Pareto-Optimal Learning from Preferences with Hidden Context

## Abstract

Abstract Ensuring AI models align with human values is essential for their safety and functionality. Reinforcement learning from human feedback (RLHF) uses human preferences to achieve this alignment. However, preferences sourced from diverse populations can result in point estimates of human values that may be sub-optimal or unfair to specific groups. We propose Pareto Optimal Preference Learning (POPL), which frames discrepant group preferences as objectives with potential trade-offs, aiming for policies that are Pareto-optimal on the preference dataset. POPL utilizes Lexicase selection, an iterative process to select diverse and Pareto-optimal solutions. Our empirical evaluations demonstrate that POPL surpasses baseline methods in learning sets of reward functions, effectively catering to distinct groups without access to group numbers or membership labels. Furthermore, we illustrate that POPL can serve as a foundation for techniques optimizing specific notions of group fairness, ensuring inclusive and equitable AI model alignment.

## 1 Introduction

For both safety and functionality, it is critical for AI models to be aligned with the values of human users and stakeholders. Reinforcement learning from human feedback (RLHF) has emerged as an effective mechanism for model alignment, using preferences as a means of capturing human values. However, when preferences are sourced from large sets of potentially diverse people, methods that rely on point estimates of human values are bound to either be sub-optimal with respect to all groups or unfair to certain groups, both problematic in their own ways.

In this work, we build upon the notion of hidden context proposed by Siththaranjan et al. [] and focus on the problem of Reinforcement Learning from Human Feedback with Hidden Context (RLHF-HC).
Hidden context is information that is unavailable to a preference learning system that affects the preferences given. For example, a person’s dominant hand might determine on what side a robotic assistant should hand them an object. Under this formulation, the goal is to build a *set*
of policies that contains the policy that is optimal under the reward function of each group of people with the same hidden context. In practice, we see two clear use cases of such a set of policies. 1) These policies can be selected from at test time to find an optimal policy for a given user without needing to train a new policy and 2) this set can be used to ensure fairness between groups. Minimizing risk with respect to this diverse distribution of policies ensures that no specific group is disregarded in the risk measurement.

Because preferences generated from individuals in different hidden context groups could be contradictory (i.e. not mutually satisfiable by a well-regularized policy or reward function), we propose to frame these preferences as objectives with potential trade-offs between each other. With this re-framing, the optimal policy for each individual hidden context group would be Pareto-optimal (non-dominated) on the dataset of preferences. With this in mind, we propose Pareto Optimal Preference Learning (POPL), where we learn a set of reward functions or policies (directly) that are optimized towards being Pareto-optimal with respect to the set of preferences given by a potentially diverse set of human annotators. To do this, we use an iterative selection process known as Lexicase selection [], which has been shown under mild assumptions to select individuals that are both Pareto-optimal and diverse. An outline of our method can be found in Figure 1.

We conduct a variety of empirical investigations to show that POPL outperforms strong baselines in learning sets of reward functions. Namely, we show in both a synthetic and large language model (LLM) domain, that POPL is able to cater reward functions for specific groups as well as limit reward hacking and jailbreaking. At the same time, we also show that this technique works for directly catering policies for distinct groups of people, without having access to the number of groups nor membership labels for annotations. Finally, we show that our technique can be used as a precursor for techniques that optimize for particular notions of group fairness.

Figure: Figure 1: Outline of our proposed method. Given a set of pairwise preferences over trajectory segments from groups with potentially different ground truth reward functions, we infer a set of reward functions or policies that captures each group’s ground truth, without group membership labels. To do this, we frame reward inference as multi-objective optimization, where each preference forms a single objective, and find a set of Pareto-optimal reward functions or policies.
Refer to caption: extracted/5676815/images/Outline.png

## 2 Related Work

Data used for RLHF systems often comes from multiple people, who are diverse in their preferences and values [, , , ]. This data, when considered in its aggregated form, can not be captured perfectly by a decision-making model that relies on a point estimate of utility [].
These models try to find a single reward function that is most likely, which is often not the optimal reward function for any one single person. When the groups are not perfectly balanced, the minority groups might be underrepresented in the inferred reward function [, , , ] or simply treated as noise []. There have been attempts at explicitly modeling different people with different levels of expertise [, , , ], but these methods generally rely on concrete ways to distinguish between groups.

In the context of RL, Myers et al. [] outlines an approach to learning a multi-modal reward function from online interaction between a human expert and a preference learning system. Ramé et al. [] learn a set of reward models by optimizing for diversity amongst the outputs. Whilst similar to our approach, we also aim to align our reward models with hidden context groups through optimizing for Pareto-optimality. For generative AI models and LLMs, there have been a variety of studies attempting to align large models with diverse human preferences. Chakraborty et al. [] and Siththaranjan et al. [] learn a mixture of preference distributions or a parameterized reward distribution, respectively. However, both these techniques operate under a contextual bandit setting which results in sub-optimal performance when used in the more general RL setting (discussed further in Section 4).
Bradley et al. [] and Ding et al. [] leverage fine-tuning to improve the diversity of model responses for better alignment and creativity, which do not directly address the ambiguity and hidden context in human preferences.
Jang et al. [] and Dai et al. [] elicit preferences specifically along different dimensions in order to cater custom reward functions for users in test time, and to be safe with respect to conflicting objectives, respectively. Whilst we also aim to cater reward functions in test time as well as optimize fairness between groups, we do not have access to labels regarding the context of the preferences generated. Finally, Rame et al. [] also generates a set of Pareto-optimal reward functions. However, in their setting, the system has access to ground truth reward functions for each group, and the Pareto-front is generated through weight interpolation between these functions.

