PLATEAU IN MONOTONIC LINEAR INTERPOLATION -A "BIASED" VIEW OF LOSS LANDSCAPE FOR DEEP NETWORKS

Abstract

Monotonic linear interpolation (MLI) -on the line connecting a random initialization with the minimizer it converges to, the loss and accuracy are monotonic -is a phenomenon that is commonly observed in the training of neural networks. Such a phenomenon may seem to suggest that optimization of neural networks is easy. In this paper, we show that the MLI property is not necessarily related to the hardness of optimization problems, and empirical observations on MLI for deep neural networks depend heavily on the biases. In particular, we show that interpolating both weights and biases linearly leads to very different influences on the final output, and when different classes have different last-layer biases on a deep network, there will be a long plateau in both the loss and accuracy interpolation (which existing theory of MLI cannot explain). We also show how the last-layer biases for different classes can be different even on a perfectly balanced dataset using a simple model. Empirically we demonstrate that similar intuitions hold on practical networks and realistic datasets.

1. INTRODUCTION

Deep neural networks can often be optimized using simple gradient-based methods, despite the objectives being highly nonconvex. Intuitively, this suggests that the loss landscape must have nice properties that allow efficient optimization. To understand the properties of loss landscape, Goodfellow et al. (2014) studied the linear interpolation between a random initialization and the local minimum found after training. They observed that the loss interpolation curve is monotonic and approximately convex (see the MNIST curve in Figure 1 ) and concluded that these tasks are easy to optimize. However, other recent empirical observations, such as Frankle (2020) observed that for deep neural networks on more complicated datasets, both the loss and the error curves have a long plateau along the interpolation path, i.e., the loss and error remain high until close to the optimum (see the CIFAR-10 curve in Figure 1 ). Does the long plateau along the linear interpolation suggest these tasks are harder to optimize? Not necessarily, since the hardness of optimization problems does not need to be related to the shape of interpolation curves (see examples in Appendix A). In this paper we give the first theory that explains the plateau in both loss and error interpolations. We attribute the plateau to simple reasons as the bias terms, the network initialization scale and the network depth, which may not necessarily be related to the difficulty of optimization. Note that there are many different theories for the optimization of overparametrized neural networks, in particular the neural tangent kernel (NTK) analysis (Jacot et al., 2018; Du et al., 2018; Allen-Zhu et al., 2019; Arora et al., 2019) and mean-field analysis (Chizat & Bach, 2018; Mei et al., 2018) . However they don't explain the plateau in both loss and error interpolations. For NTK regime, the network output is nearly linear in the parameters and the loss interpolation curve is monotonically decreasing and convex -no plateau in the loss interpolation. Mean-field regime often uses a smaller initialization on a homogeneous neural network (as considered in Chizat & Bach (2018); Mei et al. ( 2018)). In this case, the interpolated network output is basically a scaled version of the network output at the minimum and has same label predictions -no plateau in the error interpolation curve.

