论文标题
无参数的加速梯度下降，以最小化
Parameter-free accelerated gradient descent for nonconvex minimization
论文作者
论文摘要
我们提出了一种新的一阶方法，用于用Lipschitz连续梯度和Hessian最小化非凸函数。提出的方法是具有两个重新启动机制的加速梯度下降，并找到了一个解决方案，其中梯度标准小于$ O（ε^{ - 7/4}）$功能和梯度评估。与现有具有相似复杂性边界的一阶方法不同，我们的算法是无参数的，因为它不需要与问题依赖性参数（例如Lipschitz常数和目标准确性$ε$）的事先了解。实现这一优势的主要挑战是仅使用一阶信息估算Hessian的Lipschitz常数。为此，我们基于两种技术不等式开发了新的无黑森分析：梯度的詹森型不平等和梯形规则的错误。几个数值结果表明，即使没有参数调整，该提出的方法与具有相似复杂性边界的现有算法相当。
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.
