Self-Improving Robust Preference Optimization

## Abstract

Abstract Both online and offline RLHF methods such as PPO and DPO have been extremely successful in aligning AI with human preferences. Despite their success, the existing methods suffer from a fundamental problem that their optimal solution is highly task-dependent (i.e., not robust to out-of-distribution (OOD) tasks). Here we address this challenge by proposing Self-Improving Robust Preference Optimization ( SRPO ), a practical and mathematically principled offline RLHF framework that is completely robust to the changes in the task. The key idea of SRPO is to cast the problem of learning from human preferences as a self-improvement process, which can be mathematically expressed in terms of a min-max objective that aims at joint optimization of self-improvement policy and the generative policy in an adversarial fashion. The solution for this optimization problem is independent of the training task and thus it is robust to its changes.
We then show that this objective can be re-expressed in the form of a non-adversarial offline loss which can be optimized using standard supervised optimization techniques at scale without any need for reward model and online inference.
We show the effectiveness of SRPO in terms of AI Win-Rate (WR) against human (GOLD) completions. In particular, when SRPO is evaluated on the OOD XSUM dataset, it outperforms the celebrated DPO by a clear margin of 𝟏𝟓 % percent 15 \mathbf{15\%} bold_15 % after 5 5 5 5 self-revisions, achieving WR of 𝟗𝟎 % percent 90 \mathbf{90}\% bold_90 % .

## 1 Introduction

Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., ) has rapidly become a standard method to align Large Language Models (LLMs). One of the main practical issues that all the prominent existing RLHF methods (offline or online) (Ouyang et al., ; Rafailov et al., ; Azar et al., ; Zhao et al., ; Ahmadian et al., ) encounter is that their optimal solution heavily depends on the training task in terms of the distribution used to generate the preference data (behavior policy) (Munos et al., ; Azar et al., ). This makes the existing RLHF methods prone to out-of-distribution (OOD) tasks (Li et al., ; Kirk et al., ) where the evaluation distribution is significantly different from that of the behavior policy.
Also, whenever the base/SFT models significantly differ from the behavior policy, the dependency of the RLHF solutions on the behavior policy makes the preference dataset and reward model less useful (Gao et al., ) as RLHF may undo the SFT/pretraining.

To address this challenge, we introduce an alternative approach for aligning LLMs from human preferences based on more principled and robust foundations. Our goal is to find a solution that is robust to the changes in the preference dataset, meaning that changes in the distribution from which the completions are sampled do not affect the final outcome of learning significantly.
To achieve this goal, we exploit the concept of self-improving (Huang et al., ; Bai et al., ) language models. By self-improving LLM we refer to a model capable of enhancing its outputs recursively with each inference iteration. Our Self-Improving Robust Preference Optimization (SRPO) consists of two back-to-back optimization processes:

(Step 1) In-Context Self-Improving Preference Optimization: The core idea is to learn an in-context self-improving model $\pi_{\dagger}$:(^⋆⋆\star⋆^⋆⋆\star⋆$\star$From now on, a generative LLM will be considered as equivalent to a distribution or policy $\pi$ from which we can sample completions $y$ with probability $\pi(y|x)$, where $x$ is the context or prompt.) given an *in-context* completion $y$ and a context $x$, the self-improvement model, $\pi_{\dagger}$, outputs an improved completion $y^{\prime}$ with probability $\pi_{\dagger}(y^{\prime}|y,x)$
from which sampled completions are most preferred to completion $y$ according to the human preference model $p$. As explained later, it turns out that this problem, in its KL-regularized form, can be expressed as a well-defined preference optimization problem and solved analytically. Furthermore, the solution can be estimated through a supervised direct preference optimization scheme similar to the approach used by Rafailov et al. () and Azar et al. ().

(Step 2) Robust Preference Optimization of Generative Model: The next step is to exploit the self-improvement policy learned in the previous step to learn a generative LLM, $\pi$. The key idea here is that the best generative policy can be identified as a policy that generates completions requiring minimal improvement using the optimal self-improvement policy $\pi_{\dagger}$ derived in step $1$.
This goal can be achieved by minimizing the objective of step $1$ with respect to the generative policy for *in-context* completions, $y$.
Similar to step $1$, this problem, in its KL-regularized form, can also be solved analytically in terms of the optimal improvement policy $\pi_{\dagger}$ and the optimal generative policy $\pi$.
More significantly, we show that the solution for steps $1$ and $2$ can be estimated jointly through a single supervised direct preference optimization scheme using only a dataset of annotated pair-wise completions.
Thus, one can solve both for the self-improvement policy $\pi_{\dagger}$ and $\pi$ by minimizing the supervised learning objective of SRPO.
Unlike existing RLHF methods, this solution is independent of the behavior policy and is therefore robust to its changes.

