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Machine learning (ML) methods have been explored as a means of optimizing lattice field theory calculations, with increasing interest in recent years. This chapter gives an introduction and summary to the use of ML methods to optimize lattice field theory without introducing any uncontrolled approximations. We focus on those directions which are clearly established as of late 2025, marking the 50th anniversary of the beginning of the study of lattice gauge theory. This chapter is a contribution to the online book ``Lattice QCD at 50 years,'' (LQCD@50), edited by Tanmoy Bhattacharya, Maarten Golterman, Rajan Gupta, Laurent Lellouch, and Steve Sharpe.

This paper presents a comprehensive comparison of four representative Reinforcement Learning (RL) methods in autonomous vehicle (AV) decision-making, including Fitted Value Iteration (FVI), Rollout Classification Policy Iteration (RCPI), Deep Q-Network (DQN), and a discrete-action adaptation of Deep Deterministic Policy Gradient (D-DDPG). The four methods are applied to the same AV decision-making problem at an unsignalized intersection and are compared in terms of optimality, safety, training stability, and training and decision cost. Observations suggest that RCPI provides the closest estimation to the exact solution and has the most stable training curve; however, it is data-intensive and requires the largest volume of data. D-DDPG is the most data-efficient one, but at the price of the training stability. Based on the observations, we provide suggestions for the algorithm selection for AV decision making.

Public benchmark scores reflect skill, prior exposure to the questions, or both, and for most models the training data are unknown. CleanScore is a black-box audit: the estimator needs only a correct-or-incorrect score for each form, not training data or weights. Each benchmark question becomes a parent item with one public form and two independently written fresh forms that keep its numbers, facts and answer. The audit reports an interval for the public-form advantage, not a verdict, and a private control bank with an explicit transport radius provides a proxy for the ordinary form gap that a rewrite alone can produce. A registered controlled-exposure experiment, a preliminary validation, detects planted exposure in one of two model families. A registered audit of five open models on 200 GSM8K and 200 ARC-Challenge items (ARC with stem-only rewrites, scored by option log probabilities) finds no exposure-consistent advantage, and two reruns declared before they were run (GSM8K at a longer token budget, ARC scored from generated letters) leave that conclusion unchanged. For four of the ten model-benchmark pairs the upper end of the interval is about five points or less, and for none is it above 21 points. The registered claim rule is much less sensitive than the primary interval: it flags a planted ten-point advantage in 5% to 11% of audits. Registered positive controls show what a null can mean. In four models, each trained under four seeds, items leaked into fine-tuning gained 25 points on average over unleaked items on the leaked wording and 19 points on paraphrases the model never saw, so a substantial share of the effect, estimated at 76% (95% interval 50% to 113%), is invisible to a paraphrase audit. A null on surface form says much less about contamination than the name suggests.

Uncertainty bands on wind forecasts help inform reserve sizing and commitments, but often fail to deliver their stated coverage. We apply conformal prediction to native bands to see what it adds and what it costs, comparing six forecasters, from ARIMA to a tabular foundation model, at two wind farms in Great Britain (GB) under operational conditions. We show that calibration repairs empirical marginal coverage with little or no Winkler Score penalty, but this apparent success conceals under-coverage in high winds. Calibrating by forecast wind regime turns marginal coverage into regime-level coverage, cutting day-ahead out-of-band volume in high winds by over 40% relative to pooled calibration while narrowing bands overall.

Chemical reaction datasets such as USPTO suffer from substantial incompleteness, frequently missing byproducts, co-reactants, and stoichiometric coefficients. This limits their applicability and reliability in downstream applications. Here, we introduce CompleteRXN, a large-scale supervised benchmark for reaction completion under realistic missing-data conditions. We construct a dataset of aligned incomplete and atom-balanced reactions by mapping USPTO records to curated mechanistic reactions. We evaluate representative baselines, including a novel encoder-decoder reaction completion model with constrained decoding, the Constrained Reaction Balancer (CRB), and a recent algorithmic method, SynRBL. On our CompleteRXN benchmark, the CRB achieves high performance across splits of increasing difficulty, reaching 99.28% equivalence accuracy on the random split and 91.71% on the extreme out-of-distribution split. SynRBL produces many balanced and chemically plausible completions, but with lower accuracy on the benchmark test splits. Across all methods, performance degrades with increasing incompleteness. We observe a substantial drop when evaluating on reactions outside the benchmark (full uncurated USPTO), highlighting the gap between benchmark performance and practical robustness and motivating future work. Data, code and models can be found at: https://github.com/Intelligent-molecular-systems/CompleteRxn-Benchmarking.

