# A note on Is RLHF More Difficult than Standard RL? A Theoretical Perspective

Reinforcement Learning from Human Feedback (RLHF) represents a significant advancement by using human preferences as learning signals for AI agents. In this blog post, we explore the concepts from the paper "Is RLHF More Difficult than Standard RL? A Theoretical Perspective" by Yuanhao Wang, Qinghua Liu, and Chi Jin. This paper examines whether RLHF is more challenging to implement than standard RL due to the lesser amount of information contained in human preferences compared to explicit reward signals.

## Introduction

In this blog post, we explore the concepts presented in the paper "Is RLHF More Difficult than Standard RL? A Theoretical Perspective" by Yuanhao Wang, Qinghua Liu, and Chi Jin. Reinforcement Learning (RL) is a technique where agents learn to maximize cumulative rewards by interacting with their environment. However, designing effective reward functions can be complex and sometimes impossible.

To overcome this limitation, Reinforcement Learning from Human Feedback (RLHF) uses human preferences as learning signals. This method makes it easier to align the agents' goals with human values and makes the data collection process more intuitive.

The paper investigates whether RLHF is more difficult to implement than standard RL. To answer this question, the authors propose reduction approaches that convert RLHF problems into standard RL problems. They introduce the Preference-to-Reward (P2R) interface and adaptations of the OMLE algorithm, demonstrating that these methods can effectively solve a wide range of RLHF models with robust theoretical guarantees.

From a personal perspective, although my knowledge of RLHF was initially limited, this paper helped me better understand its possibilities and limitations. While it is challenging to provide an in-depth analysis, it is evident that the proposed methods can offer a powerful alternative in some RLHF cases. However, it is important to note that these methods do not address all RLHF challenges.


<p style="text-align: center;">Shoggoth with Smiley Face. Courtesy of x.com/anthrupad</p>

## Preliminaries

### Notations and Problem Formulation

The paper focuses on episodic Markov Decision Processes (MDPs), defined by a tuple $(H, S, A, P)$:

- **H**: Length of each episode.
- **S**: State space.
- **A**: Action space.
- **P**: Transition probability function. For each step $h \in [H]$ and $s, a \in S \times A$, $P_h(\cdot | s, a)$ specifies the distribution of the next state.

### Trajectory and Markov Policy

A trajectory $\tau \in (S \times A)^H$ is a sequence of interactions with the MDP, defined as $(s_1, a_1, \ldots, s_H, a_H)$. A Markov policy $\pi = \{\pi_h : S \to \Delta A\}_{h \in [H]}$ specifies a distribution of actions based on the current state at each step. In contrast, a general policy $\pi = \{\pi_h : (S \times A)^{h-1} \times S \to \Delta A\}_{h \in [H]}$ can choose an action based on the entire history up to step $h$.

### Optimization Problem

The objective of reinforcement learning is to find an optimal policy $\pi^*$ that maximizes the expected cumulative reward:
$$
\pi^* = \arg \max_{\pi} \mathbb{E} \left[ \sum_{h=1}^H r(s_h, a_h) \mid \pi \right]
$$
where $r(s_h, a_h)$ is the reward received after taking action $a_h$ in state $s_h$.

In the context of RLHF, instead of directly receiving reward signals, the algorithm interacts with a reward-free MDP and can query a **comparison oracle** (human evaluators) to obtain information about preferences between different trajectories.

### Technical Assumptions

The following assumptions are essential for the theoretical results of the paper:

1. **Link Function $\sigma$**:
- The link function $\sigma$ translates preferences into comparison probabilities. For example, if $\tau_1$ and $\tau_2$ are two trajectories, the probability that $\tau_1$ is preferred over $\tau_2$ is modeled by: $$\sigma(r^*(\tau_1) - r^*(\tau_2))$$
- **Assumption**: The function $\sigma$ is known and satisfies certain regularity properties, such as a lower bounded derivative ($\sigma'(x) \geq \alpha > 0$).