Bayesian Reward Extrapolation (B-REx) [] instead learns a distribution of reward models from pairwise human preferences.
B-REx is then able to perform Bayesian inference using MCMC [] to sample from the posterior of reward functions. With this distribution, a practitioner can establish high confidence performance bounds that can be used to assess risk in evaluated policies as well as detect reward hacking behaviors. However, B-REx and other reward inference methods often rely on a faulty assumption that humans provide preferences in a Boltzmann-rational way.

On a separate front, there have been recent efforts on direct, reward-free methods for RLHF [, ], where a policy is learned directly from the preferences. However, these methods still implicitly rely on single-point estimates of reward, making them not directly applicable for reward function curation or risk minimization in safety-critical applications with hidden context. Furthermore, the hidden context in these preferences can lead to optimization issues when using these systems [].

## 3 Preliminaries

##### Learning from Human Preferences

Reinforcement learning from human feedback considers human preferences over trajectories (or more generally, outputs of a model) in order to learn a reward model or policy that respects the preferences [, , , , ].

In order to learn meaningfully from human preferences, one must characterize how preferences are generated from some parameterized preferences model $P(\sigma_{i}\prec\sigma_{j})$. Usually, this preference model is based on the notion of Boltzmann-rationality, where humans generate preferences in accordance to the Bradley-Terry (BT) model []. The probability of pairwise preference $(\sigma_{i}\succ\sigma_{j})$ between two trajectories given some utility function $f(\sigma)$ can be written as

$$ $P(\sigma_{i}\succ\sigma_{j})=\dfrac{e^{\beta f(\sigma_{i})}}{e^{\beta f(\sigma _{j})}+e^{\beta f(\sigma_{i})}}$ (1) $$

where $\beta$ models the confidence in the preference labels. $\beta\rightarrow\infty$ signals that the preference provider is perfectly rational, and $\beta=0$ signals that preferences are random. The Bradley-Terry model is used in many fields, such as psychology [, , ], However, this model does not perfectly capture the mechanisms driving these preferences [, , , , ].

##### Contrastive Preference Learning

Contrastive Preference Learning (CPL) [] learns a policy directly from preferences without needing to learn an intermediate reward function. This method uses a regret-based model of preference rather than the standard partial return interpretation. The probability of a preference under a candidate policy can be written as the ratio of the exponentiated sum of log-likelihoods of the chosen segment to the disregarded segment.

##### Hidden Context

Siththaranjan et al. [] introduce the problem of preference learning with hidden context. This is the idea that preferences are not only generated based on the exponential utility (partial return or regret), but also on some latent hidden context variable $z$. This variable is not accessible to preference learning systems and poses a challenge as it is often the case that this variable results in breaking the assumption that preferences are generated Boltzmann-rationally.

## 4 Problem Statement and Theoretical Foundation

We operate in a common RLHF setting in which, given a dataset $D=\{\sigma_{1},\cdots,\sigma_{m}\}$ of trajectory segments and a set $\mathcal{P}=\{(i,j):\sigma_{i}\succ\sigma_{j}\}$ of pairwise preferences over these segments, we wish to infer an unknown reward function $r:\mathcal{S}\mapsto\mathbb{R}$ that respects the preferences. This reward function represents an assignment of utility $r(s)$ to each state $s$ in the state space $\mathcal{S}$. $r$ can then be used as the reward function to train a policy $\pi(a|s):\mathcal{S}\times\mathcal{A}\mapsto[0,1]$ with RL.

In order to account for hidden context in the preferences learned, [] introduce Distributional Preference Learning, which relies on a single model of utility $u(s,z)$ to output a distribution of utility assignments $u(z,\cdot)$ for each state $s\in\mathcal{S}$, marginalizing over the hidden context variable $z$. Herein, we will refer to a model that outputs a distribution per state as a Marginalized Distributional Preference Learning (MDPL) system.

However, due to the marginalization process inherent in these systems, the utility function has no concept of *persistent annotator identity*—the fact that hidden context transcends a single preference annotation. Concretely, assume that users that give the preference $\sigma_{A}\succ\sigma_{B}$, are likely to also give the preference $\sigma_{B}\succ\sigma_{C}$. Being aware of persistent annotator identity would allow for statements to be made like: “the *same people* that prefer $\sigma_{A}$ to $\sigma_{B}$ are also likely to prefer $\sigma_{B}$ to $\sigma_{C}$.” Mathematically, MDPLs represent the marginalized probability $P(R|s)$ of a specific reward value $R$ in a state $s$. However, to be sensitive to persistent annotator identity, a preference learning framework would need to maintain the full conditional $P(R|s,z)$. In a contextual bandit setting such as those often found for finetuning LLMs [], this is not an issue, as determining that an output has high risk simply depends on the distribution of rewards attributed to that specific state. However, when learning a policy using this reward function, none of the inter-segment preference dependence is captured. Below, we outline an example to help clarify why this is an issue.

