Abstract: While score-based diffusion models have achieved remarkable success in high-dimensional generative modeling, some basic theoretical questions have not been precisely resolved. In this talk, we address minimax optimality of density estimation, functional estimation, and hypothesis testing. First, we show diffusion models achieve the optimal density estimation rate over Holder balls. This result is a consequence of our sharp characterization of minimax score estimation across all noising levels. A key contribution is our lower bound argument which involves a slight…

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Abstract: Spectral gradient methods, such as the recently proposed Muon optimizer, are a promising alternative to standard gradient descent for training deep neural networks and transformers. Yet, it remains unclear in which regimes these spectral methods are expected to perform better. In this talk, I will present a simple condition that predicts when a spectral update yields a larger decrease in the loss than a standard gradient step. Informally, this criterion holds when, on the one hand, the gradient of the…

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