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Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory
Mar 18, 2024
Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges this gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models. Our analysis yields a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to our theoretical development lies in an approximation result for the conditional score function, which relies on a novel diffused Taylor approximation technique. Moreover, we demonstrate the utility of our statistical theory in elucidating the performance of conditional diffusion models across diverse applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward conditioned sample generation.
On the Role of Information Structure in Reinforcement Learning for Partially-Observable Sequential Teams and Games
Mar 01, 2024
In a sequential decision-making problem, the information structure is the description of how events in the system occurring at different points in time affect each other. Classical models of reinforcement learning (e.g., MDPs, POMDPs, Dec-POMDPs, and POMGs) assume a very simple and highly regular information structure, while more general models like predictive state representations do not explicitly model the information structure. By contrast, real-world sequential decision-making problems typically involve a complex and time-varying interdependence of system variables, requiring a rich and flexible representation of information structure. In this paper, we argue for the perspective that explicit representation of information structures is an important component of analyzing and solving reinforcement learning problems. We propose novel reinforcement learning models with an explicit representation of information structure, capturing classical models as special cases. We show that this leads to a richer analysis of sequential decision-making problems and enables more tailored algorithm design. In particular, we characterize the "complexity" of the observable dynamics of any sequential decision-making problem through a graph-theoretic analysis of the DAG representation of its information structure. The central quantity in this analysis is the minimal set of variables that $d$-separates the past observations from future observations. Furthermore, through constructing a generalization of predictive state representations, we propose tailored reinforcement learning algorithms and prove that the sample complexity is in part determined by the information structure. This recovers known tractability results and gives a novel perspective on reinforcement learning in general sequential decision-making problems, providing a systematic way of identifying new tractable classes of problems.
Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality
Feb 29, 2024We study the dynamics of gradient flow for training a multi-head softmax attention model for in-context learning of multi-task linear regression. We establish the global convergence of gradient flow under suitable choices of initialization. In addition, we prove that an interesting "task allocation" phenomenon emerges during the gradient flow dynamics, where each attention head focuses on solving a single task of the multi-task model. Specifically, we prove that the gradient flow dynamics can be split into three phases -- a warm-up phase where the loss decreases rather slowly and the attention heads gradually build up their inclination towards individual tasks, an emergence phase where each head selects a single task and the loss rapidly decreases, and a convergence phase where the attention parameters converge to a limit. Furthermore, we prove the optimality of gradient flow in the sense that the limiting model learned by gradient flow is on par with the best possible multi-head softmax attention model up to a constant factor. Our analysis also delineates a strict separation in terms of the prediction accuracy of ICL between single-head and multi-head attention models. The key technique for our convergence analysis is to map the gradient flow dynamics in the parameter space to a set of ordinary differential equations in the spectral domain, where the relative magnitudes of the semi-singular values of the attention weights determines task allocation. To our best knowledge, our work provides the first convergence result for the multi-head softmax attention model.
Double Duality: Variational Primal-Dual Policy Optimization for Constrained Reinforcement Learning
Feb 16, 2024We study the Constrained Convex Markov Decision Process (MDP), where the goal is to minimize a convex functional of the visitation measure, subject to a convex constraint. Designing algorithms for a constrained convex MDP faces several challenges, including (1) handling the large state space, (2) managing the exploration/exploitation tradeoff, and (3) solving the constrained optimization where the objective and the constraint are both nonlinear functions of the visitation measure. In this work, we present a model-based algorithm, Variational Primal-Dual Policy Optimization (VPDPO), in which Lagrangian and Fenchel duality are implemented to reformulate the original constrained problem into an unconstrained primal-dual optimization. Moreover, the primal variables are updated by model-based value iteration following the principle of Optimism in the Face of Uncertainty (OFU), while the dual variables are updated by gradient ascent. Moreover, by embedding the visitation measure into a finite-dimensional space, we can handle large state spaces by incorporating function approximation. Two notable examples are (1) Kernelized Nonlinear Regulators and (2) Low-rank MDPs. We prove that with an optimistic planning oracle, our algorithm achieves sublinear regret and constraint violation in both cases and can attain the globally optimal policy of the original constrained problem.
Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF
Feb 10, 2024Bilevel optimization has been recently applied to many machine learning tasks. However, their applications have been restricted to the supervised learning setting, where static objective functions with benign structures are considered. But bilevel problems such as incentive design, inverse reinforcement learning (RL), and RL from human feedback (RLHF) are often modeled as dynamic objective functions that go beyond the simple static objective structures, which pose significant challenges of using existing bilevel solutions. To tackle this new class of bilevel problems, we introduce the first principled algorithmic framework for solving bilevel RL problems through the lens of penalty formulation. We provide theoretical studies of the problem landscape and its penalty-based (policy) gradient algorithms. We demonstrate the effectiveness of our algorithms via simulations in the Stackelberg Markov game, RL from human feedback and incentive design.
Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems
Dec 02, 2023In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.
Empowering Autonomous Driving with Large Language Models: A Safety Perspective
Nov 28, 2023
Autonomous Driving (AD) faces crucial hurdles for commercial launch, notably in the form of diminished public trust and safety concerns from long-tail unforeseen driving scenarios. This predicament is due to the limitation of deep neural networks in AD software, which struggle with interpretability and exhibit poor generalization capabilities in out-of-distribution and uncertain scenarios. To this end, this paper advocates for the integration of Large Language Models (LLMs) into the AD system, leveraging their robust common-sense knowledge, reasoning abilities, and human-interaction capabilities. The proposed approach deploys the LLM as an intelligent decision-maker in planning, incorporating safety verifiers for contextual safety learning to enhance overall AD performance and safety. We present results from two case studies that affirm the efficacy of our approach. We further discuss the potential integration of LLM for other AD software components including perception, prediction, and simulation. Despite the observed challenges in the case studies, the integration of LLMs is promising and beneficial for reinforcing both safety and performance in AD.
Provably Efficient High-Dimensional Bandit Learning with Batched Feedbacks
Nov 24, 2023
We study high-dimensional multi-armed contextual bandits with batched feedback where the $T$ steps of online interactions are divided into $L$ batches. In specific, each batch collects data according to a policy that depends on previous batches and the rewards are revealed only at the end of the batch. Such a feedback structure is popular in applications such as personalized medicine and online advertisement, where the online data often do not arrive in a fully serial manner. We consider high-dimensional and linear settings where the reward function of the bandit model admits either a sparse or low-rank structure and ask how small a number of batches are needed for a comparable performance with fully dynamic data in which $L = T$. For these settings, we design a provably sample-efficient algorithm which achieves a $ \mathcal{\tilde O}(s_0^2 \log^2 T)$ regret in the sparse case and $ \mathcal{\tilde O} ( r ^2 \log^2 T)$ regret in the low-rank case, using only $L = \mathcal{O}( \log T)$ batches. Here $s_0$ and $r$ are the sparsity and rank of the reward parameter in sparse and low-rank cases, respectively, and $ \mathcal{\tilde O}(\cdot)$ omits logarithmic factors involving the feature dimensions. In other words, our algorithm achieves regret bounds comparable to those in fully sequential setting with only $\mathcal{O}( \log T)$ batches. Our algorithm features a novel batch allocation method that adjusts the batch sizes according to the estimation accuracy within each batch and cumulative regret. Furthermore, we also conduct experiments with synthetic and real-world data to validate our theory.
Posterior Sampling for Competitive RL: Function Approximation and Partial Observation
Oct 30, 2023This paper investigates posterior sampling algorithms for competitive reinforcement learning (RL) in the context of general function approximations. Focusing on zero-sum Markov games (MGs) under two critical settings, namely self-play and adversarial learning, we first propose the self-play and adversarial generalized eluder coefficient (GEC) as complexity measures for function approximation, capturing the exploration-exploitation trade-off in MGs. Based on self-play GEC, we propose a model-based self-play posterior sampling method to control both players to learn Nash equilibrium, which can successfully handle the partial observability of states. Furthermore, we identify a set of partially observable MG models fitting MG learning with the adversarial policies of the opponent. Incorporating the adversarial GEC, we propose a model-based posterior sampling method for learning adversarial MG with potential partial observability. We further provide low regret bounds for proposed algorithms that can scale sublinearly with the proposed GEC and the number of episodes $T$. To the best of our knowledge, we for the first time develop generic model-based posterior sampling algorithms for competitive RL that can be applied to a majority of tractable zero-sum MG classes in both fully observable and partially observable MGs with self-play and adversarial learning.
Learning Regularized Graphon Mean-Field Games with Unknown Graphons
Oct 26, 2023We design and analyze reinforcement learning algorithms for Graphon Mean-Field Games (GMFGs). In contrast to previous works that require the precise values of the graphons, we aim to learn the Nash Equilibrium (NE) of the regularized GMFGs when the graphons are unknown. Our contributions are threefold. First, we propose the Proximal Policy Optimization for GMFG (GMFG-PPO) algorithm and show that it converges at a rate of $O(T^{-1/3})$ after $T$ iterations with an estimation oracle, improving on a previous work by Xie et al. (ICML, 2021). Second, using kernel embedding of distributions, we design efficient algorithms to estimate the transition kernels, reward functions, and graphons from sampled agents. Convergence rates are then derived when the positions of the agents are either known or unknown. Results for the combination of the optimization algorithm GMFG-PPO and the estimation algorithm are then provided. These algorithms are the first specifically designed for learning graphons from sampled agents. Finally, the efficacy of the proposed algorithms are corroborated through simulations. These simulations demonstrate that learning the unknown graphons reduces the exploitability effectively.