2. **Realizability**:
- It is assumed that the true reward function $r^*$ belongs to a known class of functions $\mathcal{R}$. This means there exists an $r \in \mathcal{R}$ such that $r = r^*$.

3. **Eluder Dimension**:
- The Eluder dimension is a measure of the difficulty of identifying a function from a given function class based on possible errors. For a function class $\mathcal{F}$, the Eluder dimension, denoted $\dim_E(\mathcal{F}, \epsilon)$, quantifies the number of $\epsilon$-independent points needed to learn a function in $\mathcal{F}$.
- **$\epsilon$** represents a level of precision or tolerance in terms of the algorithm's performance.

4. **Function Approximation**:
- It is assumed that we know a class of reward functions $\mathcal{R}$ and can use function approximations to model the rewards.

### Types of Preferences

The paper distinguishes between two types of preferences for modeling human feedback:

1. **Utility-Based Preferences**:
- **Definition**: These preferences are modeled by an underlying reward function $r^*$. The comparison oracle compares two trajectories $\tau_1$ and $\tau_2$ based on the difference in their rewards.

2. **General Preferences**:
- **Definition**: For more complex preferences, the concept of a von Neumann winner is introduced. A policy $\pi^*$ is a von Neumann winner if it maximizes the average utility against any other policy, even when preferences cannot be modeled by a simple utility function. (The concept of von Neumann will be detailed in a dedicated section)

## Motivating Example

The subject is not extremely deep and involved, but an example never hurt anyone. Here’s how RLHF can be applied in a practical scenario.

### Example: Training a Home Service Robot

Imagine developing a service robot designed to help elderly people at home. Designing a precise reward function for each task, such as making tea or tidying up the kitchen, can be complex and subjective.


#### Challenges in Designing Rewards

1. **Complexity and Subjectivity of Tasks**:
- Making tea can involve multiple steps (boiling water, putting tea in the teapot, pouring water, etc.), each with its own success criteria.
- The notion of a "well-tidied kitchen" can vary significantly from person to person.

2. **User Preferences**:
- Personal preferences play a crucial role. For example, one person may prefer stronger tea, while another prefers lighter tea.
- Expectations can also vary based on the context.

#### Application of RLHF

1. **Collecting Preferences**:
- The robot performs various attempts to make tea or tidy up the kitchen.
- Users provide feedback by comparing two results of these attempts.

2. **Using the Preference-to-Reward Interface (P2R)**:
- The P2R interface converts these preferences into approximate reward signals. (We will detail this algorithm in the next section)

3. **Learning and Adapting**:
- The robot uses these estimates to improve its policies and better align its actions with user preferences.

This example shows how RLHF can transform complex and subjective tasks into manageable learning processes, aligning agents' actions with human preferences.

## Part 3: Utility-Based Preferences

### Introduction

Utility-based preferences model human preferences in terms of rewards. Instead of directly defining reward functions, Reinforcement Learning from Human Feedback (RLHF) uses trajectory comparisons to derive approximate reward functions.

For example, in the motivating example of training a household service robot, utility-based preferences can be used to capture the varying subjective preferences of different users for tasks like preparing tea or cleaning the kitchen. By comparing the outcomes of different trajectories, the robot can learn to align its actions more closely with individual user preferences.


### P2R: Preference to Reward

The P2R (Preference to Reward) algorithm converts human preferences into rewards that can be used by standard RL algorithms. It facilitates the integration of human feedback into the learning process, allowing agents to align their actions with user preferences.


### How P2R Works

1. **Confidence Set for Rewards**:

P2R maintains a confidence set $B_r$ containing possible reward functions based on the observed preferences. This confidence set is initialized to include all reward functions consistent with the expressed preferences.

2. **Comparison Oracle**:

When a new trajectory sample $\tau$ is obtained, P2R decides whether it is necessary to consult the comparison oracle for feedback. If the possible reward functions in $B_r$ sufficiently agree on the reward for $\tau$, P2R uses this estimate without querying the oracle. Otherwise, P2R queries the oracle to compare $\tau$ with a reference trajectory $\tau_0$.

