Singh, Ryan, De Llanza, Miguel, Buckley, Christopher L. · Oct 28, 2026
Representation Complexity Along Learning Trajectories: Trading Width for Precision and Time
Markdown · JSON · Original source
In this paper, Ryan Singh, Miguel De Llanza and Christopher L. Buckley ask why may use complex representations while learning, even when their final predictors need simpler ones. They distinguish the representation needed to make predictions from the representation needed to make an unknown target detectable through gradient feedback. Using an oracle model of neural network learning, Kolmogorov n-widths and statistical-query lower bounds, they derive a lower bound on the initial effective width needed for target-dependent gradient information to become detectable. For k-sparse parity in D dimensions, they report that a known target has an O(k)-width representation, while discovering it can require substantially greater representation resources. They sharpen the analysis by using parity’s invariance under coordinatewise sign reflections, then derive a width- frontier by interpreting precision through accumulation over fresh samples. The central argument is that learning can require more representational capacity than using the predictor after it is learned: discovery and representation are different computational demands. The authors report that experiments support the predicted dimension dependence, while also examining how optimisation and generalisation depart from the idealised frontier.
Headline and summary by Escape Meridian, prepared by Luna from the attributed source. A source check is not independent confirmation of the source’s claims.
Original source
cs.AI · Article
Source details
Original source date: Oct 28, 2026 (2026-10-28).
Date basis: Date basis not provided.
Added to Escape Meridian: 2026-11-11T09:38:28.914Z. First observed: 2026-10-30T06:45:37.787Z. Retrieved: 2026-10-31T11:16:26.632Z.
Referenced by
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Limits
The evidence provided is the abstract, not the paper’s full technical argument or experimental details. The abstract reports experiments supporting predicted dimension dependence but does not specify their setup or results in detail. The abstract frames the analysis around k-sparse parity and an idealised frontier; it does not establish how broadly the conclusions extend beyond that setting.
Open question
How far do the reported dimension dependence and width- frontier extend beyond k-sparse parity and the paper’s idealised learning model?
Source-stated
The authors distinguish the representation complexity required to predict from that required to make an unknown target detectable through gradient feedback.
comparing the representation complexity needed for prediction with that needed to make an unknown target statistically visible through gradient feedback
Source-stated
The authors use an oracle model, Kolmogorov n-widths and statistical-query lower bounds to derive a task-agnostic lower bound on initial effective width.
a task-agnostic lower bound on the initial effective width required for target-dependent gradient information to become detectable.
Source-stated
For k-sparse parity, the authors say a known target has an O(k)-width representation, while the general bound implies a polynomial separation between representation and discovery.
a known target admits an \(O(k)\)-width representation, while the general bound already implies a polynomial separation between representation and discovery.
Source-stated
The authors derive a sharper unsigned effective width for parity and report a critical scale at a transfer-compatible measurement resolution.
At a transfer-compatible measurement resolution, its critical scale is \(\widetilde{\Theta}_k\left( D^{k-1}\tau^2 \right)\), up to logarithmic factors.
Source-stated
The authors interpret precision through accumulation over fresh samples to obtain a corresponding width-time frontier.
Interpreting precision through accumulation over fresh samples yields a corresponding width-time frontier.
Source-stated
The authors report that experiments support the predicted dimension dependence and examine departures from the idealised frontier.
Experiments support the predicted dimension dependence and examine how optimisation and generalisation depart from this idealised frontier.
Source and contributor context
Singh, Ryan, De Llanza, Miguel, Buckley, Christopher L. · Article author
These are credits stated by the source; they do not establish a profile match or authorship of every passage.
Source media
No suitable media was found by the supported page extraction.
Terms and references
Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement.
Neural network
A system of connected simple calculations, loosely inspired by the brain, that learns patterns from data.
TIME
TIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
arXiv
arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Editorial provenance
Version 1. Publication ID: 7d96606f-c9e5-409e-85c7-26381e540bc3.
Prepared by gpt-6-luna; grounding review by gpt-6-luna.
Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-11T09:38:28.914Z. This checks fidelity to the source, not whether its claims are independently true.
Editorial selection: accept; reasons: material_development.
Observation: 1cb5881deaae4ae692c1267dd6f5d485212332ca34f8c702cf6d93d127c89dd4. Source text hash: b7c6e9808f6f098f8a848e17c550ffde2fa3eafe917417e1825702533eb93927.
Revision and source-check history
Version 1 · 2026-11-11T09:38:28.914Z
Representation Complexity Along Learning Trajectories: Trading Width for Precision and Time
In this paper, Ryan Singh, Miguel De Llanza and Christopher L. Buckley ask why may use complex representations while learning, even when their final predictors need simpler ones. They distinguish the representation needed to make predictions from the representation needed to make an unknown target detectable through gradient feedback. Using an oracle model of neural network learning, Kolmogorov n-widths and statistical-query lower bounds, they derive a lower bound on the initial effective width needed for target-dependent gradient information to become detectable. For k-sparse parity in D dimensions, they report that a known target has an O(k)-width representation, while discovering it can require substantially greater representation resources. They sharpen the analysis by using parity’s invariance under coordinatewise sign reflections, then derive a width- frontier by interpreting precision through accumulation over fresh samples. The central argument is that learning can require more representational capacity than using the predictor after it is learned: discovery and representation are different computational demands. The authors report that experiments support the predicted dimension dependence, while also examining how optimisation and generalisation depart from the idealised frontier.
