An Offline Adaptation Framework for Constrained Multi-Objective Reinforcement Learning

Authors: qian  lin
zongkai  liu
danying  mo
chao  yu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 03:23:59

Abstract: In recent years, significant progress has been made in multi-objective reinforcement learning (RL) research, which aims to balance multiple objectives by incorporating preferences for each objective. In most existing studies, specific preferences must be provided during deployment to indicate the desired policies explicitly. However, designing these preferences depends heavily on human prior knowledge, which is typically obtained through extensive observation of high-performing demonstrations with expected behaviors. In this work, we propose a simple yet effective offline adaptation framework for multi-objective RL problems without assuming handcrafted target preferences, but only given several demonstrations to implicitly indicate the preferences of expected policies. Additionally, we demonstrate that our framework can naturally be extended to meet constraints on safety-critical objectives by utilizing safe demonstrations, even when the safety thresholds are unknown. Empirical results on offline multi-objective and safe tasks demonstrate the capability of our framework to infer policies that align with real preferences while meeting the constraints implied by the provided demonstrations.

Main paper: 
 introduction: 
 In the standard reinforcement learning (RL) setting, the primary goal is to obtain a policy that maximizes a cumulative scalar reward [Sutton and Barto, 2018] . However, in many real-world applications involving multiple objectives, a desired policy must not only strike a balance among potentially conflicting objectives but also consider the constraints imposed on specific safety-critical objectives. Such requirements motivate the research of Multi-objective RL (MORL) [Liu et al., 2014 , Mossalam et al., 2016] and safe RL [Gu et al., 2022 , Achiam et al., 2017] . In addition to reward maximization, the former aims to enable policies that cater to a target preference indicating the trade-off between competing objectives, while the latter focuses on reducing the cost measures of learned policies within a given safety threshold.Despite the development of meticulous and effective mechanisms to achieve these goals, most existing MORL and safe RL algorithms rely on predefined target preferences or safety thresholds. These elements need to be carefully designed by human experts with prior knowledge, generalized from a large number of demonstrations through observation. Taking autonomous driving as an example, defining aggressive, stable, and conservative strategies through different preferences between minimizing energy consumption and increasing driving speed may require extensive human participation, which means that researchers need to categorize driving data into different styles based on human experience and observe the energy consumption-speed ratio to estimate appropriate preference weights and safety thresholds for different strategies. Moreover, it is uncertain whether policies based on manually designed preferences will exhibit expected behaviors or whether feasible policies exist under given safety constraints. The challenge of designing appropriate preferences or safety thresholds becomes more pronounced as the number of objectives increases.Compared to manually designing target preferences or safety thresholds based on human knowledge, it is more natural to infer expected behaviors through a few demonstrations that implicitly indicate the trade-off between multiple objectives and the constraints of safety. For instance, selecting demonstrations with conservative behaviors from the driving dataset can be easier than inferring preferences that lead to conservative policies. Therefore, in this work, we formulate a novel offline adaptation problem for constrained MORL, with the goal to leverage a few demonstrations with expected behaviors, rather than relying on handcrafted target preferences and safety thresholds, to generate a target policy that achieves desired trade-offs across various objectives and meets the constraints on safety-critical objectives under offline settings.To achieve this, we first focus on unconstrained MORL scenarios and propose a simple yet effective offline adaptation framework Preference Distribution Offline Adaptation (PDOA), which includes: 1) learning a set of policies that respond to various preferences during training, and 2) adapting a distribution of target preferences based on a few given demonstrations during deployment. Specifically, we initialize the first part with existing state-of-the-art offline MORL algorithms, and then in the second part, we propose to align the adapted policies with expected behaviors by modeling the posterior preference distribution regarding demonstrations. Moreover, we show that our framework can be extended to constrained MORL settings by converting a constrained RL problem into an unconstrained MORL counterpart, and incorporating a conservative estimate of preference weights on constrained objectives to mitigate the potential constraint violations. Lastly, we conduct several empirical experiments on classical MORL, safe RL tasks and a novel constrained MORL environment under offline settings, demonstrating the capability of our framework in generating policies that align with the real preferences and meet the constraints implied in demonstrations. 
related work: 
 Multi-objective RL Many existing MORL methods explicitly maintain a set of policies tailored to various given preferences to approximate the Pareto front of optimal policies. These works either apply a single-policy algorithm individually for each candidate preference [Roijers et al., 2014 , Mossalam et al., 2016] , employ evolutionary algorithms to generate a population of diverse policies [Handa, 2009 , Xu et al., 2020] or simultaneously learn a set of policies represented by a single network [Abels et al., 2019 , Basaklar et al., 2022] . Furthermore, due to the potential costs and risks associated with extensive online exploration, several studies [Zhu et al., 2023 , Lin et al., 2024] have been proposed to leverage only offline datasets for MORL by extending offline return-conditioned methods [Chen et al., 2021 , Emmons et al., 2021] or offline policy-regularized methods [Fujimoto and Gu, 2021, Wang et al., 2022] to MORL settings. Despite the ability to obtain a set of well-performing policies for various preferences, most existing MORL methods overlook the process of acquiring target preferences for identifying appropriate policies during practical deployment. In this work, we assume no online interactions and no target preferences but several demonstrations generated with expected behaviors, which are easier to access than meticulously designed target preferences.Safe RL While maximizing the expected reward, classical safe RL methods restrict the cumulative cost to stay within a predefined safety threshold through Lagrangian primal-dual methods [Stooke et al., 2020 , Chow et al., 2017] or primal approaches [Xu et al., 2021 , Sootla et al., 2022] . Recently, several studies have focused on learning a set of policies that respond to various safety thresholds in both online settings [Yao et al., 2024a] and offline settings [Liu et al., 2023a , Lin et al., 2023] . Similar to MORL, these works also assume access to well-designed safety thresholds that ensure safe behaviors. Unlike prior research, our framework relies solely on safe demonstrations to indicate the implicit constraints and offers a mechanism to mitigate constraint violations through conservatism.Several studies have applied meta-learning techniques to address MORL [Chen et al., 2019] or safe RL problems [Guan et al., 2024a] , but they require online interactions for task adaptation and suffer from low sample efficiency. Additionally, other research endeavors explore offline adaptation methods to mitigate environmental uncertainty by modelling a posterior distribution over all possible Markov Decision Processes (MDPs) [Ghosh et al., 2022] or considering varying confidence levels in conservative value estimates [Hong et al., 2022] . [Mitchell et al., 2021 , Xu et al., 2022a] focus on offline adaptation for multi-task problems, aiming to achieve fast adaptation to new downstream tasks after offline training on multi-task experience. Among these methods, offline meta RL [Mitchell et al., 2021] addresses a problem similar to ours, which aims to train a meta policy that can adapt to a new task with limited data. Prompt-DT [Xu et al., 2022a] achieves quick adaptation to new tasks by incorporating a few demonstrations as prompts into the decision transformer [Chen et al., 2021] framework. We present further discussions about the difference between multi-task RL and our setting in Appendix A.1. 
constrained multi-objective mdp (cmo-mdp): 
