What was already known
Monteiro and Svaiter's second-order monotone-VI rate was O(T^(−3/2)), with a general pth-order rate O(T^(−(p+1)/2)); a more recent faster rate applied to the narrower convex-concave minimax class. The optimal general monotone-VI complexity for p≥2 remained open.
AI involvement
The researchers posed the faster-rate conjecture. GPT 5.6 Sol supplied an initial proof and resolvent subsolver; Lesi Chen, Xinliang Zhang, Chengchang Liu and Jingzhao Zhang verified the result, and the team simplified the final algorithm.
GPT 5.6 Sol supplied the first proof of the team's conjectured faster rate, while the human researchers posed the target, checked it and simplified the final algorithm.
Importance
Improves the known higher-order convergence exponent for general smooth monotone variational inequalities.
Limitations
This is a general monotone-VI rate under the stated oracle assumptions; no matching lower bound establishes optimality. Claude’s earlier O(T^(−(p−1))) method is a separate intermediate result.
Technical verification
The paper explicitly names four human authors who verified the AI-found results and provides the initial AI algorithm and analysis beside the simplified final proof. This is source-reported human proof verification; no machine formalization or exact certificate is documented.
Formal checking
The paper explicitly names four human authors who verified the AI-found results and provides the initial AI algorithm and analysis beside the simplified final proof. This is source-reported human proof verification; no machine formalization or exact certificate is documented.
- Checker
- Lesi Chen, Xinliang Zhang, Chengchang Liu and Jingzhao Zhang
- Scope
- The Õ(T^(−p)) higher-order monotone-VI rate and its proof, developed with GPT 5.6 Sol
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors describe the Halpern iteration, resolvent subproblem and higher-order generalization, and explicitly simplify the AI's triple-loop method.
Correction and publication history
GPT 5.6 Sol found an initial proof of the faster rate; the authors verified and simplified it.
Read this published revision