Implicit Parameter-free Online Learning with Truncated Linear Models
This work addresses the challenge of improving optimization efficiency in online learning for machine learning practitioners, though it is incremental as it builds on existing parameter-free and truncated linear model concepts.
The paper tackles the problem of parameter-free online learning algorithms not accounting for loss geometry by proposing new algorithms that efficiently use truncated linear models to reduce overshooting, achieving improved performance with a single gradient per step.
Parameter-free algorithms are online learning algorithms that do not require setting learning rates. They achieve optimal regret with respect to the distance between the initial point and any competitor. Yet, parameter-free algorithms do not take into account the geometry of the losses. Recently, in the stochastic optimization literature, it has been proposed to instead use truncated linear lower bounds, which produce better performance by more closely modeling the losses. In particular, truncated linear models greatly reduce the problem of overshooting the minimum of the loss function. Unfortunately, truncated linear models cannot be used with parameter-free algorithms because the updates become very expensive to compute. In this paper, we propose new parameter-free algorithms that can take advantage of truncated linear models through a new update that has an "implicit" flavor. Based on a novel decomposition of the regret, the new update is efficient, requires only one gradient at each step, never overshoots the minimum of the truncated model, and retains the favorable parameter-free properties. We also conduct an empirical study demonstrating the practical utility of our algorithms.