# The Mechanics of n-Player Differentiable Games

arXiv: https://arxiv.org/abs/1802.05642 · HTML: https://mlsanity.com/paper/1802.05642/

## Key facts

- Cited by 118 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2018.

## Built on by (most cited first, 30 of 118)

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- [Modelling Behavioural Diversity for Learning in Open-Ended Games](https://mlsanity.com/paper/2103.07927/) (2021)
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- [Modeling the Second Player in Distributionally Robust Optimization](https://mlsanity.com/paper/2103.10282/) (2021)
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- [Policy Gradient Methods Find the Nash Equilibrium in N-player General-sum Linear-quadratic Games](https://mlsanity.com/paper/2107.13090/) (2021)
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- [ReLOAD: Reinforcement Learning with Optimistic Ascent-Descent for Last-Iterate Convergence in Constrained MDPs](https://mlsanity.com/paper/2302.01275/) (2023)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averaging](https://mlsanity.com/paper/2107.00464/) (2021)
- [Distributionally Robust Models with Parametric Likelihood Ratios](https://mlsanity.com/paper/2204.06340/) (2022)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Policy Space Diversity for Non-Transitive Games](https://mlsanity.com/paper/2306.16884/) (2023)
- [Discretization Drift in Two-Player Games](https://mlsanity.com/paper/2105.13922/) (2021)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [Contextual Games: Multi-Agent Learning with Side Information](https://mlsanity.com/paper/2107.06327/) (2021)
- [On a continuous time model of gradient descent dynamics and instability in deep learning](https://mlsanity.com/paper/2302.01952/) (2023)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [Minimax and Neyman-Pearson Meta-Learning for Outlier Languages](https://mlsanity.com/paper/2106.01051/) (2021)
- [Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems](https://mlsanity.com/paper/2112.09579/) (2021)
- [The Good Shepherd: An Oracle Agent for Mechanism Design](https://mlsanity.com/paper/2202.10135/) (2022)

## Abstract (arXiv)

The cornerstone underpinning deep learning is the guarantee that gradient descent on an objective converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, where there are multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new techniques to understand and control the dynamics in general games. The key result is to decompose the second-order dynamics into two components. The first is related to potential games, which reduce to gradient descent on an implicit function; the second relates to Hamiltonian games, a new class of games that obey a conservation law, akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in general games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- whilst at the same time being applicable to -- and having guarantees in -- much more general games.