As using the self-improvement model in SRPO is a significant departure from the existing paradigm for RLHF, we provide a high-level motivation for it in Sec. 2.
We then formalize our objective for SRPO in Sec. 3, allowing for the joint optimization of both $\pi_{\dagger}$ and $\pi$ by optimizing an adversarial min-max objective.
In Sec. 4 we present our main algorithmic/mathematical contribution: we prove that the preference probability $p$ can be expressed in terms of the log-likelihoods of the optimal self-improvement policy $\pi_{\dagger}^{*}$ and the log-likelihoods of the optimal robust generative policy $\pi^{*}$. This theoretical finding is the key result for SRPO: solving this system of equations through least-squares regression provides us with the practical supervised SRPO objective that solves for both policy and robust generative policy through a single supervised objective without any need for reward model or online inference. Our key theoretical finding is similar to the main result of DPO (Rafailov et al., ) in that both express preference probabilities in terms of the optimal policy.
However, DPO result only holds when preference probabilities conform to Bradly-Terry model (Bradley & Terry, ), whereas our key result is general as it holds across all preference models.
In Sec. 5 we further illustrate our argument on the robustness of SRPO by providing an in-depth analysis of the solution of SRPO and other direct preference optimization methods. We also showcase/analyze the robustness of SRPO on a simple synthetic example. Finally in Sec. 7 we conduct large-scale experiments on training LLMs with SRPO both on in-distribution and OOD summarization tasks and we compare the results with those of standard baselines.

## 2 Learning Self-Improvement Policy Through Preference Optimization

The goal of this section is to provide some motivation on why learning self-improvement models through preference optimization can be useful for learning a good policy. First, we start by considering a more fundamental question:

*What is the best use of human preference data?*

To answer this question, we notice that human preferences provide information on the relation between more-preferred and less-preferred completions.
This information can be used to improve the less-preferred completions towards the more preferred completions.
In other words, we can learn a model of alignment mechanics, the rules on how to improve the completions to better match human preferences.
This is arguably a more natural learning task, considering human preferences, than directly learning the highest preferred completion by humans, which is the goal of standard RLHF methods.
Note that the highest preferred answer is very unlikely to be in our completion dataset, especially when the space of possibilities is the entirety of human language, and in particular when the completions are generated from some LLM, which is subpar to humans. Instead, it is more natural to learn that given a query $x$ and a completion $y$ what would be the improved completion upon $y$, i.e., learn the model that aims at improving the output of LLM through a self-improvement process. In this case, if our model has captured the underlying rules of human preference, then it can use that to improve the subpar completions towards the best completions.

The existing self-improvement-based pipelines mostly rely on the in-context learning ability of pretrained/SFT LLMs (Bai et al., ; Wei et al., ).
In the following, we show how the self-improvement policy, $\pi_{\dagger}$, can be *optimally* trained alongside the generative policy, $\pi$, using pair-wise preferences.

## 3 SRPO Objective

We start by introducing some notations required for establishing our theoretical results.

Notations. Let $x$ and $y$ denote a context and a completion drawn from the space of all possible contexts $\mathcal{X}$ and all possible completions $\mathcal{Y}$, respectively. The large language model (LLM) is represented by the probability distribution (policy) $\pi$ where $\pi(y|x)$ denotes the probability of the completion $y$ given the context $x$. In the remainder of this article, we consider three variants of this base LLM, the trainable model $\pi_{\text{train}}$ (for which we use the short-hand notation $\pi$), the reference model $\pi_{\text{ref}}$ and the behavior model $\mu$ from which the completions in pair-wise preference dataset is sampled.

We also introduce the self-improvement $\pi(y^{\prime}|y,x)$ as a model that using a context $x$ and in-context completion (thought) $y$ aims at improving $y$ to a better completion $y^{\prime}$.
Similar to base LLM, we can define a reference model $\pi_{\text{ref}}(y^{\prime}|y,x)$ also for the self-improvement model.
Let
$\mathcal{D}=\{x,y_{1},y_{2}\}$
be a dataset of contexts and completions where $y_{1}$ and $y_{2}$ are drawn independently from $\mu(\cdot|x)$. We then present every pair $y_{1},y_{2}$ to human annotators who express preferences for one of the completions, denoted as $y_{w}\succ y_{l}$ where $y_{w}$ and $y_{l}$ denote the preferred and dis-preferred actions amongst $\{y_{1},y_{2}\}$ respectively. We then write true human preference $p(y_{1}\succ y_{2}|x)$ the probability of $y_{1}$ being preferred to $y_{2}$ knowing the context $x$. The probability comes from the randomness of the choice of the human we ask for their preference. So $p(y_{1}\succ y_{2}|x)=\mathop{\mathbb{E}}_{h}[\mathbb{I}\{h\mbox{ prefers }y_{
1}\mbox{ to }y_{2}\mbox{ given }x\}]$, where the expectation is over humans $h$.

