The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drivediscoveryofmore efficient solutions.

We introduce a variational reasoning framework for language models that treats thinking traces as latent variables and optimizes them through variational inference. Starting from the evidence lower bound (ELBO), we extend it to a multi-trace objective for tighter bounds and propose a forward-KL formulation that stabilizes the training of the variational posterior. We further show that rejection sampling finetuning and binary-reward RL, including GRPO, can be interpreted as local forward-KL objectives, where an implicit weighting by model accuracy naturally arises from the derivation and reveals a previously unnoticed bias toward easier questions. We empirically validate our method on the Qwen 2.5 and Qwen 3 model families across a wide range of reasoning tasks. Overall, our work provides a principled probabilistic perspective that unifies variational inference with RL-style methods and yields stable objectives for improving the reasoning ability of language models. Our code is available at https://github.com/sail-sg/variational-reasoning.

Similar to Eq. (12) where some validation samples can be derived from training samples, we

On-board sensors of autonomous vehicles can be obstructed, occluded, or limited by restricted fields of view, complicating downstream driving decisions. Intelligent roadside infrastructure perception systems, installed at elevated vantage points, can provide wide, unobstructed intersection coverage, supplying a complementary information stream to autonomous vehicles via vehicle-to-everything (V2X) communication. However, conventional 3D object-detection algorithms struggle to generalize under the domain shift introduced by top-down perspectives and steep camera angles. We introduce a 2.5D object detection framework, tailored specifically for infrastructure roadside-mounted cameras. Unlike conventional 2D or 3D object detection, we employ a prediction approach to detect ground planes of vehicles as parallelograms in the image frame. The parallelogram preserves the planar position, size, and orientation of objects while omitting their height, which is unnecessary for most downstream applications. For training, a mix of real-world and synthetically generated scenes is leveraged. We evaluate generalizability on a held-out camera viewpoint and in adverse-weather scenarios absent from the training set. Our results show high detection accuracy, strong cross-viewpoint generalization, and robustness to diverse lighting and weather conditions. Model weights and inference code are provided at: https://gitlab.kit.edu/kit/aifb/ATKS/public/digit4taf/2.5d-object-detection

This paper presents an intervention study on the effects of the combined methods of (1) the Socratic method, (2) Chain of Thought (CoT) reasoning, (3) simplified gamification and (4) formative feedback on university students' Maths learning driven by large language models (LLMs). We call our approach Mathematics Explanations through Games by AI LLMs (MEGA). Some students struggle with Maths and as a result avoid Math-related discipline or subjects despite the importance of Maths across many fields, including signal processing. Oftentimes, students' Maths difficulties stem from suboptimal pedagogy. We compared the MEGA method to the traditional step-by-step (CoT) method to ascertain which is better by using a within-group design after randomly assigning questions for the participants, who are university students. Samples (n=60) were randomly drawn from each of the two test sets of the Grade School Math 8K (GSM8K) and Mathematics Aptitude Test of Heuristics (MATH) datasets, based on the error margin of 11%, the confidence level of 90%, and a manageable number of samples for the student evaluators. These samples were used to evaluate two capable LLMs at length (Generative Pretrained Transformer 4o (GPT4o) and Claude 3.5 Sonnet) out of the initial six that were tested for capability. The results showed that students agree in more instances that the MEGA method is experienced as better for learning for both datasets. It is even much better than the CoT (47.5% compared to 26.67%) in the more difficult MATH dataset, indicating that MEGA is better at explaining difficult Maths problems.

Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves studying the existence and uniqueness of solutions, and their dependence to such pre-specified conditions. However, in part due to mathematical necessity, these conditions are usually specified "to arbitrary precision" only on (appropriate portions of) the boundary of the space-time domain. This does not mirror how data acquisition is performed in realistic situations, where one may observe entire "patches" of solution data at arbitrary space-time locations; alternatively one might have access to more than one solutions stemming from the same differential operator. In our short work, we demonstrate how standard tools from machine and manifold learning can be used to infer, in a data driven manner, certain well-posedness features of differential equation problems, for initial/boundary condition combinations under which rigorous existence/uniqueness theorems are not known. Our study naturally combines a data assimilation perspective with an operator-learning one.

In industrial X-ray Computed Tomography (CT), the need for rapid in-line inspection is critical. Sparse-angle tomography plays a significant role in this by reducing the required number of projections, thereby accelerating processing and conserving resources. Most existing methods aim to balance reconstruction quality and scanning time, typically relying on fixed scan durations. Adaptive adjustment of the number of angles is essential; for instance, more angles may be required for objects with complex geometries or noisier projections. The concept of optimal stopping, which dynamically adjusts this balance according to varying industrial needs, remains underutilized. Building on our previous work, we integrate optimal stopping into sequential Optimal Experimental Design (OED). We propose a novel method for computing the policy gradient within the Actor-Critic framework, enabling the development of adaptive policies for informative angle selection and scan termination. Additionally, we investigated the gap between simulation and real-world applications in the context of the developed learning-based method. Our trained model, developed using synthetic data, demonstrates reliable performance when applied to real-world data. This approach enhances the flexibility of CT operations and expands the applicability of sparse-angle tomography in industrial settings.

In this paper, we introduce **harmonic loss** as an alternative to the standard cross-entropy loss for training neural networks and large language models (LLMs). Harmonic loss enables improved interpretability and faster convergence, owing to its scale invariance and finite convergence point by design, which can be interpreted as a class center. We first validate the performance of harmonic models across algorithmic, vision, and language datasets. Through extensive experiments, we demonstrate that models trained with harmonic loss outperform standard models by: (a) enhancing interpretability, (b) requiring less data for generalization, and (c) reducing grokking. Moreover, we compare a GPT-2 model trained with harmonic loss to the standard GPT-2, illustrating that the harmonic model develops more interpretable representations. Looking forward, we believe harmonic loss has the potential to become a valuable tool in domains with limited data availability or in high-stakes applications where interpretability and reliability are paramount, paving the way for more robust and efficient neural network models.