Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

Inverse Optimal Control (IOC) aims to recover the cost function that explains observed trajectories as solutions of an optimal control problem. Classical IOC formulations rely on bilevel optimization, which repeatedly solves a nested optimal control problem and quickly becomes computationally prohibitive for realistic systems. Recent projection-based approaches offer a promising alternative but suffer from numerical instability when solved with gradient-based methods due to violations of standard constraint qualifications. In this paper, we show that these difficulties stem from the geometric structure of the IOC feasible set. We demonstrate that the set of trajectories satisfying the optimality conditions naturally forms a manifold and reformulate IOC as an optimization problem on this manifold. Based on this insight, we propose a Riemannian Inverse Optimal Control (RIOC) method that projects observed trajectories onto the manifold of optimal solutions while preserving feasibility by construction. Experiments on real human arm trajectories show that the proposed method achieves comparable or better reconstruction accuracy than classical bilevel IOC while reducing computation time by about a factor of four. These results highlight the potential of geometric optimization methods to improve the scalability and reliability of IOC for robotics and human motion analysis.

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on five of six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single proposition as its endpoint, Moonshine builds an extensible theoretical framework through conjecture generation, bridge building, and obstacle identification. This article uses Moonshine's exploration of the Jacobian conjecture as an example. It shows how the central logic of whether local nondegeneracy can force global injectivity is transferred to one-hidden-layer affine-ridge sigmoid networks. This leads to the formulation of the \emph{Neural Jacobian Conjecture} (NJC): if such a network has strictly positive Jacobian determinant on the whole space, then it must be globally injective. By invoking GPT-5.5-pro and DeepSeek-V4-pro separately, Moonshine obtained independent complete proofs for the case \(N=n+1\). In addition, with the assistance of ChatGPT through interactive use of its web interface with GPT-5.5-pro, a geometric-topological proof was developed. These results provide preliminary evidence for the plausibility of the conjecture. The general higher-width case \(N\ge n+2\), however, remains unresolved and is left for further investigation. This work illustrates Moonshine's ability to autonomously generate meaningful mathematical problems and make rigorous progress on them.

We present NOVA, an autonomous symbolic regression framework that identifies interpretable car-following and lane-change structures from raw trajectory data with minimal behavioral priors. Applied to 4,765,788 active driving observations from the NGSIM I-80 and US-101 datasets, NOVA's deterministic Rust-powered search engine evaluates over 10,000 candidate algebraic structures and identifies a compact two-term acceleration model under a forward-shifted rolling-mean prediction target. Evaluated under two complementary preprocessing pipelines, NOVA achieves $RMSE = 1.376 m/s^2$ ($R^2 = 15.57\%$) on the intent-forecasting benchmark, outperforming the best recalibrated symbolic-regression baseline (SR-LLM, PNAS~2025) by 0.135 m/s$^2$ in RMSE under an identical evaluation protocol. Across eight independent experiments, a single dominant nonlinear term emerges as a robust backbone of human car-following; a residual-guided extension further links the selected structure to an established psychophysical theory of collision avoidance. The discovered feature operators transfer zero-shot between freeway sites with under 3 pp $R^2$ loss. Extended to lane-change modelling within a multinomial logit framework, NOVA achieves 67.4\% balanced accuracy under strict vehicle-ID holdout on 502 unseen drivers, surpassing existing lane-changing baselines by +29.8 percentage points on a three-class problem.

Neural combinatorial optimization (NCO) trains autoregressive policies to solve routing problems. The standard training algorithm, REINFORCE with a rollout baseline, requires maintaining and periodically updating a frozen copy of the policy for variance reduction. This baseline introduces a structural vulnerability: on harder instances, a poor baseline produces noisy gradient estimates that can destabilize training. We evaluate Group Relative Policy Optimization (GRPO), an algorithm from large language model alignment that eliminates the baseline entirely by normalizing advantages within groups of sampled trajectories. In a controlled comparison of five RL algorithms on TSP and CVRP benchmarks within the RL4CO framework, we find that: (i) GRPO avoids the training collapse observed with REINFORCE on TSP-100, where performance degrades from cost 9.8 to 52.1 immediately after the warmup phase and does not recover under extended training; (ii) at matched gradient updates, GRPO achieves solution quality within 2% of POMO, a strong AM-based multi-start baseline, while requiring no external baseline; and (iii) P3O, a pairwise preference algorithm also from the alignment literature, is competitive on TSP but shows higher variability on CVRP. These results identify GRPO as a promising baseline-free alternative for NCO, particularly in settings where baseline-dependent training becomes fragile.

Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics. Existing methods are mainly designed for static settings and lack adaptability to streaming and time-varying environments. Adaptive filtering techniques have also been largely limited to data with scalar or vector values, leaving adaptive forecasting for matrix-valued time series inadequately understood. To bridge these gaps, we develop an adaptive tensor regression framework that includes Matrix-on-Matrix (MoM) and Tensor-on-Matrix (ToM) formulations for streaming matrix-valued prediction. The two formulations differ in whether to directly model matrix-valued outputs or to exploit temporal structure via higher-order tensor representations. For the proposed tensor regression framework, we develop stochastic gradient descent (SGD) algorithms for online learning. We show that stacking multiple responses across time into higher-order tensors improves performance; in particular, the ToM achieves lower steady-state error and stronger denoising capability than MoM, motivating our focus on the ToM model. We further characterize the tracking behavior of SGD under time-varying dynamics. From a statistical perspective, we establish fixed-time recovery guarantees for ToM under general low-dimensional structures, including sparsity, low-rankness, and their joint sparselow-rank models.

In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.

Reduced-order models (ROMs) provide an efficient surrogate for complex multiscale systems, but their predictive accuracy is often compromised by truncation errors and the inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) scales on ROM (resolved) scales is often denoted as the closure problem. In this work, we formulate ROM closure modeling as a multi-fidelity (MF) learning problem and propose an uncertainty-aware MF framework based on conditional normalizing flow to enhance ROM predictive accuracy. The proposed approach learns a probabilistic mapping from low-fidelity (LF) ROM coefficients to high-fidelity (HF) coefficients, thereby improving predictive fidelity while quantifying the uncertainty associated with the learned closure. Two correction strategies are investigated: direct learning, in which HF coefficients are predicted directly from LF inputs, and residual learning, which learns the discrepancy between LF and HF coefficients and uses it to recover the corrected HF solution. The framework is demonstrated on a vortex merging problem governed by the two-dimensional Navier Stokes equations. Results show that both correction strategies improve ROM accuracy over uncorrected ROM, with residual learning achieving consistently better performance than direct learning. Moreover, the two proposed deep generative model-based strategies provide uncertainty quantification for the corrected ROM coefficients, which is critical for assessing prediction confidence and supporting the reliable use of ROMs in practical applications.

Averaging a neural network over its random parameters and marginalizing a Gaussian sector are the same operation, the Schur complement of the eliminated block, and when that block is closed it returns a covariance and its inverse. That is all a network ensemble produces, the closed case. The open case is missing, and nuclear reaction theory has it worked out. Projecting a scattering problem onto a chosen set of channels, with the rest carrying probability irreversibly to a continuum, leaves a non-Hermitian effective generator that conserves and itemizes exactly what it loses: the nuclear optical model and its generalized optical theorem. I set the two cases side by side using only the moments of a distribution, the algebra of Gaussians, and block inversion, no field theory, and give the closed-case dictionary in full: the neural tangent kernel is the Fisher sensitivity kernel, the infinite-width Gaussian limit is the Gaussian-process emulator, and the lazy-to-feature transition is the validity boundary of a reduced-basis emulator. I then test the open export on a truncated attention map, a token-level transfer operator, and a sparse expert router, and report a mostly negative result. The conserved flux ledger ports wherever openness is genuinely present, but its distinctive content is absent, an artifact of the chosen partition, or pinned near a floor by the training objective, and the operationally useful uncertainty turns out to be epistemic, living in the closed half of the correspondence, not the open one. The negative has a structural reason this note makes precise: the open case needs an eliminated sector with a continuous spectrum and wave-like, not relaxational, dynamics, which mainstream learning's finite or dissipative objects do not supply. This is a note, not a result; its main finding is that negative one, and its value is the map that locates it.