Consider a reference policy $\pi_{\text{ref}}$, and a real positive regularisation parameter $\beta\in\mathbb{R}^{*}_{+}$. Then, we define the Self-Improving Robust Preference Optimisation objective (SRPO) for every context $x$ as

$$ $\displaystyle J^{*}(x)=\min_{\pi}\max_{\pi_{\dagger}}\mathop{\mathbb{E}}_{ \begin{subarray}{c}y_{1}\sim\pi(.|x)\\ y_{2}\sim\pi_{\dagger}(\cdot|y_{1},x)\end{subarray}}\bigg{[}p(y_{2}\succ y_{1} |x)-\beta D_{\text{KL}}(\pi_{\dagger}||\pi_{\text{ref}}|y_{1},x)+\beta D_{ \text{KL}}(\pi||\pi_{\text{ref}}|x)\bigg{]},$ (1) $$

with the KL-regularization terms are defined as:
$D_{\text{KL}}(\pi_{\dagger}||\pi_{\text{ref}}|y_{1},x)=\text{KL}(\pi_{\dagger}
(\cdot|y_{1},x)||\pi_{\text{ref}}(\cdot|y_{1},x))$ and $D_{\text{KL}}(\pi||\pi_{\text{ref}}|x)=\text{KL}(\pi(\cdot|x)||\pi_{\text{ref}
}(\cdot|x))$.

In nutshell, this objective aims at (i) finding the best self-improvement policy $\pi^{*}_{\dagger}$ that takes every $y_{1}\sim\pi$ and improves it optimally with respect to the preference distribution $p$, i.e., the improved policy is most preferred to $y_{1}$, while keeping $\pi^{*}_{\dagger}$ close to the reference policy $\pi_{\text{ref}}$, (ii) minimizing the same objective to find the best (robust) policy $\pi^{*}$ for which the generated completions can be only minimally improved by the optimal self-improvement model $\pi^{*}_{\dagger}$. the min-max nature of this objective guarantees that self-improvement is effective for all policies (close to $\pi_{\text{ref}}$) as we are optimizing $\pi_{\dagger}$ in the worst-case scenario.

## 4 Offline Solution for Optimizing SRPO Objective

The optimization problem of Eq. (1) is a non-trivial optimization problem that often requires solving a two-stage adversarial optimization problem through the game-theoretic approaches, which are often challenging and difficult to scale up, (see e.g., Munos et al., ; Rosset et al., ; Calandriello et al., , for how we can use game-theoretic approaches/objectives to train LLMs). Here, inspired by Rafailov et al. (); Azar et al. (), we aim at casting this complex optimization objective as a standard supervised learning problem that can be solved at scale given an offline pairwise preference dataset.
To derive a practical algorithm for SRPO we first notice that the inner-maximization in the objective function of Eq. (1) can be solved in closed form as follows:

$$ $\pi^{*}_{\dagger}(y_{2}|y_{1},x)=\frac{\exp\left(\frac{p(y_{2}\succ y_{1}|x)}{ \beta}\right)\pi_{\text{ref}}(y_{2}|y_{1},x)}{Z^{*}(y_{1},x)},$ (2) $$

where $Z^{*}(y_{1},x)$ is the normalization factor. One can easily show that by plugging $\pi^{*}_{\dagger}$ in the objective function of Eq. (1) we obtain:

$$ $\displaystyle J^{*}(x)=\min_{\pi}\mathop{\mathbb{E}}_{y_{1}\sim\pi(.|x)}\left[ \beta(\log(Z^{*}(y_{1},x))+D_{\text{KL}}(\pi||\pi_{\text{ref}}|x))\right].$ (3) $$

Now by solving Eq. (2) with respect to $p(y_{2}\succ y_{1}|x)$ we obtain

$$ $p(y_{2}\succ y_{1}|x)=\beta(\log(\pi^{*}_{\dagger}(y_{2}|y_{1},x))-\log(\pi_{ \text{ref}}(y_{2}|y_{1},x))+\beta\log(Z^{*}(y_{1},x))).$ (4) $$

### 4.1 Optimizing the Self-Improvement Policy π † subscript 𝜋 † \pi_{\dagger} italic_π start_POSTSUBSCRIPT † end_POSTSUBSCRIPT

