Mixture of Experts Soften the Curse of Dimensionality in Operator Learning

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a \textit{distributed} universal approximation theorem guaranteeing that any Lipschitz non-linear operator between $L^2([0,1]^d)$ spaces can be approximated uniformly over the Sobolev unit ball therein, to any given $\varepsilon>0$ accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of $\mathcal{O}(\varepsilon^{-1})$. Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies $\varepsilon$. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of $L^2([0,1]^d)$.

Breaking the Curse of Dimensionality with Distributed Neural Computation

We present a theoretical approach to overcome the curse of dimensionality using a neural computation algorithm which can be distributed across several machines. Our modular distributed deep learning paradigm, termed \textit{neural pathways}, can achieve arbitrary accuracy while only loading a small number of parameters into GPU VRAM. Formally, we prove that for every error level $\varepsilon>0$ and every Lipschitz function $f:[0,1]^n\to \mathbb{R}$, one can construct a neural pathways model which uniformly approximates $f$ to $\varepsilon$ accuracy over $[0,1]^n$ while only requiring networks of $\mathcal{O}(\varepsilon^{-1})$ parameters to be loaded in memory and $\mathcal{O}(\varepsilon^{-1}\log(\varepsilon^{-1}))$ to be loaded during the forward pass. This improves the optimal bounds for traditional non-distributed deep learning models, namely ReLU MLPs, which need $\mathcal{O}(\varepsilon^{-n/2})$ parameters to achieve the same accuracy. The only other available deep learning model that breaks the curse of dimensionality is MLPs with super-expressive activation functions. However, we demonstrate that these models have an infinite VC dimension, even with bounded depth and width restrictions, unlike the neural pathways model. This implies that only the latter generalizes. Our analysis is validated experimentally in both regression and classification tasks, demonstrating that our model exhibits superior performance compared to larger centralized benchmarks.

Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Training

Training neural networks that require adversarial optimization, such as generative adversarial networks (GANs) and unsupervised domain adaptations (UDAs), suffers from instability. This instability problem comes from the difficulty of the minimax optimization, and there have been various approaches in GANs and UDAs to overcome this problem. In this study, we tackle this problem theoretically through a functional analysis. Specifically, we show the convergence property of the minimax problem by the gradient descent over the infinite-dimensional spaces of continuous functions and probability measures under certain conditions. Using this setting, we can discuss GANs and UDAs comprehensively, which have been studied independently. In addition, we show that the conditions necessary for the convergence property are interpreted as stabilization techniques of adversarial training such as the spectral normalization and the gradient penalty.

Globally injective and bijective neural operators

Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank case, is subtler in the infinite-rank case and is proved using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not `lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may be many layers deep, are injective and surjective and provide an exact inversion from a `linearization.' This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian UQ where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness, respectively.

Fine-tuning Neural-Operator architectures for training and generalization

In this work, we present an analysis of the generalization of Neural Operators (NOs) and derived architectures. We proposed a family of networks, which we name (${\textit{s}}{\text{NO}}+\varepsilon$), where we modify the layout of NOs towards an architecture resembling a Transformer; mainly, we substitute the Attention module with the Integral Operator part of NOs. The resulting network preserves universality, has a better generalization to unseen data, and similar number of parameters as NOs. On the one hand, we study numerically the generalization by gradually transforming NOs into ${\textit{s}}{\text{NO}}+\varepsilon$ and verifying a reduction of the test loss considering a time-harmonic wave dataset with different frequencies. We perform the following changes in NOs: (a) we split the Integral Operator (non-local) and the (local) feed-forward network (MLP) into different layers, generating a {\it sequential} structure which we call sequential Neural Operator (${\textit{s}}{\text{NO}}$), (b) we add the skip connection, and layer normalization in ${\textit{s}}{\text{NO}}$, and (c) we incorporate dropout and stochastic depth that allows us to generate deep networks. In each case, we observe a decrease in the test loss in a wide variety of initialization, indicating that our changes outperform the NO. On the other hand, building on infinite-dimensional Statistics, and in particular the Dudley Theorem, we provide bounds of the Rademacher complexity of NOs and ${\textit{s}}{\text{NO}}$, and we find the following relationship: the upper bound of the Rademacher complexity of the ${\textit{s}}{\text{NO}}$ is a lower-bound of the NOs, thereby, the generalization error bound of ${\textit{s}}{\text{NO}}$ is smaller than NO, which further strengthens our numerical results.

Variational Inference with Gaussian Mixture by Entropy Approximation

Variational inference is a technique for approximating intractable posterior distributions in order to quantify the uncertainty of machine learning. Although the unimodal Gaussian distribution is usually chosen as a parametric distribution, it hardly approximates the multimodality. In this paper, we employ the Gaussian mixture distribution as a parametric distribution. A main difficulty of variational inference with the Gaussian mixture is how to approximate the entropy of the Gaussian mixture. We approximate the entropy of the Gaussian mixture as the sum of the entropy of the unimodal Gaussian, which can be analytically calculated. In addition, we theoretically analyze the approximation error between the true entropy and approximated one in order to reveal when our approximation works well. Specifically, the approximation error is controlled by the ratios of the distances between the means to the sum of the variances of the Gaussian mixture, and it converges to zero when the ratios go to infinity. This situation seems to be more likely to occur in higher dimensional weight spaces because of the curse of dimensionality. Therefore, our result guarantees that our approximation works well, for example, in neural networks that assume a large number of weights.

Spectral Pruning for Recurrent Neural Networks

Pruning techniques for neural networks with a recurrent architecture, such as the recurrent neural network (RNN), are strongly desired for their application to edge-computing devices. However, the recurrent architecture is generally not robust to pruning because even small pruning causes accumulation error and the total error increases significantly over time. In this paper, we propose an appropriate pruning algorithm for RNNs inspired by "spectral pruning", and provide the generalization error bounds for compressed RNNs. We also provide numerical experiments to demonstrate our theoretical results and show the effectiveness of our pruning method compared with existing methods.