Gaussian-process upper confidence bound (GP-UCB) and decision-estimation-coefficient (DEC) methods may appear, at first sight, to belong to different theories. This paper places the two viewpoints in a common algorithmic-information language for frequentist RKHS bandits. GP-UCB fixes an algorithmic, rather than true, Gaussian-process prior and exploits realized-trajectory complexity together with computational tractability, whereas MAMS optimizes a robust class-wide MAIR/DEC envelope. Through the unified MAIR framework and heterogeneous positive-semidefinite algorithmic priors, we generalize both the GP-UCB analysis and the MAMS algorithm, propose a safeguarded master that combines their advantages, and provide a kernel-bandit construction showing that algorithmic complexity can be more informative than class-wide minimax or DEC certificates in overparameterized models. The resulting message is that algorithmic information and class-wide minimax coefficients answer different questions and can lead to different gaps; kernel bandits provide a clean setting in which this distinction becomes mathematically visible.

The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.

Long-horizon autoregressive forecasting of oscillatory physical signals, such as seismograms, gravitational-wave strain, and similar wavefields is limited by error accumulation: as a causal model is fed its own outputs over hundreds of steps, small per-step errors compound into phase drift that pointwise metrics fail to detect. We ask when such rollout stays stable, using synthetic three-component seismograms as a physically structured testbed and the \textsc{SeismoGPT} autoregressive forecaster as the model under study. Through controlled, intra-architecture ablations evaluated on free-running rollout with paired significance tests, we isolate the contribution of each design choice. Multi-token prediction is the dominant stabilizer, accounting for almost the entire improvement over a single-token baseline ($+0.040$ median NCC); a horizon-embedding hybrid prediction head and a cross-horizon STFT-magnitude coherence loss each add a small but consistent further gain. Performance depends sharply on a context-ratio threshold near one, roughly the full P-S interval of observed signal, below which rollout generalization collapses. The dominant residual failure is a polarity inversion that a magnitude-based spectral loss cannot, by construction, penalize, identifying phase-aware objectives as the natural next step. We frame this as a controlled study of rollout stability on oscillatory wavefields, not a benchmark of forecasting architectures.

Recent advances in generative AI, especially powerful Large Language Models (LLMs) and Large Reasoning Models (LRMs), raise concerns over the interpretability, safety and sustainability of these large and opaque AI models. The power of such architectures is derived not only from the scalability of deep neural networks, but also massively parallel hardware such as GPU clusters. The diffuse nature of deep neural networks gives them great function-approximation capability when provided with sufficient training data but imposes a cost in interpretability and computational efficiency. Observing that localised machine learning (ML) models tend to be more interpretable and computationally efficient than deep neural networks on small datasets, we reason by analogy that similar advantages may apply to specific localised hardware ML architectures. We argue that localised architectures with lower bandwidth but higher expressivity per node have the potential to be fundamentally more interpretable than deep neural networks running on GPU clusters while remaining competitive for smaller datasets. We then evaluate the suitability of various hardware ML paradigms for implementing such localised architectures and evaluate their per-node expressivity, energy efficiency and practical maturity of the technology required.

In the current era of deep learning and especially generative models, there is significant investment in training very large deep neural networks. Thus far, such models have been "black boxes" that are difficult to understand in the sense that they have opaque internal mechanisms, leading to difficulties in interpretability, reliability, and control. Naturally, this lack of understanding has led to both hype and fear. This book is an attempt to "open the black box" and understand the mechanisms of large deep networks, through the perspective of representation learning, which is a major factor - arguably the single most important one - in the empirical power of deep learning models. A brief outline of this book is as follows. Chapter 1 will summarize the threads that underlie the whole text. Chapters 2, 3, 4, 5, and 6 will explain the design principles of modern neural network architectures through optimization and information theory, reducing the process of architecture development (long having been described as a sort of "alchemy") to undergraduate-level linear algebra and calculus exercises once the underlying principles are introduced. Chapters 7 and 8 will discuss applications of these principles to solve problems in more paradigmatic ways, obtaining new methods and models which are efficient, interpretable, and controllable by design, and yet no less - sometimes even more - powerful than the black-box models they resemble. Chapter 9 will discuss potential future directions for deep learning, the role of representation learning, as well as some open problems.