 Both multiple-objective RL and safe RL can be discussed based on a uniform formulation: constrained multi-objective MDP (CMO-MDP) proposed by LP3 [Huang et al., 2022] . A CMO-MDP is defined as a tuple (S, A, P, r, c, β, γ) with state space S, action space A, transition distribution P(s ′ |s, a), vector reward functions r ∈ R N for N unconstrained objectives, vector cost functions c ∈ R K + , safety thresholds β ∈ R K + for K constrained objectives and discount factor γ ∈ [0, 1]. The goal is to maximize the rewards on unconstrained objectives while ensuring the costs on constrained objectives remain within the safety threshold β. Since it is typically infeasible to maximize all task objectives simultaneously, preferences ω ∈ Ω and preference functions f ω (r) which map the reward r to a scalar utility under a specific preference ω, are introduced to control the trade-off between unconstrained objectives. Given preferences ω ∈ Ω and safety thresholds β, the goal can be formulated as follows: maxEQUATIONwhere R ω = t f ω (r t ) and C = t c t represent cumulative utility and vector cost over time t, respectively. We denote π ω,β as a policy conditioned on ω, β. In this paper, we consider the linear preference setting (i.e., f ω (r) = ω T r where ω ∈ R N and ∥ω∥ 1 = 1), which is widely studied and applied [Mossalam et al., 2016 , Abels et al., 2019] and also serves as a bridge between unconstrained and constrained MORL, as shown in Section 4.3. A CMO-MDP problem can degenerate to a standard safe RL problem when N = 1 and to a standard multi-objective problem when K = 0.Under offline MORL settings, an offline dataset D = {(s, a, s ′ , r, c, ω)} is the only data available for training, which is generated by a set of behavior policies π b (•|ω) with diverse behavioral preferences ω.One straightforward yet effective approach to learn a set of policies for various preferences is to adapt offline single-objective RL methods for MORL settings. An example of this approach is the multi-objective version of Diffusion-QL (MODF) [Lin et al., 2024] , which incorporates a preference-conditioned policy (i.e., π(a|s, ω)) and a multi-dimensional value function for N objectives (i.e., Q(s, a, ω) = Q 1 (s, a, ω), ..., Q N (s, a, ω)) into Diffusion-QL [Wang et al., 2022] :EQUATION)where i is the diffusion timestep, κ is the regularization weight, α i are pre-defined parameters of diffusion model and ϵ θ (•) is a denoiser model. The diffusion policy generates the actions by iteratively using ϵ θ (•) to recover actions from noise. The second term in the actor loss of Eq. ( 2) is a diffusion reconstruction loss, which serves as a regularization term to align the actions of diffusion policy with the behavioral actions in the dataset.Pareto-Efficient Decision Agents (PEDA) [Zhu et al., 2023] is another MORL method based on supervised RL, which trains a policy conditioned on both target preferences and vector returns through a supervised paradigm:EQUATIONwhere τ t:T = {(s t , a t , r t ), ..., (s T , a T , r T )} is a trajectory segment, g t = T t ′ =t r t ′ is the target vector return (a.k.a., return-to-go) and ω is the behavioral preference of τ t:T . It is worth noting that despite the significant performance of the above two methods in offline MORL problems, both require human-provided target preferences during deployment to achieve the desired behavior, and therefore cannot be directly applied to the setting in this paper.4 An Offline Adaptation Framework for Constrained Multi-Objective RL 4.1 Problem Formulation of Offline Adaptation for CMO-MDPIn this paper, we focus on a novel offline adaptation problem for constrained MORL, with the goal to leverage only a few demonstrations to generate the policies that exhibit expected behaviors.During training, an offline dataset D = {(s, a, s ′ , r, c, ω)} is provided for policy training. During deployment, we have access to a demonstration set corresponding to a target G, i.e.,EQUATIONwhere x i is defined as a tuple (s i , a i , s ′ i , r i , c i ) and M is the total number of transition demonstrations. The target G can be preferences in MORL problems (G = ω g ), safety thresholds in safe problems (G = β g ) or a combination of both (G = (ω g , β g )), and π * G is the expert policy that achieves the best utility and meet the constraints under the preference ω g and safety threshold β g of the target G. In our setting, the real target G is inaccessible, and the goal is to obtain an adapted policy for the target G that ensures high utility f ωg (r) and meets the constraints with safety threshold β g by leveraging demonstration set B G during the deployment phase.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Justification: The derivation of the formula is clearly presented in the paper.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
offline adaptation for the unconstrained case: 
 First, we set aside the constraints in CMO-MDP and focus on the unconstrained version. Our proposed framework, Preference Distribution Offline Adaptation (PDOA), solves the offline adaptation problem under unconstrained settings in two steps: 1) learning a set of policies that respond to various preferences during training; and then 2) adapting a distribution of target preferences based on given demonstrations during deployment.In the first part, we directly apply existing offline MORL algorithms on the dataset D to obtain a set of policies πω that respond to varying preferences ω. In the second phase, we propose to model the distribution of target preferences and then utilize this distribution to obtain a reliable estimation of target preference. Specifically, we consider the posterior probability of the target preference ω g with regard to demonstration set B ωg , i.e., P (ω g |B ωg , D) = P (Bω g |ωg,D)P (ωg|D) P (Bω g |D).Here P (B ωg |ω g , D) represents the probability that the optimal π * ωg generates samples B ωg in the real environment P(s ′ , r|s, a). Due to the inaccessibility of P and π * ω under offline settings, we replace them with the empirical dynamics Pω (s ′ |s, a) and its corresponding optimal policy π * ω , which can be obtained using offline data D during training. Therefore, the preference posterior distribution can be approximated byEQUATIONwhere (s i , a i , r i , s ′ i ) ∈ B ωg . One challenge in Eq. ( 5) is the requirement of explicitly modeling the state distribution P π * ω (s i ) and the transition probability Pω (s ′ i , r i |s i , a i ). Another concern is that demonstrations B ωg with the target preference ω g can be out-of-distribution samples for the estimation of π * ω and Mω , leading to considerable discrepancy between estimated P π * ω (s i ), Pω (s ′ i , r i |s i , a i ) and their real-world counterparts. This challenge is pronounced in multi-objective settings due to significant differences in the trajectory distributions of the optimal policy under various preferences. Therefore, following previous offline adaptation works [Ghosh et al., 2022 , Hong et al., 2022] , we opt to approximate log P π * ω (s i ) Pω (s ′ i , r i |s i , a i ) with a surrogate defined by TD error of value models of π * ω in Mω :EQUATION)where δ is a hyperparameter. This approximation makes sense because r TD ω (s, a, r, s ′ ) not only measures how likely the sample (s, a, r, s ′ ) occurred during training for preference ω but also aligns with the estimated dynamics Pω . In other words, r TD ω (s, a, r, s ′ ) has a high value if (s, a, r, s ′ ) is an in-distribution sample relative to the training datasets under preference and occurs with a high probability in Mω .Then, we fit the posterior P (ω|B ωg , D) with a Gaussian distribution N (µ, σI) with parameters µ ∈ R K and σ ∈ R K + by minimizing their Kullback-Leibler divergence, leading to the following adaptation loss:EQUATIONwhere the term η(∥µ∥ 1 -1) 2 with hyperparameter η regularizes ∥µ∥ 1 to be close to 1. We approximate the prior distribution P (ω|D) with a Gaussian distribution fit to the behavioral preferences of training data D. Eq. ( 7) indicates that ω is likely to be the target preference ω g if it corresponds to high r TD ω (s i , a i , r i , s ′ i ), log π * ω (a t |s t , ω) and P (ω|D), which means: 1) the demonstration set B ωg appears with high probability in both the training set and Mω ; 2) π * ω (a t |s t , ω) is likely to generate the actions in B ωg ; and 3) ω stays within the prior preference distribution with a high probability. We justify the effectiveness of this approach with the empirical results in Appendix A.9, where r TD ω (s i , a i , r i , s ′ i ) (called TD reward) and π * ω (a t |s t , ω) (called action likelihood reward) shows a strong correlation with the real target preference.Implementation The first part of our framework PDOA is instantiated with state-of-the-art MORL algorithms: MODF and the best algorithm MORvS in PEDA mentioned in Section 3.2, to learn a set of policies that responds to various preferences. We refer to these two instances as PDOA [MODF] and PDOA [MORvS] . The well-trained policies and their corresponding value functions obtained by MORL algorithms during training are utilized to calculate the action likelihood reward π * ω (•) and TD reward r TD ω (•) in Eq. ( 7). During deployment, the preference distribution p θ (ω) is updated through the adaptation loss (7) on the demonstration set B ωg . Finally, we obtain an adapted target preference ω a = µ/∥µ∥ 1 and policy π * ω (•|ω a ) that aligns with the real target preference ω g implied in demonstrations B ωg . More details about implementation can be found in Appendix A.5.Then, we consider a natural extension of our MORL adaptation framework to constrained settings. Under the linear preference setting, the constrained MORL problem in Eq. ( 1) can be converted to its dual form with zero duality gap [Paternain et al., 2019] :EQUATIONDenoting λ * as the solution of this problem, the dual problem 8 can be rewritten as:EQUATIONThis formulation means that the constrained MORL problem under safety threshold β is uniquely equivalent to an unconstrained problem of finding the optimal policy under an extended preferenceωg = [ω T g , λ * T ] T /∥[ω T g , λ * T ] T ∥ 1 among extended N + K objectives r = [r T t , -c T t ]T . However, solving for the extended preference is challenging and requires the real safety thresholds, which are inaccessible in our setting. Nevertheless, Section 4.2 provides an approach to infer the target preferences from the demonstration set, helping us circumvent this challenge. Therefore, we can convert a constrained MORL problem to an unconstrained MORL problem, where the vector reward is defined as r = [r t , -c t ] and dataset D is augmented to D. Here, r1:N = r corresponds to N unconstrained objectives, while rN+1:N+K = -c associated with K constrained objectives. One issue with this approach is how to set the behavior preference for augmented dataset D. We present an effective scheme for approximating behavioral preferences in Appendix A.5. However, one concern about this approach is the estimation error between the adapted preference ωa obtained through Eq. ( 7) and the real preference ωg associated with the problem 9 due to the insufficiency of demonstrations. This discrepancy can lead to constraint violations when the preference weight of the adapted preference ωa on a constrained objective is less than the one of ωg (i.e., [ ωa ] i < [ ωg ] i for any i ∈ {N + 1, ..., N + K}). Therefore, we propose to apply a conservative estimate of preference weights on constrained objectives by neglecting the minimum value with probability 1 -α in the adaptation distribution ω ∼ N (µ, σ). Specifically, the preference weights on the i th objective is estimated as follows:EQUATION)and b i = µ i for 1 ≤ i ≤ N .Here, VaR is the value of risk and CVaR is the conditional one, andφ(x) = 1 √ 2π exp(-x 22 ) is the standard normal p.d.f., and Φ(x) is the standard normal c.d.f.. The parameter α in Eq. ( 10) controls the conservatism of the estimation of preference weights. For all constrained objectives, α < 1 results in a conservative estimate b i > µ i , reducing the risk of constraint violation. Additionally, as the number of demonstrations increases, the standard deviation σ i decreases, and thus b i will move towards µ i , resulting in less conservatism. In the end, we obtain the adapted target preference ωa = b/∥b∥ 1 and the adapted policy π(s|a, ωa ).Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: The paper fully discloses all the information needed to reproduce the main experimental results of the paper. Details of our framework and implementation can be found in Section 4, 5 and Appendix A.2, A.3, A.4, A.5. Guidelines:• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways.For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. • While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 