Figure: Figure 2: An example of a situation where using POPL is preferable to using a Marginalized Distributional Preference Learning (MDPL) system. Due to the fact that these systems marginalize over the hidden context $z$ for each state, MDPLs are unable to be sensitive to persistent annotator identity. MDPLs represent the distribution of utility values in a column-wise fashion, or maintain a distribution of utilities for each state, that is decoupled from that for other states. POPL, on the other hand, represents the distribution row-wise, finding a set of utility functions that should include the ground truth for each group.
Refer to caption: extracted/5676815/images/MDPL_bad.png

##### Example 1: MDPLs fail to distinguish between trajectories with different fairness profiles.

Let us consider two groups of people that have differing utilities for states $A,B,C$, as outlined in Figure 2. MDPL systems represent these utilities column-wise, as they maintain a distribution $u(s)$ that has been marginalized over $z$ for every state $s$. Now, consider the three trajectories $AB$, $BC$ and $AC$. These trajectories have different levels of utility when considering individual groups. For example, choosing $AB$ over either of the other two trajectories would have low utility and potentially be unfair (under certain notions of fairness) to an individual from group 2. MDPL systems would not be able to make this distinction, however, ignoring the group-specific utility. Despite the fact that group labels are not available to the preference learning framework, it would be better to maintain the full (un-marginalized) distribution $u(s,z)$ in order to be sensitive to persistent annotator identity. However, as we do not have group labels, we do not have information regarding the number of groups nor the $z$ values that correspond to them. Instead, we can find utility models that satisfy a diverse set of mutually compatible preferences. With enough of these models in a set, we assume that each hidden context group will have its utilities mapped by at least one of the models in the set.

In light of this discussion, we re-frame the problem of preference learning with hidden context as follows. The goal is to learn a set $\Pi=\{\pi_{1},\pi_{2}\cdots,\pi_{n}\}$ of policies such that, for the hidden context group represented by a variable $z\in\mathcal{Z}$, there is a policy $\pi_{z}\in\Pi$ that is the optimal policy for the ground truth reward function $r_{z}$ for the group. Note that this can be accomplished by standard (reward-based) RLHF or direct (reward-free) RLHF. For the standard approach, a series of reward functions $R=\{r_{1},r_{2}\cdots,r_{n}\}$ are first learned from preferences, then used to train $n$ optimal policies. In the direct approach, the policies are learned directly from preferences such as done by Hejna et al. [] and Rafailov et al. [].

## 5 Pareto-Optimality with Lexicase Selection

In this section, we outline an algorithm that aims to generate a series of reward functions (or optimal policies under these reward functions) that align with the preferences of different groups of people. The intuition we rely on for this algorithm is that, in a setting with low noise and sufficient regularization, the set of policies or reward functions that coincide with the various ground truth reward functions (one for each hidden context group) exist on the Pareto front, where each axis corresponds to a single preference.

### 5.1 Pareto Optimality

Definition 1 (Policy passing preference). A policy $\pi(a|s):\mathcal{S}\times\mathcal{A}\mapsto[0,1]$ passes a preference $(\sigma_{i}\prec\sigma_{j})$ if the probability of the preferred segment $\prod_{(s,a)\in\sigma_{j}}\pi(a|s)$ is higher than the probability of the other segment $\prod_{(s,a)\in\sigma_{i}}\pi(a|s)$. Or, equivalently, if $\sum_{(s,a)\in\sigma_{j}}\log\pi(s,a)>\sum_{(s,a)\in\sigma_{i}}\log\pi(s,a)$.

Definition 2 (Policy-set-relative Pareto-optimality). A policy $\pi(a|s):\mathcal{S}\times\mathcal{A}\mapsto[0,1]$ is Pareto optimal with respect to a set of preferences $\mathcal{P}$ relative to a set of other policies $\Pi=\{\pi_{1},\pi_{2},\cdots,\pi_{n}\}$ if and only if there exists a preference $(\sigma_{i}\succ\sigma_{j})\in\mathcal{P}$ that $\pi$ passes but all other policies in $\Pi$ do not pass.

Definition 3 (Hidden context group). A hidden context group is a group of $m$ annotators, each with their own reward function $r_{1},r_{2},\dots,r_{m}$ that all monotonically rank the segments $\sigma\in\mathcal{D}$ the same.

Definition 4 (Optimal policy for hidden context group). An optimal policy $\pi^{*}_{g}$ for a hidden context group $g$ is the optimal policy in a given environment (MDP/R) using the group’s implicit ground truth reward function $r_{g}$ as the reward function.

Definition 5 (Contradictory preferences). A pair of preferences $(\sigma_{i}\succ\sigma_{j})$, $(\sigma_{x}\succ\sigma_{y})$ are contradictory under a specific policy regularization scheme if the likelihood of any policy that satisfies both is lower than the likelihood of a policy that satisfies either.

With no regularization, a policy that satisfies two preferences should have a higher likelihood than one that satisfies one but not the other. This is because the likelihood of a certain policy is related to the total number of preferences passed by it. Passing two preferences would therefore illicit a higher likelihood than passing just one of them. However, if by satisfying both preferences, the models would need to incur a much greater regularization loss, it is likely that these two preferences came from individuals with differing hidden context.

Theorem 1. In a completely noiseless setting, all policies that are optimal for specific HC groups are Pareto optimal with respect to the set of all preferences $\mathcal{P}$ generated from all the groups, and the space of all possible policies $\Pi=\{\pi|\pi\in\mathcal{S}\times\mathcal{A}\mapsto[0,1]\}$

A proof of Theorem 1 can be found in the appendix. In essence, a set of Pareto-optimal policies must each satisfy a unique set of mutually satisfiable preferences (ones that do not contain a contradictory preference). As such, the optimal policies for a group with hidden context would also be Pareto-optimal.