Using a language model to score or rank candidate responses has become a scalable alternative to human evaluation in reinforcement learning from human feedback (RLHF) pipelines, benchmarking, and application layer evaluations. However, output reliability depends heavily on prompting and aggregation strategy. We present an empirical investigation of four drop-in techniques -- ensemble scoring, task-specific criteria injection, calibration context, and adaptive model escalation -- for improving LLM judge accuracy on RewardBench 2, with a unifying lens of noise control on the stochastic judge: ensembling as Monte Carlo averaging over per-call noise, criteria injection as between-response discrimination sharpening, and per-response score variance as an uncertainty signal. Ensemble scoring and task-specific criteria injection (the latter virtually cost free) together reach up to 85.8% accuracy, +13.5pp over baseline. Calibration context and adaptive model escalation also improve over baseline but are dominated by criteria + ensembling on the cost-accuracy Pareto frontier. Small models benefit disproportionately from ensembling, making high-accuracy LLM judges accessible at low cost. We show that these techniques generalise across model providers, evaluating on both OpenAI GPT and Anthropic Claude families.

Steck, Ekanadham, and Kallus [arXiv:2403.05440] demonstrate that cosine similarity of learned embeddings from matrix factorization models can be rendered arbitrary by a diagonal ``gauge'' matrix $D$. Their result is correct and important for practitioners who compute cosine similarity on embeddings trained with dot-product objectives. However, we argue that their conclusion, cautioning against cosine similarity in general, conflates the pathology of an incompatible training objective with the geometric validity of cosine distance on the unit sphere. We prove that when embeddings are constrained to the unit sphere $\mathbb{S}^{d-1}$ (either during or after training with an appropriate objective), the $D$-matrix ambiguity vanishes identically, and cosine distance reduces to exactly half the squared Euclidean distance. This monotonic equivalence implies that cosine-based and Euclidean-based neighbor rankings are identical on normalized embeddings. The ``problem'' with cosine similarity is not cosine similarity, it is the failure to normalize.

This paper investigates the theoretical behavior of generative models under finite training populations. Within the stochastic interpolation generative framework, we derive closed-form expressions for the optimal velocity field and score function when only a finite number of training samples are available. We demonstrate that, under some regularity conditions, the deterministic generative process exactly recovers the training samples, while the stochastic generative process manifests as training samples with added Gaussian noise. Beyond the idealized setting, we consider model estimation errors and introduce formal definitions of underfitting and overfitting specific to generative models. Our theoretical analysis reveals that, in the presence of estimation errors, the stochastic generation process effectively produces convex combinations of training samples corrupted by a mixture of uniform and Gaussian noise. Experiments on generation tasks and downstream tasks such as classification support our theory.

This paper introduces KLong, an open-source LLM agent trained to solve extremely long-horizon tasks. The principle is to first cold-start the model via trajectory-splitting SFT, then scale it via progressive RL training. Specifically, we first activate basic agentic abilities of a base model with a comprehensive SFT recipe. Then, we introduce Research-Factory, an automated pipeline that generates high-quality training data by collecting research papers and constructing evaluation rubrics. Using this pipeline, we build thousands of long-horizon trajectories distilled from Claude 4.5 Sonnet (Thinking). To train with these extremely long trajectories, we propose a new trajectory-splitting SFT, which preserves early context, progressively truncates later context, and maintains overlap between sub-trajectories. In addition, to further improve long-horizon task-solving capability, we propose a novel progressive RL, which schedules training into multiple stages with progressively extended timeouts. Experiments demonstrate the superiority and generalization of KLong, as shown in Figure 1. Notably, our proposed KLong (106B) surpasses Kimi K2 Thinking (1T) by 11.28% on PaperBench, and the performance improvement generalizes to other coding benchmarks like SWE-bench Verified and MLE-bench.

