Optimizing Optimizers: Regret-optimal gradient descent algorithms
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The need for fast and robust optimization algorithms are of critical importance in all areas of machine learning. This paper treats the task of designing optimization algorithms as an optimal control problem. Using regret as a metric for an algorithm's performance, we derive the necessary and sufficient dynamics that regret-optimal algorithms must satisfy as a discrete-time difference equation. We study the existence, uniqueness and consistency of regret-optimal algorithms and derive bounds on rates of convergence to solutions of convex optimization problems. Though closed-form optimal dynamics cannot be obtained in general, we present fast numerical methods for approximating them, generating optimization algorithms which directly optimize their long-term regret. Lastly, these are benchmarked against commonly used optimization algorithms to demonstrate their effectiveness.
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