Are GATs Out of Balance?

While the expressive power and computational capabilities of graph neural networks (GNNs) have been theoretically studied, their optimization and learning dynamics, in general, remain largely unexplored. Our study undertakes the Graph Attention Network (GAT), a popular GNN architecture in which a node's neighborhood aggregation is weighted by parameterized attention coefficients. We derive a conservation law of GAT gradient flow dynamics, which explains why a high portion of parameters in GATs with standard initialization struggle to change during training. This effect is amplified in deeper GATs, which perform significantly worse than their shallow counterparts. To alleviate this problem, we devise an initialization scheme that balances the GAT network. Our approach i) allows more effective propagation of gradients and in turn enables trainability of deeper networks, and ii) attains a considerable speedup in training and convergence time in comparison to the standard initialization. Our main theorem serves as a stepping stone to studying the learning dynamics of positive homogeneous models with attention mechanisms.Comment: 25 pages. To be published in Advances in Neural Information Processing Systems (NeurIPS), 202

Identifying overparameterization in Quantum Circuit Born Machines

In machine learning, overparameterization is associated with qualitative changes in the empirical risk landscape, which can lead to more efficient training dynamics. For many parameterized models used in statistical learning, there exists a critical number of parameters, or model size, above which the model is constructed and trained in the overparameterized regime. There are many characteristics of overparameterized loss landscapes. The most significant is the convergence of standard gradient descent to global or local minima of low loss. In this work, we study the onset of overparameterization transitions for quantum circuit Born machines, generative models that are trained using non-adversarial gradient-based methods. We observe that bounds based on numerical analysis are in general good lower bounds on the overparameterization transition. However, bounds based on the quantum circuit's algebraic structure are very loose upper bounds. Our results indicate that fully understanding the trainability of these models remains an open question.Comment: 11 pages, 16 figure

Towards Strong Pruning for Lottery Tickets with Non-Zero Biases

The strong lottery ticket hypothesis holds the promise that pruning randomly initialized deep neural networks could offer a computationally efficient alternative to deep learning with stochastic gradient descent. Common parameter initialization schemes and existence proofs, however, are focused on networks with zero biases, thus foregoing the potential universal approximation property of pruning. To fill this gap, we extend multiple initialization schemes and existence proofs to non-zero biases, including explicit 'looks-linear' approaches for ReLU activation functions. These do not only enable truly orthogonal parameter initialization but also reduce potential pruning errors. In experiments on standard benchmark data sets, we further highlight the practical benefits of non-zero bias initialization schemes, and present theoretically inspired extensions for state-of-the-art strong lottery ticket pruning

Householder-Absolute Neural Layers For High Variability and Deep Trainability

We propose a new architecture for artificial neural networks called Householder-absolute neural layers, or Han-layers for short, that use Householder reflectors as weight matrices and the absolute-value function for activation. Han-layers, functioning as fully connected layers, are motivated by recent results on neural-network variability and are designed to increase activation ratio and reduce the chance of Collapse to Constants. Neural networks constructed chiefly from Han-layers are called HanNets. By construction, HanNets enjoy a theoretical guarantee that vanishing or exploding gradient never occurs. We conduct several proof-of-concept experiments. Some surprising results obtained on styled test problems suggest that, under certain conditions, HanNets exhibit an unusual ability to produce nearly perfect solutions unattainable by fully connected networks. Experiments on regression datasets show that HanNets can significantly reduce the number of model parameters while maintaining or improving the level of generalization accuracy. In addition, by adding a few Han-layers into the pre-classification FC-layer of a convolutional neural network, we are able to quickly improve a state-of-the-art result on CIFAR10 dataset. These proof-of-concept results are sufficient to necessitate further studies on HanNets to understand their capacities and limits, and to exploit their potentials in real-world applications

From Tight Gradient Bounds for Parameterized Quantum Circuits to the Absence of Barren Plateaus in QGANs

Barren plateaus are a central bottleneck in the scalability of variational quantum algorithms (VQAs), and are known to arise in various ways, from circuit depth and hardware noise to global observables. However, a caveat of most existing results is the requirement of t-design circuit assumptions that are typically not satisfied in practice. In this work, we loosen these assumptions altogether and derive tight upper and lower bounds on gradient concentration, for a large class of parameterized quantum circuits and arbitrary observables. By requiring only a couple of design choices that are constructive and easily verified, our results can readily be leveraged to rule out barren plateaus for explicit circuits and mixed observables, namely, observables containing a non-vanishing local term. This insight has direct implications for hybrid Quantum Generative Adversarial Networks (qGANs), a generative model that can be reformulated as a VQA with an observable composed of local and global terms. We prove that designing the discriminator appropriately leads to 1-local weights that stay constant in the number of qubits, regardless of discriminator depth. Combined with our first contribution, this implies that qGANs with shallow generators can be trained at scale without suffering from barren plateaus -- making them a promising candidate for applications in generative quantum machine learning. We demonstrate this result by training a qGAN to learn a 2D mixture of Gaussian distributions with up to 16 qubits, and provide numerical evidence that global contributions to the gradient, while initially exponentially small, may kick in substantially over the course of training