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We are pleased to accept your paper. You resolved the remaining Zen-score concern by relabelling the score as a Zen-style variant throughout: the abstract, Section 3.3, Tables 1 and 4–7, Section 6.6, Section 8 and Section 9. Section 3.3 now sets out the four ways your score departs from Lin et al. (2021) Algorithm 1, and the paper is explicit that no claim is made about the published Zen-Score. This is an honest and adequate scoping of the result. The paper's withdrawals, released data and clearly bounded claims make it a credible empirical contribution on ezkl circuit cost and on how zero-cost proxies fail for polynomial activations. Please fix the small textual errors listed in the editorial notes in the final version.
Verdict: accept. Both concerns from the round-2 minor revision are resolved in the manuscript, and the revision introduces no new correctness, validity, novelty or reproducibility problem. Concern 1, implement the published Zen-Score or relabel and scope the claims: resolved by relabelling, the second route the decision allowed. The abstract now reads 'a Zen-style score (our variant, not the published Zen-Score protocol)'. Table 1 and the row labels of Tables 4–7 read 'Zen-style (variant)'. Section 6.6 is retitled 'The Zen-Style Score'. Section 8 states 'We make no claim about the published Zen-Score', and Section 9 calls the score 'our variant of Zen-NAS'. No ZenNAS-labelled conclusion about the published proxy remains. Concern 2, state how the score differs from Lin et al. (2021) Algorithm 1: resolved. Section 3.3 lists four differences: output logits versus the pre-GAP feature map, residual links kept, a batch-std term in place of BN-variance rescaling, and fixed input and perturbation scales. It also states that no conclusion is drawn about the published proxy. No new problems: the letter states that the numbers are unchanged because the same score was computed. The Zen-style entries in Tables 4–7 and the figures in Section 6.6 match the previously reviewed values, so no result changed without explanation. References are unchanged, and Lin et al. 2021 is retained. The remaining issues are textual and are listed in the editorial notes.
Published with the paper; the authors were not required to address them before publication.
No new reviewer reports were commissioned for this round. The previous round asked for a minor revision, which the editor checks alone against the concerns it raised.
We thank the editor and reviewers. There was one substantive concern, about the Zen score, and we resolved it by the second route the editor offered: we relabel the score as a variant and remove every claim about the published proxy. Nothing else in the paper changed except the numbers it already reported.
1. "Sections 3.3 and 6.4-6.6 and Tables 4-7: either implement the published Zen-Score protocol ... or rename the score as a custom Zen-style variant throughout and remove claims about published ZenNAS from the abstract, Section 6.4 and Section 9." You are right: the score we computed is not the published Zen-Score. We did not implement the published protocol. We renamed the score "Zen-style (variant)" everywhere it appears: the abstract ("a Zen-style score (our variant, not the published Zen-Score protocol)"), Section 3.3, Table 1 (row "Zen-style (variant), after Lin et al. 2021"), the row labels of Tables 4-7, Sections 6.4 and 6.6 (now "The Zen-Style Score"), the Conclusion, and the README. Section 8 now says explicitly that we make no claim about the published Zen-Score, and the code docstring marks the function as a variant. All numbers are unchanged, because the same score was computed; only what they are claimed to measure changed.
2. "Section 3.3: state explicitly how the implemented score differs from Lin et al. (2021) Algorithm 1." Section 3.3 now lists the differences: Lin et al. compute the score on the feature map before global average pooling, after removing residual links, and add a sum of log sqrt(mean batch-norm variance) over batch-norm layers. Our score (i) is computed on the network output (logits), (ii) keeps residual links, (iii) has no batch-norm-variance term (our networks use no BatchNorm and we do not substitute a LayerNorm or GroupNorm statistic) and adds log E[std over the batch of the output] instead, and (iv) fixes the input and perturbation scales (x ~ N(0, I), eps ~ N(0, 1e-4 I)). We state that every Zen-style result concerns this variant and that we draw no conclusion about the published proxy.
We did not implement the faithful Zen-Score; doing so would require a different treatment of LayerNorm/GroupNorm networks, and we preferred to scope the claim rather than introduce an unreviewed variant of the variant. The reviewers' optional questions (logrows >= 24 for the four-resource rule, the depthwise ResNet mismatches, the sensitivity to the two retained seed-1 failures, a lower learning rate for normalized polynomial networks, release of trained weights) are left as the limitations already stated in Sections 6.9 and 8.
12 of 175 blocks of the round 3 manuscript are new or changed since round 2; 12 of 175 blocks of round 2 no longer appear. A block is a paragraph, heading, list, table or code block; a re-worded paragraph counts whole and a moved one does not count. This compares the two submitted texts and is not the response letter.
| Section | Changed | Removed |
|---|---|---|
| A Zen-style score (a variant of Zen-NAS, not the published protocol). | 2 | — |
| 4.3 Zero-Cost Proxies Evaluated | 1 | 1 |
| 6.4 Zero-Cost Proxies Across Activation Families | 5 | 5 |
| 6.6 The Zen-Style Score | 2 | — |
| What we do not claim. | 1 | 1 |
| 9. Conclusion | 1 | 1 |
| Zen-NAS. | — | 2 |
| 6.6 ZenNAS | — | 2 |
[under "A Zen-style score (a variant of Zen-NAS, not the published protocol)."]
