Training models to efficiently use test-time compute is crucial for improving the reasoning performance of LLMs. While current methods mostly do so via fine-tuning on search traces or running RL against the 0/1 outcome reward, do these approaches efficiently utilize test-time compute? Would these approaches continue to scale as the budget improves? In this paper, we try to answer these questions. We formalize the problem of optimizing test-time compute as a meta reinforcement learning (RL) problem, which provides a principled perspective on spending test-time compute from the lens of exploration and exploitation. It also motivates the use of cumulative regret to measure the efficacy of test-time compute by viewing a long output stream as consisting of several episodes from the model. While current state-of-the-art models do not optimize regret, we show that regret can be minimized by running final 0/1 reward RL regularized by a dense reward bonus, given by the "information gain" from each subsequent block in the output stream. We prescribe an approach for quantifying information gain, which measures the utility of an intermediate segment of tokens towards improving accuracy of the final answer. We instantiate this idea to develop MRT, a new class of finetuning methods for optimizing test-time compute. Fine-tuning with MRT leads to substantial improvements in both performance and token efficiency on the AIME dataset.

We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The library's ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.