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Alexander P. Kreuzer

Publications and source records attributed to Alexander P. Kreuzer.

14 recordsLinked to original sources

Single-Token Expected-Value Scoring for Cold-Start Candidate Ranking

AI-assisted sourcing streamlines candidate review, reducing the administrative burden of manual screening for recruiters. However, deploying language models as production rankers remains challenging. Zero-shot Large Language Models (LLMs) may produce unstable, non-deterministic scores and rank less accurately, while conventional deep neural rankers require millions of logged interactions that a low-traffic, niche sourcing platform does not produce. What is available instead is a few hundred thousand ordinal relevance labels -- small by ranker-training standards, but sufficient when a pretrained language model already encodes the general world knowledge the task depends on. We present single-token expected-value scoring, a ranking primitive that casts candidate-job relevance as an ordinal classification over the grade tokens {1, ..., 5} and reads the relevance score as the expectation of the first-token probability distribution. Because the score comes from a single decoding step rather than open-ended generation, it is a deterministic function of the model's logits, requires no output parsing, and serves at low latency. To learn the non-linear interdependencies of heterogeneous hiring criteria from this supervision alone, we fine-tune a Small Language Model (SLM) with a hybrid ordinal regression loss combining a Mean Squared Error term, which preserves ordinal distance, with a categorical Cross-Entropy term, which sharpens class boundaries. We evaluate along two dimensions -- Jobseeker Relevance and Employer Relevance -- using NDCG@10 and low relevance rate. Offline, our fine-tuned model outperforms a heuristic baseline and zero-shot LLMs. An end-to-end simulation shows the same direction at larger magnitude (+54.2% Jobseeker NDCG@10, -46.7% low relevance rate), and a live online experiment reduces employer low-relevance by 27.3% and raises employer keep rate by 7.07%.

cs.IR

WaLDORf: Wasteless Language-model Distillation On Reading-comprehension

Transformer based Very Large Language Models (VLLMs) like BERT, XLNet and RoBERTa, have recently shown tremendous performance on a large variety of Natural Language Understanding (NLU) tasks. However, due to their size, these VLLMs are extremely resource intensive and cumbersome to deploy at production time. Several recent publications have looked into various ways to distil knowledge from a transformer based VLLM (most commonly BERT-Base) into a smaller model which can run much faster at inference time. Here, we propose a novel set of techniques which together produce a task-specific hybrid convolutional and transformer model, WaLDORf, that achieves state-of-the-art inference speed while still being more accurate than previous distilled models.

cs.LG

On the Uniform Computational Content of Computability Theory

We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

math.LO

On the Uniform Computational Content of the Baire Category Theorem

We study the uniform computational content of different versions of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a complete (i.e., "large") metric space cannot be decomposed into countably many nowhere dense (i.e., "small") pieces. The Baire Category Theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic and likewise they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways how the sequence of closed sets is "given". Essentially, we can distinguish between positive and negative information on closed sets. We discuss all the four resulting versions of the Baire Category Theorem. Somewhat surprisingly it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire Category Theorem to notions of genericity and computably comeager sets.

math.LO

On principles between $Σ_1$- and $Σ_2$-induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between $Σ_1$-induction and $Σ_2$-induction, are equivalent to the well-foundedness of $ω^ω$. Among these principles are the iteration of partial functions ($PΣ_1$) of Hájek and Paris, the bounded monotone enumerations principle (non-iterated, BME$_1$) by Chong, Slaman, and Yang, the relativized Paris-Harrington principle for pairs, and the totality of the relativized Ackermann-Péter function. With this we show that the well-foundedness of $ω^ω$ is a far more widespread than usually suspected. Further, we investigate the $k$-iterated version of the bounded monotone iterations principle (BME$_k$), and show that it is equivalent to the well-foundedness of the $k+1$-height $ω$-tower.

math.LO

Minimal idempotent ultrafilters and the Auslander-Ellis theorem

We characterize the existence of minimal idempotent ultrafilters (on N) in the style of reverse mathematics and higher-order reverse mathematics using the Auslander-Ellis theorem and variant thereof. We obtain that the existence of minimal idempotent ultrafilters restricted to countable algebras of sets is equivalent to the Auslander-Ellis theorem (AET) and that the existence of minimal idempotent ultrafilters as higher-order objects is $Π^1_2$-conservative over a refinement of AET.