#### A Zen-style score (a variant of Zen-NAS, not the published protocol).
[under "A Zen-style score (a variant of Zen-NAS, not the published protocol)."]
Lin et al. (Lin et al. 2021) (Algorithm 1) compute the Zen-Score on the feature map before global average pooling, after removing the residual links, and add a term that sums $\log\sqrt{\text{mean batch-norm variance}}$ over the batch-norm layers. We did *not* implement that protocol. Our score is $\mathrm{Zen\text{-}style}(f)=\mathbb{E}_{x,\epsilon}\log\|f(x+\epsilon)-f(x)\|_2+\log\mathbb{E}\,[\operatorname{std}_{\text{batch}} f(x)]$ with $x\sim\mathcal{N}(0,I)$ and $\epsilon\sim\mathcal{N}(0,10^{-4}I)$, and it differs from the published score in four ways: (i) it is computed on the network *output* (logits), not the pre-pooling feature map; (ii) residual links of ResNet-style networks are *kept*; (iii) it has no batch-norm-variance term (our networks use no BatchNorm, and we do not substitute a LayerNorm or GroupNorm statistic) and instead adds $\log\mathbb{E}[\operatorname{std}_{\text{batch}}f(x)]$ as a second term; and (iv) the perturbation and input scales are fixed as stated. Every result labelled Zen-style concerns this variant and *not* the published Zen-Score; we draw no conclusion about the published proxy. In an earlier version we gave a scaling argument for its anti-correlation with accuracy on polynomial networks; because the inputs are fixed at unit scale the argument as stated is not well defined, so we withdraw it and instead report the empirical dependence of the score on depth, size and training failure (Section 6).
[under "4.3 Zero-Cost Proxies Evaluated"]
| **Proxy** | **Reference** | **Data-free** | **Poly-compatible** |
|:---|:---|:---|:---|
| Synflow | Tanaka et al. 2020 | ✓ | ✓ |
| NEAR | Husistein et al. 2025 | $\times$ (1 batch) | ✓ |
| Zen-style (variant) | after Lin et al. 2021 | ✓ (Gaussian) | ✓ |
| ZiCo | Li et al. 2023 | $\times$ (1 batch) | ✓ |
| MeCo | Jiang et al. 2023 | ✓ | ✓ |
| NASWOT | Mellor et al. 2021 | $\times$ | **No** (Proposition 2) |
| **Structural (ours)** | This work | ✓ | ✓ |
[under "6.4 Zero-Cost Proxies Across Activation Families"]
Table 4 gives Spearman $\rho$ between each proxy and mean accuracy per activation family with bootstrap intervals and Holm-adjusted significance over the 25 activation-family tests it contains. NEAR is a reliable predictor only for ReLU networks ($\rho=+0.70$, $95\%$ CI $[+0.57,+0.80]$); on the polynomial families its intervals include zero ($x^2$ -0.12 $[-0.29,+0.05]$, $x^3{+}x$ +0.01, $x^2{+}x$ -0.14). The Zen-style score (our variant, Section 3; not the published Zen-Score) is negatively correlated with accuracy in every family in this analysis (ReLU -0.23, $x^2$ -0.42, $x^3{+}x$ -0.31, $x^2{+}x$ -0.54), but not robustly: with a common training recipe its ReLU correlation is +0.19 (Table 6) and its partial correlation on ReLU is +0.02. ZiCo is the one proxy with a significant positive coefficient on a polynomial family ($x^3{+}x$, +0.45), but once failed networks are removed (Table 5) it is -0.38 on $x^3{+}x$ and -0.50 on $x^2$: its apparent success on $x^3{+}x$ comes from separating networks that train from networks that do not, not from ranking accuracy among those that train. Removing failed networks changes other conclusions too: Holm-significant entries lost are NASWOT-sign on $x^2$; NASWOT-sign on $x^2{+}x$; NASWOT-sign on $x^3{+}x$; Zen-style on $x^2$; ZiCo on $x^3{+}x$; entries gained are Synflow on $x^2$; Synflow on $x^2{+}x$; Zen-style on $x^3{+}x$; ZiCo on $x^2$; ZiCo on $x^2{+}x$. The negative correlations of the Zen-style score and of the sign-pattern NASWOT variant are therefore partly trainability effects. MeCo and Synflow are weak or inconsistent.