Original source
cs.AI · Article
Source details
Original source date: Oct 28, 2026 (2026-10-28).
Date basis: Date basis not provided.
Added to Escape Meridian: 2026-11-11T09:38:28.914Z. First observed: 2026-10-30T06:45:37.787Z. Retrieved: 2026-10-31T11:16:26.632Z.
Limits
The evidence provided is the abstract, not the paper’s full technical argument or experimental details. The abstract reports experiments supporting predicted dimension dependence but does not specify their setup or results in detail. The abstract frames the analysis around k-sparse parity and an idealised frontier; it does not establish how broadly the conclusions extend beyond that setting.
Open question
How far do the reported dimension dependence and width- frontier extend beyond k-sparse parity and the paper’s idealised learning model?
Source-stated
The authors distinguish the representation complexity required to predict from that required to make an unknown target detectable through gradient feedback.
comparing the representation complexity needed for prediction with that needed to make an unknown target statistically visible through gradient feedback
Source-stated
The authors use an oracle model, Kolmogorov n-widths and statistical-query lower bounds to derive a task-agnostic lower bound on initial effective width.
a task-agnostic lower bound on the initial effective width required for target-dependent gradient information to become detectable.
Source-stated
For k-sparse parity, the authors say a known target has an O(k)-width representation, while the general bound implies a polynomial separation between representation and discovery.
a known target admits an \(O(k)\)-width representation, while the general bound already implies a polynomial separation between representation and discovery.
Source-stated
The authors derive a sharper unsigned effective width for parity and report a critical scale at a transfer-compatible measurement resolution.
At a transfer-compatible measurement resolution, its critical scale is \(\widetilde{\Theta}_k\left( D^{k-1}\tau^2 \right)\), up to logarithmic factors.
Source-stated
The authors interpret precision through accumulation over fresh samples to obtain a corresponding width-time frontier.
Interpreting precision through accumulation over fresh samples yields a corresponding width-time frontier.
Source-stated
The authors report that experiments support the predicted dimension dependence and examine departures from the idealised frontier.
Experiments support the predicted dimension dependence and examine how optimisation and generalisation depart from this idealised frontier.
Source and contributor context
Singh, Ryan, De Llanza, Miguel, Buckley, Christopher L. · Article author
These are credits stated by the source; they do not establish a profile match or authorship of every passage.
Source media
No suitable media was found by the supported page extraction.
Terms and references
Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement.
Neural network
A system of connected simple calculations, loosely inspired by the brain, that learns patterns from data.
TIME
TIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
arXiv
arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Editorial provenance
Version 1. Publication ID: 7d96606f-c9e5-409e-85c7-26381e540bc3.
Prepared by gpt-6-luna; grounding review by gpt-6-luna.
Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-11T09:38:28.914Z. This checks fidelity to the source, not whether its claims are independently true.
Editorial selection: accept; reasons: material_development.
Observation: 1cb5881deaae4ae692c1267dd6f5d485212332ca34f8c702cf6d93d127c89dd4. Source text hash: b7c6e9808f6f098f8a848e17c550ffde2fa3eafe917417e1825702533eb93927.
uncertain · 2026-11-11T09:38:28.914Z · anthropic/claude-opus-5.5
Publication: 7d96606f-c9e5-409e-85c7-26381e540bc3
attribution — pass: The paper is correctly attributed to Ryan Singh, Miguel De Llanza and Christopher L. Buckley. The claims are framed as the authors' arguments and reported results, not as established facts.
Authors:Ryan Singh, Miguel De Llanza, Christopher L. Buckley
summary — uncertain: The reading is faithful but may not stand on its own. The paper's framing question is the role of overparameterisation (using larger networks than the final task needs), and the draft omits it. Width, k-sparse parity, oracle model and statistical-query bounds are left unexplained, and no tooltips are evident. Smallest repair: open with the overparameterisation question and say in plain words that width means network size, so readers see why discovery needing more width than representation matters.
What computational role does overparameterisation play during learning?
date — pass: originalPublished is 2026-10-28, which matches the v1 submission date on arXiv. The 2026-10-31 feed date is later and is correctly not used.
[Submitted on 28 Oct 2026]
support — pass: Each claim matches the abstract and keeps the right scope: a lower bound, the parity case, the critical scale and the experiments that support dimension dependence. The limitations correctly note that only the abstract is available and that the results are confined to an idealised setting. The question does not assume an answer.
Thus, the representation resources required to discover a predictor can substantially exceed those required to represent it once learned.
media — pass: The selected media is the verified direct arXiv abstract page for this paper. It is the most direct way to reach the source.
Challenge or correct this reading
Send a correction or an opposing source with this reading’s URL. Disagreement and uncertainty can remain unresolved.
A system of connected simple calculations, loosely inspired by the brain, that learns patterns from data.
Definition referenceTIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
Explore TIMEarXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivarXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivTIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
Explore TIMEA system of connected simple calculations, loosely inspired by the brain, that learns patterns from data.
Definition referenceTIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
Explore TIMEarXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivarXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed.
Explore arXivTIME is a news magazine publishing reporting on world affairs, business, technology and the workers behind digital services.
Explore TIME