experiment: 
 In this section, we conduct several experiments on classical MORL environments in Section 5.1 and safe RL tasks in Section 5.2 to evaluate our framework in achieving preference alignment and approaching constraint satisfaction, respectively. Then, we test our method on a set of new constrained MORL (denoted as CMORL) tasks in Section 5.3. Finally, we discuss the effectiveness of the components of our method in Section 5.4.Environments and Datasets In Section 5.1, we utilize the D4MORL datasets [Zhu et al., 2023] for MORL experiments, which involve two conflict objectives and a variety of behaviors based on preferences in multi-objective MuJoCo tasks. In Section 5.2, all algorithms are trained on the datasets from DSRL benchmark [Liu et al., 2023b] , which involve a constrained objective and an unconstrained objective in BulletSafeGym tasks [Gronauer, 2022] . In Section 5.3, we develop a set of CMORL tasks by incorporating an additional velocity constraint to the original multi-objective MuJoCo environments and collect datasets from these tasks. Thus, these environments contain two unconstrained objectives and one constrained objective. More details about these environments and datasets can be found in Appendix A.2. We construct training sets and demonstration sets based on the above datasets and present the details on Appendix A.3.Evaluation Protocols For evaluation, we define a target set T = {G}, with each target G corresponding to a demonstration set B G that meets the target G. For MORL tasks in Section 5.1, G is a target preference ω g . All target preferences are selected from the dataset's preference support set at a fixed interval 0.01 (i.e., ω g = [0.01k, 1 -0.01k], s.t. min ω ⪯ ω g ⪯ max ω where k = 0, .., 100 and ω is the behavioral preference of the dataset). For safe RL tasks in Section 5.2, G is a safety threshold β g . All safety thresholds are set to 6 equidistant points within the range of possible thresholds. The target of CMORL tasks in Section 5.3 is (ω g , β g ), the Cartesian product of the target preference spaced at 0.1 interval and 6 equidistant safety thresholds.We utilize B G to generate an adapted policy for each target G and gather 5 trajectories from environments to assess its expected vector return. For tasks without constraints, we present the average utility and Hypervolume metrics to demonstrate the overall performance across all targets and the diversity of adapted policies. For tasks with constraints, we group the adapted policies by safety thresholds and report the maximum cost return on constrained objectives and average utility and Hypervolume on unconstrained objectives for each group. We perform 3 runs with various seeds and report average performance along with 1-sigma error bar. More details about metrics can be found in Appendix A.4.The algorithms for comparison are divided into two categories. The first category is preference/threshold-agnostic baselines that have no access to the real target preferences or safety thresholds implied in the demonstrations, including: 1) BC-Finetune, where the policy is learned on training dataset via behavior cloning and is fine-tuned on demonstration sets and 2) Prompt-MODT, a multi-objective version of Prompt-DT [Xu et al., 2022a ] that transforms a multi-objective scenario into a multi-task problem via preference-based division and achieves offline adaptation for various tasks by taking demonstrations as prompts of the transformer. The second category is preference/threshold-informed baselines that have access to the implied preferences and thresholds and thus serve as the oracle benchmark, including: 1) original MODF and MORvS for MORL tasks, 2) CDT [Liu et al., 2023a] for safe RL tasks, which makes decisions based on given safety thresholds to ensure constraint satisfaction for various thresholds. For CMORL tasks, instead of adapting the preference through demonstrations, we enumerate all possible augmented preferences of MODF with a small interval and report the best performance under these preferences as the oracle. More details about comparative algorithms are placed in Appendix A.4. The average utility and Hypervolume of all algorithms are shown in Figure 1 . Our method demonstrates superior overall performance and matches the oracle performance of preference-informed methods. Moreover, we present the Pareto fronts of different algorithms in Figure 2 . Our method produces a broader and expanding Pareto front compared to BC-Finetune and Prompt-MODT, which indicates that the adapted policies of our method exhibit higher diversity and better performance. Meanwhile, we observe that the adapted policies obtained by BC-Finetune cluster around the behaviorcloned policy, and thus BC-Finetune obtains the narrow Patero fronts and low Hypervolume, which indicates the difficulty of changing the policy's preference through simple fine-tune with limited samples. Besides, Prompt-MODT also yields a restricted Pareto front, which can be attributed to the difficulty of partitioning tasks based on preferences. Fine-grained divisions result in data insufficiency for each task, while coarse-grained divisions lead to multiple preference behaviors being grouped into a single task, both of which can hurt the policy performance and diversity. In contrast to BC-Finetune and Prompt-MODT, our method explicitly learns a set of policies with various preferences through MORL, thereby ensuring policy diversity. Once the target preferences are accurately identified, we can generate policies with diverse behaviors to meet various target preferences. To verify the capability of our method in preference alignment, we show the differences between the adapted preferences obtained by our method and the real target preferences in Figure 3 , where we can observe a strong consistency between the adapted preferences and the real ones, especially on PDOA [MODF] .P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t u n e 1.50 1.75 2.00 Avg Utility 1e3 MO-Ant-v2 4 5 6 1e6 P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t u n e 0.5 1.0 1.5 1e2 MO-Swimmer-v2 0 2 1e4 P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t u n e 1.50 1.75 2.00 1e3 MO-HalfCheetah-v2 4 5 6 1e6 P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t u n e 2 4 1e3 MO-Hopper-v2 1.0 1.5 2.0 1e7 P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t uWe additionally present the performance, Pareto fronts and preference comparison on the D4MORL expert datasets in Appendix A.6.1, which are consistent with the results in this section. Here, the utility is the normalized reward, since there is only one unconstrained objective in DSRL tasks. The points above the black dashed line represent the policies that violate the constraints.Figure 11 in Appendix A.7 presents the Pareto fronts of MODF and MORvS, i.e., the expected costs and expected rewards under various preferences, for safe RL tasks, which demonstrates that MORL algorithms can learn a variety of policies that meet various safety thresholds. Then, the reward and cost of all algorithms are shown in Figure 4 . We also present the performance and Pareto fronts on other tasks in Appendix A.7. These results show that, even though the safety thresholds are inaccessible, PDOA [MODF] achieves relatively safe performance under various safety thresholds compared to other baselines, closely matching the oracle performance of the threshold-informed baseline CDT. Even with very tight safety thresholds, PDOA [MODF] can achieve constraint satisfaction or experience few constraint violations. Meanwhile, its reward performance is comparable to or even exceeds that of CDT. However, PDOA [MORvS] performs poorly on the DSRL datasets because MORvS requires accurate predictions of the target return for each preference, which is challenging due to the abundance of suboptimal trajectories in the DSRL datasets. Additionally, BC-Finetune and Prompt-MODT, which align behaviors without considering the constraints on constrained objectives, exhibit insufficient policy diversity and constraint violations in most environments.The results on CMO datasets are shown in Figure 5 . The CMO tasks involve more objectives than previous MORL and safe RL tasks, posing a challenge in identifying and aligning behaviors. This is because behaviors with various preferences increase exponentially as the number of objectives increases. We can observe that in most environments, the performance of BC-Finetune and Prompt-MODT shows low Hypervolume and a weak correlation with the safety threshold, indicating that they cannot align with the desired behaviors and exhibit high policy diversity. Nevertheless, PDOA [MODF] approaches the oracle performance with high average utility, Hypervolume, and few constraint violations, which is consistent with the results presented in Sections 5.1 and 5.2. In this section, we aim to figure out: 1) the impact of the quantity of demonstrations on adaptation performance, and 2) the influence of the conservatism parameter α of Eq. ( 10) on safety performance. We conduct a set of ablation experiments for PDOA [MODF] on several MORL and safe RL tasks and present the results in Figure 6 . In Figure 6 (a)(b), as the demonstration size (DS) increases, our method achieves better average utility, Hypervolume for MORL tasks and higher reward for safe RL tasks. Nevertheless, even with a very small number of demonstrations (DS = 16), the adapted policies generated by our method still exhibit sufficient diversity and safety. Figure 6 (c)(d) demonstrate that by increasing the conservatism parameters, we can further reduce constraint violations and ultimately achieve constraint satisfaction, which validates the effectiveness of our conservatism mechanism in enhancing safety.Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: Codes and instructions are provided in supplemental material to generate the dataset used and reproduce the main results in the paper. Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
