distribution A(m),+spindle, A(m),−spindle in Equation (6) to match these interactions, i.e., computing parameters α and β that best matched the distribution of these interactions. Figure 3 shows that the theoretical distributions A(m),+spindle, A(m),−spindle well matched the true distribution of non-generalizable interactions in a real DNN, which partially proved the faithfulness of our theory."

The "theoretical" spindle distribution is obtained by fitting the free parameters α and β to the very ∆A(m),+ and ∆A(m),− curves shown in Figure 3. The reported agreement is therefore a two-parameter least-squares fit, not an out-of-sample prediction or an independent derivation. Moreover, the curves being fit are the noise-injected interactions that Section 2.2.2 labeled "non-generalizable" by assumption, so the experiment re-describes its own construction rather than testing Hypothesis 1 against an independent measure of generalization.

2 more flagged steps

fitted input called prediction [Section 2.3.3, Eq. (9); Section 3, 'Aspect 2: matching error' (Figure 4)]
"Then, the distributions of generalizable interactions A(m),+decay, A(m),−decay and non-generalizable interactions A(m),+spindle, A(m),−spindle can be disentangled as the optimization of the following parameter model: min α,β;δ,γ Σ_m (A(m),+ − A(m),+theory)^2 + (A(m),− − A(m),−theory)^2. ... Figure 4 shows that strength of the matching error was small, which indicated that the disentangling algorithm effectively extracted the distribution of generalizable interactions and the distribution of non-generalizable interactions."

Equation (9) chooses α, β, δ, and γ to minimize the squared difference between the measured A(m) and the fitted A_theory, and Aspect 2 then reports that this minimized difference is small. A minimized training objective is small by construction for a four-parameter model, so the small residual cannot confirm that the decay-plus-spindle decomposition is the true explanation of generalization. The same fitting procedure absorbs whatever departure from the assumed shapes exists into the free parameters, and the fitted residual is then presented as evidence of the faithfulness of the theory.
ansatz smuggled in via citation [Section 2.3.2, 'Modeling the decay-shaped distribution', Eq. (8) and closing paragraph]
"In this way, we formulate the strength of positive interaction effect and that of negative interaction effect of each m-th order, as follows. ... More importantly, despite the complexity of Theorem 2.3, later experiments will demonstrate the success of our choice, i.e., using the penalized interactions under noises to mimic the decay-shaped distribution of interactions."

The decay-shaped distribution, which is the generalizable half of Hypothesis 1, is not derived from a definition or measurement of generalization; it is an explicitly adopted "choice" that models generalizable interactions as "penalized interactions under noises." This model is imported from the authors' own prior work, Ren et al. [2024b], which shares authors with the present paper (Qihan Ren, Junpeng Zhang, Quanshi Zhang). The promised demonstration comes from the in-sample fits of Eq. (9) and Appendix H's curve matches, so the generalizable-decay claim rests on a self-citation chain plus fitted residuals rather than on a Jaccard-based measurement linking the fitted decay component to actual generalization.

full rationale

The paper's central Hypothesis 1—that generalizable interactions are decay-shaped and non-generalizable interactions are spindle-shaped—is not validated through the paper's own definition of generalization (Definition 2.2, Jaccard similarity between training and testing interaction distributions). Instead, "generalizable" is operationalized as the interactions remaining after the normal learning phase, and "non-generalizable" is operationalized as interactions that newly emerge under Gaussian parameter noise, FGSM perturbations, or overfitting. These operationalizations are never checked against Sim(Dtrain, Dtest), so the two observed shapes are properties of the chosen operations rather than of the stated definition. In addition, the confirmations in Figures 3, 4, and 7 are largely in-sample: Eq. (9) minimizes the exact squared matching error that Section 3 later reports as small, and Eq. (6) is fit to the same ∆A(m) curves it is said to match. The decay model is imported from the same authors' Ren et al. [2024b] as an explicit "choice," so the generalizable-decay half also rests on a self-citation chain. Some independent content remains: the loss-gap division into normal and overfitting phases is externally measured, Claim 1's low-order Jaccard advantage is directly measured in Appendix G, and the empirical regularity that overfitting adds medium-to-high-order interactions is a real observation. These give the paper moderate empirical grounding, but the central dichotomy as stated is only partially supported, making the circularity substantial yet not total. Score 6.

