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We study online convex optimization under stochastic sub-gradient observationfaults, where we introduce adaptive algorithms with minimax optimal regretguarantees. We specifically study scenarios where our sub-gradient observationscan be noisy or even completely missing in a stochastic manner. To this end, wepropose algorithms based on sub-gradient descent method, which achieve tightminimax optimal regret bounds. When necessary, these algorithms utilizeproperties of the underlying stochastic settings to optimize their learningrates (step sizes). These optimizations are the main factor in providing theminimax optimal performance guarantees, especially when observations arestochastically missing. However, in real world scenarios, these properties ofthe underlying stochastic settings may not be revealed to the optimizer. Forsuch a scenario, we propose a blind algorithm that estimates these propertiesempirically in a generally applicable manner. Through extensive experiments, weshow that this empirical approach is a natural combination of regularstochastic gradient descent and the minimax optimal algorithms (which work bestfor randomized and adversarial function sequences, respectively).

Minimax Optimal Online Stochastic Learning for Sequences of Convex Functions under Sub-Gradient Observation Failures