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Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system.

GPT-5.4 mini brings some of the smarts of OpenAI's latest model to ChatGPT Free and Go users The new model offers performance improvements in reasoning, multimodal understanding and more. The ChatGPT icon, as seen on iPhone 12 running iOS. When OpenAI released GPT-5.4 at the start of March, the company said the new model was designed primarily for professional work like programming and data analysis. Now OpenAI is launching GPT-5.4 mini and nano, and while it is once again highlighting the usefulness of these new systems for tasks like coding, one of the new models is available to Free and Go users . What's more, that model, GPT-5.4 mini, even offers performance that approaches GPT-5.4 in a handful of areas.

Both the discounted and total reward episodic settings are amenable to function approximation.

As a valuable by-product, the regret analysis used in this paper can improve several existing results by a factor ofO(logK).

Many important datasets contain samples that are missing one or more feature values. Maintaining the interpretability of machine learning models in the presence of such missing data is challenging. Singly or multiply imputing missing values complicates the model's mapping from features to labels. On the other hand, reasoning on indicator variables that represent missingness introduces a potentially largenumber ofadditional terms, sacrificing sparsity.

Large language models (LLMs) excel at explicit reasoning, but their implicit computational strategies remain underexplored. Decades of psychophysics research show that humans intuitively process and integrate noisy signals using near-optimal Bayesian strategies in perceptual tasks. We ask whether LLMs exhibit similar behaviour and perform optimal multimodal integration without explicit training or instruction. Adopting the psychophysics paradigm, we infer computational principles of LLMs from systematic behavioural studies. We introduce a behavioural benchmark - BayesBench: four magnitude estimation tasks (length, location, distance, and duration) over text and image, inspired by classic psychophysics, and evaluate a diverse set of nine LLMs alongside human judgments for calibration. Through controlled ablations of noise, context, and instruction prompts, we measure performance, behaviour and efficiency in multimodal cue-combination. Beyond accuracy and efficiency metrics, we introduce a Bayesian Consistency Score that detects Bayes-consistent behavioural shifts even when accuracy saturates. Our results show that while capable models often adapt in Bayes-consistent ways, accuracy does not guarantee robustness. Notably, GPT-5 Mini achieves perfect text accuracy but fails to integrate visual cues efficiently. This reveals a critical dissociation between capability and strategy, suggesting accuracy-centric benchmarks may over-index on performance while missing brittle uncertainty handling. These findings reveal emergent principled handling of uncertainty and highlight the correlation between accuracy and Bayesian tendencies. We release our psychophysics benchmark and consistency metric (https://bayes-bench.github.io) as evaluation tools and to inform future multimodal architecture designs.