### 5.2 Lexicase Selection

To obtain a set of Pareto optimal policies, we adopt the idea of lexicase selection [, ], which performs selection based on a lexicographic ordering of metrics. These metrics are shuffled into a random order (and therefore a random prioritization) for each selection event, and only the candidates that perform exactly the best on each metric are kept around. This process is repeated until we run out of metrics or are left with a single candidate individual.

A key property of lexicase selection is that it selects candidates that are Pareto-optimal relative to a starting set of candidates that tend to spread the *corners* of the Pareto front and thus be diverse []. This idea has been utilized in many machine learning optimization problems for improving generalization, as shown in recent work [, , , , ]. In the context of preference-based reward learning, preferences are used as binary metrics that can be generated in one of two ways: either a pass/fail approach or a likelihood approach.

In the pass/fail approach, each metric is passed if the candidate policy correctly ranks the pair of segments in the preference corresponding to that metric. Note that the content of the preference does not change the chance of passing a metric, and so this framework does not take into account Boltzmann rationality, where the probability of giving a preference depends on the difference in the exponentiated utility between the two alternatives. To account for the fact that people might give noisy preferences *within* their hidden context group, we use a likelihood passing approach, where a policy passes a metric with probability determined by the likelihood of that preference under the policy, with a specific regularization mechanism.

An important quality presented by lexicase selection is the notion that selected individuals are assessed with respect to their current pool. If all candidates in the pool fail a given preference (either deterministically or in the stochastic selection approach), then no individuals are filtered out. This idea helps overcome the fact that there might be contradictory preferences found in the dataset. When this happens, preferences that come earlier in the shuffle take priority over those that come later. For example, if the preference that comes first in the shuffle is contradictory to the preference that comes second, no individuals that pass the first preference will pass the second, and so no individuals will be filtered out (equivalent to the second preference not being in the shuffle). With the random orderings that lexicase selection uses, different preferences will be prioritized at different times, selecting policies or reward functions that pass diverse sets of compatible preferences.

It is important to note that lexicase selection does not guarantee that the individuals are globally pareto-optimal (i.e. pareto-optimal with respect to every policy or reward function possible). Instead, it guarantees that the individuals are Pareto-optimal with respect to the other candidates in the same pool. Whilst this distinction is important, we do not believe it significantly degrades our ability to recover the optimal policy for each hidden context group. This is because, with a large enough population, lexicase selection is able to continually search for individuals on the Pareto-frontier through successive generations.

## 6 Experiments

In this section, we detail our experimental results to validate the proposed POPL method. Further implementation details are provided in Appendix C.

### 6.1 Baselines

Throughout our experiments, we will use 3 main baselines. In the experiments on reward function inference, we use Bayesian Reward Extrapolation (B-REx) [] as a baseline, as it generates a large set of reward function hypotheses based on a Boltzmann-rational likelihood function, and has demonstrated efficacy in RL domains.

For our policy inference experiments, we compare to Contrastive Preference Learning [] as it is a leading RLHF algorithm for arbitrary sequential tasks. We also use a naive method of learning a *set* of policies based on CPL that we call Multi-CPL. In this approach, after pretraining, we fine-tune the last layer using the CPL objective multiple times to generate a large set of reward functions. Although we could do full network fine-tuning, we wanted to hold constant the trainable parameters available to each approach to ensure a fair comparison to POPL, which only uses last-layer fine-tuning. A secondary reason for the policy inference experiments is to verify whether our proposed technique is sensitive to the choice of preference generation model (regret vs. partial return).

For our language model (contextual bandit) experiments, we compare to both B-REx and Distributional Preference Learning (DPL) [], as well as standard the standard RLHF paradigm [], as these present a variety of approaches for generating reward models that can be used to ensure fairness across groups.

### 6.2 Metrics

Given a set of reward functions or policies, we can verify how well they perform on the two downstream tasks we have identified for this work: personalization and fairness. For personalization, we inspect the content of the personalized policies or reward functions to verify their alignment with each hidden context group’s preferences. For fairness, we measure how well a new policy $\pi_{\text{fair}}$ performs with respect to all hidden context groups. Given a set of policies $\Pi$, one can generate $\pi_{\text{fair}}$ by picking actions that have the highest lower (10%) quantile probability across the set. In other words, we ensure that no single group is having its values undermined (by taking low-probability actions) in an attempt to satisfy a different group. Although this is a relatively simple notion of fairness, this method could be extended to be compatible with other fairness optimization approaches [].

### 6.3 Synthetic Stateless Experiment

The first set of experiments we perform will test whether POPL is able to recover a set of reward functions from a series of preferences generated with hidden context in a very simple stateless domain. Doing this, we are testing whether the fact that the outputs of lexicase selection are an approximation of the global Pareto-front significantly degrades the quality of reward functions generated. Then, we will select a personalized reward function for each group and compare them to the ground truth reward functions used to generate the preferences.

Figure: (a) B-REx Distribution
Refer to caption: extracted/5676815/images/synthetic_experiments/1d_identity_0_brex_all.png

Following the synthetic experiments outlined by Siththaranjan et al. [], we compare B-REx and lexicographic reward inference methods on learning a distribution of reward functions based on preferences from two distinct groups of humans defined by a hidden context variable $z\sim\mathcal{B}(0.5)$ where $\mathcal{B}(0.5)$ is a Bernoulli distribution. We compare the distributions based on their representation of the various hidden context groups that might have different preferences.

