arXiv subjects

Martin Anthony

Publications and source records attributed to Martin Anthony.

4 recordsLinked to original sources

arXiv2026

Finite-Sample Coverage Audits for High-Recall Candidate Generation: Certification and Learning-Theoretic Design

An initial high-recall stage in an empirical pipeline decides which items pass to later review, labelling, or modelling, and relevant items it misses are lost to every subsequent stage. We study how many audit labels are needed to certify, with finite-sample validity, that this missed relevant mass is small, and our main results characterise the label complexity of this problem. We first show that no procedure using only labels from inside the candidate set can certify any non-trivial bound on the missed mass: the audit must sample the excluded pool, the only region where unrecovered relevant items can lie. We then prove a matching finite-corpus lower bound. Any valid audit that certifies fewer than $m$ missed relevant items with high probability when none are present, even if adaptive and permitted to label the entire included pool, must inspect on the order of $N_0/m$ excluded-pool labels. Excluded-pool auditing is therefore minimax rate-optimal, not merely convenient, for missed-mass certification in the zero-miss regime. Building on this characterisation, we develop an exact finite-sample toolkit, using binomial and hypergeometric inversion rather than asymptotic approximation, that certifies missed mass, converts it to recall through a two-pool design, certifies pre-specified families of nested candidate generators simultaneously, and produces stress-test certificates against declared perturbation mechanisms. These certificates can be paired with observable review burden to select the least burdensome pre-specified candidate generator meeting a missed-mass target. Every guarantee holds under one discipline: the candidate generator, or the pre-specified family from which it is selected, and the audit rule are fixed before the certification labels are examined.

cs.LG↗

arXiv2026

Chamber geometry and specification numbers of Boolean threshold functions

The specification number $\sigma_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions. Its essential points form the unique minimum such set. We develop Zuev's geometric interpretation: the threshold functions are the chambers of a central hyperplane arrangement in the $(n+1)$-dimensional space of weights and thresholds, and the essential points of a function correspond exactly to the facets of its chamber, so the specification number is the chamber's facet number. The lower bound $\sigma_n(f)\ge n+1$ becomes the fact that a pointed full-dimensional cone has at least $n+1$ facets, with equality for simplicial chambers. The average specification number $\overline\sigma_n$ becomes an average facet count. We evaluate this average exactly via the resonance arrangement and bound it through a theorem of Fukuda, Tamura, and Tokuyama, obtaining $\overline\sigma_n\le 2n$; hence $\overline\sigma_n=\Theta(n)$. This settles a question of Gutekunst, M\'esz\'aros, and Petersen. The method also extends to polynomial threshold functions. The same geometry links threshold functions with a threshold zonotope, whose vertices are modified Chow vectors. Its one-skeleton is the one-inclusion graph, and a vertex's degree is the specification number of that function. Finally, we treat the operations of Lozin et al. on functions of minimum specification number. Adding a variable and extending on a variable both take the product of a chamber closure with a half-line, preserving simpliciality. For the symmetric-variables extension we give an exact thresholdness criterion and show that minimum specification number is preserved whenever the extension is a threshold function. We also resolve a question they pose concerning a fourth operation.

cs.DM↗

arXiv2026

Generalised Eigenvalue Geometry of Semantic Adversarial Attacks

Recent empirical work shows that semantically equivalent paraphrases can fool financial sentiment classifiers: although a paraphrase remains close to the original under a strong reference embedding, it may shift the target model's representation enough to change the predicted class. Existing robustness theory either assumes a single-model threat model or focuses mainly on empirical attack algorithms. We develop a continuous local model of semantic paraphrase perturbations that captures this two-model structure. We show that the worst-case local displacement of the target representation, subject to a proxy-model budget, is governed by the largest generalised eigenvalue of a matrix pencil $(A,B)$ constructed from the Jacobians of the two embedding maps. The resulting attackability index $\lambda^*(x)$ is intrinsic to the local paraphrase geometry and the chosen embedders, yields a closed-form prediction-flip condition for affine readouts, and supports conservative population and finite-sample attackability certificates. For uniform control over classes of affine readouts, we derive a distribution-free VC bound for binary attackability indicators and a scale-sensitive margin bound based on an attackability-adjusted margin that subtracts a local geometric penalty from the standard classifier margin. We also connect the continuous theory to discrete paraphrase search, identify an asymmetry between successful and unsuccessful finite searches, and give a covering condition under which the discrete and continuous settings agree. Finally, we propose an empirical verification framework using soft-token relaxations and generated paraphrase sets to assess the local eigenvalue geometry, prediction-flip condition, and finite-search approximation on a deployed financial-text classifier.

stat.ML↗

arXiv2014

Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).

math.OC↗