Advancements in deep learning have demonstrated that the performance of neural networks scales predictably as a function of model size, data size, and computational resources [Hestness et al., 2017, Kaplan et al., 2020a, Rosenfeld et al., 2020, Henighan et al., 2020a]. These trends, known as neural scaling laws, have motivated research into understanding how scaling influences model performance in a range of domains, in particular, Large Language Models (LLMs) [Brown et al., 2020, Hoffmann et al., 2022]. However, scaling during inference--the process by which a trained model makes predictions on new data--has received less attention. Recent works have shown empirically that LLMs can gain substantial benefits from repeated prompts to perform better on difficult tasks such as coding and formal proofs, where verification of the correct answer can be done [Brown et al., 2024, Snell et al., 2024, Bansal et al., 2024]. These works demonstrate that the performance of weaker models can be amplified without further training, by simply repeating inference trials. A natural question then arises: Can we interpret, or predict the inference scaling behavior of a model with repeated attempts? To answer this question, we propose a simple toy model that isolates the inference scaling laws which dictate how certain performance metrics improve as a function of the number of inference attempts. Inspired by the work of Hutter [2021], which introduced a model to study scaling behavior for memorization and generalization, we devise a simple setting to capture the effect of repeated inference attempts, focusing on the coverage metric, also known as pass@k. In this work, we present analytical predictions for coverage from a probabilistic perspective and demonstrate how inference improves with the number of repeated trials in a predictable way, which matches the observed behavior in Brown et al. [2024] and Snell et al. [2024].

Scaling the amount of compute used to train language models has dramatically improved their capabilities. However, when it comes to inference, we often limit the amount of compute to only one attempt per problem. Here, we explore inference compute as another axis for scaling by increasing the number of generated samples. Across multiple tasks and models, we observe that coverage - the fraction of problems solved by any attempt - scales with the number of samples over four orders of magnitude. In domains like coding and formal proofs, where all answers can be automatically verified, these increases in coverage directly translate into improved performance. When we apply repeated sampling to SWE-bench Lite, the fraction of issues solved with DeepSeek-V2-Coder-Instruct increases from 15.9% with one sample to 56% with 250 samples, outperforming the single-attempt state-of-the-art of 43% which uses more capable frontier models. Moreover, using current API pricing, amplifying the cheaper DeepSeek model with five samples is more cost-effective and solves more issues than paying a premium for one sample from GPT-4o or Claude 3.5 Sonnet. Interestingly, the relationship between coverage and the number of samples is often log-linear and can be modelled with an exponentiated power law, suggesting the existence of inference-time scaling laws. Finally, we find that identifying correct samples out of many generations remains an important direction for future research in domains without automatic verifiers. When solving math word problems from GSM8K and MATH, coverage with Llama-3 models grows to over 95% with 10,000 samples. However, common methods to pick correct solutions from a sample collection, such as majority voting or reward models, plateau beyond several hundred samples and fail to fully scale with the sample budget.

Deep neural networks (DNNs) are becoming a key component in diverse systems across the board. However, despite their success, they often err miserably; and this has triggered significant interest in formally verifying them. Unfortunately, DNN verifiers are intricate tools, and are themselves susceptible to soundness bugs. Due to the complexity of DNN verifiers, as well as the sizes of the DNNs being verified, debugging such errors is a daunting task. Here, we present a novel tool, named DelBugV, that uses automated delta debugging techniques on DNN verifiers. Given a malfunctioning DNN verifier and a correct verifier as a point of reference (or, in some cases, just a single, malfunctioning verifier), DelBugV can produce much simpler DNN verification instances that still trigger undesired behavior -- greatly facilitating the task of debugging the faulty verifier. Our tool is modular and extensible, and can easily be enhanced with additional network simplification methods and strategies. For evaluation purposes, we ran DelBugV on 4 DNN verification engines, which were observed to produce incorrect results at the 2021 neural network verification competition (VNN-COMP'21). We were able to simplify many of the verification queries that trigger these faulty behaviors, by as much as 99%. We regard our work as a step towards the ultimate goal of producing reliable and trustworthy DNN-based software.