Separation and Bias of Deep Equilibrium Models on Expressivity and Learning Dynamics

Abstract

The deep equilibrium model (DEQ) generalizes the conventional feedforward 1 neural network by fixing the same weights for each layer block and extending 2 the number of layers to infinity. This novel model directly finds the fixed points 3 of such a forward process as features for prediction. Despite empirical evidence 4 showcasing its efficacy compared to feedforward neural networks, a theoretical 5 understanding for its separation and bias is still limited. In this paper, we take a 6 step by proposing some separations and studying the bias of DEQ in its expressive 7 power and learning dynamics. The results include: (1) A general separation is 8 proposed, showing the existence of a width-m DEQ that any fully connected neural 9 networks (FNNs) with depth O ( m α ) for α ∈ (0 , 1) cannot approximate unless 10 its width is sub-exponential in m ; (2) DEQ with polynomially bounded size and 11 magnitude can efficiently approximate certain steep functions (which has very large 12 derivatives) in L ∞ norm, whereas FNN with bounded depth and exponentially 13 bounded width cannot unless its weights magnitudes are exponentially large; (3) 14 The implicit regularization caused by gradient flow from a diagonal linear DEQ 15 is characterized, with specific examples showing the benefits brought by such 16 regularization. From the overall study, a high-level conjecture from our analysis 17 and empirical validations is that DEQ has potential advantages in learning certain 18 high-frequency components. 19