Lower Bounds for Stepsize-Based Acceleration of Gradient Descent
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Statement
Carefully designed stepsize schedules alone accelerate plain gradient descent beyond its textbook O(1/T) rate, without momentum. Whether they can reach the optimal O(T^-2) was open. A lower bound of Omega(T^-1.9319) for last-iterate convergence under predetermined nonnegative stepsize schedules says they cannot.
Context
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
An open direction in a currently active corner of convex optimization, a few years old and confined to that community.
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The abstract closes with it: "The proof was developed by GPT-5.6 Sol Pro under the authors' guidance." The authors added material to make the proof correct and readable, and separately used Codex to formalize the proof in Lean 4.
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
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