We notice that using the convention $p(y_{1}\succ y_{1}|x)=\frac{1}{2}$ Eq. (4) implies

$$ $\frac{1}{2}=\beta(\log(\pi^{*}_{\dagger}(y_{1}|y_{1},x))-\log(\pi_{\text{ref}} (y_{1}|y_{1},x)))+\beta\log(Z^{*}(y_{1},x)).$ (5) $$

Now by subtracting Eq. (4) from Eq. (5) we derive

$$ $p(y_{2}\succ y_{1}|x)=\frac{1}{2}+\beta\left[\log\left(\frac{\pi^{*}_{\dagger} (y_{2}|y_{1},x)}{\pi_{\text{ref}}(y_{2}|y_{1},x))}\right)-\log\left(\frac{\pi^ {*}_{\dagger}(y_{1}|y_{1},x)}{\pi_{\text{ref}}(y_{1}|y_{1},x)}\right)\right].$ (6) $$

This is our first key result that express preference $p(y_{2}\succ y_{1}|x)$ in terms of the optimal self-improvement policy $\pi^{*}_{\dagger}$. So we can enforce this equation for all $y_{1}$ and $y_{2}$ through following $\ell_{2}$ loss:

$$ $L_{\dagger}(\pi_{\dagger})=\mathop{\mathbb{E}}_{\begin{subarray}{c}y_{1},y_{2} \sim\mu(\cdot|x)\\ x\sim\rho\end{subarray}}\left[p(y_{2}\succ y_{1}|x)-\frac{1}{2}-\beta\left[ \log\left(\frac{\pi_{\dagger}(y_{2}|y_{1},x)}{\pi_{\text{ref}}(y_{2}|y_{1},x)) }\right)-\log\left(\frac{\pi_{\dagger}(y_{1}|y_{1},x)}{\pi_{\text{ref}}(y_{1}| y_{1},x)}\right)\right]\right]^{2}.$ (7) $$

Using the standard properties of $\ell_{2}$-norm to replace $P(y_{2}\succ y_{1}|x)$ with $\mathbf{1}(y_{1}\succ y_{2}|x)$, as $P(y_{2}\succ y_{1}|x)=\mathop{\mathbb{E}}(\mathbf{1}(y_{1}\succ y_{2}|x))$, in the objective of Eq. (15) allows us to derive the following sample loss for the improvement model:

$$ $\displaystyle\widehat{L}_{\dagger}(\pi_{\dagger})=$ $\displaystyle\mathop{\mathbb{E}}_{\begin{subarray}{c}(y_{l},y_{w},x)\sim \mathcal{D}\end{subarray}}\left[\frac{1}{2}-\beta\left[\log\left(\frac{\pi_{ \dagger}(y_{w}|y_{l},x)}{\pi_{\text{ref}}(y_{w}|y_{l},x))}\right)-\log\left( \frac{\pi_{\dagger}(y_{l}|y_{l},x)}{\pi_{\text{ref}}(y_{l}|y_{l},x)}\right) \right]\right]^{2}$ (8) $\displaystyle+\mathop{\mathbb{E}}_{\begin{subarray}{c}(y_{l},y_{w},x)\sim \mathcal{D}\end{subarray}}\left[\frac{1}{2}-\beta\left[\log\left(\frac{\pi_{ \dagger}(y_{w}|y_{w},x)}{\pi_{\text{ref}}(y_{w}|y_{w},x))}\right)-\log\left( \frac{\pi_{\dagger}(y_{l}|y_{w},x)}{\pi_{\text{ref}}(y_{l}|y_{w},x)}\right) \right]\right]^{2}$ $$

### 4.2 Optimizing the Robust Generative Policy π 𝜋 \pi italic_π

In this section we want to derive an offline objective to optimize for generative model $\pi$ as well as improvement model $\pi_{\dagger}$. We start by collecting terms in Eq. (5) we obtain

$$ $\beta\log(Z^{*}(y_{1},x))=\beta(\log(\pi_{\text{ref}}(y_{1}|y_{1},x))-\log(\pi ^{*}_{\dagger}(y_{1}|y_{1},x)))-\frac{1}{2}.$ $$

Thus the objective of Eq. (3) can be expressed in terms of $\log(\pi^{*}_{\dagger}(y_{1}|y_{1},x))$ (up to an additive and multiplicative constant) as follows:

$$ $\displaystyle J^{*}(x)\propto\min_{\pi}\mathop{\mathbb{E}}_{y_{1}\sim\pi(.|x)} \left[\log\left(\frac{\pi_{\text{ref}}(y_{1}|y_{1},x)}{\pi^{*}_{\dagger}(y_{1} |y_{1},x)}\right)+D_{\text{KL}}(\pi||\pi_{\text{ref}}|x))\right]\,.$ (9) $$

