Analysis of neural network optimization in the mean-field regime is important as the setting allows for feature learning. Existing theory has been developed mainly for neural networks in finite dimensions, i.e., each neuron has a finite-dimensional parameter. However, the setting of infinite-dimensional input naturally arises in machine learning problems such as nonparametric functional data analysis and graph classification. In this paper, we develop a new mean-field analysis of two-layer neural network in an infinite-dimensional parameter space. We first give a generalization error bound, which shows that the regularized empirical risk minimizer properly generalizes when the data size is sufficiently large, despite the neurons being infinite-dimensional. Next, we present two gradient-based optimization algorithms for infinite-dimensional mean-field networks, by extending the recently developed particle optimization framework to the infinite-dimensional setting. We show that the proposed algorithms converge to the (regularized) global optimal solution, and moreover, their rates of convergence are of polynomial order in the online setting and exponential order in the finite sample setting, respectively. To our knowledge this is the first quantitative global optimization guarantee of neural network on infinite-dimensional input and in the presence of feature learning.

Our results heavily rely on the specific nature of the periodic activation function, so a natural question is to which extent our results can be extended beyond the single periodic neuron class. For lower bounds, a challenging but very interesting generalization would be to establish the cryptographic-hardness of learning certain family of GLMs whose activation function does not need to be periodic. A potentially easier route forward on this direction, would be to consider the Hermite decomposition of the activation function, similar to [A3], and establish lower bounds on the performance of low-degree methods [A23], of SGD [A3], or of local search methods methods [A15], for activation functions whose low-degree Hermite coefficients are exponentially small. For upper bounds, we believe that our proposed LLL-based algorithm may be extended beyond learning even periodic activation functions, such as the cosine activation, by appropriately post-processing the measurements, but leave this for future work. Furthermore, it would be interesting to better understand (empirically or analytically) the noise tolerance of our LLL-based algorithm for "low-frequency" activation functions, such as the absolute value underlying the phase retrieval problem which has "zero" frequency.

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Diffusion models have achieved state-of-the-art performance in generative modeling tasks across various domains. Prior works on time series diffusion models have primarily focused on developing conditional models tailored to specific forecasting or imputation tasks. In this work, we explore the potential of taskagnostic, unconditional diffusion models for several time series applications. We propose TSDiff, an unconditionally-trained diffusion model for time series. Our proposed self-guidance mechanism enables conditioning TSDiff for downstream tasks during inference, without requiring auxiliary networks or altering the training procedure. We demonstrate the effectiveness of our method on three different time series tasks: forecasting, refinement, and synthetic data generation. First, we show that TSDiff is competitive with several task-specific conditional forecasting methods (predict). Second, we leverage the learned implicit probability density of TSDiff to iteratively refine the predictions of base forecasters with reduced computational overhead over reverse diffusion (refine). Notably, the generative performance of the model remains intact -- downstream forecasters trained on synthetic samples from TSDiff outperform forecasters that are trained on samples from other state-of-the-art generative time series models, occasionally even outperforming models trained on real data (synthesize).

Elon Musk Boosts New Yorker's Sam Altman Exposé on X as Trial Begins The move comes as the trial for Elon Musk's lawsuit against OpenAI kicks off in federal court in Oakland. Elon Musk is boosting a post on X promoting The New Yorker's extensive investigation into Sam Altman's allegedly deceptive behavior, WIRED has confirmed. The move comes just as Musk's lawsuit against OpenAI and Altman heads to a jury trial in a federal courtroom on Monday morning. People scrolling X on Monday reported seeing an April 6 post from Ronan Farrow, a coauthor on the New Yorker article, promoting the investigation. A pop-up on the post on X's mobile app says it was boosted by @elonmusk, who also owns the platform.

Normalization techniques have become a basic component in modern convolutional neural networks (ConvNets). In particular, many recent works demonstrate that promoting the orthogonality of the weights helps train deep models and improve robustness. For ConvNets, most existing methods are based on penalizing or normalizing weight matrices derived from concatenating or flattening the convolutional kernels. These methods often destroy or ignore the benign convolutional structure of the kernels; therefore, they are often expensive or impractical for deep ConvNets. In contrast, we introduce a simple and efficient "Convolutional Normalization" (ConvNorm) method that can fully exploit the convolutional structure in the Fourier domain and serve as a simple plug-and-play module to be conveniently incorporated into any ConvNets. Our method is inspired by recent work on preconditioning methods for convolutional sparse coding and can effectively promote each layer's channel-wise isometry. Furthermore, we show that our ConvNorm can reduce the layerwise spectral norm of the weight matrices and hence improve the Lipschitzness of the network, leading to easier training and improved robustness for deep ConvNets. Applied to classification under noise corruptions and generative adversarial network (GAN), we show that the ConvNorm improves the robustness of common ConvNets such as ResNet and the performance of GAN. We verify our findings via numerical experiments on CIFAR and ImageNet.

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