Logistic Regression: Tight Bounds for Stochastic and Online Optimization
This resolves an open problem in learning theory, providing foundational insights for optimization in machine learning, though it is incremental in refining asymptotic bounds.
The paper tackles the question of whether the logistic loss offers advantages over non-smooth losses like the hinge loss in optimization, showing that for sub-exponential iterations, it provides no improvement and establishing tight polynomial convergence bounds for stochastic logistic optimization.
The logistic loss function is often advocated in machine learning and statistics as a smooth and strictly convex surrogate for the 0-1 loss. In this paper we investigate the question of whether these smoothness and convexity properties make the logistic loss preferable to other widely considered options such as the hinge loss. We show that in contrast to known asymptotic bounds, as long as the number of prediction/optimization iterations is sub exponential, the logistic loss provides no improvement over a generic non-smooth loss function such as the hinge loss. In particular we show that the convergence rate of stochastic logistic optimization is bounded from below by a polynomial in the diameter of the decision set and the number of prediction iterations, and provide a matching tight upper bound. This resolves the COLT open problem of McMahan and Streeter (2012).