Graph similarity computation is an important task in graph mining with many real-world applications, such as chemical compound analysis, program dependence comparison, and social network analysis. The problem is computationally challenging: Graph Edit Distance (GED) and Maximum Common Subgraph (MCS), which quantify the distance between graphs and identify their largest shared subgraph, respectively, are NP-hard to compute exactly. To address these computational challenges, Graph Neural Networks (GNNs) have been applied to this problem. However, existing GNN-based approaches have primarily focused on small-scale graph datasets, while scalability to larger scales remains underexplored. In this work, we propose FiLSim, a scalable GNN-based framework, and investigate whether a simplified GNN pipeline can maintain competitive performance on standard benchmarks while scaling to large graphs. We additionally propose a sequential graph-editing strategy to construct large-scale graph datasets with reliable GED labels to evaluate scalability. The proposed framework is evaluated on four benchmark datasets and four newly created large-scale datasets, demonstrating its effectiveness and scalability. FiLSim remains accurate on graphs with more than 150{,}000 nodes, where the baseline methods run out of memory or fail to learn, and scores all test pairs within seconds on a single GPU.

Representations are routinely used across machine learning, psychology, and neuroscience to draw inferences about the computations of biological and artificial systems. Such inferences presume that the geometry of these representations reflects the computation being performed. For artificial neural networks, however, the extent to which the function a network computes constrains this geometry remains unclear. One key obstacle is that these networks admit parameter symmetries: changes in parameterization that preserve function exactly while reshaping representational geometry. Here, we show that a broad class of parameter symmetries acts on representations through just three primitive feature transformations: addition, duplication, and scaling. This feature-level characterization yields a closed-form decomposition of representational geometry into essential and auxiliary components, revealing how representational degeneracy can exist without and grows with overparameterization even when function is held fixed. Finally, we establish conditions under which implementation-level selection rules resolve this degeneracy, yielding identifiable geometries in which features are weighted according to their contributions to the network's function. Together, our results delineate when representational geometry can support inferences about computation, and when it cannot.

External lexical resources can help large language models (LLMs) translate texts in low-resource languages and specialized domains, but these resources may conflict with models' prior lexical knowledge. To isolate this conflict, we construct a controlled Mandarin-to-English translation task in which familiar source words are assigned new meanings within and across semantic categories. We treat these prompt-supplied meanings as contextual knowledge (CK) and systematically evaluate the effects of word frequency, sentence context, and prompt framing on the translated outputs. Across nine models, CK reliance is greater for less frequent source words and in sentence contexts that support the supplied meanings. Prompt framing can reduce CK reliance, but seven of nine models favor CK even when explicitly instructed to ignore the supplied meanings. Within model families, larger models are more likely than smaller models to utilize CK in their outputs. Three follow-up experiments further show that modal weakening lowers CK reliance, the effect of repetition depends on prompt layout, and the effect of reasoning varies across models. Overall, our controlled study reveals how CK reliance varies across conditions and models, motivating new methods to detect and evaluate lexical conflicts and guide models' lexical choices in LLM-based translation.

We study planning for finite-horizon partially observable Markov decision processes (POMDPs) with finite state, action, and observation spaces. Both exact and approximate planning are known to be computationally hard in this setting, so prior work has explored natural assumptions that make planning easier. Recent work [Golowich et al., 2022] gave quasipolynomial-time approximate planning algorithms for observable POMDPs. These are POMDPs whose observation channels satisfy an $\ell_1$-observability condition, but notably, no assumptions are required on the transition dynamics. In a similar vein, for every fixed $ρ>0$, we give quasipolynomial-time approximate planning algorithms for $ρ$-decodable POMDPs. We show that decodability is implied by both $\ell_1$- and $\ell_2$-observability and is strictly weaker than observability. Moreover, we show that the optimal decodability constant can be computed in polynomial time, whereas computing the optimal $\ell_1$-observability constant is NP-hard in the worst case. We also show that the fast filter-forgetting guarantee underlying the approach of [Golowich et al., 2022] does not extend directly from $\ell_1$-observability to constant $\ell_2$-observability. Thus, decodability is an easy-to-check, unifying assumption under which quasipolynomial-time approximate planning for POMDPs is possible.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as budgeted heteroskedastic multi-judge estimation under independent, unbiased score observations. Given $K$ prompt-response pairs and $J$ judges with known costs but unknown query-judge variances, our goal is to estimate the bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its leading error bound. Since this allocation depends on the unknown variances, we then address the unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and uses optimistic variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a local minimax lower bound matching the leading allocation-dependent rate, establishing asymptotic instance-optimality up to logarithmic factors. An Assouad-type construction based on local, variance-preserving perturbations yields this bound both in expectation and with constant probability. A complementary Fano-type change-of-measure argument captures the logarithmic dependence on the confidence level with a different (suboptimal) allocation geometry. Finally, we numerically validate our approach on synthetic data and resampled HelpSteer2 evaluations, demonstrating improvements over naïve uniform allocation.

Operations research practitioners typically tackle NP-hard combinatorial problems using large neighborhood search (LNS), a scalable heuristic that iteratively refines a current solution by locally re-optimizing subsets of its variables. In contrast, most existing approaches for integrating combinatorial optimization layers into neural networks still assume access to an exact global solution, which is computationally intractable. We bridge this gap by introducing regularized LNS (RLNS). By regularizing or perturbing local subproblems, we turn the LNS heuristic into an efficient MCMC sampler over the combinatorial set of feasible solutions, with associated Fenchel-Young losses. Under entropic regularization, we prove that RLNS performs exact block Gibbs sampling. Furthermore, adjusting the number of RLNS iterations allows us to interpolate between pseudolikelihood and exact maximum likelihood estimation, for end-to-end learning without global solvers. We demonstrate our approach on $k$-subset selection, generalized assignment, and stochastic vehicle scheduling problems.