The utility in this scenario can be modeled as

$$ $u(a,z)=\begin{cases}a&\text{if }a<0.8\\ 2az&\text{otherwise}\end{cases}$ (2) $$

Figure 3 shows the resulting reward functions from performing reward inference on a dataset of preferences with underlying hidden context using POPL and B-REx. For each of these methods, we display all the reward functions created through each approach. As there are two equally proportioned groups, each with overlapping preferences, B-REx tends to concentrate all of the probability mass on a single one of these groups. POPL, on the other hand, is able to learn two distinct equivalence classes of reward functions. The reward functions inferred are monotonically increasing in the range $a\in[0,0.8]$. After this range, where the hidden context has an effect, some of the reward functions predict a decreased utility to reflect the first group’s preferences, and other reward functions predict increased utility to predict the other group’s preferences.

In order to test whether POPL achieves our goal of having at least one reward function in the set that covers each hidden context group, we inspect some selected reward functions for each group. We use a smaller set of the preferences that all have a shared hidden context, and attempt to select a reward function from the distribution that is catered to a user in this group. Figure 3 also shows the results of catering a reward function for each of the hidden context classes $z=0$ and $z=1$. Due to B-REx using the Boltzmann rationality assumption, it concentrates much of the distribution on the $z=1$ case, and does not capture the preferences given by the $z=0$ group. POPL, on the other hand, is able to recover the reward functions for both groups from the learned distributions.

### 6.4 Minigrid Policy Inference

Now that we have demonstrated POPL’s efficacy in reward inference from preferences with hidden context, we perform a second set of experiments to verify whether 1) POPL is able to generate *policies* directly from preferences, and 2) POPL is able to perform in a sequential RL domain, where annotators’ hidden context is persistent (i.e. potentially affects more than one segment preference annotation).The domain used in these experiments is outlined in Figure 4(a). The hidden context groups in this scenario delineate whether the annotator inherently prefers the bottom or top door to be used to get to the goal. The preferences were labeled according to the regret preference model from members of both groups (extracted from the optimal policy for each group’s ground truth reward model).
After running this optimization, catered policies for each group can be found in Figures 4(c) and 4(d) for POPL, and 4(e) and 4(f) for MultiCPL. We find that POPL is able to successfully cater policies for both groups of people, despite not having labels regarding their group membership.

Figure: (a) Domain
Refer to caption: extracted/5676815/images/minigrid/minigrid.png

### 6.5 Language Model Experiments

A final set of experiments that we perform tests the ability of POPL to scale to domains involving human annotations. We investigate whether POPL can be sensitive to hidden context in whether annotators prefer *harmless* or *helpful* responses []. When reward models are trained on the entire set of preferences, whether they were generated based on helpfulness or harmlessness is hidden context, as this information affects preferences but is not available to a reward inference system.

Importantly, preferences based on helpfulness and harmlessness can often be contradictory. In fact, Wei et al. [] find examples of user prompts that directly pit these objectives against each other, leading a language model to output harmful outputs, a phenomenon known as *jailbreaking*. An RLHF system built with hidden context in mind would help detect jailbreaking before a harmful output would be given to a user. In the context of this work, a model that is susceptible to jailbreaking would be unfair to certain groups (compromising its efficacy for the harmlessness group in order to optimize for the helpfulness group).

Table 1 presents the jailbreak rates and helpfulness accuracy for standard RLHF, B-REx, DPL, and our proposed POPL.
For B-REx and POPL, we generate a set of reward functions by extrapolating the last layer of a fine-tuned Llama-2-7b [] preference model.
Default settings use the mean reward across the entire set. For fairness optimization, we use the 10${}^{\text{th}}$ percentile of reward values across all the reward functions in the set.

The results indicate that B-REx’s performance is inferior to standard RLHF, even when employing fairness-focused strategies using the lower quantile of rewards. This suggests that the likelihood estimated by the BT model does not adequately accommodate scenarios where preferences are in conflict, and B-REx fails to accurately approximate the distribution of rewards. POPL performs the best out of all methods without employing any fairness optimization. Given the high-dimensional nature of reward features in LLM tasks, a population-based approach is essential for accurately modeling and enhancing the diversity of reward hypotheses.

When compared to the current state-of-the-art, POPL outperforms Mean & Var DPL and competes closely with Categorical DPL. Notably, unlike DPL which requires training a new reward model with different outputs, POPL efficiently extrapolates directly from the last layer of pre-trained RLHF reward models, making it highly efficient and broadly applicable. For example, POPL can be applied to a pre-trained 7b-LLM reward model in under an hour on a single NVIDIA A100 GPU.
Another advantage of POPL is its independence from assumptions about the distribution of reward hypotheses. In contrast, DPL methods require a predefined reward distribution, such as the assumption of normally distributed rewards for Mean & Var DPL, or correctly sized bins for Categorical DPL.