[under "6.4 Zero-Cost Proxies Across Activation Families"]
| **Proxy** | **ReLU** ($n{=}111$) | **$x^2$** ($n{=}121$) | **$x^3{+}x$** ($n{=}121$) | **$x^2{+}x$** ($n{=}121$) |
|:---|---:|---:|---:|---:|
| NEAR | $+0.70^{*}$ \[+0.57,+0.80\] | $-0.12$ \[-0.29,+0.05\] | $+0.01$ \[-0.17,+0.19\] | $-0.14$ \[-0.30,+0.03\] |
| Synflow | $+0.22$ \[+0.03,+0.40\] | $-0.24$ \[-0.41,-0.04\] ($n{=}99$) | $+0.13$ \[-0.14,+0.43\] ($n{=}70$) | $-0.18$ \[-0.38,+0.03\] ($n{=}99$) |
| ZiCo | $+0.24$ \[+0.07,+0.40\] | $-0.10$ \[-0.29,+0.09\] | $+0.45^{*}$ \[+0.29,+0.60\] | $-0.04$ \[-0.22,+0.16\] |
| MeCo | $+0.16$ \[-0.02,+0.32\] | $-0.09$ \[-0.26,+0.08\] ($n{=}109$) | $+0.05$ \[-0.16,+0.27\] ($n{=}79$) | $-0.06$ \[-0.24,+0.14\] ($n{=}109$) |
| Zen-style (variant) | $-0.23$ \[-0.42,-0.04\] | $-0.42^{*}$ \[-0.54,-0.27\] ($n{=}109$) | $-0.31$ \[-0.49,-0.11\] ($n{=}79$) | $-0.54^{*}$ \[-0.64,-0.42\] ($n{=}109$) |
| NASWOT (ReLU patterns) | $+0.60^{*}$ \[+0.44,+0.75\] | n/a | n/a | n/a |
| NASWOT (pre-activation signs) | $+0.60^{*}$ \[+0.44,+0.75\] | $-0.27^{*}$ \[-0.43,-0.11\] | $-0.29^{*}$ \[-0.45,-0.12\] | $-0.35^{*}$ \[-0.49,-0.19\] |
[under "6.4 Zero-Cost Proxies Across Activation Families"] [block of 1001 characters not shown; it begins "| **Proxy** | **ReLU** ($n{=}111$) | **$x^2$** ($n{=}71$) | **$x^3{+}x$** ($n{=}…"]
[under "6.4 Zero-Cost Proxies Across Activation Families"] [block of 1062 characters not shown; it begins "| **Proxy** | **ReLU** ($n{=}72$) | **$x^2$** ($n{=}77$) | **$x^3{+}x$** ($n{=}7…"]
[under "6.4 Zero-Cost Proxies Across Activation Families"]
| **Proxy** | **ReLU** | $\mathbf{x^2}$ | $\mathbf{x^3{+}x}$ | $\mathbf{x^2{+}x}$ |
|:---|---:|---:|---:|---:|
| NEAR | $+0.22^{*}$ | $+0.05$ | $+0.03$ | $-0.01$ |
| Synflow | $+0.40^{*}$ | $-0.37^{*}$ | $-0.19$ | $-0.31^{*}$ |
| ZiCo | $+0.13$ | $-0.06$ | $+0.48^{*}$ | $+0.01$ |
| MeCo | $+0.23^{*}$ | $-0.05$ | $-0.01$ | $-0.01$ |
| Zen-style (variant) | $+0.02$ | $-0.32^{*}$ | $-0.12$ | $-0.36^{*}$ |
| NASWOT (ReLU patterns) | $+0.00$ | – | – | – |
| NASWOT (pre-activation signs) | $-0.00$ | $-0.07$ | $-0.06$ | $-0.20^{*}$ |
…and 4 more blocks not shown.[under "Zen-NAS."]
#### Zen-NAS.
[under "Zen-NAS."]
We implement the Zen score as $\mathrm{Zen}(f)=\mathbb{E}_{x,\epsilon}\log\|f(x+\epsilon)-f(x)\|_2
+\log\mathbb{E}\,[\operatorname{std}_{\text{batch}} f(x)]$ with $x\sim\mathcal{N}(0,I)$ and $\epsilon\sim\mathcal{N}(0,10^{-4}I)$. In an earlier version we gave a scaling argument for its anti-correlation with accuracy on polynomial networks; because the inputs are fixed at unit scale the argument as stated is not well defined, so we withdraw it and instead report the empirical dependence of the score on depth, size and training failure (Section 6).
[under "4.3 Zero-Cost Proxies Evaluated"]
| **Proxy** | **Reference** | **Data-free** | **Poly-compatible** |
|:---|:---|:---|:---|
| Synflow | Tanaka et al. 2020 | ✓ | ✓ |
| NEAR | Husistein et al. 2025 | $\times$ (1 batch) | ✓ |
| ZenNAS | Lin et al. 2021 | ✓ (Gaussian) | ✓ |
| ZiCo | Li et al. 2023 | $\times$ (1 batch) | ✓ |
| MeCo | Jiang et al. 2023 | ✓ | ✓ |
| NASWOT | Mellor et al. 2021 | $\times$ | **No** (Proposition 2) |
| **Structural (ours)** | This work | ✓ | ✓ |
[under "6.4 Zero-Cost Proxies Across Activation Families"] [block of 1506 characters not shown; it begins "Table 4 gives Spearman $\rho$ between each proxy and mean accuracy per activatio…"]
[under "6.4 Zero-Cost Proxies Across Activation Families"] [block of 1061 characters not shown; it begins "| **Proxy** | **ReLU** ($n{=}111$) | **$x^2$** ($n{=}121$) | **$x^3{+}x$** ($n{=…"]
…and 7 more blocks not shown.