Akshay Kumar

Akshay Kumar, Jarvis Haupt

This paper examines gradient flow dynamics of two-homogeneous neural networks for small initializations, where all weights are initialized near the origin. For both square and logistic losses, it is shown that for sufficiently small initializations, the gradient flow dynamics spend sufficient time in the neighborhood of the origin to allow the weights of the neural network to approximately converge in direction to the Karush-Kuhn-Tucker (KKT) points of a neural correlation function that quantifies the correlation between the output of the neural network and corresponding labels in the training data set. For square loss, it has been observed that neural networks undergo saddle-to-saddle dynamics when initialized close to the origin. Motivated by this, this paper also shows a similar directional convergence among weights of small magnitude in the neighborhood of certain saddle points.

Akshay Kumar, Jarvis Haupt

This paper studies the gradient flow dynamics that arise when training deep homogeneous neural networks assumed to have locally Lipschitz gradients and an order of homogeneity strictly greater than two. It is shown here that for sufficiently small initializations, during the early stages of training, the weights of the neural network remain small in (Euclidean) norm and approximately converge in direction to the Karush-Kuhn-Tucker (KKT) points of the recently introduced neural correlation function. Additionally, this paper also studies the KKT points of the neural correlation function for feed-forward networks with (Leaky) ReLU and polynomial (Leaky) ReLU activations, deriving necessary and sufficient conditions for rank-one KKT points.

Akshay Kumar, Jarvis Haupt

Recent works exploring the training dynamics of homogeneous neural network weights under gradient flow with small initialization have established that in the early stages of training, the weights remain small and near the origin, but converge in direction. Building on this, the current paper studies the gradient flow dynamics of homogeneous neural networks with locally Lipschitz gradients, after they escape the origin. Insights gained from this analysis are used to characterize the first saddle point encountered by gradient flow after escaping the origin. Also, it is shown that for homogeneous feed-forward neural networks, under certain conditions, the sparsity structure emerging among the weights before the escape is preserved after escaping the origin and until reaching the next saddle point.

Akshay Kumar, Jarvis Haupt

The first part of this paper studies the evolution of gradient flow for homogeneous neural networks near a class of saddle points exhibiting a sparsity structure. The choice of these saddle points is motivated from previous works on homogeneous networks, which identified the first saddle point encountered by gradient flow after escaping the origin. It is shown here that, when initialized sufficiently close to such saddle points, gradient flow remains near the saddle point for a sufficiently long time, during which the set of weights with small norm remain small but converge in direction. Furthermore, important empirical observations are made on the behavior of gradient descent after escaping these saddle points. The second part of the paper, motivated by these results, introduces a greedy algorithm to train deep neural networks called Neuron Pursuit (NP). It is an iterative procedure which alternates between expanding the network by adding neuron(s) with carefully chosen weights, and minimizing the training loss using this augmented network. The efficacy of the proposed algorithm is validated using numerical experiments.

Vinay Damodaran, Sharanya Chakravarthy, Akshay Kumar et al.

Visual Question Answering (VQA) is of tremendous interest to the research community with important applications such as aiding visually impaired users and image-based search. In this work, we explore the use of scene graphs for solving the VQA task. We conduct experiments on the GQA dataset which presents a challenging set of questions requiring counting, compositionality and advanced reasoning capability, and provides scene graphs for a large number of images. We adopt image + question architectures for use with scene graphs, evaluate various scene graph generation techniques for unseen images, propose a training curriculum to leverage human-annotated and auto-generated scene graphs, and build late fusion architectures to learn from multiple image representations. We present a multi-faceted study into the use of scene graphs for VQA, making this work the first of its kind.