3. **Updating the Confidence Set**:

With each oracle query, the set $B_r$ is updated to include only those reward functions consistent with the new comparisons. This update process uses maximum likelihood techniques to adjust the estimates of the true reward function.

### Utility of P2R


1. **Query Efficiency**:
- P2R minimizes the number of oracle queries by using reward estimates when possible. This reduces the workload for human evaluators and makes learning more efficient.

2. **Compatibility with Standard RL Algorithms**:
- By converting preferences into approximate rewards, P2R allows direct use of standard RL algorithms. RL algorithms such as Q-learning, SARSA, or policy-based methods can thus benefit from human preferences without significant reengineering.

3. **Theoretical Robustness**:
- P2R provides theoretical guarantees on the learning efficiency, ensuring that the learned policies are close to optimal with a reasonable number of samples.

### Instantiations of P2R

1. **Tabular MDPs**:
- **Definition**: A tabular MDP is a type of MDP where the state and action spaces are discrete and relatively small, allowing transitions and rewards to be represented in tables.
- **Algorithm**: For tabular MDPs, P2R can be used with the UCBVI-BF algorithm (Upper Confidence Bound for Value Iteration with Bonus Function).

2. **RL Problems with Low Bellman-Eluder Dimension**:
- **Definition**: The Bellman-Eluder dimension measures the complexity of an RL problem in terms of state and action dependence. A problem with a low Bellman-Eluder dimension has a structure that facilitates learning.
- **Algorithm**: For RL problems with a low Bellman-Eluder dimension, P2R can be used with the GOLF algorithm (Gradient-Optimistic Linear Function).

3. **MDPs with Generalized Eluder Dimension**:
- **Definition**: The generalized Eluder dimension is an extension of the Eluder dimension that applies to more complex and general function classes.
- **Algorithm**: For MDPs with a generalized Eluder dimension, P2R can be used with the OMLE algorithm (Optimistic Model-based Learning).

### Theoretical Analysis of P2R

The theoretical analysis of P2R shows that this algorithm allows learning an $\epsilon$-optimal policy by effectively converting human preferences into rewards usable by standard RL algorithms.

**Sample Complexity**: P2R has a sample complexity proportional to the size of the state and action space and the episode length, ensuring efficient learning even in complex environments.

**Query Complexity**: P2R minimizes oracle queries by using reward estimates when possible. The query complexity depends on the desired precision but remains manageable through the use of confidence sets and targeted queries.

**Theoretical Robustness**: The theoretical guarantees of P2R include convergence to $\epsilon$-optimal policies with a reasonable number of samples and queries, ensuring robustness comparable to standard RL algorithms.

P2R thus optimizes sample and query complexity while guaranteeing the learning of $\epsilon$-optimal policies from human preferences.

### P-OMLE: Optimistic Model-based Learning from Preferences

The P-OMLE (Preference-based Optimistic Model-based Learning) method is an adaptation of the OMLE algorithm to directly handle trajectory preferences. It aims to reduce query complexity while maintaining the effectiveness of learning optimal policies from human feedback.

### OMLE Algorithm: Optimistic Model-based Learning with Exploration

The OMLE (Optimistic Model-based Learning with Exploration) algorithm comes from another paper *(Optimistic MLE—A Generic Model-based Algorithm for Partially Observable Sequential Decision Making)* and is not detailed in the primary paper we are discussing. Here is a brief explanation to provide context.

OMLE uses optimistic planning and promotes exploration to learn optimal policies. It starts by defining a confidence set for the reward and transition functions. At each step, it plans optimistically using the current best estimates, executes the policy to collect data, and updates the model estimates based on new observations. This process repeats until convergence.

The algorithm is advantageous for its efficient exploration, adaptability, and robust theoretical guarantees, ensuring convergence to optimal policies with a reasonable number of samples.