In this study, we propose an overlapped wavelet diffusion framework for Low-Light Image Enhancement (LLIE), which incorporates two complementary components to achieve blocking artifact-free and detail-preserving enhancement. Although recent diffusion-based LLIE methods have demonstrated remarkable performance compared with traditional approaches, DiffLL still suffers from blocking artifacts caused by the Haar Wavelet Transform (WT) and blurred edges or over-smoothed textures due to the limitations of its High-Frequency Restoration Module (HFRM). To overcome these issues, we introduce an Overlapped WT (OWT) that incorporates correlations across neighboring regions, thereby structurally preventing blocking artifacts. Furthermore, we integrate a low-frequency-guided High-Frequency Enhance Block (HFEBlock) to strengthen detail recovery, yielding sharper edges and more reliable textures. Extensive experiments on the LOLv1 and LOLv2-real datasets demonstrate that our framework, termed OWDiff, consistently outperforms existing LLIE methods both qualitatively and quantitatively, achieving superior visual quality while maintaining computational efficiency. OWDiff effectively addresses the structural limitations of the Haar WT and the HFRM, achieving an average PSNR gain of 0.58 dB, along with a 1.64% relative improvement in SSIM and a 5.9% relative reduction in LPIPS, compared to DiffLL across both the LOLv1 and LOLv2-real datasets.

Microservice systems are widely used to build cloud applications, yet their complexity makes failures inevitable, degrading user experience and causing economic loss. Automated anomaly detection and root cause analysis (RCA) are now active research areas, but existing techniques share five limitations. First, most treat anomaly detection and RCA separately, assuming anomalies are detected correctly, and falter when detection is imprecise due to noise or delay. Second, they focus on metrics, logs, and traces, leaving event data such as API calls and configuration changes underexplored. Third, many require a given service call graph and cannot diagnose without one. Fourth, the field lacks standardised datasets and evaluation frameworks, so methods are hard to compare fairly. Fifth, although causal inference-based RCA has become dominant, its effectiveness, efficiency, and robustness remain unclear. This thesis addresses these limitations through two groups of contributions. The first introduces methods that exploit observability data both independently and collectively. BARO is an end-to-end anomaly detection and RCA approach for metric data. EventADL is an end-to-end framework for event data. TORAI is a multimodal RCA framework that requires no service call graph. Extensive experiments on real microservice systems demonstrate their effectiveness and robustness. The second group delivers benchmarking datasets, an evaluation framework, and systematic evaluation efforts. RCAEval is a comprehensive benchmark providing ready-to-use datasets and reproducible baselines for future research. A systematic evaluation of existing RCA methods, especially causal inference-based approaches, offers insights that guide future directions. This thesis thereby advances automated anomaly detection and RCA for microservice failures, enabling future research on incident mitigation and remediation.

The boundary between program execution and gradient-based optimization has long limited the use of code itself as a learnable scientific model. We present a compiler that translates a self-hosting subset of Scheme into differentiable computation graphs for autograd backends. Because the subset can compile its own evaluator, this yields differentiable meta-circular interpretation (DMCI): a compiled Scheme interpreter executes programs supplied as data, while reverse-mode autodiff propagates gradients to continuous constants embedded in those programs. The interpreter is compiled once, so new programs inherit differentiability without recompilation or custom gradient machinery, while retaining closures, recursion, and data structures. We prove that gradients through the compiled interpreter are correct almost everywhere and show that they match direct compilation to numerical precision across 171 recursive and higher-order program-seed pairs. We then use DMCI for program-and-parameter co-search, where a large language model proposes Scheme programs and exact gradients calibrate their continuous parameters through a single frozen interpreter. This enables OpenEvolve-style program search in which an outer loop proposes discrete program structures and DMCI supplies exact gradient-based calibration of each candidate's continuous parameters. On battery capacity-fade data, the search recovers a knee-like degradation structure and improves held-out extrapolation over hand-crafted baselines on the harder early-extrapolation split, matching them on the later split. On a high-dimensional El Nino inverse problem, DMCI optimizes an interpreted Kalman-filter likelihood where gradient-free search fails. These results extend symbolic regression and neurosymbolic search from closed-form expressions to executable, stateful programs, making model-generated code directly optimizable against data.

