Ensuring safe, comfortable, and efficient planning is crucial for autonomous driving systems. While end-to-end models trained on large datasets perform well in standard driving scenarios, they struggle with complex low-frequency events. Recent Large Language Models (LLMs) and Vision Language Models (VLMs) advancements offer enhanced reasoning but suffer from computational inefficiency. Inspired by the dual-process cognitive model "Thinking, Fast and Slow", we propose $\textbf{FASIONAD}$ -- a novel dual-system framework that synergizes a fast end-to-end planner with a VLM-based reasoning module. The fast system leverages end-to-end learning to achieve real-time trajectory generation in common scenarios, while the slow system activates through uncertainty estimation to perform contextual analysis and complex scenario resolution. Our architecture introduces three key innovations: (1) A dynamic switching mechanism enabling slow system intervention based on real-time uncertainty assessment; (2) An information bottleneck with high-level plan feedback that optimizes the slow system's guidance capability; (3) A bidirectional knowledge exchange where visual prompts enhance the slow system's reasoning while its feedback refines the fast planner's decision-making. To strengthen VLM reasoning, we develop a question-answering mechanism coupled with reward-instruct training strategy. In open-loop experiments, FASIONAD achieves a $6.7\%$ reduction in average $L2$ trajectory error and $28.1\%$ lower collision rate.

Ensuring safe, comfortable, and efficient navigation is a critical goal for autonomous driving systems. While end-to-end models trained on large-scale datasets excel in common driving scenarios, they often struggle with rare, long-tail events. Recent progress in large language models (LLMs) has introduced enhanced reasoning capabilities, but their computational demands pose challenges for real-time decision-making and precise planning. This paper presents FASIONAD, a novel dual-system framework inspired by the cognitive model "Thinking, Fast and Slow." The fast system handles routine navigation tasks using rapid, data-driven path planning, while the slow system focuses on complex reasoning and decision-making in challenging or unfamiliar situations. A dynamic switching mechanism based on score distribution and feedback allows seamless transitions between the two systems. Visual prompts generated by the fast system enable human-like reasoning in the slow system, which provides high-quality feedback to enhance the fast system's decision-making. To evaluate FASIONAD, we introduce a new benchmark derived from the nuScenes dataset, specifically designed to differentiate fast and slow scenarios. FASIONAD achieves state-of-the-art performance on this benchmark, establishing a new standard for frameworks integrating fast and slow cognitive processes in autonomous driving. This approach paves the way for more adaptive, human-like autonomous driving systems.

Large Language Models (LLMs) and Reinforcement Learning (RL) are two powerful approaches for building autonomous agents. However, due to limited understanding of the game environment, agents often resort to inefficient exploration and trial-and-error, struggling to develop long-term strategies or make decisions. We propose a method that extracts experience from interaction records to model the underlying laws of the game environment, using these experience as internal motivation to guide agents. These experience, expressed in language, are highly flexible and can either assist agents in reasoning directly or be transformed into rewards for guiding training. Our evaluation results in Crafter demonstrate that both RL and LLM agents benefit from these experience, leading to improved overall performance.

Symmetry is ubiquitous in many real-world phenomena and tasks, such as physics, images, and molecular simulations. Empirical studies have demonstrated that incorporating symmetries into generative models can provide better generalization and sampling efficiency when the underlying data distribution has group symmetry. In this work, we provide the first theoretical analysis and guarantees of score-based generative models (SGMs) for learning distributions that are invariant with respect to some group symmetry and offer the first quantitative comparison between data augmentation and adding equivariant inductive bias. First, building on recent works on the Wasserstein-1 ($\mathbf{d}_1$) guarantees of SGMs and empirical estimations of probability divergences under group symmetry, we provide an improved $\mathbf{d}_1$ generalization bound when the data distribution is group-invariant. Second, we describe the inductive bias of equivariant SGMs using Hamilton-Jacobi-Bellman theory, and rigorously demonstrate that one can learn the score of a symmetrized distribution using equivariant vector fields without data augmentations through the analysis of the optimality and equivalence of score-matching objectives. This also provides practical guidance that one does not have to augment the dataset as long as the vector field or the neural network parametrization is equivariant. Moreover, we quantify the impact of not incorporating equivariant structure into the score parametrization, by showing that non-equivariant vector fields can yield worse generalization bounds. This can be viewed as a type of model-form error that describes the missing structure of non-equivariant vector fields. Numerical simulations corroborate our analysis and highlight that data augmentations cannot replace the role of equivariant vector fields.