#### How P-OMLE Works


1. **Initialization**

**Initial Confidence Set $B_1$**: P-OMLE starts by defining an initial confidence set $B_1$ for the reward and transition functions, initialized to include all functions consistent with the observed preferences.

2. **Optimistic Planning**

**Optimism in the Face of Uncertainty**: At each step $t$, P-OMLE performs optimistic planning to determine the policy $\pi^t$ and the associated reward and transition functions $(r^t, p^t)$ that maximize the estimated value. This planning uses an optimism-based approach, assuming the current best estimates are correct.

3. **Data Collection**

**Policy Execution**: P-OMLE executes the policy $\pi^t$ to collect a new trajectory $\tau$. This trajectory is then compared with a reference trajectory $\tau_0$ using the comparison oracle, which provides feedback on the preferences between $\tau$ and $\tau_0$.

4. **Updating the Confidence Set**

**Preference-Based Updating**: The confidence set $B_t$ is updated based on the new comparison data, using maximum likelihood techniques to adjust the reward and transition function estimates. Each new comparison provides additional information about the reward function and allows excluding certain functions from $B_t$ that are no longer consistent with the observations.

#### Utility of P-OMLE

P-OMLE presents several advantages:

1. **Query Complexity Reduction**
- **Query Efficiency**: P-OMLE reduces the number of necessary oracle queries by limiting inquiries to only when uncertainty is high.

2. **Adaptability**
- **Flexibility to Complex Preferences**: P-OMLE adapts to trajectory preferences, allowing for handling more complex and varied human feedback.

3. **Theoretical Robustness**
- **Performance Guarantees**: P-OMLE is supported by robust theoretical guarantees. Specifically, the learned policies converge to optimality with a sample complexity proportional to the generalized Eluder dimension of the model and an improved query complexity, transitioning from cubic to linear dependence on $d_R$ (measure of the complexity of the reward function class).

### Instantiations of P-OMLE

The paper proposes several instantiations of P-OMLE for different types of MDPs and reward function classes:

1. **Adversarial Tabular MDPs**
- **Definition**: An adversarial tabular MDP is an MDP where states and actions are discrete, but rewards can be chosen adversarially for each episode, making learning more challenging.
- **Algorithm**: P-OMLE uses an algorithm based on optimistic planning methods.
- **Sample Complexity**: $O(|S|^2 |A| H^3 / \epsilon^2)$.
- **Query Complexity**: Optimized proportionally to the complexity of the state and action space.

2. **Adversarial Linear MDPs**
- **Definition**: An adversarial linear MDP is an MDP where transitions can be modeled by linear functions, but rewards can be chosen adversarially.
- **Algorithm**: P-OMLE uses linear planning methods to estimate reward and transition functions.
- **Sample Complexity**: $O(d H^2 K^{6/7})$.
- **Query Complexity**: Reduced through the use of linear models for reward and transition estimates.

### Extension to K-Element Comparison

To improve P-OMLE's efficiency, the paper proposes an extension to handle K-element comparisons, where the oracle evaluates multiple trajectories simultaneously.

#### K-Element Comparison Operation

1. **Oracle Query**

**Multiple Comparisons**: Instead of comparing two trajectories at a time, the oracle evaluates a set of $K$ trajectories simultaneously.

2. **Updating the Confidence Set**

**Incorporating Multiple Comparisons**: The information obtained from K-element comparisons is used to update the confidence set $B_t$.

#### Advantages of K-Element Comparison

1. **Query Complexity Reduction**

**Query Efficiency**: By querying the oracle with multiple trajectories simultaneously, the algorithm reduces the total number of necessary queries.

2. **Learning Efficiency**

**Data Enrichment**: K-element comparison allows gathering richer and more diverse information with each query, thus improving overall learning efficiency.

#### Associated Theorem

**Theorem 10**: The query complexity with K-element comparison is reduced by a factor of $\min\{K, m\}$, where $m$ is the number of exploratory policies needed. This means the number of oracle queries decreases proportionally to the number of elements compared simultaneously.

