In the field of machine learning, understanding how to solve two-player games using iterative numerical
optimization methods to find the Nash Equilibrium solution has become crucial. These methods are essential for enhancing model robustness, especially when dealing with competing training objectives in two-player scenarios. Moreover, many machine learning problems inherently involve multiple players, such as multi-agent reinforcement learning, collaborative robots, and competing drones. In the first part of this talk, we will introduce the Variational Inequality framework, which can model all these scenarios and facilitate a unified discussion on solving them through iterative gradient-based methods. We will highlight the unique aspects of the learning dynamics compared to standard minimization and explain why using different optimization methods is vital for achieving good performance. Additionally, we will discuss how noise from the stochasticity of gradient updates affects convergence as well as present methods to address such issues in both unconstrained and constrained settings.