conclusion: 
 In this paper, we present an offline adaptation framework Preference Distribution Offline Adaptation (PDOA) for constrained MORL problems where we assume no access to real target preferences or safety thresholds and only a few demonstrations with expected behaviors are available in our framework. For unconstrained MORL scenarios, we propose to 1) employ MORL methods to train a set of preference-varying policies, and then 2) align the preferences of adapted policies with expected behaviors. Furthermore, we expand our framework to accommodate constraints on specific objectives by transforming constrained problems into unconstrained counterparts. Additionally, we introduce a conservatism mechanism for preference estimation on constrained objectives to mitigate potential constraint violations. Empirical results on MORL and safe RL tasks illustrate the capability of our framework in generating diverse policies aligning with expected behaviors and approach constraint satisfaction through conservative preference estimation.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes] Justification: All the training and testing details necessary to understand the results can be found in Section 5 and Appendix A.2, A.3, A.4 and A.5.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes] Justification: The paper reports error bars suitably.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
 Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?Answer: [Yes] Justification: We present the information of compute resources used in this paper in Appendix A.5.Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
code of ethics: 
 Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?Answer: [Yes] Justification: The research conducted in the paper conforms, in every respect, with the NeurIPS Code of Ethics.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [Yes] Justification: The paper discusses both potential positive societal impacts and negative societal impacts of the work performed, which can be found in Section A.11.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: The PDOA framework involves multiple steps of preference sampling and gradient updates, which can lead to additional computational burden and latency during deployment. Besides, although the manual design of preferences and safety thresholds is not necessary, the process of constructing demonstrations may still involve human expertise.A Appendix / supplemental material A.1 Connections and Differences to Other Fields Multi-task RL While the constrained MORL setting we focus on can align with the multi-task framework by treating policy learning under various preferences or thresholds as distinct tasks, several key differences distinguish our study from previous multi-task RL settings: (1) One primary motivation of our work is to circumvent the handcrafted design of target preferences, whereas previous studies [Mitchell et al., 2021 , Xu et al., 2022a] on multi-task RL mainly focus on policy generalization to new tasks. (2) We achieve offline adaptation by updating the target preference distribution rather than the policies themselves, which distinguishes our framework from offline meta-RL methods. This adaptation pipeline not only avoids potential performance degradation due to policy parameter shifts but also allows us to incorporate conservatism to mitigate constraint violations.(3) Accurately identifying various preferences and constraints implied in demonstrations during adaptation is challenging as these demonstrations could come from the same dynamics and policies with similar preferences.Multi-constraint RL Existing research on multi-constraint RL typically explores the correlation between multiple constraints from gradient perspectives [Yao et al., 2024b , Kim et al., 2024] . Some works [Guan et al., 2024b] attempt to meet different types of constraints simultaneously. In contrast, our approach, from the policy diversity perspective, aims to find appropriate policies for various safety requirements and thus places less focus on constraint correlation for a specific threshold. Nevertheless, previous studies on multiple constraint RL will provide important insights into managing potential correlations between multiple objectives during preference-conditioned policy learning, thereby improving overall performance.Inverse RL Existing IRL methods [Arora and Doshi, 2021] typically assume no knowledge about the reward function and focus on learning reward functions from mere demonstrations through apprenticeship learning, maximum entropy optimization or adversarial learning. Some of these methods have been applied in MORL [Takayama and Arai, 2022] or safe RL [Malik et al., 2021] . In contrast to these studies, our work assumes knowledge of the reward information on different objectives and pays more attention to identifying the preferences of demonstrations during deployment than learning a specific decisive reward during training.Bayesian RL Bayesian RL [Ghavamzadeh et al., 2015] is widely used for promoting online exploration and dealing with uncertainty in model-based RL or offline RL, while it serves as a cornerstone of our theoretical derivation, allowing us to incorporate prior preference information from datasets and infer the expected preferences by estimating the preference posterior based on a few demonstrations. Ghosh's work [Ghosh et al., 2022] is the most related Bayesian RL study that inspires our approach. Nevertheless, it aims to reduce the dynamics uncertainty caused by limited offline samples, while our approach focuses on utilizing Bayesian inference to solve the uncertainty of the target preference distribution based on a few demonstrations.Offline Imitation Learning Each adaptation can be viewed as the imitation of a expert policy with a certain preference. Therefore, offline Imitation Learning (IL) is a potential approach to solving the problem by applying an off-the-shelf offline IL algorithm [Xu et al., 2022b , Kim et al., 2022] , which enables the combination of sub-optimal trajectories in the offline dataset with expert demonstrations provided by users to imitate expert behaviors. However, applying offline IL to our problem presents several challenges. We conduct comparative experiments in Appendix A.8 to further compare algorithms and illustrate these challenges.Environments and Datasets for Multi-objective RL We utilize the D4MORL dataset [Zhu et al., 2023] collected from multi-objective MuJoCo environments by a set of preference-varying policies [Xu et al., 2020] . We consider 5 MuJoCo tasks including HalfCheetah, Ant, Hopper, Walker2d, and Swimmer. Among them, the HalfCheetah, Walker2d, and Swimmer tasks involve conflicting objectives of running speed and energy saving. The Ant task includes two speed objectives in both the x and y axes, while the Hopper considers running and jumping objectives. For each task, we test algorithms on two types of datasets: Expert and Amateur, which are collected by pure expert policies and noise-injected expert policies, respectively.Environments and Datasets for Safe RL We utilize the datasets in DSRL benchmark [Liu et al., 2023b] that are collected by a set of behavior policies trained under various safe thresholds. We select 8 BulletSafetyGym tasks [Gronauer, 2022] for safe RL experiments, which involve 4 agents (Ball, Car, Drone, Ant) and 2 task goals (Run, Circle). In these tasks, the goal of the reward function is to navigate to target positions or complete circuits as quickly as possible, while the costs are incurred when the agent enters risky areas.Environments and Datasets for Constrained Multi-objective RL We develop a set of CMORL tasks, namely CMO-Hopper, CMO-HalfCheetah, CMO-Ant and CMO-Swimmer. These tasks are constructed based on the multi-objective MuJoCo environments [Xu et al., 2020] and augmented by an additional constraint on robotics' velocity, which is consistent with the constraint setting of Safety Gymnasium tasks in the DSRL benchmark. When the agent's moving velocity exceeds a specific value, it will receive 1 cost. To gather datasets for these tasks, we create a behavioral preference set and utilize TRPO+Lagrangian [Liu et al., 2023b] to train individual policies with the goal of maximizing utility under each preference in this set while accounting for a specified velocity constraint. The safety threshold of the velocity constraint is adjusted throughout the training process to encompass a range of potential safety thresholds. The details for each CMO task are shown in Table 1 . The replay buffers collected during the training of these policies constitute the CMO datasets. We present Pareto fronts of CMO datasets under different cost return intervals in Figure 7 , illustrating that CMO datasets encompass a wide range of behaviors catering for various preferences and safety thresholds. We denote R(π G ) as the estimated return vector of the policy π G adapted for target G in the target set T .Average Utility For unconstrained scenarios, G = ω g and average utility is calculated by:EQUATIONFor constrained scenarios, G = (ω g , β). We denote T βg = {ω g |(ω g , β g ) ∈ T } as