Nicholas J. A. Harvey, Christopher Liaw, Sikander Randhawa
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引用次数: 0
摘要
考虑最小化函数的问题，这些函数具有 Lipschitz 特性和凸性，但不一定可微。我们从这类函数中构造出一个函数，对于这个函数，子梯度下降的第 T 次迭代误差为 Ω(log( T ) / √ T ) 。这与已知的 O (log( T ) / √ T ) 上限相吻合。我们将证明强凸函数的类似结果。存在这样一个函数，它的子梯度下降的第 T 次迭代的误差为 Ω(log( T ) /T ) ，与已知的上界 O (log( T ) /T ) 匹配。这些结果解决了沙米尔（2012）提出的一个问题
Tight analyses for subgradient descent I: Lower bounds
Consider the problem of minimizing functions that are Lipschitz and convex, but not necessarily differentiable. We construct a function from this class for which the T th iterate of subgradient descent has error Ω(log( T ) / √ T ) . This matches a known upper bound of O (log( T ) / √ T ) . We prove analogous results for functions that are additionally strongly convex. There exists such a function for which the error of the T th iterate of subgradient descent has error Ω(log( T ) /T ) , matching a known upper bound of O (log( T ) /T ) . These results resolve a question posed by Shamir (2012)
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