Modelling Mean-Field Games with Neural Ordinary Differential Equations

Mean-field game theory relies on approximating games that would otherwise have been intractable to model. While the games can be solved analytically via the associated system of partial derivatives, this approach is not model-free, can lead to the loss of the existence or uniqueness of solutions and may suffer from modelling bias. To reduce the dependency between the model and the game, we combine mean-field game theory with deep learning in the form of neural ordinary differential equations. The resulting model is data-driven, lightweight and can learn extensive strategic interactions that are hard to capture using mean-field theory alone. In addition, the model is based on automatic differentiation, making it more robust and objective than approaches based on finite differences. We highlight the efficiency and flexibility of our approach by solving three mean-field games that vary in their complexity, observability and the presence of noise. Using these results, we show that the model is flexible, lightweight and requires few observations to learn the distribution underlying the data.

MLGym: A New Framework and Benchmark for Advancing AI Research Agents

We introduce Meta MLGym and MLGym-Bench, a new framework and benchmark for evaluating and developing LLM agents on AI research tasks. This is the first Gym environment for machine learning (ML) tasks, enabling research on reinforcement learning (RL) algorithms for training such agents. MLGym-bench consists of 13 diverse and open-ended AI research tasks from diverse domains such as computer vision, natural language processing, reinforcement learning, and game theory. Solving these tasks requires real-world AI research skills such as generating new ideas and hypotheses, creating and processing data, implementing ML methods, training models, running experiments, analyzing the results, and iterating through this process to improve on a given task. We evaluate a number of frontier large language models (LLMs) on our benchmarks such as Claude-3.5-Sonnet, Llama-3.1 405B, GPT-4o, o1-preview, and Gemini-1.5 Pro. Our MLGym framework makes it easy to add new tasks, integrate and evaluate models or agents, generate synthetic data at scale, as well as develop new learning algorithms for training agents on AI research tasks. We find that current frontier models can improve on the given baselines, usually by finding better hyperparameters, but do not generate novel hypotheses, algorithms, architectures, or substantial improvements. We open-source our framework and benchmark to facilitate future research in advancing the AI research capabilities of LLM agents.

Soft Condorcet Optimization for Ranking of General Agents

A common way to drive progress of AI models and agents is to compare their performance on standardized benchmarks. Comparing the performance of general agents requires aggregating their individual performances across a potentially wide variety of different tasks. In this paper, we describe a novel ranking scheme inspired by social choice frameworks, called Soft Condorcet Optimization (SCO), to compute the optimal ranking of agents: the one that makes the fewest mistakes in predicting the agent comparisons in the evaluation data. This optimal ranking is the maximum likelihood estimate when evaluation data (which we view as votes) are interpreted as noisy samples from a ground truth ranking, a solution to Condorcet's original voting system criteria. SCO ratings are maximal for Condorcet winners when they exist, which we show is not necessarily true for the classical rating system Elo. We propose three optimization algorithms to compute SCO ratings and evaluate their empirical performance. When serving as an approximation to the Kemeny-Young voting method, SCO rankings are on average 0 to 0.043 away from the optimal ranking in normalized Kendall-tau distance across 865 preference profiles from the PrefLib open ranking archive. In a simulated noisy tournament setting, SCO achieves accurate approximations to the ground truth ranking and the best among several baselines when 59\% or more of the preference data is missing. Finally, SCO ranking provides the best approximation to the optimal ranking, measured on held-out test sets, in a problem containing 52,958 human players across 31,049 games of the classic seven-player game of Diplomacy.

States as Strings as Strategies: Steering Language Models with Game-Theoretic Solvers

Game theory is the study of mathematical models of strategic interactions among rational agents. Language is a key medium of interaction for humans, though it has historically proven difficult to model dialogue and its strategic motivations mathematically. A suitable model of the players, strategies, and payoffs associated with linguistic interactions (i.e., a binding to the conventional symbolic logic of game theory) would enable existing game-theoretic algorithms to provide strategic solutions in the space of language. In other words, a binding could provide a route to computing stable, rational conversational strategies in dialogue. Large language models (LLMs) have arguably reached a point where their generative capabilities can enable realistic, human-like simulations of natural dialogue. By prompting them in various ways, we can steer their responses towards different output utterances. Leveraging the expressivity of natural language, LLMs can also help us quickly generate new dialogue scenarios, which are grounded in real world applications. In this work, we present one possible binding from dialogue to game theory as well as generalizations of existing equilibrium finding algorithms to this setting. In addition, by exploiting LLMs generation capabilities along with our proposed binding, we can synthesize a large repository of formally-defined games in which one can study and test game-theoretic solution concepts. We also demonstrate how one can combine LLM-driven game generation, game-theoretic solvers, and imitation learning to construct a process for improving the strategic capabilities of LLMs.

