Deterministic Dynamics of Sampling Processes in Score-Based Diffusion Models with Multiplicative Noise Conditioning

Score-based diffusion models generate new samples by learning the score function associated with a diffusion process. While the effectiveness of these models can be theoretically explained using differential equations related to the sampling process, previous work by Song and Ermon (2020) demonstrated that neural networks using multiplicative noise conditioning can still generate satisfactory samples. In this setup, the model is expressed as the product of two functions: one depending on the spatial variable and the other on the noise magnitude. This structure limits the model's ability to represent a more general relationship between the spatial variable and the noise, indicating that it cannot fully learn the correct score. Despite this limitation, the models perform well in practice. In this work, we provide a theoretical explanation for this phenomenon by studying the deterministic dynamics of the associated differential equations, offering insight into how the model operates.

On the convergence result of the gradient-push algorithm on directed graphs with constant stepsize

Gradient-push algorithm has been widely used for decentralized optimization problems when the connectivity network is a direct graph. This paper shows that the gradient-push algorithm with stepsize $\alpha>0$ converges exponentially fast to an $O(\alpha)$-neighborhood of the optimizer under the assumption that each cost is smooth and the total cost is strongly convex. Numerical experiments are provided to support the theoretical convergence results.