### Theoretical Analysis of P-OMLE

The theoretical analysis of P-OMLE shows that this algorithm allows learning an $\epsilon$-optimal policy with reduced sample and query complexity. By combining optimistic planning with preference-based updates, P-OMLE ensures efficient convergence to optimal policies while minimizing oracle queries.

**Sample Complexity**: The method has a sample complexity proportional to the size of the state and action space, as well as the episode length, ensuring that the number of samples needed to achieve $\epsilon$-optimality is manageable.

**Query Complexity**: By using K-element comparisons and optimistic updates, P-OMLE significantly reduces the number of necessary oracle queries, which is crucial for making human feedback-based learning feasible in real-world scenarios.

**Theoretical Robustness**: The theoretical guarantees associated with P-OMLE ensure that this method is reliable and effective for reinforcement learning from human feedback.

### Potential Modifications for UCBVI-BF and GOLF

The UCBVI-BF and GOLF algorithms can also be adapted to better integrate human preferences. By adjusting the update and planning mechanisms to account for human preferences and feedback, these algorithms can

enhance their efficiency and robustness in the context of RLHF. This could include incorporating techniques to reduce query complexity and optimizing confidence sets to better handle complex preferences.

### Differences between P-OMLE and P2R

P2R and P-OMLE primarily differ in their approach to learning and optimization:

**P2R**: Provides a straightforward interface for converting preferences into rewards usable by standard RL algorithms. While effective, P2R may lead to high query complexity due to its "black-box" nature.

**P-OMLE**: Uses a "white-box" modification of the OMLE algorithm, allowing for specialized analysis and significant query complexity reduction. P-OMLE focuses on optimistic planning and confidence set updates based on preferences directly, enhancing efficiency.

## Part 4: Learning from General Preferences

### Introduction

Section 4 of the paper addresses reduction methods to handle general preferences, i.e., preferences that cannot be directly modeled by a utility function. The authors show how these preferences can be approached by reducing them to learning problems in Factorized Independent Markov Games (FI-MGs) or adversarial MDPs. This section also details the use of the OMLE algorithm adapted to these preferences.

For instance, in the context of the motivating example of a household service robot, general preferences might involve more complex tasks that require coordination between multiple agents, such as a robot and a human working together to organize a room. These general preferences can be captured by comparing different collaborative trajectories and learning the optimal policies for such interactions. This section also details the use of the OMLE algorithm adapted to these preferences.


### Reduction to Markov Games

#### Factorized Independent Markov Games (FI-MGs)

A **Factorized Independent Markov Game (FI-MG)** is a zero-sum Markov game (a game where the total gains and losses are always zero) with the following characteristics:

- **State and Action Spaces**: The state space $S$ is factorized into two subspaces $S^{(1)}$ and $S^{(2)}$, and the action space $A$ is factorized into $A^{(1)}$ and $A^{(2)}$.
- **Factorized Transition**: The transition between states is also factorized into two independent components:
$$
P_h(s_{h+1} | s_h, a_h) = P_h(s_{h+1}^{(1)} | s_h^{(1)}, a_h^{(1)}) \times P_h(s_{h+1}^{(2)} | s_h^{(2)}, a_h^{(2)})
$$
where $s_h = (s_h^{(1)}, s_h^{(2)})$ and $a_h = (a_h^{(1)}, a_h^{(2)})$.

- **Restricted Policies**: The policy classes $\Pi^{(1)}$ and $\Pi^{(2)}$ contain policies that map a partial trajectory to a distribution over actions, respectively for subspaces $S^{(1)}$ and $S^{(2)}$.

#### Finding the von Neumann Winner

**von Neumann Winner**:

**Definition**: A policy $\pi^*$ is a von Neumann winner if it maximizes the average utility against any other policy. Formally, in a zero-sum game, a policy $\pi^*$ maximizes the expected gain compared to any other adversarial policy. This means that, for any adversarial policy $\pi'$, the expected gain of following $\pi^*$ is at least as high as the expected gain of following $\pi'$. In other words, $\pi^*$ guarantees the best possible result against the most unfavorable adversary.