subset of T that groups all target preference ω g by safety threshold β g . The average utility is a function of β g :EQUATION)This metric reflects the overall performance of the adapted policies across all target preferences.Hypervolume We first consider the unconstrained settings and introduce the adapted Pareto set P that contains all adapted policies that are not dominated by any other adapted policy. Hypervolume measures the volume enclosed by the returns of policies in the adapted Pareto set P :EQUATIONwhere H(P, r 0 ) = {z ∈ R n |∃π ∈ P : r 0 ⪯ z ⪯ R(π)} and 1 H(P,r0) equals 1 if z ∈ H(P ) and 0 otherwise. For constrained scenarios, we group all target preferences by safety threshold and report Hypervolume on each group. The Hypervolume reflects not only the overall performance across all target preferences but also the diversity of the policies, as a high diversity of policies corresponds to a broad Pareto front and consequently a high Hypervolume.Sparsity, which measures the density of policies in the approximated Pareto set, is also commonly used for evaluating MORL performance. A lower sparsity indicates a denser Pareto front, which Update µ, σ through gradient descent (Eq. ( 7)) 12: end for 13: if no constraint then is generally more desirable. However, the sparsity metric is not applicable in our setting. This is because, given that the number of adapted policies is fixed, an increase in utility and Hypervolume performance can result in a worse sparsity metric. Moreover, better sparsity metrics encourage a lower diversity of adapted policies, which diverges from our intended goal.The pseudocode for the training and adaptation process is provided in Algorithm 1.In constrained settings, the vector reward is defined as r = [r t , -c t ] and dataset D is augmented to D. For behavioral preferences in D, we utilize ω = U (τ )/∥U (τ )∥ 1 as behavioral preferences for all transitions in trajectory τ . Here,U (τ ) = [R 1 , ..., R N , C max 1 -C 1 , ..., C max K -C K ],where R i and C i represent the accumulations of the i th reward and cost of τ , respectively, and C max i is the maximum of C i among all trajectories. The formulation of ω aims to approximate the normal vector of the Pareto front that consists of (R 1 , ..., R N , -C 1 , ..., -C K ).Preference Distribution Offline Adaptation (PDOA) We utilize the policy and value function obtained by MODF and PEDA to calculate the TD reward and action likelihood reward in Eq. ( 7). For MODF, except the state-action value function Q(s, a, ω) = Q 1 (s, a, ω), ..., Q N (s, a, ω) obtained during training, we additionally train a state value function V (s, ω) = E π(a|s,ω) [Q(s, a, ω)] to derive TD reward. As for the action likelihood reward, the action likelihood of the diffusion policy model is intractable but can be approximated as log π * ω (a|s, ω) ≈ E i∼U ,ϵ∼N (0,I) ∥a -â0 ∥ 2 2 [Kang et al., 2024] , where â0 is reconstruction of a i and a i is obtained by corrupting a with noise ϵ. For PEDA, which does not involve value functions, we disregard the TD reward when calculating the adaptation loss. The prior preference distribution P (ω|D) in Eq. ( 7) is approximated with a Gaussian distribution N (µ D , σ D ), where the expectation µ D and standard deviation σ D are estimated on behavioral preferences of the training dataset D. The weight η of the regularization term (∥µ∥ 1 -1) 2 in Eq. ( 7) is set to 1.0.During adaptation, we model the preference distribution using a Gaussian model and perform gradient updates on this distribution according to Eq. ( 7). For each target, the number of gradient updates is set to 1000, with 64 preferences sampled from the distribution for each gradient update. All samples in the demonstration set are used for gradient updates within one batch. We use the Adam optimizer with a learning rate of 0.05. The conservatism weight α in Eq. ( 10) is set to 1.0 for MORL tasks and 0.7 for safe RL and CMORL tasks. The weight of the TD reward in Eq. ( 6) is set to 0.01 for PDOA [MODF] .For MODF, we use the original implementation in https: //github.com/qianlin04/PRMORL. We remove the Regularization Weight Adaptation component in the original MODF since it requires additional online interactions during evaluation. We follow the default hyperparameters except for the regularization weight, which is set to 200 for MORL and CMORL tasks and 20 for safe RL tasks. For PEDA, we use the original implementation in https://github.com/baitingzbt/PEDA. PEDA utilizes a linear regression model to predict target vector returns given target preferences. For D4MORL datasets, it fits this model using Expert datasets. In DSRL and CMO datasets, which contain suboptimal samples, we construct a Pareto set that consists of all near-undominated trajectories with a small tolerance (5%) and use it to fit the linear regression model. For Prompt-MODT, we implement it based on the MODT algorithm in PEDA by incorporating prompts into the decision transformer. The target vectors predicted by the above linear regression model are used as the initialized return-to-go for Prompt-MODT. Besides, to apply Prompt-MODT, we need to transform the multi-objective problem into a multi-task problem, for which we divide trajectories with different behavioral preferences into separate tasks with a preference interval of 0.05. For example, trajectories with behavior preferences ranging from [0.00, 1.00] to [0.05, 0.95] are classified as task 1, and trajectories with behavior preferences ranging from [0.05, 0.95] to [0.10, 0.90] are classified as task 2. For BC-Finetune, we model the policy using a Gaussian model and employ behavioral cloning through MSE loss. During adaptation, behavior cloning is performed on the demonstration set with 0.01 learning rate and 1000 gradient update steps.The training and testing were conducted on 1 NVIDIA GeForce RTX 3090 GPU. The testing utilizes 5 CPU threads to simultaneously collect test data from multiple environments. The total time for training and testing does not exceed 10 hours. The memory usage for a single run depends on the size of the dataset used, but generally does not exceed 10 GB. 9, and 10, respectively. In Figure 8 , our method demonstrates overall competitive performance in terms of average utility and Hypervolume metrics. Figure 9 illustrates that our method achieves a broader Pareto front, indicating a high diversity of adapted policies. Figure 10 shows that our method can obtain expected behaviors due to the consistency between our adapted preferences and real target preferences. These results are consistent with those obtained on the Amateur Datasets in Section 5.1, which demonstrates that our framework achieves consistent advantages across datasets of varying quality.P D O A [ M O D F ] P D O A [ M O R v S ] P r o m p t -M O D T B C -F in e t uFigures 11 and 12 show the performance of MODF and MORvS under various target preferences, which demonstrate the capability of these MORL algorithms in obtaining policies that satisfy different safety thresholds. Figure 13 illustrates the normalized cost and reward for additional DSRL tasks, including CarCircle, DroneCircle, CarRun, and DroneRun. The results are consistent with those in Section 5.2, demonstrating that MORL methods can effectively learn a set of policies that meet various safety thresholds, and that PDOA [MODF] can achieve very few constraint violations. involve a limited number of demonstrations (128 transitions per target preference) compared to the millions of samples in the offline dataset, leading to significant demo sufficiency and data imbalance.We apply an OIL baseline DWBC [Xu et al., 2022b] in our MORL and safe RL experiments and present the results in Figure 14 and 15, where we limit the number of tested preferences to 10 in the D4MORL tasks because the original evaluation involves dozens of target preferences per task, and training a policy for each target preference using DWBC would be too time-consuming. In both MORL and safe RL environments, DWBC demonstrates unstable performance. It is competitive in tasks such as MO-Ant, MO-Swimmer, but performs poorly in other tasks, including MO-Walker2d. It even struggles to learn effective policies in some cases, such as MO-Hopper and AntCircle. These issues can be attributed to demo sufficiency and data imbalance. Additionally, DWBC experiences high constraint violations in most safe RL tasks, as it does not incorporate safety considerations into its policy learning. Our research introduces a novel framework for addressing constrained multi-objective reinforcement learning (CMORL), which is widely applicable to many real-world scenarios, such as autonomous driving, healthcare, where behavioral preferences and safety criteria are difficult to define precisely. These approaches liberate researchers from the labor required to align preferences and safety thresholds. Besides, the offline training paradigm eliminates the expenses and dangers associated with online exploration. For instance, in autonomous driving, online interactions with the environment have the risk of accidents and injuries; however, our method can mitigate this risk by learning from a pre-recorded driving dataset generated by safe and target behavior policies.Nonetheless, our method may not be suitable for domains requiring rapid responses and frequent updates due to the additional computational load and latency caused by gradient updates during adaptation. Besides, providing personalized services with this framework necessitates a certain amount of user data as demonstrations, raising concerns about privacy breaches and data misuse.Despite the risks and challenges mentioned, we believe that CMORL holds great promise for automating and enhancing sequential decision-making in highly impactful domains. Additional work is required to make this framework robust enough for application in multi-objective and safety-critical scenarios.Answer: [Yes] Justification: The paper discusses the limitations of the work, which can be found in Section 6.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.For Prompt-MODT, which required trajectory segments as prompts during adaptation, we sample a trajectory segment with length=M to constitute the demonstration set. 


