Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain.

Transformers exhibit in-context learning (ICL), enabling adaptation to various tasks via prompts without the need for computationally intensive fine-tuning. Recent research investigates ICL’s mechanisms under analytically tractable models, with some conjecturing that ICL with linear attention implements one step of gradient descent for simple linear regression tasks. This paper reevaluates this claim, revealing it relies on strong assumptions like feature independence. Relaxing these assumptions, we prove that ICL with linear attention resembles preconditioned gradient descent, with a pre-conditioner that depends on the data covariance. Our experiments support this finding. We also empirically explore softmax-attention and find that increasing the number of attention heads better approximates gradient descent. Our work offers a nuanced perspective on the connection between ICL and gradient descent, emphasizing data assumptions.

Transformers have become the go-to architecture for language and vision tasks, yet their theoretical properties, especially memorization capacity, remain elusive. This paper investigates the memorization abilities of multi-head attention mechanisms, examining how many example sequences they can memorize, as a function of the number of heads and sequence length. Motivated by experimental findings on vision transformers, we introduce novel assumptions about the linear independence of input data, distinct from the commonly used general-position assumption. Under these assumptions, we demonstrate that an attention layer with $H$ heads, dimension $d$, and context size $n < d,$ featuring $\Theta(Hd^2)$ parameters, can memorize $\Omega(Hn)$ examples. Our analysis sheds light on how different attention heads handle various example sequences, aided by the softmax operator’s saturation property. We validate our findings through experiments on synthetic data.

Large Language Models (LLMs) have shown remarkable performance in various natural language tasks, but they often struggle with planning problems that require structured reasoning. To address this limitation, the conversion of planning problems into the Planning Domain Definition Language (PDDL) has been proposed as a potential solution, enabling the use of automated planners. However, generating accurate PDDL files typically demands human inputs or correction, which can be time-consuming and costly. In this paper, we propose a novel approach that leverages LLMs and environment feedback to automatically generate PDDL domain and problem description files without the need for human intervention. Our method introduces an iterative refinement process that generates multiple problem PDDL candidates and progressively refines the domain PDDL based on feedback obtained from interacting with the environment. To guide the refinement process, we develop an Exploration Walk (EW) metric, which provides rich feedback signals for LLMs to update the PDDL file. We evaluate our approach on $10$ PDDL environments. We achieve an average task solve rate of 66\% compared to a 29\% solve rate by GPT-4's intrinsic planning with chain-of-thought prompting. Our work enables the automated modeling of planning environments using LLMs and environment feedback, eliminating the need for human intervention in the PDDL translation process and paving the way for more reliable LLM agents in challenging problems. Our code is available at https://github.com/BorealisAI/llm-pddl-planning