Efficient Methods for Online Multiclass Logistic Regression
This resolves an open problem for researchers and practitioners in online learning by providing a computationally efficient method for multiclass logistic regression, with applications to bandit prediction and boosting.
The paper tackles the computational intractability of online multiclass logistic regression by developing a new algorithm, FOLKLORE, which runs significantly faster with quadratic time per iteration, at the cost of linear dependence on predictor norms in regret, making it the first practical algorithm for this problem.
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.