B.7 Additional Details on the Running Time In this section, we provide additional details on the running time of the algorithms.

Deploying reinforcement learning (RL) in robotics, industry, and health care is blocked by two obstacles: the difficulty of specifying accurate rewards and the risk of unsafe, data-hungry exploration. We address this by proposing a two-stage framework that first learns a safe initial policy from a reward-free dataset of expert demonstrations, then fine-tunes it online using preference-based human feedback. We provide the first principled analysis of this offline-to-online approach and introduce BRIDGE, a unified algorithm that integrates both signals via an uncertainty-weighted objective. We derive regret bounds that shrink with the number of offline demonstrations, explicitly connecting the quantity of offline data to online sample efficiency. We validate BRIDGE in discrete and continuous control MuJoCo environments, showing it achieves lower regret than both standalone behavioral cloning and online preference-based RL. Our work establishes a theoretical foundation for designing more sample-efficient interactive agents.

B.7 Additional Details on the Running Time In this section, we provide additional details on the running time of the algorithms.

This formulation of RL has had significant empirical success in the recent past [24, 23, 33, 32].

Anomaly detection (AD) is a critical task across domains such as cybersecurity and healthcare. In the unsupervised setting, an effective and theoretically-grounded principle is to train classifiers to distinguish normal data from (synthetic) anomalies. We extend this principle to semi-supervised AD, where training data also include a limited labeled subset of anomalies possibly present in test time. We propose a theoretically-grounded and empirically effective framework for semi-supervised AD that combines known and synthetic anomalies during training. To analyze semi-supervised AD, we introduce the first mathematical formulation of semi-supervised AD, which generalizes unsupervised AD. Here, we show that synthetic anomalies enable (i) better anomaly modeling in low-density regions and (ii) optimal convergence guarantees for neural network classifiers -- the first theoretical result for semi-supervised AD. We empirically validate our framework on five diverse benchmarks, observing consistent performance gains. These improvements also extend beyond our theoretical framework to other classification-based AD methods, validating the generalizability of the synthetic anomaly principle in AD.

We consider offline reinforcement learning (RL) in $H$-horizon Markov decision processes (MDPs) under the linear $q^\pi$-realizability assumption, where the action-value function of every policy is linear with respect to a given $d$-dimensional feature function. The hope in this setting is that learning a good policy will be possible without requiring a sample size that scales with the number of states in the MDP. Foster et al. [2021] have shown this to be impossible even under $\textit{concentrability}$, a data coverage assumption where a coefficient $C_\text{conc}$ bounds the extent to which the state-action distribution of any policy can veer off the data distribution. However, the data in this previous work was in the form of a sequence of individual transitions. This leaves open the question of whether the negative result mentioned could be overcome if the data was composed of sequences of full trajectories. In this work we answer this question positively by proving that with trajectory data, a dataset of size $\text{poly}(d,H,C_\text{conc})/\epsilon^2$ is sufficient for deriving an $\epsilon$-optimal policy, regardless of the size of the state space. The main tool that makes this result possible is due to Weisz et al. [2023], who demonstrate that linear MDPs can be used to approximate linearly $q^\pi$-realizable MDPs. The connection to trajectory data is that the linear MDP approximation relies on "skipping" over certain states. The associated estimation problems are thus easy when working with trajectory data, while they remain nontrivial when working with individual transitions. The question of computational efficiency under our assumptions remains open.