gpt 6 astra ultra attempt on CS-729

Regression without dimensional curse: a sampling obstruction, an additive exception, and validation bounds

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For squared-loss regression, the usual target is the conditional mean $m(x)=\mathbb E[Y\mid X=x]$, when defined, and approximation error may be measured by $\mathbb E[(\widehat m(X)-m(X))^2]$. Other losses or norms define other targets. A PDE method approximates a solution $u$ of a specified differential equation, with initial or boundary data, in a chosen error norm. The curse of dimensionality refers to sample or computation requirements growing exponentially with dimension, for example $\epsilon^{-c d}$ to obtain error $\epsilon$ in dimension $d$. Training data fit the estimator; validation data estimate its performance under a specified sampling distribution.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. Source-record limitation: This is a broad deep-learning research agenda without a specified regression class, PDE, sampling model, error criterion, or dimensional dependence. There is no determinate mathematical target to preserve.

Statement recorded in the supplied source.

What error bounds can be obtained for regression of high-dimensional nonsmooth functions, how can PDEs be solved without the curse of dimensionality, and what training and validation data sizes and distributions are efficient and effective for deep learning? [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The source asks about high-Dimensional Regression and PDE Learning Without Dimensional Curse.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004682 (OWR-7155444-003). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mini-Workshop: Analysis of Data-driven Optimal Control $2022$, Oberwolfach Reports, DOI 10.4171/owr/2021/23, pp. 1209--1257 https://doi.org/10.4171/owr/2021/23.
• S3 Jentzen et al., feedforward networks and compositional functions https://epubs.siam.org/doi/10.1137/21M1391596.
• S4 Duan et al., deep neural network approximation of composition functions https://arxiv.org/abs/2304.07947.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

What error bounds can be obtained for regression of high-dimensional nonsmooth functions, how can PDEs be solved without the curse of dimensionality, and what training and validation data sizes and distributions are efficient and effective for deep learning?

Source scope and attempted route

The report, printed pages 1233--1235, discusses compositional approximation theorems and then poses a research agenda about nonsmooth regression, PDEs, and training/validation distributions. It does not quantify a common target class or error norm. We attempt to make the role of structure precise by comparing uniform approximation of unrestricted Lipschitz functions with an additive subclass, followed by a separate validation calculation. All results below are explicitly scoped auxiliary facts, not a solution of the original agenda or a new PDE complexity theorem.

An exponential point-query obstruction

Equip $[0,1]^d$ with $\ell_\infty$ distance. For an integer $m\ge1$ put $h=1/(2m)$ and place $M=m^d$ centers at points whose coordinates are $(2j-1)/(2m)$, $1\le j\le m$. Set \[ b_c(x)=\max(0,h-\|x-c\|_\infty),\qquad f_\sigma(x)=\sum_c\sigma_c b_c(x),\quad \sigma_c\in\{-1,1\}. \] The positive supports of distinct bumps are disjoint. Every $f_\sigma$ is $1$-Lipschitz. Within the same bump this follows from the reverse triangle inequality; if one point is outside all bumps it follows by distance to that bump's boundary. If two points lie in different bumps with centers $c,c'$, some coordinate of the centers differs by at least $2h$. Writing $r=\|x-c\|_\infty$, $r'=\|x'-c'\|_\infty$ gives \[ \|x-x'\|_\infty\ge2h-r-r'=b_c(x)+b_{c'}(x'), \] which bounds even the sum of the two absolute function values. This proves the claim for either choice of signs.

An adaptive randomized algorithm making fewer than $M$ exact point evaluations cannot guarantee uniform error less than $h$ with success probability greater than $1/2$ for every member of this family. To see this, choose the signs independently and uniformly, independently of the algorithm's random seed. Each query reveals at most one previously unknown sign; boundary queries reveal none. Conditional on the complete transcript and seed, all unqueried signs remain independent fair signs, because query choices depend only on already revealed signs. Choose an unqueried center by a fixed ordering. Its reported prediction $z$ is independent of its sign, whereas its true value is $h$ or $-h$. At least one of $|z-h|,|z+h|$ is at least $h$. Thus the conditional probability that the uniform error is at least $h$ is at least $1/2$. Average first over transcripts and then over sign vectors. At least one fixed input function has failure probability at least $1/2$.

Consequently a worst-case point-evaluation guarantee at any error $\epsilon<h$ and success probability exceeding $1/2$ requires at least $m^d$ evaluations. This is a concrete curse for this oracle, norm, and function class; it is not an i.i.d. regression lower bound under an unstated distribution. The same input-information obstruction applies to uniform approximation at time $T$ for the trivial PDE $\partial_tu=0$, $u(0,x)=f(x)$, whose solution is exactly $u(t,x)=f(x)$. It says nothing against specified parabolic PDE classes with stronger regularity or compositional promises.