Heavy tails are ubiquitous, emerging in various fields such as extreme events in ocean waves [9], floods [21], social sciences [27, 16], human activities [17, 35], biology [18] and computer sciences [29]. Learning to generate heavy-tailed target distributions has been explored using GANs through tail estimation [10, 15, 1]. While estimating the tail behavior of a heavy-tailed distribution is important, selecting objectives that measure discrepancies between these distributions and facilitate stable learning is equally crucial. In generative modeling, the goal is to generate samples that mimic those from an underlying data distribution, typically by designing algorithms that minimize a probability divergence between the generated and target distributions. Thus, it is crucial to choose a divergence that flexibly and accurately respects the behavior of the data distribution.

Group-invariant generative adversarial networks (GANs) are a type of GANs in which the generators and discriminators are hardwired with group symmetries. Empirical studies have shown that these networks are capable of learning group-invariant distributions with significantly improved data efficiency. In this study, we aim to rigorously quantify this improvement by analyzing the reduction in sample complexity for group-invariant GANs. Our findings indicate that when learning group-invariant distributions, the number of samples required for group-invariant GANs decreases proportionally with a power of the group size, and this power depends on the intrinsic dimension of the distribution's support. To our knowledge, this work presents the first statistical estimation for group-invariant generative models, specifically for GANs, and it may shed light on the study of other group-invariant generative models.

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized $\alpha$-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

We study the implicit bias of gradient flow on linear equivariant steerable networks in group-invariant binary classification. Our findings reveal that the parameterized predictor converges in direction to the unique group-invariant classifier with a maximum margin defined by the input group action. Under a unitary assumption on the input representation, we establish the equivalence between steerable networks and data augmentation. Furthermore, we demonstrate the improved margin and generalization bound of steerable networks over their non-invariant counterparts.

Spectral methods which represent data points by eigenvectors of kernel matrices or graph Laplacian matrices have been a primary tool in unsupervised data analysis. In many application scenarios, parametrizing the spectral embedding by a neural network that can be trained over batches of data samples gives a promising way to achieve automatic out-of-sample extension as well as computational scalability. Such an approach was taken in the original paper of SpectralNet (Shaham et al. 2018), which we call SpecNet1. The current paper introduces a new neural network approach, named SpecNet2, to compute spectral embedding which optimizes an equivalent objective of the eigen-problem and removes the orthogonalization layer in SpecNet1. SpecNet2 also allows separating the sampling of rows and columns of the graph affinity matrix by tracking the neighbors of each data point through the gradient formula. Theoretically, we show that any local minimizer of the new orthogonalization-free objective reveals the leading eigenvectors. Furthermore, global convergence for this new orthogonalization-free objective using a batch-based gradient descent method is proved. Numerical experiments demonstrate the improved performance and computational efficiency of SpecNet2 on simulated data and image datasets.

As an important population-based algorithm, ant colony optimization (ACO) has been successfully applied into various combinatorial optimization problems. However, much existing work in ACO focuses on solving centralized problems. In this paper, we present a novel algorithm that takes the power of ants to solve Distributed Constraint Optimization Problems (DCOPs), called ACO_DCOP. In ACO_DCOP, a new mechanism that captures local benefits is proposed to compute heuristic factors and a new method that considers the cost structure of DCOPs is proposed to compute pheromone deltas appropriately. Moreover, pipelining technique is introduced to make full use of the computational capacity and improve the efficiency. In our theoretical analysis, we prove that ACO_DCOP is an anytime algorithm. Our empirical evaluation indicates that ACO_DCOP is able to find solutions of equal or significantly higher quality than state-of-the-art DCOP algorithms.