Efficient improper learning for online logistic regression
This work addresses a computational bottleneck in online learning for logistic regression, offering a practical solution for applications requiring efficient real-time predictions.
The paper tackles the problem of exponential constants in regret for online logistic regression by designing an efficient improper algorithm that achieves O(B log(Bn)) regret with O(d^2) per-round complexity, avoiding the prohibitive computational cost of prior methods.
We consider the setting of online logistic regression and consider the regret with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014]) that any proper algorithm which has logarithmic regret in the number of samples (denoted n) necessarily suffers an exponential multiplicative constant in B. In this work, we design an efficient improper algorithm that avoids this exponential constant while preserving a logarithmic regret. Indeed, [Foster et al., 2018] showed that the lower bound does not apply to improper algorithms and proposed a strategy based on exponential weights with prohibitive computational complexity. Our new algorithm based on regularized empirical risk minimization with surrogate losses satisfies a regret scaling as O(B log(Bn)) with a per-round time-complexity of order O(d^2).