Solving this objective with respect to $\pi$ we obtain:

$$ $\pi^{*}(y|x)=\frac{\frac{\pi_{\text{ref}}(y|x)}{\pi_{\text{ref}}(y|y,x)}\pi^{* }_{\dagger}(y|y,x)}{Z^{*}(x)}$ (10) $$

where $Z^{*}(x)$ is the normalization factor. Again by taking the logarithm from both side we obtain

$$ $\log(\pi^{*}(y|x))=\log\left(\frac{\pi_{\text{ref}(y|x)}}{\pi_{\text{ref}}(y|y ,x)}\pi^{*}_{\dagger}(y|y,x)\right)-\log(Z^{*}(x)).$ $$

Now by collecting terms in Eq. (4) and solving for $\log(\pi^{*}_{\dagger}(y_{2}|y_{1},x)))$ we obtain

$$ $\log(\pi^{*}_{\dagger}(y_{2}|y_{1},x))=\frac{p(y_{2}\succ y_{1}|x)}{\beta}- \log(Z^{*}(y_{1},x)))-\log(\pi_{\text{ref}}(y_{2}|y_{1},x))$ (11) $$

Now by plugging Eq. (2) into Eq. (10) we deduce

$$ $\displaystyle\pi^{*}(y|x)=\frac{\exp\left(-\log(Z^{*}(y,x))\right)\pi_{\text{ ref}}(y|x)}{Z^{*}(x)}.$ (12) $$

Solving this equation with respect to $\log(Z^{*}(y,x))$ implies

$$ $\displaystyle\log(Z^{*}(y,x))=\log(\pi_{\text{ref}}(y|x))-\log(\pi^{*}(y|x))- \log(Z^{*}(x)).$ (13) $$

Combining Eq. (11) and Eq. (13) we have for any $y_{1}$ and $y_{2}$

$$ $\displaystyle\frac{p(y_{2}\succ y_{1}|x)}{\beta}-\log\left(\frac{\pi^{*}_{ \dagger}(y_{2}|y_{1},x)}{\pi_{\text{ref}}(y_{2}|y_{1},x)}\right)$ $\displaystyle=\log\left(\frac{\pi_{\text{ref}}(y_{1}|x))}{\pi^{*}(y_{1}|x)} \right)-\log(Z^{*}(x)),$ $\displaystyle\frac{p(y_{1}\succ y_{2}|x)}{\beta}-\log\left(\frac{\pi^{*}_{ \dagger}(y_{1}|y_{2},x)}{\pi_{\text{ref}}(y_{1}|y_{2},x)}\right)$ $\displaystyle=\log\left(\frac{\pi_{\text{ref}}(y_{2}|x))}{\pi^{*}(y_{2}|x)} \right)-\log(Z^{*}(x)).$ $$

Subtracting these two Equations and collecting terms leads to our *key result* in which we express the preference $p$ in terms of the optimal self-improvement policy $\pi^{*}_{\dagger}$ and the optimal robust policy $\pi^{*}$.

$$ $\displaystyle p(y_{2}\succ y_{1}|x)=$ $\displaystyle\frac{1}{2}+\frac{\beta}{2}\Bigg{[}\log\left(\frac{\pi^{*}_{ \dagger}(y_{2}|y_{1},x)}{\pi_{\text{ref}}(y_{2}|y_{1},x)}\right)-\log\left( \frac{\pi^{*}(y_{1}|x)}{\pi_{\text{ref}}(y_{1}|x))}\right)$ (14) $\displaystyle-\left(\log\left(\frac{\pi^{*}_{\dagger}(y_{1}|y_{2},x)}{\pi_{ \text{ref}}(y_{1}|y_{2},x)}\right)-\log\left(\frac{\pi^{*}(y_{2}|x)}{\pi_{ \text{ref}}(y_{2}|x))}\right)\right)\Bigg{]}.$ $$

###### Remark 1 .

One may notice the similarity of this result and Equation 6 of DPO paper (Rafailov et al., ). Both results express $p(y_{2}\succ y_{1}|x)$ in terms of the optimal policy $\pi^{*}$. However the result of DPO only holds under the assumption that $p$ conforms to the Bradly-Terry model, whereas our result is general and holds for all $p$.

