Scientific reasoning is an increasingly important capability of large language models, yet improving the robustness and efficiency of training such reasoning remains a key open challenge. We study this problem in instruction-based molecular optimization, where answer-only supervised fine-tuning (SFT) collapses multi-step reasoning and reinforcement learning with verifiable rewards (RLVR) suffers from sparse feedback. Reference-guided Policy Optimization (RePO) mitigates both by anchoring policy updates to dataset-provided references, but its effectiveness is tightly coupled to reference quality: weak or misaligned references impose a performance ceiling. To overcome this ceiling, we propose active reasoning, a paradigm in which the policy actively decides, on a per-instance basis, when to imitate a reference and when to reinforce its own discoveries, while continuously upgrading what it imitates. We instantiate this paradigm as Active Group Relative Policy Optimization (Active-GRPO), realized through two coupled mechanisms: active imitate-reinforce and active referencing. The former performs imitation learning when the reference still outperforms the policy's own candidates, and shifts to self-improvement via reinforcement learning once the policy has generated molecules that surpass the reference. The latter continuously upgrades the reference itself by replacing it with the best policy-generated candidate discovered so far, progressively raising the imitation target and ensuring that reference guidance remains informative--rather than restrictive--throughout training. Across TOMG-Bench MOLOPT, Active-GRPO improves average SR Sim from 0.0959 for GRPO and 0.1665 for RePO to 0.1773 under matched three-seed evaluation, with statistically significant gains on LogP, MR, and QED.

This paper introduces the R package spca, which provides a computational framework for least squares sparse principal component analysis (LS-SPCA). Unlike other SPCA methods, LS-SPCA generates uncorrelated sparse principal components (sPCs) that effectively maximize the explained variance while maintaining strong correlations with standard principal components (PCs). The framework also includes more computationally efficient variants that produce mildly correlated sPCs, which often have lower cardinality while explaining equal or greater variance than the LS-SPCA optimal sPCs. The spca package is built on an efficient C++ backend for matrix computations, with distinct engines for tall and fat matrices, and a flexible R frontend. The user interface offers several options for computing sPCs, such as deciding whether sparsification should stop when a threshold for cumulative variance explained or R2 with the PCs is reached, and choosing between simple forward selection, stepwise forward selection, or backward elimination for variable selection. In addition to the print(), summary(), and plot() methods, the package includes tools for comparing different "spca" solutions, grouping sparse loadings, and representing foreign SPCA solutions as "spca" objects. This article demonstrates with real datasets the use of the package in a typical LS-SPCA workflow and briefly contrasts LS-SPCA with conventional SPCA solutions . Then it compares different LS-SPCA solutions obtained from the dataset. Finally, the performance of spca on large tall and fat matrices is discussed, showing that spca offers a computationally efficient alternative for computing interpretable sPCs.

Prediction sets should have high coverage to be useful, but some coverage notions are more practically relevant than others. In the classification setting, class-conditional coverage requires that the prediction set (i.e., the set of candidate labels for a new test point) must achieve the target accuracy level within each class, which may be challenging to satisfy when many classes are rare and have few calibration points. At the other extreme, marginal coverage requires only that coverage holds on average over the distribution of all classes, which can lead to low-probability labels being essentially ignored. To find a middle ground, recent work has introduced macro-coverage, defined as the unweighted average of class-conditional coverages. Macro-coverage offers a compromise between marginal coverage and class-conditional coverage that is particularly appropriate for long-tailed settings. In this work, we show that label-weighted conformal prediction can be used to produce prediction sets with a finite-sample macro-coverage guarantee, and more generally a guarantee on a family of generalized macro-coverage objectives that aggregate coverage at the level of arbitrary class groupings and take a weighted average. We further characterize the form of the smallest prediction sets satisfying a given generalized macro-coverage objective and propose a corresponding conformal score function. We validate our theoretical results on two large-scale image classification datasets.

Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $τ\in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $τ$ provides.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

Modern engineering, spanning electrical, mechanical, aerospace, civil, and computer disciplines, stands as a cornerstone of human civilization and the foundation of our society. However, engineering design poses a fundamentally different challenge for large language models (LLMs) compared with traditional textbook-style problem solving or factual question answering. Although existing benchmarks have driven progress in areas such as language understanding, code synthesis, and scientific problem solving, real-world engineering design demands the synthesis of domain knowledge, navigation of complex trade-offs, and management of the tedious processes that consume much of practicing engineers' time. Despite these shared challenges across engineering disciplines, no benchmark currently captures the unique demands of engineering design work. In this work, we introduce EngDesign, an Engineering Design benchmark that evaluates LLMs' abilities to perform practical design tasks across nine engineering domains. Unlike existing benchmarks that focus on factual recall or question answering, EngDesign uniquely emphasizes LLMs' ability to synthesize domain knowledge, reason under constraints, and generate functional, objective-oriented engineering designs. Each task in EngDesign represents a real-world engineering design problem, accompanied by a detailed task description specifying design goals, constraints, and performance requirements. EngDesign pioneers a simulation-based evaluation paradigm that moves beyond textbook knowledge to assess genuine engineering design capabilities and shifts evaluation from static answer checking to dynamic, simulation-driven functional verification, marking a crucial step toward realizing the vision of engineering Artificial General Intelligence (AGI).

We prove a single algebraic mixed coincidence identity that unifies a broad swath of information-theoretic variational results. For any family of priors $\{π_i\}$ and real exponents $\{ α_i \}$, the log of the mixed count $E_{x\simν}\!\left[\prod_{i=1}^W π_i^{α_i}(x)\right]$ is simultaneously a Boltzmann coincidence weight, an exponential-family normalizer, a maximum-entropy value, and a KL-barycenter optimum. The identity yields a unified derivation of classical cornerstones of information theory: concentration of empirical distributions (Sanov-type decompositions and Gibbs conditioning), hypothesis-testing error exponents (Chernoff information and its multi-way analogue), change-of-measure inequalities (Donsker-Varadhan and PAC-Bayes), and laws governing rare-pattern coincidences (Erdos-Renyi run-length, iterative guesswork, rate-distortion, and birthday thresholds). Each is recovered as a specialization of the same algebraic equality. It strictly generalizes the classical Renyi entropy and divergence variational formulas (one and two priors respectively) to a $W$-prior simplex, and holds for unnormalized and continuum-indexed priors. Among its consequences are an exact multi-prior PAC-Bayes penalty that subtracts an explicit "coincidence bonus" from the usual single-prior posterior penalty, and the asymptotic MAP error exponent for $W$-ary hypothesis testing as an edge-restricted simplex optimum. We demonstrate the calculus at scale on two large alphabets encoding richly modeled sequential languages: on language-model next-token predictives where we recover contrastive decoding, and on human genomic regulatory sequence where it separates correlated from diverse prior families along a sliding-window trace.

Causal discovery in the form of a directed acyclic graph (DAG) for dynamic time series data has been widely studied in various applications. In this work, we propose a dynamic DAG discovery algorithm, Meta-D2AG, based on online metalearning. Meta-D2AG is designed to learn dynamic DAG structures from potentially nonlinear and non-stationary time series datasets, accounting for changes in both parameters and graph structures. Unlike most of the existing work focusing on observational, offline, and/or stationary settings, Meta-D2AG explicitly treats data collected at different time points with distribution shifts as distinct domains, which is assumed to occur as a result of external interventions. Moreover, MetaD2AG involves a new online meta-learning framework to take advantage of the temporal transition among existing domains such that it can quickly adapt to new domains with few measurements. A first-order optimization approach is utilized to efficiently solve the meta-learning framework, and theoretical analysis establishes the identifiability conditions and the convergence of the learning process. We demonstrate the promising performance of the proposed meta learning framework through better accuracy on benchmark datasets against state-of-the-art baselines.

Warning: This method may be misused for malicious purposes. Large Language Models (LLMs) are increasingly deployed in high-risk domains.

Neural combinatorial optimization (NCO) has achieved remarkable performance, yet its learned model representations and decision rationale remain a black box. This impedes both academic research and practical deployment, since researchers and stakeholders require deeper insights into NCO models. In this paper, we take the first critical step towards interpreting NCO models by investigating their representations through various probing tasks. Moreover, we introduce a novel probing tool named Coefficient Significance Probing (CS-Probing) to enable deeper analysis of NCO representations by examining the coefficients and statistical significance during probing. Extensive experiments and analysis reveal that NCO models encode low-level information essential for solution construction, while capturing high-level knowledge to facilitate better decisions. Using CS-Probing, we find that prevalent NCO models impose varying inductive biases on their learned representations, uncover direct evidence related to model generalization, and identify key embedding dimensions associated with specific knowledge. These insights can be potentially translated into practice, for example, with minor code modifications, we improve the generalization of the analyzed model. Our work represents a first systematic attempt to interpret black-box NCO models, showcasing probing as a promising tool for analyzing their internal mechanisms and revealing insights for the NCO community. The source code is publicly available 2.