Decision: The paper studies a well-motivated, significant problem (multi-objective reinforcement learning) and proposes an offline adaptation approach that does not require preferences between objectives to be indicated during deployment time (instead they are inferred from offline demonstrations).
The reviewers agreed the proposed approach is technically sound, novel, and the empirical evaluations and ablations convincingly show the merits of the proposal. During the reviewer-author discussion, the authors clarified several weaknesses pointed out by the reviewers; incorporating them into a revision will substantially strengthen the paper.

Reviews: review: paper_summary: paper_summary: The paper tackles the important problem of adapting multi-objective policies to diverse preferences and constraints at deployment. This work uses state-of-the-art multi-objective RL methods to learn a set of policies that can align with diverse preferences. At deployment time, it doesn't assume access to the explicit hand-crafted preferences but introduces an adaptation method that uses demos to infer the target preferences. Additionally, the authors formulate this method to take into account explicit safety constraints at deployment time, inferred using only the demonstrations, without any online interaction. The approach is tested extensively on standard multi-objective and safe RL benchmarks, and they show that it can achieve higher performance, incur lower costs, and generate more diverse policies in these environments.
strength_weakness: strength_weakness: 1. At training time, the method assumes access to the explicit preferences and demonstrations responding to those preferences. However, in practical settings, this information is not available even at training time, and the preferences and their clusters have to be learned in an unsupervised / self-supervised setting. I would be curious to know what the authors think about the effectiveness of this approach to learn the preferences and the policies in an end2end framework.
2. I would recommend the authors to present a clear distinction between the past work, and the approach finally used in this work to generate the multi-objective policies. In this current presentation, it is slightly confusing to understand the particular contributions/adaptations of past work  to incorporate this with the posterior approximation framework. A pseudocode explaining the entire algorithm would make the method clearer to the readers.
3. In the background, it would be nice to see some discussion contrasting the work in multi-objective adaptation to inverse RL, because in the introduction the authors mention - “it is more natural to infer expected behaviors through a few demos that implicitly indicate….” Comparing this to the approaches in inverse RL, and further, in Bayesian RL would be very useful to highlight the contributions. 
4. Under expert demos, the prompt-MODT baseline performs comparably to the methods introduced in this work. I would recommend the authors provide a wider discussion of the effectiveness of explicitly modeling human preferences using the adaptation module and if using the distributional Bayesian approach makes it robust to noisy samples.
questions: questions: 1. The method is based on settings where the total reward is a linear combination of preferences and individual rewards. How can this setting be adapted to settings where we have access only to the total reward and demos and not the individual components? 
2. In the evaluation metrics, I would be curious to know why is the hypervolume and therefore, the diversity in the policies important. For the baselines, we observe that the policies with low hypervolume still have considerably close performance to this work in terms of utility and cost. A paragraph along these lines would allow a broader audience to understand the significance of these experiments and results.
3. In certain environments, despite violating constraints, the returns are high, which indicates that the cost and the rewards are independent of each other. This is slightly confusing and I would appreciate some clarification in this respect.
4. In Figure 6, it would be nice to analyze why higher demos incur a higher cost.
limitations: limitations: 1. The method assumes access to the preferences at training time, which might not be practical for real-world settings, where the individual objectives and the preference distribution are unknown. This could be the reason that the method outperforms the prompt-MODT baseline i.e. the error introduced due to implicitly modeling the preferences, and not having access to the target distribution could potentially decrease the effectiveness of PDOA.
2. The authors mention that the adaptation step is not real-time, which makes it difficult to transfer this system to real-world RL systems such as LLMs and robots.
3. The evaluation is limited to locomotion environments, so, I suggest the authors include a discussion on the implications of the method and evaluation towards adaptation and safety in real-world RL problems.
paper_summary: paper_summary: This paper addresses the issue of handcrafted target preferences in multi-objective RL (MORL) by proposing an offline adaptation framework called Preference Distribution Offline Adaptation (PDOA), which implicitly learns the preferences from a small number of demonstrations. Specifically, PDOA involves two steps: (i) It learns a set of policies for various preferences during training using an off-the-shelf MORAL algorithm; (ii) It then adapts a distribution of target preferences based on the demonstrations at deployment. To update the posterior of target preferences in a tractable manner, PDOA adopts multiple approximation schemes for both unconstrained and constrained settings. Moreover, in the constrained setting, to address the discrepancy between the true preference and the estimated one, PDOA also proposes to use a conservative estimate with the help of CVaR. The PDOA is evaluated on multiple MOD4RL locomotion tasks and has strong performance in terms of average utility and hypervolume. Empirical studies on the number of demonstrations and the conservatism parameter are also provided.
strength_weakness: strength_weakness: - One issue is the lack of stronger offline imitation learning baselines. If we already have a number of demonstrations of the target preferences at test time, then one reasonable solution is to view this as a single-objective problem and then apply an off-the-shelf offline imitation learning (IL) algorithm with these demonstrations, such as DWBC [Xu et al., 2022] and DemoDICE [Kim et al., 2022], which both focus on offline imitation with a small number of expert demonstrations along with other sub-optimal trajectories. In the setting of PDOA, one can simply do offline IL by using $\mathcal{B}_\omega$ as the expert demonstrations and $\mathcal{D}$ as the sub-optimal samples.
- Another concern is on approximation of the posterior in (6). While I can get the intuition that in-distribution samples tend to have higher $r^{TD}$, the surrogate in (6) appears to be quite heuristic and not very principled. Specifically, the $\pi^*$ in the original posterior expression could be different from the policy (whose Q and V are used in Equation 6) learned by MORL at the first stage. More theoretical justification of this approximation is needed.  
- For the constrained setting, there is an additional parameter $\alpha$ to determine to strike a balance between constraint violation and achieved utility. However, my concern is that $\alpha$ is a parameter to be used at test time, and hence there is very little one can do in terms of parameter tuning. While Figure 6 did provide a study on the influence of $\alpha$, we can see that this value does matter a lot in the balance between constraint violation and utility. As a result, this could make PDOA less attractive in practice.