To optimize for $\pi$ and $\pi_{\dagger}$ using (14) we enforce this equation for all $y_{1}$ and $y_{2}$ through following $\ell_{2}$ loss:

$$ $\displaystyle L(\pi,\pi_{\dagger})=$ $\displaystyle\mathop{\mathbb{E}}_{\begin{subarray}{c}y_{1},y_{2}\sim\mu(\cdot| x)\\ x\sim\rho\end{subarray}}\Bigg{[}p(y_{2}\succ y_{1}|x)-\frac{1}{2}-\frac{\beta} {2}\bigg{[}\log\left(\frac{\pi_{\dagger}(y_{2}|y_{1},x)}{\pi_{\text{ref}}(y_{2 }|y_{1},x)}\right)-\log\left(\frac{\pi(y_{1}|x)}{\pi_{\text{ref}}(y_{1}|x))}\right)$ (15) $\displaystyle-\left(\log\left(\frac{\pi_{\dagger}(y_{1}|y_{2},x)}{\pi_{\text{ ref}}(y_{1}|y_{2},x)}\right)-\log\left(\frac{\pi(y_{2}|x)}{\pi_{\text{ref}}(y_ {2}|x))}\right)\right)\bigg{]}\Bigg{]}^{2}.$ $$

Using the standard properties of $\ell_{2}$-norm to replace $P(y_{2}\succ y_{1}|x)$ with $\mathbf{1}(y_{1}\succ y_{2}|x)$, as $P(y_{2}\succ y_{1}|x)=\mathop{\mathbb{E}}(\mathbf{1}(y_{1}\succ y_{2}|x))$, in the loss of Eq. (15) allows us to derive the following sample loss for SRPO:

$$ $\displaystyle\widehat{L}(\pi,\pi_{\dagger})=$ $\displaystyle\mathop{\mathbb{E}}_{(y_{l},y_{w},x)\sim\mathcal{D}}\Bigg{[}\beta \bigg{[}\log\left(\frac{\pi_{\dagger}(y_{w}|y_{l},x)}{\pi_{\text{ref}}(y_{w}|y _{l},x)}\right)+\log\left(\frac{\pi(y_{w}|x)}{\pi_{\text{ref}}(y_{w}|x))}\right)$ (16) $\displaystyle-\left(\log\left(\frac{\pi_{\dagger}(y_{l}|y_{w},x)}{\pi_{\text{ ref}}(y_{l}|y_{w},x)}\right)+\log\left(\frac{\pi(y_{l}|x)}{\pi_{\text{ref}}(y_ {l}|x))}\right)\right)\bigg{]}-1\Bigg{]}^{2}.$ $$

###### Remark 1 .

### 4.3 Full (Combination) Loss for SRPO

We note that both (16) and (8) are aligned in the sense that both losses are optimizing the same objective of (1). So one can use the convex combination of these two losses as the full loss of SRPO. Also one can use a single LLM (denoted by $\pi$) to represent both $\pi$ and $\pi_{\dagger}$ by exploiting the in-context learning power of LLMs (Brown et al., ) such that $\pi_{\dagger}(y^{\prime}|y,x)=\pi(y^{\prime}|y,x)$. So for every $\alpha\in[0,1]$ we define the full sample loss of SRPO as follows

$$ $\displaystyle\widehat{L}_{\alpha}(\pi)=$ $\displaystyle(1-\alpha)\widehat{L}(\pi,\pi_{\dagger}=\pi)+\alpha\widehat{L}_{ \dagger}(\pi_{\dagger}=\pi).$ (17) $$

The following pseudo-code can be used to train the LLM policy using SRPO objective:

Figure: Algorithm 1 Sampled SRPO

## 5 Robustness of SRPO

We provide an in depth comparison between SRPO and prior work on direct preference optimization in terms of their robustness to the behavior policy $\mu$. In particular we consider as a point of reference DPO (PPO (^⋆⋆\star⋆^⋆⋆\star⋆$\star$As it is shown by Azar et al. () the optimal solutions of DPO and PPO are identical. So in the remainder of this section we only focus on DPO.)) and IPO for which we have a good understanding of the underlying mathematical foundation.