math.LO

A lower bound on Gowers' FIN_k theorem

Gowers' FIN$_k$ theorem, also called Gowers' pigeonhole principle or Gowers' theorem, is a Ramsey-type theorem. It first occurred in the study of Banach space theory and is a natural generalization of Hindman's theorem. In this short note, we will show that Gowers' FIN$_k$ theorem does not follow from ACA$_0$.

math.LO

Measure theory and higher order arithmetic

We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor space exists. As base system we take $\mathsf{ACA}_0^ω+ (μ)$. The system $\mathsf{ACA}_0^ω$ is the higher order extension of Friedman's system $\mathsf{ACA}_0$, and $(μ)$ denotes Feferman's $μ$, that is a uniform functional for arithmetical comprehension defined by $f(μ(f))=0$ if $\exists n f(n)=0$ for $f\in \mathbb{N}^\mathbb{N}$. Feferman's $μ$ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reasons $\mathsf{ACA}_0^ω+ (μ)$ is the weakest fragment of higher order arithmetic where $σ$-additive measures are directly definable. We obtain that over $\mathsf{ACA}_0^ω+ (μ)$ the existence of the Lebesgue measure is $Π^1_2$-conservative over $\mathsf{ACA}_0^ω$ and with this conservative over $\mathsf{PA}$. Moreover, we establish a corresponding program extraction result.

math.LO

Bounded variation and the strength of Helly's selection theorem

We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.

math.LO

On idempotent ultrafilters in higher-order reverse mathematics

We analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics. Let (Uidem) be the statement that an idempotent ultrafilter on the natural numbers exists. We show that over ACA_0^w, the higher-order extension of ACA_0, the statement (Uidem) implies the iterated Hindman's theorem (IHT), and we show that ACA_0^w + (Uidem) is Pi^1_2-conservative over ACA_0^w + IHT and thus over ACA_0^+.

math.LO

From Bolzano-Weierstraß to Arzelà-Ascoli

We show how one can obtain solutions to the Arzelà-Ascoli theorem using suitable applications of the Bolzano-Weierstraß principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the Arzelà-Ascoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] --> [0,1] contains a subsequence that converges uniformly with the rate 2^-k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instance-wise equivalent over RCA_0 to the Bolzano-Weierstraß principle BW and that AA_weak is instance-wise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent.

math.LO

Non-principal ultrafilters, program extraction and higher order reverse mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let U be the statement that a non-principal ultrafilter exists and let ACA_0^ω be the higher order extension of ACA_0. We show that ACA_0^ω+U is Π^1_2-conservative over ACA_0^ω and thus that ACA_0^ω+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly Π^1_2 statement \forall f \exists g A(f,g) in ACA_0^ω+U a realizing term in Gödel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f)).

math.LO

On the strength of weak compactness

We study the logical and computational strength of weak compactness in the separable Hilbert space \ell_2. Let weak-BW be the statement the every bounded sequence in \ell_2 has a weak cluster point. It is known that weak-BW is equivalent to ACA_0 over RCA_0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to the Π^0_2-CA. This means that for each Π^0_2 sentence A(n) there is a sequence (x_i) in \ell_2, such that one can define the comprehension functions for A(n) recursively in a cluster point of (x_i). As consequence we obtain that the Turing degrees d > 0" are exactly those degrees that contain a weak cluster point of any computable, bounded sequence in \ell_2. Since a cluster point of any sequence in the unit interval [0,1] can be computed in a degree low over 0', this show also that instances of weak-BW are strictly stronger than instances of BW. We also comment on the strength of weak-BW in the context of abstract Hilbert spaces in the sense of Kohlenbach and show that his construction of a solution for the functional interpretation of weak compactness is optimal.

math.LO

The cohesive principle and the Bolzano-Weierstraß principle

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle (BW) and a weak variant of it. We show that BW is instance-wise equivalent to the weak König's lemma for $Σ^0_1$-trees ($Σ^0_1$-WKL). This means that from every bounded sequence of reals one can compute an infinite $Σ^0_1$-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d >> 0' are exactly those containing an accumulation point for all bounded computable sequences. Let BW_weak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BW_weak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and - using this - obtain a classification of the computational and logical strength of BW_weak. Especially we show that BW_weak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BW_weak.

math.LO