In contrast to the advances in characterizing the sample complexity for solving Markov decision processes (MDPs), the optimal statistical complexity for solving constrained MDPs (CMDPs) remains unknown. We resolve this question by providing minimax upper and lower bounds on the sample complexity for learning near-optimal policies in a discounted CMDP with access to a generative model (simulator). In particular, we design a model-based algorithm that addresses two settings: (i) relaxed feasibility, where small constraint violations are allowed, and (ii) strict feasibility, where the output policy is required to satisfy the constraint.

Large-scale driving datasets such as Waymo Open Dataset and nuScenes substantially accelerate autonomous driving research, especially for perception tasks such as 3D detection and trajectory forecasting. Since the driving logs in these datasets contain HD maps and detailed object annotations that accurately reflect the realworld complexity of traffic behaviors, we can harvest a massive number of complex traffic scenarios and recreate their digital twins in simulation. Compared to the handcrafted scenarios often used in existing simulators, data-driven scenarios collected from the real world can facilitate many research opportunities in machine learning and autonomous driving.

We will use the well known Performance Difference Lemma [16] in our analysis. We can obtain a performance difference lemma for the meta-policies as follows. Here, we get (a)is from Assumption 3.1 from which we have P In this section, we describe all the simulation and real-world environments in detail. B.1 Simulation Environments Point 2DNavigation: Point 2DNavigation [9] is a 2 dimensional goal reaching environment with S R2, A R2, and the following dynamics, xt+1 = xt +dxt, yt+1 = xt +dyt, such that dx2t +dy2t 0.12 Where xt and yt are the x and y location of the agent, dxt and dyt are the actions taken which correspond to the displacement in the x and y direction respectively, all taken at time step t. The goals are located on a semi circle of radius 2, and the episode terminates when the agent reaches the goal or spends more than 100time steps in the environment.

Recently, auto-bidding technique has become an essential tool to increase the revenue of advertisers. Facing the complex and ever-changing bidding environments in the real-world advertising system (RAS), state-of-the-art auto-bidding policies usually leverage reinforcement learning (RL) algorithms to generate realtime bids on behalf of the advertisers. Due to safety concerns, it was believed that the RL training process can only be carried out in an offline virtual advertising system (VAS) that is built based on the historical data generated in the RAS. In this paper, we argue that there exists significant gaps between the VAS and RAS, making the RL training process suffer from the problem of inconsistency between online and offline (IBOO). Firstly, we formally define the IBOO and systematically analyze its causes and influences. Then, to avoid the IBOO, we propose a sustainable online RL (SORL) framework that trains the auto-bidding policy by directly interacting with the RAS, instead of learning in the VAS. Specifically, based on our proof of the Lipschitz smooth property of the Q function, we design a safe and efficient online exploration (SER) policy for continuously collecting data from the RAS. Meanwhile, we derive the theoretical lower bound on the safety degree of the SER policy. We also develop a variance-suppressed conservative Q-learning (V-CQL) method to effectively and stably learn the auto-bidding policy with the collected data.