Probably Approximately Correct (PAC) bounds on the generalization error (the difference between the empirical prediction error computed on the training data and its expectation) are an established tool in statistical machine learning, and there have been several extensions of these bounds to dynamical systems. In parallel, in recent years, there has been growing interest in finite-sample bounds for parameter estimation error in system identification. The latter bounds tend to be algorithm-specific, relying on reformulating the identification algorithm as a least-squares regression problem. This is in contrast to classical PAC bounds, which are algorithm-agnostic and can be applied to any identification algorithm based on minimizing a cost function involving the empirical loss. In this paper, we show how oracle bounds derived from PAC bounds can be used to derive finite-sample bounds for dynamical systems. As an illustration of this approach, we present finite-sample parameter-estimation bounds for linear and bilinear systems with bounded inputs and noise. These bounds apply to identification algorithms based on minimizing a regularized empirical loss function. The key ingredient is to establish an oracle inequality, i.e., an inequality between the true prediction error of the learned model and the minimum achievable true prediction error, and then use either the convexity of the true loss or system-theoretic properties of the loss function to derive bounds on the parameter estimation error. We also analyze the rates of the obtained bounds and compare them with those of classical asymptotic results.

Chain-of-thought (CoT) reasoning has emerged as a powerful paradigm for extending the computational capabilities of transformer-based models. A growing body of recent work has begun to uncover the fine-grained computational power of transformers augmented with CoT, as well as their connections to classical complexity theory. In this work, we advance this line of research by establishing the first hierarchy theorem for CoT-based computation. Our main result shows that for any reasonable resource bound $r(n) \geq n$ and any $\varepsilon > 0$, \[ \mathrm{CoT}(r(n)) \subsetneq \mathrm{CoT}(r(n)^{1+\varepsilon}), \] where $\mathrm{CoT}(r(n))$ denotes the class of decision problems solvable by transformers using $O(r(n))$ CoT reasoning steps on inputs of length $n$. Our result shows that increasing the number of CoT reasoning steps provably yields strictly greater computational power, establishing CoT steps as a fundamental resource in transformer-based computation. Unlike classical hierarchy theorems, which are typically proved via diagonalization, our approach leverages known coarse-grained simulations between CoT-augmented transformers and Turing machines in both directions. Building on these connections, we adapt the classical padding technique from complexity theory to this setting to obtain a tight hierarchy. A key technical contribution of our work is the implementation of padding within the transformer-based CoT model.

The success of modern score-based generative models is largely driven by diffusion-based training, which learns a collection of time-indexed score fields corresponding to progressively noisier versions of the data distribution. While such models achieve impressive sample quality, leveraging them for sound probabilistic inference---particularly sampling from arbitrary conditionals of the model---remains challenging. This difficulty arises because conditioning information is only available for clean data and not for the higher noise levels required to generate exact conditional samples. In this paper, we introduce DIffusion-free SCOre matching (DISCO), a principled approach that sidesteps the need for time-dependent score fields altogether. Our method is based on an objective that, while reminiscent of diffusion-based training, estimates only a single score of the (slightly perturbed) data distribution, for which the exact conditional score is readily obtained. In experiments, DISCO is competitive with state-of-the-art diffusion models in terms of sample quality. More importantly, DISCO yields a more faithful representation of the underlying data distribution and---crucially---enables accurate sampling from arbitrary conditionals, outperforming standard heuristic samplers. This capability opens the door to principled and flexible probabilistic reasoning with score-based models.

OpenAI's agents breached Hugging Face in July 2026, and the same company released more than seven hundred AI-generated manuscripts of mathematical proofs in October 2026. The first event is read as a failure in security, while the second is read as a success in mathematics. We read both through the hybrid framework of our paper (Kim and Kang, 2026, arXiv:2611.01914), in which coding, proving, and learning from data are derivations by one and the same deductive mechanism, distinguished by the kinds of premises from which they are derived. Under this framework, the AI systems behind the two events ran the same loop, in which a statistical proposer produces candidate exploits or proofs and a deductive verifier settles each one by a derivation. What separates the events is the logical form of the statement wanted by each of the two domains, security and mathematics. The safety of a program is a $Π_1$ statement, for it says that no exploit exists, which is not assured by any finite list of tests that have been passed by the program. By contrast, the provability of a sentence is a $Σ_1$ statement, for it says that a proof exists, which is assured by a single instance, namely a proof that has been checked by the deductive verifier. The theorem behind both is Turing's Undecidability Theorem: the halting set is $Σ_1$ but not decidable, so its complement is $Π_1$ and not $Σ_1$, which is to say that membership has a finite witness but non-membership has none. We prove that attaching a statistical proposer to a deductive verifier does not change this undecidability result, however good the proposer is. We then discuss what does assure safety and where the residual risk goes.