In the case of both IPO and DPO the analytical solution is already well-established and analyzed for both algorithms (Azar et al., ; Rafailov et al., ; Tang et al., ). In particular the optimal solution for both IPO and DPO can be expressed explicitly in terms of the soft-max of the expected preference as follows (Azar et al., ):

$$ $\pi^{*}(y|x)=\frac{\exp\left(\frac{\mathop{\mathbb{E}}_{y^{\prime}\sim\mu( \cdot|x)}(\Psi(p(y\succ y^{\prime}|x)))}{\beta}\right)\pi_{\text{ref}}(y|x)}{Z ^{*}(x)},$ (18) $$

with the choice of $\Psi=I(\cdot)$ and $\Psi=\sigma^{-1}(\cdot)$ for IPO and DPO, respectively, where $\sigma^{-1}$ denotes the inverse-sigmoid (logit function). Thus, based on (18), we can see that the solution for both IPO and DPO has strong dependency on $\mu$ in the form of expected preference under the distribution $\mu$. Thus it may not be robust to changes in $\mu$. This dependency on $\mu$ can be especially problematic when we evaluate the model on out-of-distribution tasks where the desired behavior is very different from $\mu$ and the expected preference under the distribution $\mu$ is not a good measure of performance. SRPO solution on the other hand has no dependency on the behavior policy $\mu$: from (2) we observe that the optimal self-improvement policy $\pi^{*}_{\dagger}$ is independent of $\mu$ and, unlike DPO and IPO cases, is expressed in terms of softmax of $P(y_{2}\succ y_{1}|x)$ for any pair of completions $(y_{1},y_{2})$. Also the 0 0-revision policy $\pi^{*}$ is also completely independent of $\mu$ as it is evident from (10) (i.e., it is proportional to $\pi^{*}(y|y,x)$ which itself is independent of $\mu$). Thus, from a mathematical point of view, SRPO provides a robust solution for the problem of direct preference optimization that does not depend on the behavior policy $\mu$.

To illustrate further the differences between SRPO and DPO/IPO with regard to robustness to $\mu$ we consider a simple bandit example. For simplicity we assume there is no context $x$, i.e., we are in a standard bandit setting. Consider the simple case where we have 3 actions (completions) $y_{0}$, $y_{1}$ and $y_{2}$, for which the preference model is given as follows:

$$ $\displaystyle P=\begin{pmatrix}0.5&0.99&0.3\\ 0.01&0.5&0.25\\ 0.7&0.75&0.5\end{pmatrix},$ $$

in which $p(y_{i}\succ y_{j})=P_{ij}$. In this case $y_{2}$ is clearly the preferred outcome as it dominates both $y_{1}$ and $y_{0}$ by probability larger than $0.5$. On the other hand, if we only consider preference with respect to $y_{1}$ then arm $y_{0}$ is a better outcome as it is preferred to $y_{1}$ with higher probability than $y_{2}$. Therefore, if a preference optimization method is robust to changes in $\mu$ one might expect that the optimal solution should be independent of the frequency of $y_{2}$ in the preference dataset (i.e., $\mu(y_{2})$).

To test this hypothesis we consider two synthetic dataset of actions generated from distributions $\mu_{0}$ and $\mu_{1}$: We set $\mu_{0}$ to be a uniform behavior policy ($\mu_{0}(y_{0})=\mu_{0}(y_{1})=\mu_{0}(y_{2})=\frac{1}{3}$) and $\mu_{1}$ skewed towards $y_{1}$ ($\mu_{1}(y_{1})=0.7,\mu_{1}(y_{0})=\mu_{1}(y_{2})=0.15$). We then generate a dataset of $10000$ pairs from $\mu_{0}$ and $\mu_{1}$ and rate them according to the preference model $p$ (for any pair $(y_{1},y_{2})$ we assign the preference by sampling from $p(y_{1}\succ y_{2})$, that is $y_{1}$ is preferred to $y_{2}$ with probability $p(y_{1}\succ y_{2})$). This provides us with two dataset of rated completions $\mathcal{D}_{0}$ and $\mathcal{D}_{1}$ for $\mu_{0}$ and $\mu_{1}$. We then use these two datasets to train the policy $\pi$ using SRPO, DPO and IPO using a simple Adam optimizer. In the case of IPO and DPO we optimize only the 0 0-revision policy $\pi(y)$ where as for SRPO we also optimize the self-improvement policy $\pi(y|y^{\prime})$ as well. We set the regularization constant $\beta$ for all methods to $1$. We consider a uniform distribution $\pi_{\text{ref}}(y)=1/3$ for all algorithms and all $y$s. In the case of SRPO we set the self-improvement reference policy $\pi_{\text{ref}}(y|y^{\prime})=1/3$ for all $y$ and $y^{\prime}$. Also for SRPO we set the combination coefficient $\alpha=0$ for simplicity.

We observe that in the case of using uniform $\mu_{0}$ as a behavior policy all methods do the right thing and their policies converge to solutions in which $y_{2}$ dominates $y_{1}$ and $y_{0}$ (Fig. 1(a)). However, when we use the behavior policy $\mu_{1}$ which is skewed towards $y_{1}$, both DPO and IPO converge to a solution in which $y_{0}$ dominates $y_{1}$ and $y_{2}$, while the policy of SRPO remains intact (Fig. 1(b)). Notice that the SRPO policy is slightly different in both cases. This is to be expected, we are in a finite data setting, and the sampling distribution will have some influence on the empirical preference model (defining the empirical solution of SRPO).