**Table 1: Results on LLM jailbreaks. POPL has the lowest jailbreak rate across all methods without any fairness optimization. For fairness optimization, POPL has a lower jailbreak rate than B-REx, standard RLHF, as well as Mean & var. DPL, and is competitive with categorical DPL.**
| Method | Training data | Jailbreak rate (%) | Helpfulness acc. (%) |
| --- | --- | --- | --- |
| Standard | Helpful | 52.4 | 72.6 |
| Standard | Harmless | 3.7 | 49.5 |
| Standard | Combined | 25.1 | 68.2 |
| Mean & var. DPL | Combined | 30.5 | 68.4 |
| $\Lsh$<br>Fair |  | 20.3 | 66.4 |
| Categorical DPL | Combined | 32.1 | 66.2 |
| $\Lsh$<br>Fair |  | 13.4 | 66.2 |
| Bayesian REx | Combined | 28.3 | 67.5 |
| $\Lsh$<br>Fair |  | 27.8 | 50.4 |
| POPL | Combined | 17.6 | 66.1 |
| $\Lsh$<br>Fair |  | 15.0 | 65.7 |

## 7 Conclusion

When learning from human preferences for the sake of aligning to human values, one must first decide *which* humans to pay attention to. As these systems rely on point estimates of return or regret, they can at most align with a single human’s preferences. We have formalized this as
the problem of reinforcement learning from human preferences with hidden context. Under this conception, a set of policies must be generated that contains the optimal policy for each group of people.

To solve this problem, we relied on the concept of Pareto-optimality to generate a series of reward functions and/or policies that are optimal with respect to unique sub-sets of preferences. To optimize towards Pareto-optimality, we used a technique known as lexicase selection, that selects individuals from a large set based on a randomized (lexicographic) prioritization of the training data.

We verified that lexicase selection can be used to generate diverse distributions of either reward functions or policies that align with the diverse preferences that human annotators have. We evaluated and verified the performance of POPL in a variety of domains, including a synthetic stateless domain, a sequential RL domain, and even language model jailbreak detection. Across these domains, we have demonstrated POPL’s efficacy when compared to contemporary algorithms in dealing with hidden context in the preferences.

One major limitation of this work is the lack of use of gradients in training policies. The optimization procedure used after lexicase selection relies on random variations and repeated selections, which allows for effective trade-offs between exploration and exploitation of the preference landscape. However, it may be possible to augment the core idea in future work to allow it to utilize gradients. Furthermore, a study into the conditions required for the output of the procedure to be globally Pareto-optimal could be instrumental.

#### Acknowledgments

This work has taken place in the Safe, Correct, and Aligned Learning and Robotics Lab (SCALAR) at The University of Massachusetts Amherst. SCALAR research is supported in part by the NSF (IIS-2323384), AFOSR (FA9550-20-1-0077), and the Center for AI Safety (CAIS).

LD and LS were supported by the National Science Foundation under Grant No. 2117377.
This work was performed in part using high-performance computing equipment at Amherst College obtained under National Science Foundation Grant No. 2117377. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the funding agencies.
The authors would like to thank Philip Thomas, members of the SCALAR lab at UMass Amherst, members of the PUSH lab at Amherst College, University of Massachusetts Amherst, and Hampshire College for helpful discussions, and anonymous reviewers for their thoughtful feedback.

#### Author Contributions

RB conducted preliminary experimentation, conceived and framed the core ideas, implemented the core algorithm, ran the synthetic and Minigrid experiments, and generated the figures. RB also contributed significantly to the writing and organization of the manuscript. LD implemented the algorithm in an efficient framework optimized for scalability, conducted the LLM experiments, and formulated the associated analyses. LD also contributed to the ideation and provided feedback on multiple drafts of the manuscript.

LS provided oversight on the high-level conceptualization of lexicase selection and its application. LS offered continuous advice, reviewed the manuscript, and contributed to the theoretical framing of the work. SN oversaw the project from inception to completion, contributed significantly to ideation and conceptual framing, and ensured the project stayed aligned with its broader goals. SN also helped with securing funding and provided extensive feedback on the manuscript.

## Appendix A Full Algorithm

Algorithm 1 gives an outline of a single step of Pareto Optimal Preference Learning (POPL).

Figure: Algorithm 1 One Step of Pareto Optimal Preference Learning

## Appendix B Proofs

Proof of Theorem 1 (Contradiction).
Let us assume that there is a policy $\pi^{*}_{z}$ that is the optimal policy according to a hidden context group $z$. This means $\pi_{z}$ passes all the preferences compatible with the values of group $z$ and fails only the preferences that are not compatible with preferences given by group $z$. For sake of contradiction, we assert that this reward function is not Pareto optimal with respect to all other reward function candidates. This means there exists another reward function $\pi^{\prime}$ that performs better than or equal to $\pi^{*}_{z}$ across every preference in $\mathcal{P}$, including those generated by group $z$. This is a clear contradiction as that would imply $\pi^{\prime}$ is the optimal policy for group $z$ instead of $\pi_{z}$.

## Appendix C Implementation Details

In this section, we include more implementation details of our experiments.

### C.1 Synthetic Experiments

We follow the experimental procedure of Siththaranjan et al. [] in generating preferences, except modify their code such that we ensure that annotator identity is held constant for each preference. We use last layer finetuning on a neural network that is randomly initialized. We did not include any pre-training here to ensure that we are not pushing our reward models towards any modes before starting to train. We use a batch size of 2048 preferences, a step size of 0.1 and 10000 steps of MCMC for B-Rex. For POPL, we use a population size of 100 and a generation count of 100. We use a $\beta$ (confidence) value of 10, although have found that changing this value does not significantly affect B-REx’s performance.

### C.2 Gridworld Model Experiments

For the Gridworld model experiments, we base our environment on the Minigrid package []. Demonstrations were generated by rolling out many checkpointed policies at different levels of performance, trained using Proximal Policy Optimization (PPO). Then, these demonstrations were annotated based on a high performing policy’s action selection probabilities.

