Using the method of auxiliary coordinates, we derive a training algorithm that works by alternating steps that train an auxiliary embeddingwith steps that train the mapping.

Neural Information Processing Systems http://nips.cc/

The objective(1) can be extended to a weighted average, but we do not write out the weights and consider them as part ofℓn(x,ξ)andFn(x).

Neural tangent kernel (NTK) is a powerful tool to analyze training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but it is incomplete for neural networks with Hadamard products (NNs-Hp), e.g., StyleGAN and polynomial neural networks (PNNs). In this work, we derive the finite-width NTK formulation for a special class of NNs-Hp, i.e., polynomial neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our results, we elucidate the separation of PNNs over standard neural networks with respect to extrapolation and spectral bias. Our two key insights are that when compared to standard neural networks, PNNs can fit more complicated functions in the extrapolation regime and admit a slower eigenvalue decay of the respective NTK, leading to a faster learning towards high-frequency functions. Besides, our theoretical results can be extended to other types of NNs-Hp, which expand the scope of our work. Our empirical results validate the separations in broader classes of NNs-Hp, which provide a good justification for a deeper understanding of neural architectures.

We show that most local optima of PNN optimizations are global while we have a characterization of regions where bad local optimizers may exist.

Physical Neural Networks (PNN) are promising platforms for next-generation computing systems. However, recent advances in digital neural network performance are largely driven by the rapid growth in the number of trainable parameters and, so far, demonstrated PNNs are lagging behind by several orders of magnitude in terms of scale. This mirrors size and performance constraints found in early digital neural networks. In that period, efficient reuse of parameters contributed to the development of parameter-efficient architectures such as convolutional neural networks. In this work, we numerically investigate hardware-friendly weight-tying for PNNs. Crucially, with many PNN systems, there is a time-scale separation between the fast dynamic active elements of the forward pass and the only slowly trainable elements implementing weights and biases. With this in mind,we propose the Reuse of Layers for eXpanding a Neural Network (ReLaX-Net) architecture, which employs a simple layer-by-layer time-multiplexing scheme to increase the effective network depth and efficiently use the number of parameters. We only require the addition of fast switches for existing PNNs. We validate ReLaX-Nets via numerical experiments on image classification and natural language processing tasks. Our results show that ReLaX-Net improves computational performance with only minor modifications to a conventional PNN. We observe a favorable scaling, where ReLaX-Nets exceed the performance of equivalent traditional RNNs or DNNs with the same number of parameters.

The enormous energy demand of artificial intelligence is driving the development of alternative hardware for deep learning. Physical neural networks try to exploit physical systems to perform machine learning more efficiently. In particular, optical systems can calculate with light using negligible energy. While their computational capabilities were long limited by the linearity of optical materials, nonlinear computations have recently been demonstrated through modified input encoding. Despite this breakthrough, our inability to determine if physical neural networks can learn arbitrary relationships between data -- a key requirement for deep learning known as universality -- hinders further progress. Here we present a fundamental theorem that establishes a universality condition for physical neural networks. It provides a powerful mathematical criterion that imposes device constraints, detailing how inputs should be encoded in the tunable parameters of the physical system. Based on this result, we propose a scalable architecture using free-space optics that is provably universal and achieves high accuracy on image classification tasks. Further, by combining the theorem with temporal multiplexing, we present a route to potentially huge effective system sizes in highly practical but poorly scalable on-chip photonic devices. Our theorem and scaling methods apply beyond optical systems and inform the design of a wide class of universal, energy-efficient physical neural networks, justifying further efforts in their development.

By replacing standard non-linearities with polynomial activations, Polynomial Neural Networks (PNNs) are pivotal for applications such as privacy-preserving inference via Homomorphic Encryption (HE). However, training PNNs effectively presents a significant challenge: low-degree polynomials can limit model expressivity, while higher-degree polynomials, crucial for capturing complex functions, often suffer from numerical instability and gradient explosion. We introduce a robust and versatile training framework featuring two synergistic innovations: 1) a novel Boundary Loss that exponentially penalizes activation inputs outside a predefined stable range, and 2) Selective Gradient Clipping that effectively tames gradient magnitudes while preserving essential Batch Normalization statistics. We demonstrate our framework's broad efficacy by training PNNs within deep architectures composed of HE-compatible layers (e.g., linear layers, average pooling, batch normalization, as used in ResNet variants) across diverse image, audio, and human activity recognition datasets. These models consistently achieve high accuracy with low-degree polynomial activations (such as degree 2) and, critically, exhibit stable training and strong performance with polynomial degrees up to 22, where standard methods typically fail or suffer severe degradation. Furthermore, the performance of these PNNs achieves a remarkable parity, closely approaching that of their original ReLU-based counterparts. Extensive ablation studies validate the contributions of our techniques and guide hyperparameter selection. We confirm the HE-compatibility of the trained models, advancing the practical deployment of accurate, stable, and secure deep learning inference.