**Proposition 11**: Finding a restricted Nash equilibrium in an FI-MG is equivalent to finding a von Neumann winner in the original RLHF problem.

### Learning from Final-State Preferences via Adversarial MDPs

An **Adversarial MDP** is a framework in which the algorithm interacts with a series of MDPs having the same transitions but rewards chosen adversarially for each episode.

#### Formal Definition

**Regret**: Regret is defined as the gap between the expected gain of the algorithm and the best possible gain with a fixed Markov policy:

$$
\text{Regret}_K(A) = \max_{\pi \in \Pi_{\text{Markov}}} \sum_{k=1}^K \mathbb{E}^\pi \left[ \sum_{h=1}^H r_h^k(s_h, a_h) \right] - \sum_{k=1}^K \mathbb{E}^{\pi^k} \left[ \sum_{h=1}^H r_h^k(s_h, a_h) \right]
$$
where $K$ is the number of episodes, $\pi$ is a Markov policy, and $r_h^k$ is the reward function for episode $k$.

#### Algorithm for Adversarial MDPs

**Algorithm 4**: Implementation of learning the von Neumann winner via adversarial MDPs.


- **Steps**:
1. **Creating Independent Copies**: Create two independent copies of the original MDP, each controlled by adversarial MDP algorithms $A^{(1)}$ and $A^{(2)}$.
2. **Bernoulli Rewards**: Provide Bernoulli-type rewards based on the observed final states ($s_H^{(1)}$ and $s_H^{(2)}$).
3. **Updating Policies**: Update policies based on adversarial rewards.

**Theorem 12**: If the adversarial MDP algorithm $A$ has sub-linear regret, this algorithm can find an approximate von Neumann winner using efficient sample and query complexity.

### Learning from Trajectory Preferences via OMLE

For general trajectory-based preferences, the OMLE algorithm is adapted to learn optimal policies in contexts where preferences do not follow a simple utility model.

#### How OMLE Works

1. **Model Class Assumption**:
- It is assumed that the learner has a class of preference models $\mathcal{M}$ and a class of transition functions $\mathcal{P}$.

2. **Optimistic Model-based Learning**:
- OMLE uses an optimistic approach to plan and evaluate trajectories, choosing policies that maximize expected rewards under observed preferences.

**Algorithm 5: OMLE for Trajectory Preferences**


- **Steps**:
1. **Initialization**: Define an initial confidence set for the reward and transition functions.
2. **Optimistic Planning**: Choose the reward and transition functions that maximize expected rewards.
3. **Data Collection**: Execute optimistic policies to collect trajectory data and preference comparisons.
4. **Updating the Confidence Set**: Adjust estimates of the reward and transition functions.

**Theorem 13**: By using OMLE, an approximate von Neumann winner can be learned with a sample complexity of $O(H^2 d_P |Π_{exp}|^2 \ln |P| / \epsilon^2 + H d_R |Π_{exp}| / \epsilon)$.

### Comparison of Methods

1. **FI-MG vs Adversarial MDPs**:
- **FI-MG**: Used to model factorized independent Markov games, where transitions and actions are divided into distinct subspaces.
- **Adversarial MDPs**: Models adversarial interactions in MDPs, with rewards chosen adversarially for each episode, allowing for finding von Neumann winners in more competitive contexts.

2. **OMLE vs Algorithm for Adversarial MDPs**:
- **OMLE**: Uses an optimistic approach to plan and evaluate trajectories, particularly suited for general trajectory-based preferences.
- **Adversarial MDPs**: Uses independent copies of the original MDP and Bernoulli-type rewards, effective for learning optimal policies with sub-linear regret.

