The Benefits of Over-parameterization at Initialization in Deep ReLU Networks

Abstract

It has been noted in existing literature that over-parameterization in ReLU networks generally improves performance. While there could be several factors involved behind this, we prove some desirable theoretical properties at initialization which may be enjoyed by ReLU networks. Specifically, it is known that He initialization in deep ReLU networks asymptotically preserves variance of activations in the forward pass and variance of gradients in the backward pass for infinitely wide networks, thus preserving the flow of information in both directions. Our paper goes beyond these results and shows novel properties that hold under He initialization: i) the norm of hidden activation of each layer is equal to the norm of the input, and, ii) the norm of weight gradient of each layer is equal to the product of norm of the input vector and the error at output layer. These results are derived using the PAC analysis framework, and hold true for finitely sized datasets such that the width of the ReLU network only needs to be larger than a certain finite lower bound. As we show, this lower bound depends on the depth of the network and the number of samples, and by the virtue of being a lower bound, over-parameterized ReLU networks are endowed with these desirable properties. For the aforementioned hidden activation norm property under He initialization, we further extend our theory and show that this property holds for a finite width network even when the number of data samples is infinite. Thus we overcome several limitations of existing papers, and show new properties of deep ReLU networks at initialization.

The Separation Capacity of Random Neural Networks

“…Theorem 4.1 is new in its own right and improves on a previous result by Giryes et al [GSB16], see also [GSB20]. It is also closely related to work of Arpit and Bengio [AB19], who investigated the capability of a random ReLU layer as in Definition 1.2, but with bias b = 0, to preserve Euclidean norms.…”

Correlated Initialization for Correlated Data

“…Altering the derivation including rectified linear units (ReLU) led to the "He" initialization (He et al, 2015). "He" initialization seems also to be favorably related to over-parameterization of networks (Arpit & Bengio, 2019). Similar to these early works, more recent work has also considered activation scale (Hanin & Rolnick, 2018) and gradients (Balduzzi et al, 2017) as well as dynamical isometry properties (Saxe et al, 2013;Yang & Schoenholz, 2017).…”

Max-Affine Spline Insights Into Deep Network Pruning

“…1 (a), it is clear that overparametrization increases the number of subdivision lines and the chance to well position some of them. In addition, overparametrization has also been used favorably as a way to help gradient descent based techniques facilitating optimization (Arpit & Bengio, 2019); lastly, overparametrized also positions the initial parameters close to good local minima (Allen-Zhu et al, 2019;Zou & Gu, 2019;Kawaguchi et al, 2019) reducing the amount of updated needed during training. We formalize those points in the remark below.…”