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pith:DFNHQBQF

pith:2026:DFNHQBQFWPBQDUKLOU25IOCEIG

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Estimating the expected output of wide random MLPs more efficiently than sampling

George Robinson, Jacob Hilton, Michael Winer, Paul Christiano, Victor Lecomte, Wilson Wu

The expected output of wide random MLPs can be estimated without sampling using layer-wise cumulant and Hermite approximations.

arxiv:2605.05179 v2 · 2026-05-06 · cs.LG · cond-mat.dis-nn · stat.ML

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Claims

C1strongest claim

We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling.

C2weakest assumption

The claim depends on the networks being sufficiently wide for the cumulant and Hermite approximations of activation distributions to remain accurate; the abstract qualifies the result with this width condition but does not specify the scaling or error bounds that would make the assumption hold for a given target MSE.

C3one line summary

For sufficiently wide random MLPs, cumulant and Hermite approximations of layer-wise activation distributions yield expected outputs at lower computational cost than Monte Carlo sampling, with good performance on rare-event probabilities.

References

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[1] URLhttps://www.alignment.org/blog/competing-with-sampling/. J. Pennington, S. S. Schoenholz, and S. Ganguli. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. 2017 · doi:10.1090/mcom/4127

[2] Our algorithms perform poorly at low width, especially when the number of hidden layers L is large, and can even getworsewith the maximum cumulant order K in this setting

[3] , ϕ(Z n) from the cumu- lants of Z1,

[4] When the maximum cumulant order K is odd, the basic version of our algorithm tracks the full trace of the (K+ 1) th-order cumulant tensor, as defined in Section 4.1. The ablated version of our algorit 2016

[5] However, it performs very poorly once the probability drops below 1/(number of samples), since the squared error is very large in the unlikely event that there is a positive sample

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Receipt and verification

First computed	2026-05-20T00:00:40.855454Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

195a780605b3c301d14b7535d4384441a92e5447f7ca4f02f98abb3e4c54a2f9

Aliases

arxiv: 2605.05179 · arxiv_version: 2605.05179v2 · doi: 10.48550/arxiv.2605.05179 · pith_short_12: DFNHQBQFWPBQ · pith_short_16: DFNHQBQFWPBQDUKL · pith_short_8: DFNHQBQF

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Canonical record JSON

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    "abstract_canon_sha256": "bfa6caa99c84bfde05d21320502be24c830a92578b16dd19e8fc560f1eb10b03",
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      "stat.ML"
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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-06T17:46:12Z",
    "title_canon_sha256": "e83d27cd0f0820321d5c976495742f2ea233ec3502cc7ea1a34c9f6db17548e4"
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