[Xu et al., 2022] Haoran Xu, Xianyuan Zhan, Honglei Yin, and Huiling Qin, “Discriminator-Weighted Offline Imitation Learning from Suboptimal Demonstrations,” ICML 2022.

[Kim et al., 2022] Geon-Hyeong Kim, Seokin Seo, Jongmin Lee, Wonseok Jeon, HyeongJoo Hwang, Hongseok Yang, and Kee-Eung Kim, “DemoDICE: Offline Imitation Learning with Supplementary Imperfect Demonstrations,” ICLR 2022.
questions: questions: Please see the comments above. Some additional questions:
- How many trajectories (in the demonstration set) are needed for target preference adaptation in the main experiments (Figures 1-2)? 
- In Equation 5, during the preference estimation phase, the dataset $\mathcal{D}$ only takes part in the prior $P(\omega| \mathcal{D})$. Under PDOA, it could be possible to leverage the information from $\mathcal{D}$ to further improve the accuracy of the target preference. Can the authors comment on this?
limitations: limitations: The paper discusses one limitation about the need for human expertise in constructing demonstrations in Section 6.
paper_summary: paper_summary: The paper presents a new multi-objective RL setup to learn a policy offline that can adapt to unknown target preference online, utilizing posterior estimates of the true target preference.
strength_weakness: strength_weakness: After carefully thinking through this setup, I'm a bit worried about the core contribution of this paper. 

