This note discusses proofs of convergence for gradient methods (also called “first-order methods”) based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants. We hope the structure and presentation of these amortized-analysis proofs will be useful as a guiding principle in learning and using these proofs.

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Theory of Computing
Continuous Methods in Discrete Optimization
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bansal, N., & Gupta, A. (2019). Potential-function proofs for gradient methods. Theory of Computing, 15. doi:10.4086/toc.2019.v015a004