Implicit Midpoint Gradient Descent: Fast and Learning rate free convergence for Zero-Sum Games

What the math gives to ML

The paper derives an implicit midpoint discretization for unconstrained bilinear zero-sum dynamics, yielding a Cayley-transform update instead of the expansive Euler step used by simultaneous gradient descent. Because the game operator is skew-symmetric, the exact discrete dynamics preserve norms for every positive step size and decompose into independent planar rotations. The strongest transfer target is a low-dimensional adversarial head in GANs or actor-critic systems, where the implicit linear solve is feasible. A second transferable mechanism is averaging successive iterates, which attenuates rotational modes before evaluation.