Input Space Mode Connectivity in Deep Neural Networks

We extend the concept of loss landscape mode connectivity to the input space of deep neural networks. Mode connectivity was originally studied within parameter space, where it describes the existence of low-loss paths between different solutions (loss minimizers) obtained through gradient descent. We present theoretical and empirical evidence of its presence in the input space of deep networks, thereby highlighting the broader nature of the phenomenon. We observe that different input images with similar predictions are generally connected, and for trained models, the path tends to be simple, with only a small deviation from being a linear path. Our methodology utilizes real, interpolated, and synthetic inputs created using the input optimization technique for feature visualization. We conjecture that input space mode connectivity in high-dimensional spaces is a geometric effect that takes place even in untrained models and can be explained through percolation theory. We exploit mode connectivity to obtain new insights about adversarial examples and demonstrate its potential for adversarial detection. Additionally, we discuss applications for the interpretability of deep networks.

Weak Correlations as the Underlying Principle for Linearization of Gradient-Based Learning Systems

Deep learning models, such as wide neural networks, can be conceptualized as nonlinear dynamical physical systems characterized by a multitude of interacting degrees of freedom. Such systems in the infinite limit, tend to exhibit simplified dynamics. This paper delves into gradient descent-based learning algorithms, that display a linear structure in their parameter dynamics, reminiscent of the neural tangent kernel. We establish this apparent linearity arises due to weak correlations between the first and higher-order derivatives of the hypothesis function, concerning the parameters, taken around their initial values. This insight suggests that these weak correlations could be the underlying reason for the observed linearization in such systems. As a case in point, we showcase this weak correlations structure within neural networks in the large width limit. Exploiting the relationship between linearity and weak correlations, we derive a bound on deviations from linearity observed during the training trajectory of stochastic gradient descent. To facilitate our proof, we introduce a novel method to characterise the asymptotic behavior of random tensors.