For MultiCPL and POPL, we use behavior cloning directly on the demonstrations for 1000 iterations with a batch size of 64 and a learning rate of 0.001 with the Adam optimizer [] as pretraining. The model architecture was a simple convolutional neural network that takes input from the agent’s view window, and has a single fully connected layer with 128 nodes to output the 7 actions from the environment. For both MultiCPL and POPL, we use last layer fine-tuning. For MultiCPL, we use the CPL objective, a learning rate of 0.001, where each model in a population of 500 models is trained for 20 iterations. For POPL, we use a learning rate of 0.2, and 1000 total steps. We sample 640 preferences every 10 iterations (as we can cache the last layer features for this examples for improved performance), and sub-sample a batch of size 64 for each step of lexicase selection. For a fair comparison between these two approaches, we approximately hold constant total wall clock time on the same hardware. Given a final population of policies generated by POPL or MultiCPL, we select the top 10 models for each hidden context class as the catered policy for that group.

### C.3 Language Model Experiments

In the LLM experiments, we assess the performance of reward learning by examining preference accuracy on the test set. To investigate vulnerabilities to jailbreak, we analyze pairs of responses to jailbreak prompts designed by Wei et al. [] to deceive the model into giving a harmful response. We calculate the percentage of prompts where it assigns a higher reward to the jailbroken response (”jailbreak rate”). Additionally, we evaluate the reward function’s ability to assess helpfulness on non-harmful prompts, i.e., the reward function predicts higher rewards on the more helpful response. We compare our method to normal RLHF with an LLM-based preference model, Bayesian Reward Extrapolation (B-REx), and distributional preference learning (DPL). DPL methods predict parameters of the distribution of reward values for each response, rather than a single reward value, in order to better account for hidden context in human preferences.

For standard RLHF, we use the pre-trained Llama-2-7b [] preference model by Siththaranjan et al. [], which is fine-tuned on the HH-RLHF dataset using LoRA []. We implement B-REx by performing linear reward extrapolation on the last layer of the pre-trained Llama-2-7b preference model. Following the B-REx implementation in [], we run 200,000 steps of MCMC with a step size of 0.05. We use a burn-in of 5000 and a skip every 20 samples to reduce auto-correlation. For POPL, we run lexicase selection for 100 generations with a population size of 1000, and randomly sample 100 reward functions in the last generation.

Because the ranking likelihood is invariant to affine transformations of the rewards, we normalize the rewards by subtracting the median reward calculated on the training set over all the responses. This ensures that the reward values are comparable when calculating the lower quantile of rewards in risk-averse optimization.

## Appendix D Broader Societal Impacts

The proposed work on Pareto Optimal Preference Learning (POPL) aims to enhance the alignment of AI systems with diverse human values, thereby addressing critical issues of fairness and representation. By focusing on learning from human preferences with hidden context, our method seeks to ensure that AI models do not disproportionately favor or disadvantage specific groups, making them more equitable and just. This has the potential to significantly improve the societal acceptance and trust in AI systems, particularly in sensitive applications such as healthcare, education, and law enforcement, where fairness and inclusivity are critical.

However, there are potential negative societal impacts to consider. The deployment of AI systems that can cater to specific groups might inadvertently reinforce existing biases if the hidden context reflects social prejudices or discriminatory practices. Therefore, it is crucial to incorporate safeguards and robust validation mechanisms to detect and mitigate any biased outcomes. As researchers and developers, we must be vigilant about the sources of our training data and continually audit AI systems for unintended consequences.

Moreover, the computational work required for training these models can have environmental impacts, given the high-energy consumption associated with large-scale AI computations. Researchers should consider optimizing algorithms to be more efficient and exploring the use of renewable energy sources to mitigate this impact.

By considering these factors, we aim to advance AI technologies in a direction that promotes fairness, inclusiveness, and sustainability, ensuring that they serve the broader interests of society.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 3
presentation: 1
questions: I am unsure if I fully understand the paper. If I have any misunderstandings, please let me know. I would be happy to receive clarification.

1. As I understand it, there is no learning in Algorithm 1, which represents one step of POPL. In my understanding, it solely relies on the random initialization of hypotheses and the selection of good hypotheses. With a large number of trials, it might eventually find good policy hypotheses, but I wonder if this method is truly practical.
2. In general, there are many policies that passes exactly the same preference subsets. However, Algorithm 1 only checks whether the candidates pass at least on preference or not. Then, how to determine a Pareto-optimal policy among these policies?
3. In Algorithm 1, if we use POPL to learn a set of reward functions, how can we specify the dimensionality of these reward functions and learn them from the given demonstrations?
4. Compared to the previous works, such as preference-driven MORL [1] or multi-objective alignment in LLM [2], what is the main difference and advantage of POPL?
5. I cannot understand the left side of Figure 2 (titled “MDPLs”). In this case, what does z indicate?

### References

[1] Basaklar, Toygun, Suat Gumussoy, and Umit Y. Ogras. "Pd-morl: Preference-driven multi-objective reinforcement learning algorithm." ICLR 2023.

