Parameter-Free Accelerated Gradient Descent for Nonconvex Minimization
提出一种无需预设问题参数（如Lipschitz常数和精度）的加速梯度下降法，能在O(ε^{-7/4})次评估内找到梯度范数小于ε的解，数值实验表明其性能与需调参的同类算法相当。
We propose a new first-order method for minimizing nonconvex functions with a Lipschitz continuous gradient and Hessian.The proposed method is an accelerated gradient descent with two restart mechanisms and finds a solution where the gradient norm is less than ε in O(ε -7/4 ) function and gradient evaluations.Unlike existing first-order methods with similar complexity bounds, our algorithm is parameter-free because it requires no prior knowledge of problem-dependent parameters, e.g., the Lipschitz constants and the target accuracy ε.The main challenge in achieving this advantage is estimating the Lipschitz constant of the Hessian using only first-order information.To this end, we develop a new Hessian-free analysis based on two technical inequalities: a Jensen-type inequality for gradients and an error bound for the trapezoidal rule.Several numerical results illustrate that the proposed method performs comparably to existing algorithms with similar complexity bounds, even without parameter tuning.