PokéChamp: an Expert-level Minimax Language Agent

We introduce Pok\'eChamp, a minimax agent powered by Large Language Models (LLMs) for Pok\'emon battles. Built on a general framework for two-player competitive games, Pok\'eChamp leverages the generalist capabilities of LLMs to enhance minimax tree search. Specifically, LLMs replace three key modules: (1) player action sampling, (2) opponent modeling, and (3) value function estimation, enabling the agent to effectively utilize gameplay history and human knowledge to reduce the search space and address partial observability. Notably, our framework requires no additional LLM training. We evaluate Pok\'eChamp in the popular Gen 9 OU format. When powered by GPT-4o, it achieves a win rate of 76% against the best existing LLM-based bot and 84% against the strongest rule-based bot, demonstrating its superior performance. Even with an open-source 8-billion-parameter Llama 3.1 model, Pok\'eChamp consistently outperforms the previous best LLM-based bot, Pok\'ellmon powered by GPT-4o, with a 64% win rate. Pok\'eChamp attains a projected Elo of 1300-1500 on the Pok\'emon Showdown online ladder, placing it among the top 30%-10% of human players. In addition, this work compiles the largest real-player Pok\'emon battle dataset, featuring over 3 million games, including more than 500k high-Elo matches. Based on this dataset, we establish a series of battle benchmarks and puzzles to evaluate specific battling skills. We further provide key updates to the local game engine. We hope this work fosters further research that leverage Pok\'emon battle as benchmark to integrate LLM technologies with game-theoretic algorithms addressing general multiagent problems. Videos, code, and dataset available at https://sites.google.com/view/pokechamp-llm.

Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving

We introduce Goedel-Prover, an open-source large language model (LLM) that achieves the state-of-the-art (SOTA) performance in automated formal proof generation for mathematical problems. The key challenge in this field is the scarcity of formalized math statements and proofs, which we tackle in the following ways. We train statement formalizers to translate the natural language math problems from Numina into formal language (Lean 4), creating a dataset of 1.64 million formal statements. LLMs are used to check that the formal statements accurately preserve the content of the original natural language problems. We then iteratively build a large dataset of formal proofs by training a series of provers. Each prover succeeds in proving many statements that the previous ones could not, and these new proofs are added to the training set for the next prover. The final prover outperforms all existing open-source models in whole-proof generation. On the miniF2F benchmark, it achieves a 57.6% success rate (Pass@32), exceeding the previous best open-source model by 7.6%. On PutnamBench, Goedel-Prover successfully solves 7 problems (Pass@512), ranking first on the leaderboard. Furthermore, it generates 29.7K formal proofs for Lean Workbook problems, nearly doubling the 15.7K produced by earlier works.

MATH-Perturb: Benchmarking LLMs' Math Reasoning Abilities against Hard Perturbations

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations -- modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycksmath et. al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models.

Generative Diffusion Modeling: A Practical Handbook

This handbook offers a unified perspective on diffusion models, encompassing diffusion probabilistic models, score-based generative models, consistency models, rectified flow, and related methods. By standardizing notations and aligning them with code implementations, it aims to bridge the "paper-to-code" gap and facilitate robust implementations and fair comparisons. The content encompasses the fundamentals of diffusion models, the pre-training process, and various post-training methods. Post-training techniques include model distillation and reward-based fine-tuning. Designed as a practical guide, it emphasizes clarity and usability over theoretical depth, focusing on widely adopted approaches in generative modeling with diffusion models.

DOLLAR: Few-Step Video Generation via Distillation and Latent Reward Optimization

Diffusion probabilistic models have shown significant progress in video generation; however, their computational efficiency is limited by the large number of sampling steps required. Reducing sampling steps often compromises video quality or generation diversity. In this work, we introduce a distillation method that combines variational score distillation and consistency distillation to achieve few-step video generation, maintaining both high quality and diversity. We also propose a latent reward model fine-tuning approach to further enhance video generation performance according to any specified reward metric. This approach reduces memory usage and does not require the reward to be differentiable. Our method demonstrates state-of-the-art performance in few-step generation for 10-second videos (128 frames at 12 FPS). The distilled student model achieves a score of 82.57 on VBench, surpassing the teacher model as well as baseline models Gen-3, T2V-Turbo, and Kling. One-step distillation accelerates the teacher model's diffusion sampling by up to 278.6 times, enabling near real-time generation. Human evaluations further validate the superior performance of our 4-step student models compared to teacher model using 50-step DDIM sampling.

