What, How, Where, and How Well? A Survey on Test-Time Scaling in Large Language Models

As enthusiasm for scaling computation (data and parameters) in the pretraining era gradually diminished, test-time scaling (TTS), also referred to as ``test-time computing'' has emerged as a prominent research focus. Recent studies demonstrate that TTS can further elicit the problem-solving capabilities of large language models (LLMs), enabling significant breakthroughs not only in specialized reasoning tasks, such as mathematics and coding, but also in general tasks like open-ended Q&A. However, despite the explosion of recent efforts in this area, there remains an urgent need for a comprehensive survey offering a systemic understanding. To fill this gap, we propose a unified, multidimensional framework structured along four core dimensions of TTS research: what to scale, how to scale, where to scale, and how well to scale. Building upon this taxonomy, we conduct an extensive review of methods, application scenarios, and assessment aspects, and present an organized decomposition that highlights the unique functional roles of individual techniques within the broader TTS landscape. From this analysis, we distill the major developmental trajectories of TTS to date and offer hands-on guidelines for practical deployment. Furthermore, we identify several open challenges and offer insights into promising future directions, including further scaling, clarifying the functional essence of techniques, generalizing to more tasks, and more attributions.

Learning Functional Dependencies with Sparse Regression

We study the problem of discovering functional dependencies (FD) from a noisy dataset. We focus on FDs that correspond to statistical dependencies in a dataset and draw connections between FD discovery and structure learning in probabilistic graphical models. We show that discovering FDs from a noisy dataset is equivalent to learning the structure of a graphical model over binary random variables, where each random variable corresponds to a functional of the dataset attributes. We build upon this observation to introduce AutoFD a conceptually simple framework in which learning functional dependencies corresponds to solving a sparse regression problem. We show that our methods can recover true functional dependencies across a diverse array of real-world and synthetic datasets, even in the presence of noisy or missing data. We find that AutoFD scales to large data instances with millions of tuples and hundreds of attributes while it yields an average F1 improvement of 2 times against state-of-the-art FD discovery methods.