Deep generative models, particularly diffusion models, have achieved remarkable success but face significant challenges when trained on real-world, long-tailed datasets-where few "head" classes dominate and many "tail" classes are underrepresented. This paper develops a theoretical framework for long-tailed learning via diffusion models through the lens of deep mutual learning. We introduce a novel regularized training objective that combines the standard diffusion loss with a mutual learning term, enabling balanced performance across all class labels, including the underrepresented tails. Our approach to learn via the proposed regularized objective is to formulate it as a multi-player game, with Nash equilibrium serving as the solution concept. We derive a non-asymptotic first-order convergence result for individual gradient descent algorithm to find the Nash equilibrium.

Deep generative models, particularly diffusion models, have achieved remarkable success but face significant challenges when trained on real-world, long-tailed datasets-where few head classes dominate and many tail classes are underrepresented. This paper develops a theoretical framework for long-tailed learning via diffusion models through the lens of deep mutual learning. We introduce a novel regularized training objective that combines the standard diffusion loss with a mutual learning term, enabling balanced performance across all class labels, including the underrepresented tails. Our approach to learn via the proposed regularized objective is to formulate it as a multi-player game, with Nash equilibrium serving as the solution concept. We derive a non-asymptotic first-order convergence result for individual gradient descent algorithm to find the Nash equilibrium. We show that the Nash gap of the score network obtained from the algorithm is upper bounded by $\mathcal{O}(\frac{1}{\sqrt{T_{train}}}+\beta)$ where $\beta$ is the regularizing parameter and $T_{train}$ is the number of iterations of the training algorithm. Furthermore, we theoretically establish hyper-parameters for training and sampling algorithm that ensure that we find conditional score networks (under our model) with a worst case sampling error $\mathcal{O}(\epsilon+1), \forall \epsilon> 0$ across all class labels. Our results offer insights and guarantees for training diffusion models on imbalanced, long-tailed data, with implications for fairness, privacy, and generalization in real-world generative modeling scenarios.

Efficiently sampling from un-normalized target distributions is a fundamental problem in scientific computing and machine learning. Traditional approaches such as Markov Chain Monte Carlo (MCMC) guarantee asymptotically unbiased samples from such distributions but suffer from computational inefficiency, particularly when dealing with high-dimensional targets, as they require numerous iterations to generate a batch of samples. In this paper, we introduce an efficient and scalable neural implicit sampler that overcomes these limitations. The implicit sampler can generate large batches of samples with low computational costs by leveraging a neural transformation that directly maps easily sampled latent vectors to target samples without the need for iterative procedures. To train the neural implicit samplers, we introduce two novel methods: the KL training method and the Fisher training method.

B.5 Langevin dynamics Under mild assumptions on r log p

Traditional denoising diffusion models are defined on finite-dimension Euclidean spaces (Song and Ermon, 2019; Song et al., 2021; Ho et al., 2020; Dhariwal and Nichol, 2021).

Diffusion models have recently gained a lot of attention both from academia and industry.

This paper introduces a data-adaptive nonlinear diffusion process for score-based diffusion models.

Generative models have many applications in machine learning. To list a few, they have been usedtogenerate high-fidelity images [26,6],synthesize realistic speech andmusic fragments [58], improve the performance of semi-supervised learning [28, 10], detect adversarial examples and other anomalous data [54], imitation learning [22], and explore promising states in reinforcement learning [41].

In this supplementary material, we first present the details of the conditional score network in Sec. 2. Sec. 4. Finally, we conduct experiments on Chinese MER dataset CH-SIMS [ I) which is subsequently fixed for the model (i.e., not learnable). Table 1: Hyperparameter settings in IMDer.Hyperparameter CMU-MOSI CMU-MOSEI Optimizer Adam Adam Batch size 32 128 Learning rate 0.001 0.002 σ used in our stochastic differential equation 25 25 Number of iterations for Euler-Maruyama solver 500 500 Shallow Feature Extractor Kernel size for E CH-SIMS contains 2281 refined video segments with fine-grained annotations of modalities. For vision modality, we use MultiComp OpenFace2.0 The experimental results are listed in the Tab. 3. Obviously, our proposed IMDer consistently achieves better results than MMIN or GCNet under random missing protocol.