Robust decision-making requires compression. A system that forms a rich support state cannot usually preserve its full structure at the point of action. It must retain only those distinctions needed to act, verify, abstain, or defer under the current consequence geometry. This paper formalizes support sufficiency as action-sufficient compression. Let $H$ denote a full support state, $\mathcal{A}$ a finite action set, and $Z$ a consequence geometry specifying payoff structure. For fixed $Z$, the coarsest exactly action-sufficient compression is the quotient of support space by policy equivalence. Two support states may be merged exactly when they require the same optimal action. This clarifies why content-only and scalar-confidence-only arbitration fail whenever their induced partitions cross action boundaries. Approximate sufficiency is then defined by bounded expected policy regret. In the finite single-cycle setting, this yields a rate-regret problem with source $H$, reproduction alphabet $\mathcal{A}$, and distortion given by consequence-sensitive regret. The optimal stochastic action channel inherits the standard rate-distortion Gibbs form, applied here to support states with regret distortion. The contribution is interpretive: action adequacy is distinguished from reconstruction fidelity, information-bottleneck prediction, and rational inattention. Robust single-cycle arbitration does not require preserving all support, but it does require preserving the distinctions that consequence geometry makes action-relevant.

Current LLM coding agents are predominantly trained on composite benchmarks (e.g., bug fixing), which often leads to task-specific overfitting and limited generalization. To address this, we propose a novel scaling paradigm that shifts the focus from task-level optimization to atomic skill mastery. We first formalize five fundamental atomic skills, code localization, code editing, unit-test generation, issue reproduction, and code review, that serve as the basis vectors for complex software engineering tasks. Compared with composite coding tasks, these atomic skills are more generalizable and composable. Then, we scale coding agents by performing joint RL over atomic skills. In this manner, atomic skills are consistently improved without negative interference or trade-offs between them. Notably, we observe that improvements in these atomic skills generalize well to other unseen composite coding tasks, such as bug-fixing, code refactoring, machine learning engineering, and code security. The observation motivates a new scaling paradigm for coding agents by training with atomic skills. Extensive experiments demonstrate the effectiveness of our proposed paradigm. Notably, our joint RL improves average performance by 18.7% on 5 atomic skills and 5 composite tasks.

We build upon a simple micro-founded model of asset trading proposed by Kyle (1985) to study under what conditions a trader who is privately informed of the future return of the asset may want to share her information with other traders. Despite what conventional wisdom suggests, we show that in the unique equilibrium of the game the informed trader reveals her information with positive probability. A consequence of it is that, in contrast with the corresponding no-communication benchmark, the equilibrium price need not be fully revealing of the asset's return, even if traders are risk neutral. This, in turn, has significant implications on the distribution of the social surplus. While our model initially assumes that inter-agent communication is restricted by an arbitrarily given social network, we also study which such networks arise when links are endogenously formed through traders' prior connection decisions.

The purpose of this paper is to consider whether and how panel data can be used to estimate individual demand, as opposed to market-level demand, while accounting for simultaneity resulting from prices being determined in markets. We consider linear demand models and random coefficient demand models, together with linear supply models. We find that the bias of individual demand estimates obtained using familiar panel data methods, like differencing, disappears as the number of consumers in each market grows, as long as the time-varying, i.e. idiosyncratic, component of preferences is orthogonal to the unobserved, time-varying component of supply. This approximate control is assumed in many panel discrete choice models and is plausible in other models where idiosyncratic preferences represent random variation in preferences over time. Macroeconomic effects can be allowed for by including regressors characterizing time effects, such as trends and time period dummies, or fixed time effects.

Brandt et al. (2016) characterized a probabilistic social choice function known as maximal lotteries within a framework based on fractional preference profiles, which abstracts away from individual voters. While this modeling assumption enables a more elegant and transparent proof, it complicates comparison with other results in the literature. The purpose of this note is to transfer their results to the standard model of social choice, where each preference profile is defined for a finite number of voters. Along the way, we prove a slightly stronger version of their main theorem that uses a weaker continuity condition and allows for real-valued (rather than only rational-valued) probabilities.

The Baldwin and Nanson rules are two voting rules proposed to identify the Condorcet winner whenever one exists. Both rules operate as recursive Borda elimination procedures: the Baldwin rule successively eliminates the alternatives with the lowest Borda score, whereas the Nanson rule eliminates all alternatives whose Borda scores do not exceed the mean. This paper investigates the axiomatic properties of the Baldwin and Nanson rules and provides unified axiomatic characterizations. In particular, our axioms are closely comparable to Young's (1974) characterization of the Borda rule.