Learning linear predictors with the logistic loss---both in stochastic and online settings---is a fundamental task in learning and statistics, with direct connections to classification and boosting. Existing "fast rates" for this setting exhibit exponential dependence on the predictor norm, and Hazan et al. (2014) showed that this is unfortunately unimprovable. Starting with the simple observation that the logistic loss is 1-mixable, we design a new efficient improper learning algorithm for online logistic regression that circumvents the aforementioned lower bound with a regret bound exhibiting a doubly-exponential improvement in dependence on the predictor norm. This provides a positive resolution to a variant of the COLT 2012 open problem of McMahan and Streeter (2012) when improper learning is allowed. This improvement is obtained both in the online setting and, with some extra work, in the batch statistical setting with high probability. We also show that the improved dependency on predictor norm is also near-optimal. Leveraging this improved dependency on the predictor norm yields the following applications: (a) we give algorithms for online bandit multiclass learning with the logistic loss with an $\tilde{O}(\sqrt{n})$ relative mistake bound across essentially all parameter ranges, thus providing a solution to the COLT 2009 open problem of Abernethy and Rakhlin (2009), and (b) we give an adaptive algorithm for online multiclass boosting with optimal sample complexity, thus partially resolving an open problem of Beygelzimer et al. (2015) and Jung et al. (2017). Finally, we give information-theoretic bounds on the optimal rates for improper logistic regression with general function classes, thereby characterizing the extent to which our improvement for linear classes extends to other parameteric and even nonparametric settings.

通过发现logistic loss是1-mixable，我们设计了一个新的高效的improper learning algorithm，解决了COLT 2012 open problem variant中的问题，并在预测器范数上具有双重指数提高的遗憾限制，这提供了一种解决在线学习领域中的问题的方法，我们还展示了这种改进对多类学习和在线提升等应用的影响。

逻辑回归：偏差性的重要性