Online Optimization of Smoothed Piecewise Constant Functions
This addresses the challenge of optimizing non-Lipschitz functions in online learning, which is incremental as it builds on prior smoothed analysis frameworks.
The paper tackles the problem of online optimization of smoothed piecewise constant functions, motivated by adaptively picking parameters in learning algorithms, and presents algorithms that achieve sublinear regret in full information and bandit settings.
We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. 1 This is with good reason---any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden---in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.