There are several major challenges to offline learning of CMO-MDP:
1. Policy learning conditioned on preference vector (solved by previous works and used in this paper) (MODF, MORvS)
2. Trajectory distributions shift under the optimal policy and approximate policy, leading to failures of estimating preference vector w (solved by [Ghosh et al., Hong et al., 2022], Eq 6, used in this paper)
3. Preference vector online estimation (through posterior update) (Eq 5) (this is a straightforward application), and the actual update is Eq 7.

It seems to me that the paper's core contribution is:
1. Proposed a new problem.
2. Proposed a good preference target vector estimation (Eq 7).
3. Cast constrained problem as unconstrained and added a conservative estimate to the preference vector so that constraints are not violated during adaptation.

I'm not fully able to determine how significant these contributions are.
questions: questions: For Eq 5, can the author comment on -- what is the estimation quality of the target preference policy if the sampling policy pi-hat-star-w is sub-optimal? I can break this question down to two parts:
- How does the sub-optimality of the approximate policy to optimal policy affect the estimation of w?
- How does the estimation quality of w affect the convergence of the approximate policy to the true optimal policy?
limitations: limitations: The authors addressed the limitation.
paper_summary: paper_summary: This paper presents a novel framework for multi-objective RL  that eliminates the need for handcrafted target preferences. Instead, it uses demonstrations to implicitly indicate the preferences of expected policies. The proposed offline adaptation framework can also handle safety-critical objectives by utilizing safe demonstrations, even when safety thresholds are unknown. The authors provide empirical results showing that the framework effectively infers policies that align with real preferences and meet the constraints implied by the demonstrations.
strength_weakness: strength_weakness: (1) Preference is closely related to the reward function definition. How does this definition compare to other metrics such as the ones based on the states/state-action/latent variable distribution? For some tasks, there exists some correlation between reward functions, and this will potentially affect the preference estimation results.

For other potential weaknesses, please see my questions.
questions: questions: (1) Preference is closely related to the reward function definition, and correlation among reward functions will result in preference calculation issues. For example, what if $r_1 = 0.5 r_2 = 0.2 r_3$, and $r_4$ is independent to $r_1$? In this setting, we would get a small preference mismatch score between two demonstrations even if they have significant differences in the reward returns for $r_4$. Do you have any methods to deal with this?

(2) In section 4.2, the first step of your method is to learn a set of policies that respond to various preferences during training. How do you ensure that the learned policy is subject to specific preferences?

(3) Based on the description in section 4.3, are High-reward-high-cost and low-reward-low-cost demonstrations categorized as similar behaviors?

(4) How does the proposed method perform in other DSRL datasets/tasks such as Car-Circle and Drone-Circle?

(5) Where are CDT results discussed in section 5.2? I did not them in Figures 3 and 4.

(6) How do your method and problem formulation relate to multi-constraint safe RL since multiple constraints can also be viewed as multiple objectives? Some related works:

[1] Kim, D., Lee, K., & Oh, S. (2024). Trust region-based safe distributional reinforcement learning for multiple constraints. Advances in neural information processing systems, 36.

[2] Yao, Y., Liu, Z., Cen, Z., Huang, P., Zhang, T., Yu, W., & Zhao, D. (2023). Gradient Shaping for Multi-Constraint Safe Reinforcement Learning. arXiv preprint arXiv:2312.15127.

[3] Guan, J., Shen, L., Zhou, A., Li, L., Hu, H., He, X., ... & Jiang, C. (2024). POCE: Primal Policy Optimization with Conservative Estimation for Multi-constraint Offline Reinforcement Learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 26243-26253).
limitations: limitations: Yes, the authors addressed the limitations.

Comments: title: Thanks for the response
comment: Thanks for insightful suggestions. If there are still confusions, please let us know. We would be happy to provide further clarifications.
Apologies for the confusion. We would like to supplement the related explanations about W2 and W3.

**W2: Justification of TD-reward surrogate**

We will use an example to illustrate the effectiveness of the TD-error approximation. Let's assume there are three types of transitions: $(s,a,r_1,s_1),(s,a,r_2,s_2),(s,a,r_3,s_3)$, with the proportions $p$, $1-p$ and $0$ in the dataset, respectively. 

For the out-of-distribution transition $(s,a,r_3,s_3)$, it is expected to associate with high uncertainty and a large TD-error, according to previous research [1,2,3] and our empirical results in Figure 14. 

For the in-distribution transitions $(s,a,r_1,s_1),(s,a,r_2,s_2)$, the Q-value can be expressed as $Q(s,a)=p\cdot (r_1+V(s_1))+(1-p)\cdot(r_2+V(s_2))$. Thus, $(s,a,r_1,s_1)$ corresponds to a low TD-error $||Q(s,a)-r_1-V(s_1)||^2_2$ when it has a high probability of occurring in the dataset (i.e., when $p$ is close to $1$).

Therefore, we expect to obtain $TD(s,a,r_1,s_1)<TD(s,a,r_2,s_2)<TD(s,a,r_3,s_3)$ when $p$ is high. In this case, TD reward aligns with the estimated dynamics $\hat P(s',r'|s,a)$ in the posterior for in-distribution transitions, where a higher occurrence probability leads to a higher TD reward. For out-of-distribution transitions, while the TD reward might not align with the estimated dynamics, it remains low to mitigate the adverse effects of inaccuracies in dynamics estimation. Similarly, for state distribution, a higher occurrence probability of states corresponds to a more accurate value function estimate and a higher TD reward, whereas out-of-distribution states typically result in a smaller TD reward.

I hope this example provides an intuitive insight into this approximation. Rigorous theoretical derivations would involve the generalization of Q-value neural networks and their relationship to dynamics uncertainty, which are relatively complex and challenging. We would leave further theoretical analysis to our future work. Another potential direction is to adopt alternative approaches, such as explicit dynamic modeling, which could offer more straightforward theoretical results.

[1] Bai, Chenjia, et al. "Pessimistic Bootstrapping for Uncertainty-Driven Offline Reinforcement Learning." ICLR 2021.

[2] Rigter, Marc, Bruno Lacerda, and Nick Hawes. "One risk to rule them all: A risk-sensitive perspective on model-based offline reinforcement learning." NIPS 2024.

[3] Ghosh, D., Ajay, A., Agrawal, P., & Levine, S.. Offline rl policies should be trained to be adaptive. ICML 2022.

 

**W3: Conservative parameter $\alpha$**

Our statement that "a roughly designed $\alpha$ will not significantly undermine our approach" was based on comparisons with the baselines, where our approach showed relatively fewer violations and a higher correlation between cost and reward (indicating the high policy diversity we expected) than baselines under various $\alpha$. We know this isn't a direct comparation shown in Figure 6 and apologize for any confusion caused by some of our claims in the response.

We acknowledge that the influence of this parameter on the final performance cannot be overlooked, and we appreciate the reviewer for highlighting this. We believe that fine-tuning this parameter using a limited iteration of online interaction and re-adaptation could indeed address this issue. Additionally, the tasks used in the ablation experiments were randomly selected, and we plan to conduct these experiments across a broader set of tasks and report the results in a future version.





Thank you again for the constructive feedback. We are happy to provide further explanations if there are still any confusions.
Thank the authors for the detailed response. My concerns about the baselines and the demonstration set have been nicely addressed. Some follow-up comments / questions:

- About W2: My question about the justification for $r^{TD}$ remains unresolved. To justify the use of (6), one would need to roughly show that $r^{TD}$ is generally larger when the posterior is larger. Somehow I do not see why the explanation in the rebuttal is relevant. Please let me know if I missed anything.

- About W3: It is mentioned that “...a roughly designed $\alpha$ will not significantly undermine our approach.” However, in Figure 6(c)-(d), it appears that a difference of 0.2 in $\alpha$ can indeed have a significant impact on both the cost and the utility. Moreover, as the ablation study on $\alpha$ is only conducted on two tasks (and not evaluated extensively), the influence of this parameter can possibly be even more substantial. While I can see tuning $\alpha$ might be possible in some cases and understand what the authors try to say by ``Figure 6 demonstrates that our approach maintains relatively few constraint violations even when the conservatism mechanism is removed (i.e., $\alpha=1$),” I do feel that the statement can be an over-claim.
I thank the authors for their response. I have read the rebuttal and maintain my score.
Thanks again for all the insightful comments and suggestions. They are helpful in improving our paper.
Thanks for the additional explanation. All my concerns and questions have been successfully addressed. I have raised my score from 5 to 6 in favor of acceptance.