A structural exception with an explicit network

Assume instead \[ f(x)=\frac1d\sum_{i=1}^d g_i(x_i), \] where each $g_i:[0,1]\to\mathbb R$ is $1$-Lipschitz. No smoothness or differentiability is assumed. Evaluate $f(0)$ and $f((j/m)e_i)$ for each $i$ and $j=1,\ldots,m$, for a total of $1+dm$ queries. They determine \[ q_i(j/m)=f((j/m)e_i)-f(0) =\frac{g_i(j/m)-g_i(0)}d. \] Let $\widehat q_i$ be the linear interpolant of these values, including $q_i(0)=0$, and define $\widehat f(x)=f(0)+\sum_i\widehat q_i(x_i)$. For a $1$-Lipschitz function on an interval of length $1/m$, write $x=(1-t)a+tb$. Its linear interpolation error is at most \[ (1-t)|g(x)-g(a)|+t|g(x)-g(b)| \le 2t(1-t)/m\le1/(2m). \] Divide by $d$ for each $q_i$ and sum to obtain $\|\widehat f-f\|_\infty\le1/(2m)$.

This interpolant has an exact ReLU representation. If $s_{ij}$ is the slope of $\widehat q_i$ on interval $[j/m,(j+1)/m]$, then \[ \widehat q_i(t)=s_{i0}t+ \sum_{j=1}^{m-1}(s_{ij}-s_{i,j-1})\max(t-j/m,0). \] The derivative on each open interval telescopes to the required slope, the value at zero is correct, and continuity proves equality at every knot as well. On $[0,1]$, $t=\max(t,0)$, so including one linear term per coordinate requires at most $dm$ hidden units, with a constant output bias. Query and representation counts are polynomial in $d$ and $1/\epsilon$. This positive result depends on the additive promise and exact designed samples; it does not establish statistical training efficiency for arbitrary deep networks.

Validation with a stated distribution and selection rule

Let $H\ge1$ candidate predictors be fixed independently of $N$ fresh validation samples drawn from a common distribution. Suppose each validation loss lies in $[0,1]$, and write $R_j$ and $\widehat R_j$ for its population and empirical means. Then \[ \Pr\{\max_{j\le H}|\widehat R_j-R_j|>\eta\} \le 2H e^{-2N\eta^2}. \] Here is a complete exponential-moment proof. For a random variable $Z\in[0,1]$, the second derivative of $\log\mathbb E e^{tZ}$ is the variance under its exponentially tilted law. Any law on $[0,1]$ has variance at most $1/4$, since its variance is no larger than $\mathbb E(Z-1/2)^2\le1/4$. Integrating the derivative bound from zero gives $\mathbb E e^{t(Z-\mathbb EZ)}\le e^{t^2/8}$ for every real $t$. Independence, Markov's inequality and $t=4\eta$ bound either tail of a mean by $e^{-2N\eta^2}$. Union over both tails and all candidates proves the display.

In particular, taking $N\ge\log(2H/\delta)/(2\eta^2)$ controls all candidates with probability at least $1-\delta$. Selecting the smallest validation loss then has true risk at most the smallest candidate risk plus $2\eta$, by applying the simultaneous bounds twice. Candidates may depend on a separate training sample, since the argument can condition on that sample. It does not cover repeatedly adapting new candidates to the same validation observations.

Nor can validation from one distribution certify another without assumptions: a function observed only at $x=0$ can take either value $1$ or $-1$ at an unobserved $x=1$, with identical observations at zero. For any prediction $z$ at one, $\max(|z-1|,|z+1|)\ge1$. Thus a distribution concentrated at zero cannot resolve test error concentrated at one.

Adversarial audit and remaining scope

The audit checked the Lipschitz bound across different bump supports, adaptive sign independence, strictness of the error threshold, the use of function queries rather than inaccessible component queries, the ReLU count at $m=1$, and the independence required by validation. No computations or theorem prover were used. The attempt establishes why structural promises and data access matter, but the source's unspecified regression/PDE/data agenda remains unresolved. Its original partially-solved metadata is preserved separately from this attempt's zero completion claim.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. The coordinating Codex (GPT-6; exact serving revision not exposed) also provided a separate model-assisted written-proof review of this manuscript. This is not independent human verification. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. No general regression class, PDE class, sampling law or error target is supplied by the source. The Lipschitz oracle obstruction, additive construction and fresh-validation theorem apply only to their explicit auxiliary models.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• OWR23/2021, Analysis of Data-driven Optimal Control; workshop discussion p.1211 and Kang/Gong compositional approximation discussion pp.1233–1235 inspected from the cached primary report. Historical claims are not a current-status survey. https://doi.org/10.4171/owr/2021/23.

Final status

Unresolved in this attempt. No general regression class, PDE class, sampling law or error target is supplied by the source. The Lipschitz oracle obstruction, additive construction and fresh-validation theorem apply only to their explicit auxiliary models.