Evaluating Agents using Social Choice Theory

We argue that many general evaluation problems can be viewed through the lens of voting theory. Each task is interpreted as a separate voter, which requires only ordinal rankings or pairwise comparisons of agents to produce an overall evaluation. By viewing the aggregator as a social welfare function, we are able to leverage centuries of research in social choice theory to derive principled evaluation frameworks with axiomatic foundations. These evaluations are interpretable and flexible, while avoiding many of the problems currently facing cross-task evaluation. We apply this Voting-as-Evaluation (VasE) framework across multiple settings, including reinforcement learning, large language models, and humans. In practice, we observe that VasE can be more robust than popular evaluation frameworks (Elo and Nash averaging), discovers properties in the evaluation data not evident from scores alone, and can predict outcomes better than Elo in a complex seven-player game. We identify one particular approach, maximal lotteries, that satisfies important consistency properties relevant to evaluation, is computationally efficient (polynomial in the size of the evaluation data), and identifies game-theoretic cycles.

Using Cooperative Game Theory to Prune Neural Networks

We show how solution concepts from cooperative game theory can be used to tackle the problem of pruning neural networks. The ever-growing size of deep neural networks (DNNs) increases their performance, but also their computational requirements. We introduce a method called Game Theory Assisted Pruning (GTAP), which reduces the neural network's size while preserving its predictive accuracy. GTAP is based on eliminating neurons in the network based on an estimation of their joint impact on the prediction quality through game theoretic solutions. Specifically, we use a power index akin to the Shapley value or Banzhaf index, tailored using a procedure similar to Dropout (commonly used to tackle overfitting problems in machine learning). Empirical evaluation of both feedforward networks and convolutional neural networks shows that this method outperforms existing approaches in the achieved tradeoff between the number of parameters and model accuracy.

TacticAI: an AI assistant for football tactics

Identifying key patterns of tactics implemented by rival teams, and developing effective responses, lies at the heart of modern football. However, doing so algorithmically remains an open research challenge. To address this unmet need, we propose TacticAI, an AI football tactics assistant developed and evaluated in close collaboration with domain experts from Liverpool FC. We focus on analysing corner kicks, as they offer coaches the most direct opportunities for interventions and improvements. TacticAI incorporates both a predictive and a generative component, allowing the coaches to effectively sample and explore alternative player setups for each corner kick routine and to select those with the highest predicted likelihood of success. We validate TacticAI on a number of relevant benchmark tasks: predicting receivers and shot attempts and recommending player position adjustments. The utility of TacticAI is validated by a qualitative study conducted with football domain experts at Liverpool FC. We show that TacticAI's model suggestions are not only indistinguishable from real tactics, but also favoured over existing tactics 90% of the time, and that TacticAI offers an effective corner kick retrieval system. TacticAI achieves these results despite the limited availability of gold-standard data, achieving data efficiency through geometric deep learning.

Explainability Techniques for Chemical Language Models

Explainability techniques are crucial in gaining insights into the reasons behind the predictions of deep learning models, which have not yet been applied to chemical language models. We propose an explainable AI technique that attributes the importance of individual atoms towards the predictions made by these models. Our method backpropagates the relevance information towards the chemical input string and visualizes the importance of individual atoms. We focus on self-attention Transformers operating on molecular string representations and leverage a pretrained encoder for finetuning. We showcase the method by predicting and visualizing solubility in water and organic solvents. We achieve competitive model performance while obtaining interpretable predictions, which we use to inspect the pretrained model.

Combining Tree-Search, Generative Models, and Nash Bargaining Concepts in Game-Theoretic Reinforcement Learning

Multiagent reinforcement learning (MARL) has benefited significantly from population-based and game-theoretic training regimes. One approach, Policy-Space Response Oracles (PSRO), employs standard reinforcement learning to compute response policies via approximate best responses and combines them via meta-strategy selection. We augment PSRO by adding a novel search procedure with generative sampling of world states, and introduce two new meta-strategy solvers based on the Nash bargaining solution. We evaluate PSRO's ability to compute approximate Nash equilibrium, and its performance in two negotiation games: Colored Trails, and Deal or No Deal. We conduct behavioral studies where human participants negotiate with our agents ($N = 346$). We find that search with generative modeling finds stronger policies during both training time and test time, enables online Bayesian co-player prediction, and can produce agents that achieve comparable social welfare negotiating with humans as humans trading among themselves.

Developing, Evaluating and Scaling Learning Agents in Multi-Agent Environments

The Game Theory & Multi-Agent team at DeepMind studies several aspects of multi-agent learning ranging from computing approximations to fundamental concepts in game theory to simulating social dilemmas in rich spatial environments and training 3-d humanoids in difficult team coordination tasks. A signature aim of our group is to use the resources and expertise made available to us at DeepMind in deep reinforcement learning to explore multi-agent systems in complex environments and use these benchmarks to advance our understanding. Here, we summarise the recent work of our team and present a taxonomy that we feel highlights many important open challenges in multi-agent research.