Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) loss can be misaligned with pass@N in that pass@N accuracy decreases with longer training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

Reward models play a critical role in guiding large language models toward outputs that align with human expectations. However, an open challenge remains in effectively utilizing test-time compute to enhance reward model performance. In this work, we introduce Reward Reasoning Models (RRMs), which are specifically designed to execute a deliberate reasoning process before generating final rewards. Through chain-of-thought reasoning, RRMs leverage additional test-time compute for complex queries where appropriate rewards are not immediately apparent. To develop RRMs, we implement a reinforcement learning framework that fosters self-evolved reward reasoning capabilities without requiring explicit reasoning traces as training data. Experimental results demonstrate that RRMs achieve superior performance on reward modeling benchmarks across diverse domains. Notably, we show that RRMs can adaptively exploit test-time compute to further improve reward accuracy.

We propose two simple, principled and practical algorithms that enjoy provable scaling laws for the test-time compute of large language models (LLMs). The first one is a two-stage knockout-style algorithm: given an input problem, it first generates multiple candidate solutions, and then aggregate them via a knockout tournament for the final output. Assuming that the LLM can generate a correct solution with non-zero probability and do better than a random guess in comparing a pair of correct and incorrect solutions, we prove theoretically that the failure probability of this algorithm decays to zero exponentially or by a power law (depending on the specific way of scaling) as its test-time compute grows. The second one is a two-stage league-style algorithm, where each candidate is evaluated by its average win rate against multiple opponents, rather than eliminated upon loss to a single opponent. Under analogous but more robust assumptions, we prove that its failure probability also decays to zero exponentially with more test-time compute. Both algorithms require a black-box LLM and nothing else (e.g., no verifier or reward model) for a minimalistic implementation, which makes them appealing for practical applications and easy to adapt for different tasks. Through extensive experiments with diverse models and datasets, we validate the proposed theories and demonstrate the outstanding scaling properties of both algorithms.

Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) can be with pass@N in that pass@N accuracy with longer CE training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions both with and without Chain-of-Thought reasoning traces; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

Chain of thought is a natural inference-time method for increasing the computational power of transformer-based large language models (LLMs), but comes at the cost of sequential decoding. Are there more efficient alternatives to expand a transformer's expressive power without adding parameters? We consider transformers with *padding* tokens as a form of parallelizable test-time compute. We show that averaging-hard-attention, masked-pre-norm transformers with polynomial padding recognize precisely the class $\mathsf{FO}$-uniform $\mathsf{TC}^0$ of extremely parallelizable problems. While the $\mathsf{TC}^0$ upper bound was known, proving a matching lower bound had been elusive. Further, our novel analysis reveals the precise expanded power of padded transformers when coupled with another form of inference-time compute, namely dynamically increasing depth via *looping*.

Parallel sampling promises substantial gains in test-time scaling, but its effectiveness is sharply limited by diversity collapse, where models concentrate on a few modes and repeated samples produce the same mistakes. We propose the mode-conditioning (ModC) framework, which explicitly allocates test-time compute across reasoning modes using either specialist models or mode-specific prefixes. ModC consistently improves scaling across controlled graph-search tasks and large-scale reasoning benchmarks, spanning model families and sizes from 0.5B to 7B. On OpenThoughts, fine-tuning Qwen2.5-7B with ModC achieves a 4x efficiency gain over standard training while also improving the maximum attainable Pass@k. We further show that gradient clustering enables ModC without explicit mode labels, yielding up to 10% gains on datasets such as NuminaMath. Finally, we show that ModC improves reinforcement learning (RL) and can further boost diversity-inducing RL methods. These results demonstrate that standard training underutilizes the diversity in data, and that ModC provides a simple, effective remedy for unlocking the full benefits of diversity in test-time scaling.

Recurrent-depth transformers scale test-time compute by iterating latent computations before emitting tokens. We study the geometry of these iterates and argue for a simple, two-scale operational picture: (i) within a looped block, updates act as small-scale refinements; (ii) across consecutive blocks, states undergo a larger-scale drift. Across training, our measurements show that loop steps become smaller and increasingly orthogonal to one another, indicating better local modeling of fine structure rather than merely pushing in a single direction. These dynamics motivate an early-exit mechanism based on the model's second-order difference in step-size, which we show is superior in terms of performance, stability and time-efficiency, when compared to the KL-divergence exit strategy of Geiping et al. and its naive first-order counterpart.

Large language models learn from their vast pre-training corpora, gaining the ability to solve an ever increasing variety of tasks; yet although researchers work to improve these datasets, there is little effort to understand how efficient the pre-training apparatus is at extracting ideas and knowledge from the data. In this work, we use retrieval augmented generation along with test-time compute as a way to quantify how much dataset value was left behind by the process of pre-training, and how this changes across scale. We demonstrate that pre-training then retrieving from standard and largely open-sourced datasets results in significant accuracy gains in MMLU, Math-500, and SimpleQA, which persist through decontamination. For MMLU we observe that retrieval acts as a ~5x compute multiplier versus pre-training alone. We show that these results can be further improved by leveraging additional compute at test time to parse the retrieved context, demonstrating a 10 percentage point improvement on MMLU for the public LLaMA 3.1 8B model. Overall, our results suggest that today's pre-training methods do not make full use of the information in existing pre-training datasets, leaving significant room for progress.

We study the source of uncertainty in DeepSeek R1-32B by analyzing its self-reported verbal confidence on question answering (QA) tasks. In the default answer-then-confidence setting, the model is regularly over-confident, whereas semantic entropy - obtained by sampling many responses - remains reliable. We hypothesize that this is because of semantic entropy's larger test-time compute, which lets us explore the model's predictive distribution. We show that granting DeepSeek the budget to explore its distribution by forcing a long chain-of-thought before the final answer greatly improves its verbal score effectiveness, even on simple fact-retrieval questions that normally require no reasoning. Furthermore, a separate reader model that sees only the chain can reconstruct very similar confidences, indicating the verbal score might be merely a statistic of the alternatives surfaced during reasoning. Our analysis concludes that reliable uncertainty estimation requires explicit exploration of the generative space, and self-reported confidence is trustworthy only after such exploration.

The performance of Large Language Models (LLMs) and the associated dollar costs of API calls can fluctuate over time, potentially invalidating conclusions drawn in prior research. To address this, we propose a Fair Evaluation protocol for Test-Time Compute (FEval-TTC), designed to ensure consistent assessment of test-time compute (TTC) methods, regardless of such fluctuations. FEval-TTC focuses on the evaluation of TTC methods that utilize underlying Chains-of-Thought (CoT). It supports evaluations across multiple LLMs on a diverse set of mathematical and commonsense reasoning datasets. The few-shot prompting and answer extraction processes are standardized across datasets, reducing both time and monetary overhead for researchers. Furthermore, we provide a cost modelling procedure that estimates both the token and dollar cost per query, facilitating equitable comparisons of prevalent TTC methods. We open-source FEval-TTC for public use at https://github.com/networkslab/feval_ttc .