On the complexity of finding small subgradients in nonsmooth optimization
G Kornowski, O Shamir - arXiv preprint arXiv:2209.10346, 2022 - arxiv.org
arXiv preprint arXiv:2209.10346, 2022•arxiv.org
We study the oracle complexity of producing $(\delta,\epsilon) $-stationary points of Lipschitz
functions, in the sense proposed by Zhang et al.[2020]. While there exist dimension-free
randomized algorithms for producing such points within $\widetilde {O}(1/\delta\epsilon^ 3) $
first-order oracle calls, we show that no dimension-free rate can be achieved by a
deterministic algorithm. On the other hand, we point out that this rate can be derandomized
for smooth functions with merely a logarithmic dependence on the smoothness parameter …
functions, in the sense proposed by Zhang et al.[2020]. While there exist dimension-free
randomized algorithms for producing such points within $\widetilde {O}(1/\delta\epsilon^ 3) $
first-order oracle calls, we show that no dimension-free rate can be achieved by a
deterministic algorithm. On the other hand, we point out that this rate can be derandomized
for smooth functions with merely a logarithmic dependence on the smoothness parameter …
We study the oracle complexity of producing -stationary points of Lipschitz functions, in the sense proposed by Zhang et al. [2020]. While there exist dimension-free randomized algorithms for producing such points within first-order oracle calls, we show that no dimension-free rate can be achieved by a deterministic algorithm. On the other hand, we point out that this rate can be derandomized for smooth functions with merely a logarithmic dependence on the smoothness parameter. Moreover, we establish several lower bounds for this task which hold for any randomized algorithm, with or without convexity. Finally, we show how the convergence rate of finding -stationary points can be improved in case the function is convex, a setting which we motivate by proving that in general no finite time algorithm can produce points with small subgradients even for convex functions.
arxiv.org
以上显示的是最相近的搜索结果。 查看全部搜索结果