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

The central claim rests on four prior theorems/results from the same group (universal matching, sparsity, Gaussian random interactions, M(δ) dynamics), four fitted parameters, and an unvalidated assumption that noise injection produces non-generalizable interactions. The paper does not offer a parameter-free prediction; every quantitative match is a fit.

free parameters (5)

α (spindle effective variable ratio) = fitted in Eq (9)
Scales the binomial coefficient C(αn, m) to model spindle-shaped distribution in trained DNNs; not derived from first principles.
β (spindle magnitude) = fitted in Eq (9)
Overall magnitude of the spindle-shaped interaction distribution.
δ (decay noise magnitude) = fitted in Eq (9), set ≤ 1e-3 in Appendix H experiments
Controls how strongly high-order interactions are penalized in the decay model M(δ).
γ (decay magnitude) = fitted in Eq (9)
Overall magnitude of the decay-shaped interaction distribution.
τ (salient interaction threshold) = τ = 0.02 · E_x[|v(x) - v(x_∅)|]
Chosen by hand to select salient interactions; affects which interactions enter the measured distributions.

assumptions (5)

domain assumption Universal matching property: an AND-OR interaction model can exactly reproduce a DNN's outputs on all 2^n masked samples (Theorem 2.1).
Invoked in Section 2.1 to justify treating interactions as inference patterns; proven in prior work by Chen et al. 2024 and in Appendix C.
domain assumption Sparsity: well-trained DNNs encode only O(n^p) salient interactions (p in [1.5, 2]) under three smoothness conditions.
Used in Section 2.1 to restrict attention to salient interactions; cited from Ren et al. 2024a.
domain assumption Non-generalizable interaction effects follow a Gaussian distribution in randomly initialized DNNs.
Foundation for the binomial spindle model in Section 2.3.1; cited from Zhang et al. 2024, not re-derived here.
domain assumption Theorem 2.3: injecting noise δ into training reduces high-order interactions via matrix M(δ), giving the decay-shaped distribution.
Used to formulate A_decay in Section 2.3.2; M(δ) formula is given in Appendix H and credited to Ren et al. 2024b.
ad hoc to paper Adding Gaussian noise to parameters or adversarial perturbations to inputs creates non-generalizable representations.
Section 2.2.2 uses noise-injected DNNs as ground truth for non-generalizable interactions; not independently validated.

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Cite this review

Pith. "Pith review of Revisiting Generalization Power of a DNN in Terms of Symbolic Interactions." pith.science (2026). https://pith.science/paper/NSQDS4NL

@misc{pith2026250210162,
  author       = {Pith},
  title        = {Pith review of: Revisiting Generalization Power of a DNN in Terms of Symbolic Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSQDS4NL}},
  note         = {Machine review of arXiv:2502.10162}
}

read the original abstract

This paper aims to analyze the generalization power of deep neural networks (DNNs) from the perspective of interactions. Unlike previous analysis of a DNN's generalization power in a highdimensional feature space, we find that the generalization power of a DNN can be explained as the generalization power of the interactions. We found that the generalizable interactions follow a decay-shaped distribution, while non-generalizable interactions follow a spindle-shaped distribution. Furthermore, our theory can effectively disentangle these two types of interactions from a DNN. We have verified that our theory can well match real interactions in a DNN in experiments.

Figures

Figures reproduced from arXiv: 2502.10162 by the authors.

**Figure 2.** The two-stage dynamics of interactions in the learning of a DNN. In the first stage, noise interactions [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The distributions of newly emerged interactions ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The decay-shaped distribution of generalizable interactions ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The distributions of generalizable interactions ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The averaged Jaccard similarity between interactions extracted from training samples and those extracted [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Comparison between the theoretical distribution of interactions [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: The decay-shaped distribution of generalizable interactions ( [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: The distributions of generalizable interactions ( [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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We trained different DNNs, including LeNet[LeCun et al., 1998], VGG-11, VGG-13, and AlexNet, on different datasets, including MNIST, CIFAR-10, CUB200-2011, and Tiny-ImageNet

We computed the average Jaccard similarity EcSim(Dtrain, Dtest)(introduced in Define2.2) between interactions ex- tracted from training samples and those extracted from testing samples across 10 categories in the datasets. We trained different DNNs, including LeNet[LeCun et al., 1998], VGG-11, VGG-13, and AlexNet, on different datasets, including MNIST, C...

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Their experiments further confirmed that most randomly initialized DNNs align well with this proposition

proposed that DNNs with entirely random weights mainly encode medium-order and high-order interactions under certain assumptions. Their experiments further confirmed that most randomly initialized DNNs align well with this proposition. However, this may not fully apply to Bert-Medium. In the first phase, Bert-Medium primarily encodes extremely low-order i...

work page 2011

[7] [14]

residual Figure 8: The decay-shaped distribution of generalizable interactions (A(m),+ decay , A(m),− decay ) and the spindle-shaped distribu- tion of non-generalizable interactions (A(m),+ spindle , A(m),− spindle ) disentangled by our method. The significance of interactions in the spindle-shaped distribution increased when we injected more non-generali...

work page 2011

[8] [2009]

The Caltech-UCSD Birds-200-2011 Dataset

Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The Caltech-UCSD Birds-200-2011 Dataset

work page 2011