[2] Yang, Rui, et al. "Rewards-in-context: Multi-objective alignment of foundation models with dynamic preference adjustment." arXiv preprint arXiv:2402.10207 (2024).
soundness: 2
strengths: POPL can capture the Pareto optimal reward functions.
summary: This paper introduces a reinforcement learning from human feedback with hidden context (RLHF-HC) framework called Pareto optimal preference learning (POPL). While marginalized distributional preference learning (MDPL) marginalizes $u(s\vert z)$ over the hidden context variable $z$ to produce a distribution $u(s)$ for each state $s\in\mathcal{S}$, POPL preserves the full conditional distributions. By leveraging these conditional probabilities, POPL can effectively capture Pareto optimal reward functions.
weaknesses: - The validity of the proposed algorithm is not theoretically explained.
- The scalability of the proposed algorithm remains uncertain.
- There is no way to determine the dimensionality of the underlying reward functions or the number of Pareto optimal policies.

---

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: See the weakness part above.
soundness: 2
strengths: 1. This work concentrates on an import problem that human feedback involves hidden information. The Pareto-optimal is indeed one possible solution.

2. Experiments on different tasks are given.
summary: This work concentrates on the issue that human feedback data might involve noises caused by hidden information. It proposes a method named POPL that aims to learn a Pareto-optimal policy. POPL uses lexicase selection and conducts experiments on different tasks.
weaknesses: 1. I found the presentation for the method is quite hard for me to understand. Lexicase selection, as key idea of the method, is not introduced clearly. I am not clear about how this selection method is conducted. Also, I think a figure for process might be better for readers to understand.

2. Similarly, I found that many concepts are used without a clear explanation. For example, I am not clear what "hypotheses" refers to as it first shows in Sec. 6.1 and also in Alg. 1. Also, as MDPL is mentioned, this formal calculation for this method is not give. I cannot under stand the illustration of Fig.3.

3. Further, if more intuition about the method is provided, it might be better for readers to understand this method.

4. Since this method concentrates on the RLHF alignment process, I am curious about the comparison between POPL and more RLHF methods like DPO. Also, the detailed information for the standard RLHF is not given.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 2
questions: -  What's the formal definition of $\sigma$, is it always a $(s,a)$ pair?
- Could authors provides more clearer highlight of the technical contributions?
- For me, even though the definitions are provided, the concept is still confused. Could authors gives the mathemitical definitions of concepts and theorems in Section 5.1, e.g., optimality?
soundness: 3
strengths: - This paper provide a novel way to deal with the heterogenity among different sourses
- Various experiments under different settings are conducted to evaluate the performance
summary: POPL enhances alignment by addressing diverse human preferences as multiple objectives, achieving solutions that are Pareto-optimal and fair across groups. It uses lexicase selection to find policies that respect varied preferences across. Empirical results show POPL’s effectiveness in aligning AI to human values safely and equitably across applications like robotics and language models.
weaknesses: - Noations and concepts are sometimes not well-defined. For instance, $\sigma$ first occur in Section 3 without any definitions. I would suggest the authors to more clearly defined notations.
- The presentation of experiment results is quite unclear. For example, Figure 3(b) is really chaotic and I can hardly tell the information here.
- Lack of comparision with other methods. From my interpretation, many methods are proposed to solve the similar issue, e,g., Nash learning for RLHF and general preference framework. The methods should be more properly evaluated and compared.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. By formulating the RLHF-HC problem as a multi-objective problem, how will the method deal with (non-)**transitivity** of real-world human preference ? As this could be a problem for real-world human feedback (See ,e.g., section 7.3 of [1]).
2. ‘Theory Foundation’ in the title of Section 4 seems to be missing, or it has been formulated in Section 5 ? A re-organization may be needed.
3. Why the probability of segments in Definition 1 does not include the distributions of the states, such as $\prod_{(s,a)\in\sigma}\pi(a|s)d(s)$ ?
4. In line 258, is the word ‘illicit’ a typo ? It may be ‘elicit’.
5. In the proofs in Appendix B, what does $\pi_z$ refer to ? Is it just $\pi_z^*$ ?
soundness: 3
strengths: 1. The problems of previous methods are clarified with a simple yet illustrative example.
2. The intuitions and theoretical formulations in Section 5.1 are interesting and may be useful for future research.
3. Experiments are diverse and persuasive, ranging from analytic environments to language model experiments, which show the proposed method’s efficiency and applicability.
summary: The paper tries to solve the problem of learning from complex human preference by first formulating it as the problem of preference learning with hidden context, and then relying on the framework and techniques of multi-objective optimization. The Pareto Optimal Preference Learning (POPL) method, based on the lexicase selection algorithm, is proposed to efficiently learn and generate diverse rewards and policies. The extensive and diverse experiments are conducted to illustrate the efficiency and applicability of the method.
weaknesses: 1. It would be better to concisely introduce the theory proposed in Section 4 in the introduction Section.
2. Paragraph headings in Section 2 can make them more clear.
3. In line 147, the comma ahead of ‘However’ should be a full stop.
4. The process of lexicase selection for preference learning in Section 5.2 can be made more clear with figures, formulas, or diagrams.

Meta-review: This paper proposes Pareto Optimal Preference Learning (POPL) to enable pluralistic alignment by framing discrepant group preferences as objectives with potential trade-offs, aiming for policies that are Pareto-optimal on the preference dataset.

The proposed Pareto-optimal solution is a promising direction for improving RLHF alignment and The concept of Pareto optimality is well-integrated into the framework. It also includes a wide range of experiments across different tasks.

The main concern is the validity of the proposed algorithm is not sufficiently supported by theoretical explanations. There is a need for more formal justification to back up the claims made about the algorithm's effectiveness. Moreover, the scalability of the proposed algorithm is not clearly addressed. It remains uncertain how well the method will perform as the complexity of the task or the number of reward functions increases.

Decision: Reject