Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernel

Neural operators are aiming at approximating operators mapping between Banach spaces of functions, achieving much success in the field of scientific computing. Compared to certain deep learning-based solvers, such as Physics-Informed Neural Networks (PINNs), Deep Ritz Method (DRM), neural operators can solve a class of Partial Differential Equations (PDEs). Although much work has been done to analyze the approximation and generalization error of neural operators, there is still a lack of analysis on their training error. In this work, we conduct the convergence analysis of gradient descent for the wide shallow neural operators within the framework of Neural Tangent Kernel (NTK). The core idea lies on the fact that over-parameterization and random initialization together ensure that each weight vector remains near its initialization throughout all iterations, yielding the linear convergence of gradient descent. In this work, we demonstrate that under the setting of over-parametrization, gradient descent can find the global minimum regardless of whether it is in continuous time or discrete time.

Adaptive Refinement Protocols for Distributed Distribution Estimation under $\ell^p$-Losses

Consider the communication-constrained estimation of discrete distributions under $\ell^p$ losses, where each distributed terminal holds multiple independent samples and uses limited number of bits to describe the samples. We obtain the minimax optimal rates of the problem in most parameter regimes. An elbow effect of the optimal rates at $p=2$ is clearly identified. To show the optimal rates, we first design estimation protocols to achieve them. The key ingredient of these protocols is to introduce adaptive refinement mechanisms, which first generate rough estimate by partial information and then establish refined estimate in subsequent steps guided by the rough estimate. The protocols leverage successive refinement, sample compression and thresholding methods to achieve the optimal rates in different parameter regimes. The optimality of the protocols is shown by deriving compatible minimax lower bounds.

Seed-Music: A Unified Framework for High Quality and Controlled Music Generation

We introduce Seed-Music, a suite of music generation systems capable of producing high-quality music with fine-grained style control. Our unified framework leverages both auto-regressive language modeling and diffusion approaches to support two key music creation workflows: \textit{controlled music generation} and \textit{post-production editing}. For controlled music generation, our system enables vocal music generation with performance controls from multi-modal inputs, including style descriptions, audio references, musical scores, and voice prompts. For post-production editing, it offers interactive tools for editing lyrics and vocal melodies directly in the generated audio. We encourage readers to listen to demo audio examples at https://team.doubao.com/seed-music .

Component Fourier Neural Operator for Singularly Perturbed Differential Equations

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations.

Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks

First-order methods, such as gradient descent (GD) and stochastic gradient descent (SGD), have been proven effective in training neural networks. In the context of over-parameterization, there is a line of work demonstrating that randomly initialized (stochastic) gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. However, the learning rate of GD for training two-layer neural networks exhibits poor dependence on the sample size and the Gram matrix, leading to a slow training process. In this paper, we show that for the $L^2$ regression problems, the learning rate can be improved from $\mathcal{O}(\lambda_0/n^2)$ to $\mathcal{O}(1/\|\bm{H}^{\infty}\|_2)$, which implies that GD actually enjoys a faster convergence rate. Furthermore, we generalize the method to GD in training two-layer Physics-Informed Neural Networks (PINNs), showing a similar improvement for the learning rate. Although the improved learning rate has a mild dependence on the Gram matrix, we still need to set it small enough in practice due to the unknown eigenvalues of the Gram matrix. More importantly, the convergence rate is tied to the least eigenvalue of the Gram matrix, which can lead to slow convergence. In this work, we provide the convergence analysis of natural gradient descent (NGD) in training two-layer PINNs, demonstrating that the learning rate can be $\mathcal{O}(1)$, and at this rate, the convergence rate is independent of the Gram matrix.

Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks

Optimization algorithms is crucial in training physics-informed neural networks (PINNs), unsuitable methods may lead to poor solutions. Compared to the common gradient descent algorithm, implicit gradient descent (IGD) outperforms it in handling some multi-scale problems. In this paper, we provide convergence analysis for the implicit gradient descent for training over-parametrized two-layer PINNs. We first demonstrate the positive definiteness of Gram matrices for general smooth activation functions, like sigmoidal function, softplus function, tanh function and so on. Then the over-parameterization allows us to show that the randomly initialized IGD converges a globally optimal solution at a linear convergence rate. Moreover, due to the different training dynamics, the learning rate of IGD can be chosen independent of the sample size and the least eigenvalue of the Gram matrix.

Localization and Discrete Beamforming with a Large Reconfigurable Intelligent Surface

In millimeter-wave (mmWave) cellular systems, reconfigurable intelligent surfaces (RISs) are foreseeably deployed with a large number of reflecting elements to achieve high beamforming gains. The large-sized RIS will make radio links fall in the near-field localization regime with spatial non-stationarity issues. Moreover, the discrete phase restriction on the RIS reflection coefficient incurs exponential complexity for discrete beamforming. It remains an open problem to find the optimal RIS reflection coefficient design in polynomial time. To address these issues, we propose a scalable partitioned-far-field protocol that considers both the near-filed non-stationarity and discrete beamforming. The protocol approximates near-field signal propagation using a partitioned-far-field representation to inherit the sparsity from the sophisticated far-field and facilitate the near-field localization scheme. To improve the theoretical localization performance, we propose a fast passive beamforming (FPB) algorithm that optimally solves the discrete RIS beamforming problem, reducing the search complexity from exponential order to linear order. Furthermore, by exploiting the partitioned structure of RIS, we introduce a two-stage coarse-to-fine localization algorithm that leverages both the time delay and angle information. Numerical results demonstrate that centimeter-level localization precision is achieved under medium and high signal-to-noise ratios (SNR), revealing that RISs can provide support for low-cost and high-precision localization in future cellular systems.

tSF: Transformer-based Semantic Filter for Few-Shot Learning

Few-Shot Learning (FSL) alleviates the data shortage challenge via embedding discriminative target-aware features among plenty seen (base) and few unseen (novel) labeled samples. Most feature embedding modules in recent FSL methods are specially designed for corresponding learning tasks (e.g., classification, segmentation, and object detection), which limits the utility of embedding features. To this end, we propose a light and universal module named transformer-based Semantic Filter (tSF), which can be applied for different FSL tasks. The proposed tSF redesigns the inputs of a transformer-based structure by a semantic filter, which not only embeds the knowledge from whole base set to novel set but also filters semantic features for target category. Furthermore, the parameters of tSF is equal to half of a standard transformer block (less than 1M). In the experiments, our tSF is able to boost the performances in different classic few-shot learning tasks (about 2% improvement), especially outperforms the state-of-the-arts on multiple benchmark datasets in few-shot classification task.