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.

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.

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.

FightLadder: A Benchmark for Competitive Multi-Agent Reinforcement Learning

Recent advances in reinforcement learning (RL) heavily rely on a variety of well-designed benchmarks, which provide environmental platforms and consistent criteria to evaluate existing and novel algorithms. Specifically, in multi-agent RL (MARL), a plethora of benchmarks based on cooperative games have spurred the development of algorithms that improve the scalability of cooperative multi-agent systems. However, for the competitive setting, a lightweight and open-sourced benchmark with challenging gaming dynamics and visual inputs has not yet been established. In this work, we present FightLadder, a real-time fighting game platform, to empower competitive MARL research. Along with the platform, we provide implementations of state-of-the-art MARL algorithms for competitive games, as well as a set of evaluation metrics to characterize the performance and exploitability of agents. We demonstrate the feasibility of this platform by training a general agent that consistently defeats 12 built-in characters in single-player mode, and expose the difficulty of training a non-exploitable agent without human knowledge and demonstrations in two-player mode. FightLadder provides meticulously designed environments to address critical challenges in competitive MARL research, aiming to catalyze a new era of discovery and advancement in the field. Videos and code at https://sites.google.com/view/fightladder/home.

Tuning-Free Stochastic Optimization

Large-scale machine learning problems make the cost of hyperparameter tuning ever more prohibitive. This creates a need for algorithms that can tune themselves on-the-fly. We formalize the notion of "tuning-free" algorithms that can match the performance of optimally-tuned optimization algorithms up to polylogarithmic factors given only loose hints on the relevant problem parameters. We consider in particular algorithms that can match optimally-tuned Stochastic Gradient Descent (SGD). When the domain of optimization is bounded, we show tuning-free matching of SGD is possible and achieved by several existing algorithms. We prove that for the task of minimizing a convex and smooth or Lipschitz function over an unbounded domain, tuning-free optimization is impossible. We discuss conditions under which tuning-free optimization is possible even over unbounded domains. In particular, we show that the recently proposed DoG and DoWG algorithms are tuning-free when the noise distribution is sufficiently well-behaved. For the task of finding a stationary point of a smooth and potentially nonconvex function, we give a variant of SGD that matches the best-known high-probability convergence rate for tuned SGD at only an additional polylogarithmic cost. However, we also give an impossibility result that shows no algorithm can hope to match the optimal expected convergence rate for tuned SGD with high probability.

Maximum Likelihood Estimation is All You Need for Well-Specified Covariate Shift

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization -- generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the minimax optimality for covariate shift under the well-specified setting. That is, no algorithm performs better than MLE in this setting (up to a constant factor), justifying MLE is all you need. Our result holds for a very rich class of parametric models, and does not require any boundedness condition on the density ratio. We illustrate the wide applicability of our framework by instantiating it to three concrete examples -- linear regression, logistic regression, and phase retrieval. This paper further complement the study by proving that, under the misspecified setting, MLE is no longer the optimal choice, whereas Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in certain scenarios.

ZeroSwap: Data-driven Optimal Market Making in DeFi

Automated Market Makers (AMMs) are major centers of matching liquidity supply and demand in Decentralized Finance. Their functioning relies primarily on the presence of liquidity providers (LPs) incentivized to invest their assets into a liquidity pool. However, the prices at which a pooled asset is traded is often more stale than the prices on centralized and more liquid exchanges. This leads to the LPs suffering losses to arbitrage. This problem is addressed by adapting market prices to trader behavior, captured via the classical market microstructure model of Glosten and Milgrom. In this paper, we propose the first optimal Bayesian and the first model-free data-driven algorithm to optimally track the external price of the asset. The notion of optimality that we use enforces a zero-profit condition on the prices of the market maker, hence the name ZeroSwap. This ensures that the market maker balances losses to informed traders with profits from noise traders. The key property of our approach is the ability to estimate the external market price without the need for price oracles or loss oracles. Our theoretical guarantees on the performance of both these algorithms, ensuring the stability and convergence of their price recommendations, are of independent interest in the theory of reinforcement learning. We empirically demonstrate the robustness of our algorithms to changing market conditions.

Consistency Models as a Rich and Efficient Policy Class for Reinforcement Learning

Score-based generative models like the diffusion model have been testified to be effective in modeling multi-modal data from image generation to reinforcement learning (RL). However, the inference process of diffusion model can be slow, which hinders its usage in RL with iterative sampling. We propose to apply the consistency model as an efficient yet expressive policy representation, namely consistency policy, with an actor-critic style algorithm for three typical RL settings: offline, offline-to-online and online. For offline RL, we demonstrate the expressiveness of generative models as policies from multi-modal data. For offline-to-online RL, the consistency policy is shown to be more computational efficient than diffusion policy, with a comparable performance. For online RL, the consistency policy demonstrates significant speedup and even higher average performances than the diffusion policy.

Is RLHF More Difficult than Standard RL?

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games under a restricted set of policies. The latter case can be further reduce to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.