Benign Overfitting in Out-of-Distribution Generalization of Linear Models

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee [Tsigler and Bartlett, 2023], as well as under-parameterized OOD guarantee [Ge et al., 2024] when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $O(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $O(1/n)$, where $n$ is the number of samples.

Building Math Agents with Multi-Turn Iterative Preference Learning

Recent studies have shown that large language models' (LLMs) mathematical problem-solving capabilities can be enhanced by integrating external tools, such as code interpreters, and employing multi-turn Chain-of-Thought (CoT) reasoning. While current methods focus on synthetic data generation and Supervised Fine-Tuning (SFT), this paper studies the complementary direct preference learning approach to further improve model performance. However, existing direct preference learning algorithms are originally designed for the single-turn chat task, and do not fully address the complexities of multi-turn reasoning and external tool integration required for tool-integrated mathematical reasoning tasks. To fill in this gap, we introduce a multi-turn direct preference learning framework, tailored for this context, that leverages feedback from code interpreters and optimizes trajectory-level preferences. This framework includes multi-turn DPO and multi-turn KTO as specific implementations. The effectiveness of our framework is validated through training of various language models using an augmented prompt set from the GSM8K and MATH datasets. Our results demonstrate substantial improvements: a supervised fine-tuned Gemma-1.1-it-7B model's performance increased from 77.5% to 83.9% on GSM8K and from 46.1% to 51.2% on MATH. Similarly, a Gemma-2-it-9B model improved from 84.1% to 86.3% on GSM8K and from 51.0% to 54.5% on MATH.

Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization

We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_\textbf{x} \max_{\textbf{y} \in Y} f(\textbf{x}, \textbf{y})$, where the objective function $f(\textbf{x}, \textbf{y})$ is nonconvex in $\textbf{x}$ and concave in $\textbf{y}$, and the constraint set $Y \subseteq \mathbb{R}^n$ is convex and bounded. In the convex-concave setting, the single-timescale GDA achieves strong convergence guarantees and has been used for solving application problems arising from operations research and computer science. However, it can fail to converge in more general settings. Our contribution in this paper is to design the simple deterministic and stochastic TTGDA algorithms that efficiently find one stationary point of the function $\Phi(\cdot) := \max_{\textbf{y} \in Y} f(\cdot, \textbf{y})$. Specifically, we prove the theoretical bounds on the complexity of solving both smooth and nonsmooth nonconvex-concave minimax optimization problems. To our knowledge, this is the first systematic analysis of TTGDA for nonconvex minimax optimization, shedding light on its superior performance in training generative adversarial networks (GANs) and in solving other real-world application problems.

On Limitation of Transformer for Learning HMMs

Despite the remarkable success of Transformer-based architectures in various sequential modeling tasks, such as natural language processing, computer vision, and robotics, their ability to learn basic sequential models, like Hidden Markov Models (HMMs), is still unclear. This paper investigates the performance of Transformers in learning HMMs and their variants through extensive experimentation and compares them to Recurrent Neural Networks (RNNs). We show that Transformers consistently underperform RNNs in both training speed and testing accuracy across all tested HMM models. There are even challenging HMM instances where Transformers struggle to learn, while RNNs can successfully do so. Our experiments further reveal the relation between the depth of Transformers and the longest sequence length it can effectively learn, based on the types and the complexity of HMMs. To address the limitation of transformers in modeling HMMs, we demonstrate that a variant of the Chain-of-Thought (CoT), called $\textit{block CoT}$ in the training phase, can help transformers to reduce the evaluation error and to learn longer sequences at a cost of increasing the training time. Finally, we complement our empirical findings by theoretical results proving the expressiveness of transformers in approximating HMMs with logarithmic depth.

Towards Principled Superhuman AI for Multiplayer Symmetric Games

Multiplayer games, when the number of players exceeds two, present unique challenges that fundamentally distinguish them from the extensively studied two-player zero-sum games. These challenges arise from the non-uniqueness of equilibria and the risk of agents performing highly suboptimally when adopting equilibrium strategies. While a line of recent works developed learning systems successfully achieving human-level or even superhuman performance in popular multiplayer games such as Mahjong, Poker, and Diplomacy, two critical questions remain unaddressed: (1) What is the correct solution concept that AI agents should find? and (2) What is the general algorithmic framework that provably solves all games within this class? This paper takes the first step towards solving these unique challenges of multiplayer games by provably addressing both questions in multiplayer symmetric normal-form games. We also demonstrate that many meta-algorithms developed in prior practical systems for multiplayer games can fail to achieve even the basic goal of obtaining agent's equal share of the total reward.