Convergence in min-max optimization
Lai, Kevin A.
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Min-max optimization is a classic problem with applications in constrained optimization, robust optimization, and game theory. This dissertation covers new convergence rate results in min-max optimization. We show that the classic fictitious play dynamic with lexicographic tiebreaking converges quickly for diagonal payoff matrices, partly answering a conjecture by Karlin from 1959. We also show that linear last-iterate convergence rates are possible for the Hamiltonian Gradient Descent algorithm for the class of “sufficiently bilinear” min-max problems. Finally, we explore higher-order methods for min-max optimization and monotone variational inequalities, showing improved iteration complexity compared to first-order methods such as Mirror Prox.