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The theory of deep learning focuses almost exclusively on supervisedlearning, non-convex optimization using stochastic gradient descent, andoverparametrized neural networks. It is common belief that the optimizerdynamics, network architecture, initialization procedure, and other factors tietogether and are all components of its success. This presents theoreticalchallenges for analyzing state-based and/or online deep learning.  Motivated by applications in control, we give a general black-box reductionfrom deep learning to online convex optimization. This allows us to decoupleoptimization, regret, expressiveness, and derive agnostic online learningguarantees for fully-connected deep neural networks with ReLU activations. Wequantify convergence and regret guarantees for any range of parameters andallow any optimization procedure, such as adaptive gradient methods and secondorder methods. As an application, we derive provable algorithms for deepcontrol in the online episodic setting.

Provable Regret Bounds for Deep Online Learning and Control