Papers › Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians
Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians
Naoki Marumo, Akiko Takeda
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We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and H\"older continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than ϵ in O(H_ν^(1/(2 + 2 ν)) ϵ^(- (4 + 3 ν)/(2 + 2 ν))) function and gradient evaluations, where ν∈[0, 1] and H_ν are the H\"older exponent and constant, respectively. Our algorithm is ν-independent and thus universal; it automatically achieves the above complexity bound with the optimal ν∈[0, 1] without knowledge of H_ν. In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient's Lipschitz constant or the target accuracy ϵ. Numerical results illustrate that the proposed method is promising.
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