  Despite the success of deep neural networks (DNNs) for real-world
applications over time-series data such as mobile health, little is known about
how to train robust DNNs for time-series domain due to its unique
characteristics compared to images and text data. In this paper, we propose a
novel algorithmic framework referred as RObust Training for Time-Series (RO-TS)
to create robust DNNs for time-series classification tasks. Specifically, we
formulate a min-max optimization problem over the model parameters by
explicitly reasoning about the robustness criteria in terms of additive
perturbations to time-series inputs measured by the global alignment kernel
(GAK) based distance. We also show the generality and advantages of our
formulation using the summation structure over time-series alignments by
relating both GAK and dynamic time warping (DTW). This problem is an instance
of a family of compositional min-max optimization problems, which are
challenging and open with unclear theoretical guarantee. We propose a
principled stochastic compositional alternating gradient descent ascent
(SCAGDA) algorithm for this family of optimization problems. Unlike traditional
methods for time-series that require approximate computation of distance
measures, SCAGDA approximates the GAK based distance on-the-fly using a moving
average approach. We theoretically analyze the convergence rate of SCAGDA and
provide strong theoretical support for the estimation of GAK based distance.
Our experiments on real-world benchmarks demonstrate that RO-TS creates more
robust DNNs when compared to adversarial training using prior methods that rely
on data augmentation or new definitions of loss functions. We also demonstrate
the importance of GAK for time-series data over the Euclidean distance. The
source code of RO-TS algorithms is available at
https://github.com/tahabelkhouja/Robust-Training-for-Time-Series
