On the Impossibility of Global Convergence in Multi-Loss Optimization
This is a foundational result for researchers in optimization and game theory, showing inherent limitations in multi-loss settings like GANs or multi-agent RL, though it is incremental in addressing a specific theoretical bottleneck.
The paper tackles the problem of global convergence in multi-loss optimization, proving that no algorithm can simultaneously achieve desirable convergence properties, as demonstrated by constructing a two-player zero-sum game where the only simultaneous critical point is a strict maximum.
Under mild regularity conditions, gradient-based methods converge globally to a critical point in the single-loss setting. This is known to break down for vanilla gradient descent when moving to multi-loss optimization, but can we hope to build some algorithm with global guarantees? We negatively resolve this open problem by proving that desirable convergence properties cannot simultaneously hold for any algorithm. Our result has more to do with the existence of games with no satisfactory outcomes, than with algorithms per se. More explicitly we construct a two-player game with zero-sum interactions whose losses are both coercive and analytic, but whose only simultaneous critical point is a strict maximum. Any 'reasonable' algorithm, defined to avoid strict maxima, will therefore fail to converge. This is fundamentally different from single losses, where coercivity implies existence of a global minimum. Moreover, we prove that a wide range of existing gradient-based methods almost surely have bounded but non-convergent iterates in a constructed zero-sum game for suitably small learning rates. It nonetheless remains an open question whether such behavior can arise in high-dimensional games of interest to ML practitioners, such as GANs or multi-agent RL.