This paper studies the reinforcement learning-based distributed secondary frequency control and active power allocation of islanded microgrids under event-triggered mechanism. First, a novel bandwidth-conscious memory-event-triggered mechanism is proposed to reduce communication and computation burdens, which introduces the mean of memory signals to smooth the measured outputs perturbed by disturbances and noises. Meanwhile, a dynamic triggering threshold depending on real-time bandwidth status is constructed to adaptively adjust the data transmission rate according to the changes in bandwidth status and system responses. Second, a new reinforcement learning-based distributed secondary controller using Q-learning algorithm is presented to dynamically regulate the frequency controller gains in response to complex system environments, which helps in achieving better frequency restoration performance. Third, the system stability is analyzed by some linear matrix inequality conditions. Then, simulation outcomes based on load variation and plug-and-play test illustrate that our strategy achieves satisfactory performance in frequency restoration and active power sharing. In addition, some comparison results show the merits of the constructed event-triggered scheme and Q-learning-based distributed secondary frequency controller.

Diffusion Transformer, the backbone of Sora for video generation, successfully scales the capacity of diffusion models, pioneering new avenues for high-fidelity sequential data generation. Unlike static data such as images, sequential data consists of consecutive data frames indexed by time, exhibiting rich spatial and temporal dependencies. These dependencies represent the underlying dynamic model and are critical to validate the generated data. In this paper, we make the first theoretical step towards bridging diffusion transformers for capturing spatial-temporal dependencies. Specifically, we establish score approximation and distribution estimation guarantees of diffusion transformers for learning Gaussian process data with covariance functions of various decay patterns. We highlight how the spatial-temporal dependencies are captured and affect learning efficiency. Our study proposes a novel transformer approximation theory, where the transformer acts to unroll an algorithm. We support our theoretical results by numerical experiments, providing strong evidence that spatial-temporal dependencies are captured within attention layers, aligning with our approximation theory.

We study the inductive biases of diffusion models with a conditioning-variable, which have seen widespread application as both text-conditioned generative image models and observation-conditioned continuous control policies. We observe that when these models are queried conditionally, their generations consistently deviate from the idealized "denoising" process upon which diffusion models are formulated, inducing disagreement between popular sampling algorithms (e.g. DDPM, DDIM). We introduce *Schedule Deviation*, a rigorous measure which captures the rate of deviation from a standard denoising process, and provide a methodology to compute it. Crucially, we demonstrate that the deviation from an idealized denoising process occurs irrespective of the model capacity or amount of training data. We posit that this phenomenon occurs due to the difficulty of bridging distinct denoising flows across different parts of the conditioning space and show theoretically how such a phenomenon can arise through an inductive bias towards smoothness.

Diffusion models have achieved tremendous success in generating high-dimensional data like images, videos and audio. These models provide powerful data priors that can solve linear inverse problems in zero shot through Bayesian posterior sampling. However, exact posterior sampling for diffusion models is intractable. Current solutions often hinge on approximations that are either computationally expensive or lack strong theoretical guarantees. In this work, we introduce an efficient diffusion sampling algorithm for linear inverse problems that is guaranteed to be asymptotically accurate. We reveal a link between Bayesian posterior sampling and Bayesian filtering in diffusion models, proving the former as a specific instance of the latter. Our method, termed filtering posterior sampling, leverages sequential Monte Carlo methods to solve the corresponding filtering problem. It seamlessly integrates with all Markovian diffusion samplers, requires no model re-training, and guarantees accurate samples from the Bayesian posterior as particle counts rise. Empirical tests demonstrate that our method generates better or comparable results than leading zero-shot diffusion posterior samplers on tasks like image inpainting, super-resolution, and deblurring.

Diffusion models have revolutionized various application domains, including computer vision and audio generation. Despite the state-of-the-art performance, diffusion models are known for their slow sample generation due to the extensive number of steps involved. In response, consistency models have been developed to merge multiple steps in the sampling process, thereby significantly boosting the speed of sample generation without compromising quality. This paper contributes towards the first statistical theory for consistency models, formulating their training as a distribution discrepancy minimization problem. Our analysis yields statistical estimation rates based on the Wasserstein distance for consistency models, matching those of vanilla diffusion models. Additionally, our results encompass the training of consistency models through both distillation and isolation methods, demystifying their underlying advantage.

Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data points and is a critical component for achieving high classification accuracy. Many existing deep learning approaches directly impose the fusion loss when training neural networks. In this work, inspired by the convection-diffusion ordinary differential equations (ODEs), we propose a novel diffusion residual network (Diff-ResNet), internally introduces diffusion into the architectures of neural networks. Under the structured data assumption, it is proved that the proposed diffusion block can increase the distance-diameter ratio that improves the separability of inter-class points and reduces the distance among local intra-class points. Moreover, this property can be easily adopted by the residual networks for constructing the separable hyperplanes. Extensive experiments of synthetic binary classification, semi-supervised graph node classification and few-shot image classification in various datasets validate the effectiveness of the proposed method.

We consider the well-studied problem of learning a linear combination of $k$ ReLU activations with respect to a Gaussian distribution on inputs in $d$ dimensions. We give the first polynomial-time algorithm that succeeds whenever $k$ is a constant. All prior polynomial-time learners require additional assumptions on the network, such as positive combining coefficients or the matrix of hidden weight vectors being well-conditioned.Our approach is based on analyzing random contractions of higher-order moment tensors. We use a multi-scale clustering procedure to argue that sufficiently close neurons can be collapsed together, sidestepping the conditioning issues present in prior work. This allows us to design an iterative procedure to discover individual neurons.

As one of the most popular methods in the field of reinforcement learning, Q-learning has received increasing attention. Recently, there have been more theoretical works on the regret bound of algorithms that belong to the Q-learning class in different settings. In this paper, we analyze the cumulative regret when conducting Nash Q-learning algorithm on 2-player turn-based stochastic Markov games (2-TBSG), and propose the very first gap dependent logarithmic upper bounds in the episodic tabular setting. This bound matches the theoretical lower bound only up to a logarithmic term. Furthermore, we extend the conclusion to the discounted game setting with infinite horizon and propose a similar gap dependent logarithmic regret bound. Also, under the linear MDP assumption, we obtain another logarithmic regret for 2-TBSG, in both centralized and independent settings.