The assumption that data are independently and identically distributed (IID) is staple in statistical machine learning. It suggests that a hypothesis selected by an algorithm, after observing several training samples, should perform effectively on test samples from the same unknown distribution.

We present an algorithm for learning switched linear dynamical systems in discrete time from noisy observations of the system's full state or output. Switched linear systems use multiple linear dynamical modes to fit the data within some desired tolerance. They arise quite naturally in applications to robotics and cyberphysical systems. Learning switched systems from data is a NP-hard problem that is nearly identical to the k-linear regression problem of fitting k > 1 linear models to the data. A direct mixed-integer linear programming (MILP) approach yields time complexity that is exponential in the number of data points. In this paper, we modify the problem formulation to yield an algorithm that is linear in the size of the data while remaining exponential in the number of state variables and the desired number of modes. To do so, we combine classic ideas from the ellipsoidal method for solving convex optimization problems, and well-known oracle separation results in non-smooth optimization. We demonstrate our approach on a set of microbenchmarks and a few interesting real-world problems. Our evaluation suggests that the benefits of this algorithm can be made practical even against highly optimized off-the-shelf MILP solvers.

Now, one could try to translate the constraint as|xk x0k| α for all k = 1,...,r. Here, we investigate the impact of the balance parameterγ from Equation (5) on the accuracyfairnesstradeoff. Note thatourmethod canincrease bothaccuracy, albeit only by a small amount, and fairness for certain values ofγ (e.g., γ = 2). Across all datasets and values ofγ the largest increase in certification for adversarial training is roughly 7%, with a simultaneous accuracy drop of 0.5%, and the largest accuracy drop is roughly 1%, with a simultaneous increaseincertification of2.9%. Wenote that although Fischer et al.[15]support terms with real-valued functions, we only consider linear functions since nonlinear constraints, e.g.,x2 < 3, cannot be encoded exactly as MILP.

ROUGE-N scores are the overlapping of n-grams [2] between the generated and referencesequence. Those scores are then averaged overthe whole corpus toreach anoverall quality. For both proposed MMMMI attack methods, shadow models are indispensable. The first hidden layer in the attack model has 256 units and the second hidden layer has20units, bothactivatedbyReLU function. We used resnet-LSTM architecture as the target model architecture.