## Discussions and Critiques

### Main Contributions

1. **Reduction to Standard RL**:
- The authors show how RLHF problems can be converted into standard RL problems, enabling the use of existing RL algorithms with robustness guarantees. This proves that it is possible to extend RLHF methods to RL without much difficulty, allowing the use of traditional algorithms like P2R, P-OMLE, UCBVI-BF, and GOLF with few modifications. This approach facilitates the integration of human preferences into reinforcement learning processes, making the methods more accessible and applicable to a wider range of problems.
- This simplifies the practical implementation of RLHF in complex environments by reusing proven standard RL algorithms.
- The reduction also allows leveraging recent advancements in standard RL to improve RLHF performance.

2. **General Approaches**:
- The proposed methods offer an elegant and practical approach to leveraging existing RL techniques in the context of human feedback, adapting to different types of preferences, whether utility-based or more general.

3. **Theoretical Guarantees**:
- The paper provides rigorous proofs to demonstrate the effectiveness of the proposed approaches in terms of sample and query complexity. The robust theoretical guarantees ensure that the learned policies converge to optimality with manageable sample and query complexity. This includes results showing that P2R and P-OMLE can learn $\epsilon$-optimal policies with a reasonable number of samples and queries, even in complex environments.

### Intuitions Behind the Theorems and Proofs

1. **Preference-to-Reward Interface (P2R)**:
- P2R converts preferences into rewards usable by standard RL algorithms, reducing learning complexity and benefiting from advancements in traditional RL algorithms.

2. **OMLE and P-OMLE**:
- OMLE and P-OMLE use an optimistic approach to plan and evaluate trajectories, ensuring that the learned policies are close to optimal. This approach maximizes the use of available preference information and minimizes the number of oracle queries.

### Potential Weaknesses

1. **Strong Assumptions**:
- Some assumptions, such as the precise knowledge of the link function $\sigma$, may not be realistic in all contexts. In practice, human preferences can be more complex and difficult to model precisely.

2. **Query Complexity**:
- Although query complexity is reduced, it can still be high for very complex problems or nuanced preferences. This could limit the applicability of the proposed methods in scenarios where query resources are limited.

3. **Lack of Details**:
- The paper could benefit from more details on the theorems and proofs to clarify certain steps and assumptions. A deeper explanation of key concepts and theoretical justifications would strengthen the understanding of the proposed methods.

### Personal Reflection

As an international student with no experience with research papers before my semester at NYCU, it is difficult to provide an in-depth analysis. However, I believe these methods can be a powerful alternative in some RLHF cases, even though they do not address all RLHF challenges. If I were the author, I would design something similar, while seeking to further simplify the assumptions and explore methods to empirically estimate the link function $\sigma$. This could make the approaches more robust and applicable in a wider variety of contexts.

## Conclusion

### Summary of Results

The paper demonstrates that RLHF can be reduced to standard RL problems, simplifying the integration of human preferences into RL algorithms. The main contributions include:

- **Reduction Approaches**: Converting RLHF into standard RL problems or factorized independent Markov games.
- **Effective Algorithms**: Introduction of P2R and P-OMLE.
- **Theoretical Guarantees**: Proofs of convergence and performance.

### Implications and Applications

The proposed methods are applicable to various RL contexts, such as robotics, games, and fine-tuning language models, offering practical and robust solutions for integrating human preferences.

### Final Thoughts

In conclusion, the results of this paper show that RLHF is not inherently more complex than standard RL, opening up new perspectives for the application of RLHF in various fields. By effectively integrating human preferences, these methods offer practical solutions for aligning agent actions with human values and expectations.


## References

1. Yuanhao Wang, Qinghua Liu, Chi Jin. "Is RLHF More Difficult than Standard RL? A Theoretical Perspective."
2. Qinghua Liu, Praneeth Netrapalli, Csaba Szepesvári, Chi Jin. "Optimistic MLE—A Generic Model-based Algorithm for Partially Observable Sequential Decision Making."
3. Huyen Chip. "Reinforcement Learning from Human Feedback." [Huyen Chip Blog](https://huyenchip.com/2023/05/02/rlhf.html). May 2, 2023.