Agnostic Learnability of Halfspaces via Logistic Loss.
International Conference on Machine Learning(2022)
摘要
We investigate approximation guarantees provided by logistic regression for the fundamental problem of agnostic learning of homogeneous halfspaces. Previously, for a certain broad class of “well-behaved” distributions on the examples, Diakonikolas et al. (2020) proved an tilde{Omega}(OPT) lower bound, while Frei et al. (2021) proved an tilde{O}(sqrt{OPT}) upper bound, where OPT denotes the best zero-one/misclassification risk of a homogeneous halfspace. In this paper, we close this gap by constructing a well-behaved distribution such that the global minimizer of the logistic risk over this distribution only achieves Omega(sqrt{OPT}) misclassification risk, matching the upper bound in (Frei et al., 2021). On the other hand, we also show that if we impose a radial-Lipschitzness condition in addition to well-behaved-ness on the distribution, logistic regression on a ball of bounded radius reaches tilde{O}(OPT) misclassification risk. Our techniques also show for any well-behaved distribution, regardless of radial Lipschitzness, we can overcome the Omega(sqrt{OPT}) lower bound for logistic loss simply at the cost of one additional convex optimization step involving the hinge loss and attain tilde{O}(OPT) misclassification risk. This two-step convex optimization algorithm is simpler than previous methods obtaining this guarantee, all of which require solving O(log(1/OPT)) minimization problems.
更多查看译文
关键词
halfspaces,agnostic learnability,logistic loss
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要