LLM-based agents in Lean solved 9 of 353 open Erdős problems and proved 44 of 492 OEIS conjectures at a few hundred dollars each.
An Improved Last-Iterate Convergence Rate for Anchored Gradient Descent Ascent
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abstract
We analyze the last-iterate convergence of the Anchored Gradient Descent Ascent algorithm for smooth convex-concave min-max problems. While previous work established a last-iterate rate of $\mathcal{O}(1/t^{2-2p})$ for the squared gradient norm, where $p \in (1/2, 1)$, it remained an open problem whether the improved exact $\mathcal{O}(1/t)$ rate is achievable. In this work, we resolve this question in the affirmative. This result was discovered autonomously by an AI system capable of writing formal proofs in Lean. The Lean proof can be accessed at https://github.com/google-deepmind/formal-conjectures/pull/3675/commits/a13226b49fd3b897f4c409194f3bcbeb96a08515
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2026 2verdicts
UNVERDICTED 2representative citing papers
Anchored gradient descent achieves O(1/sqrt(T)) last-iterate convergence for monotone inclusions 0 in F(z) + A(z) by extending prior unconstrained proofs.
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Advancing Mathematics Research with AI-Driven Formal Proof Search
LLM-based agents in Lean solved 9 of 353 open Erdős problems and proved 44 of 492 OEIS conjectures at a few hundred dollars each.
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Last-Iterate Convergence of Anchored Gradient Descent
Anchored gradient descent achieves O(1/sqrt(T)) last-iterate convergence for monotone inclusions 0 in F(z) + A(z) by extending prior unconstrained proofs.