Sahel Vahedi Noori

Sahel Vahedi Noori, Bin Hu, Geir Dullerud et al.

This paper derives a complete set of quadratic constraints (QCs) for the repeated ReLU. The complete set of QCs is described by a collection of matrix copositivity conditions. We also show that only two functions satisfy all QCs in our complete set: the repeated ReLU and flipped ReLU. Thus our complete set of QCs bounds the repeated ReLU as tight as possible up to the sign invariance inherent in quadratic forms. We derive a similar complete set of incremental QCs for repeated ReLU, which can potentially lead to less conservative Lipschitz bounds for ReLU networks than the standard LipSDP approach. The basic constructions are also used to derive the complete sets of QCs for other piecewise linear activation functions such as leaky ReLU, MaxMin, and HouseHolder. Finally, we illustrate the use of the complete set of QCs to assess stability and performance for recurrent neural networks with ReLU activation functions. We rely on a standard copositivity relaxation to formulate the stability/performance condition as a semidefinite program. Simple examples are provided to illustrate that the complete sets of QCs and incremental QCs can yield less conservative bounds than existing sets.

Sahel Vahedi Noori, Bin Hu, Geir Dullerud et al.

This paper presents a new dynamic integral quadratic constraint (IQC) for the repeated Rectified Linear Unit (ReLU). These dynamic IQCs can be used to analyze stability and induced $\ell_2$-gain performance of discrete-time, recurrent neural networks (RNNs) with ReLU activation functions. These analysis conditions can be incorporated into learning-based controller synthesis methods, which currently rely on static IQCs. We show that our proposed dynamic IQCs for repeated ReLU form a superset of the dynamic IQCs for repeated, slope-restricted nonlinearities. We also prove that the $\ell_2$-gain bounds are nonincreasing with respect to the horizon used in the dynamic IQC filter. A numerical example using a simple (academic) RNN shows that our proposed IQCs lead to less conservative bounds than existing IQCs.

Sahel Vahedi Noori, Bin Hu, Geir Dullerud et al.

This paper presents sufficient conditions for the stability and $\ell_2$-gain performance of recurrent neural networks (RNNs) with ReLU activation functions. These conditions are derived by combining Lyapunov/dissipativity theory with Quadratic Constraints (QCs) satisfied by repeated ReLUs. We write a general class of QCs for repeated RELUs using known properties for the scalar ReLU. Our stability and performance condition uses these QCs along with a "lifted" representation for the ReLU RNN. We show that the positive homogeneity property satisfied by a scalar ReLU does not expand the class of QCs for the repeated ReLU. We present examples to demonstrate the stability / performance condition and study the effect of the lifting horizon.