Multiclass logistic regression is a fundamental task in machine learning with applications in classification and boosting. Previous work (Foster et al., 2018) has highlighted the importance of improper predictors for achieving "fast rates" in the online multiclass logistic regression problem without suffering exponentially from secondary problem parameters, such as the norm of the predictors in the comparison class. While Foster et al. (2018) introduced a statistically optimal algorithm, it is in practice computationally intractable due to its run-time complexity being a large polynomial in the time horizon and dimension of input feature vectors. In this paper, we develop a new algorithm, FOLKLORE, for the problem which runs significantly faster than the algorithm of Foster et al.(2018) -- the running time per iteration scales quadratically in the dimension -- at the cost of a linear dependence on the norm of the predictors in the regret bound. This yields the first practical algorithm for online multiclass logistic regression, resolving an open problem of Foster et al.(2018). Furthermore, we show that our algorithm can be applied to online bandit multiclass prediction and online multiclass boosting, yielding more practical algorithms for both problems compared to the ones in Foster et al.(2018) with similar performance guarantees. Finally, we also provide an online-to-batch conversion result for our algorithm.

翻译：多级物流回归是机器学习的基本任务,包括应用分类和提升。 先前的工作( Foster 等人, 2018年)已经强调了在不因次要问题参数(如比较类中的预测者标准)而导致指数化变化的情况下,在网上多级物流回归问题中实现“ 快速率”的不适当预测器的重要性。 Foster 等人( 2018年) 引入了一个统计上最优化的算法, 在实践中, 由于其运行时的复杂性, 在时间范围与输入特性矢量的尺寸, 是一个巨大的多级多功能化问题, 而在本文中, 我们开发了一种新的算法, FOLKLORE, 其运行速度大大快于Foster等人的算法( 2018年) -- -- 其运行时间在规模上不因二次问题而成倍增速。 尽管Fosteration 等值的运行时间, 却以直线线性依赖在遗憾圈中的预测者标准为代价。 这产生了第一个实际的在线多级物流回归算法, 解决Foster et al. (20) 。 此外, 我们的算算法可以应用于在线多级多级多级预测和多级的在线推进系统转换, 。 最后, 将产生结果, 与我们的软化结果。