Figure: (a) SRPO vs IPO and DPO for uniform behavior $\mu_{0}$.
Refer to caption: x1.png

## 6 Related works

Our work lies in offline preference optimization, a vivid area of research since the introduction of DPO (Rafailov et al., ). Some of the core concepts of this research topic was generalized and formalized by Azar et al. (). In particular they characterized the underlying optimal solution for a generic preference optimization objective and introduced IPO for addressing some of the related shortcomings of DPO. SLiC-HF (Zhao et al., ) was introduced around the same time, from a less RL-centric point of view. All these approaches have been abstracted later by Tang et al. (), the general recipe being to build a contrastive loss function from a convex classification function and to make use of the analytical solution of the RL problem to learn directly the policy. A common underlying assumption is that the related RL problem is KL-regularized. This has been generalized to more general $f$-divergences by Wang et al. (). These are just a few among many works on direct alignment. However, they all share the fact of not considering self-improvement policies, contrary to SRPO. This has a strong incidence on the related solution concept, making SRPO the sole direct alignment approach being robust to the sampling distribution $\mu$, as showcased in Sec. 5.

Offline preference optimization was introduced as an alternative to more classic RLHF approaches, such as PPO (Schulman et al., ; Ouyang et al., ) or more generally policy-gradient-based approaches (Roit et al., ; Ahmadian et al., ). These methods require training a *reward model* on a preference dataset, usually with a Bradley-Terry model (Bradley & Terry, ). The reward model is then used to fine-tune the LLM via online RL, requiring many generations from the model. This reward model shares the common issue of DPO and other direct preference alignment methods, it is dependent on the sampling distribution $\mu$ used for constructing the preference dataset, contrary to SRPO. Moreover, classic RLHF is online, while SRPO is offline and thus more easily scalable.

Some similarities also exist between SRPO and Nash-MD (Munos et al., ). Indeed, if in Eq. 1 we replace the self-improvement policy $\pi_{\dagger}(\cdot|y,x)$ by a classic policy $\pi(\cdot|x)$, then we obtain the saddle-point optimization problem that Nash-MD solves. However, considering a self-improvement policy is a core contribution of our work, and it is not anecdotal. From a technical viewpoint, this is critical for simplifying the minimax problem of Eq. (1) and obtaining a simple offline optimization problem. NashMD on the other hand adapts algorithms from the game-theory literature and can only be solved online with all the stability issues of online methods and large inference costs. From practical point of view self-improvement provides a boost in performance by refining the original generations of LLM. The feature that Nash-MD is missing. Finally, even though the Nash equilibrium of Nash-MD does not depend on the sampling distribution $\mu$, it relies on a learned reward function, with the possible associated caveats mentioned earlier, which is not the case of SRPO.

Our work is also obviously related to the concept of chain of thoughts (Wei et al., ; Yao et al., ), self-improvement (Huang et al., ) and self-refining LLMs (Madaan et al., ). However, it is very often used as a way of prompting a model to obtain better results, and less often as a component of a learning paradigm (Liu et al., ; Huang et al., ). To our best knowledge, we propose the first approach that combines training self-improvement LLMs and offline preference optimization through a single supervised objective, moreover in a theoretically grounded manner and showing the robustness to $\mu$.

## 7 Experiments

Setup. In our experiments, we consider the offline direct preference optimization setup to learn from human preferences  (Rafailov et al., ). In the offline setting, the goal is to train the LLM policy directly from a dataset $\mathcal{D}$ of pairwise completions $(y_{l},y_{w})$ sampled from a behavior policy $\mu$ and annotated by human raters without using a reward model or online inference/RL. We empirically test the effectiveness of SRPO against two offline preference learning methods, namely Direct Preference Optimisation (DPO) (Rafailov et al., ) and Identity Preference Optimisation (IPO) (Azar et al., ) as baselines. We make this choice since both these baselines, like SRPO, are mathematically well-grounded offline methods. Also, they have been widely used by the AI community in solving different language tasks (Tunstall et al., ; Wallace et al., ; Yuan et al., ; Pang et al., ; Lin et al., ).

Implementation details. SRPO trains simultaneously both the standard generative policy $\pi$ and the self-improvement policy $\pi_{\dagger}$ used for revising the completions of models through a single optimization process. As explained earlier we only use a single LLM to represent both $\pi$ and $\pi_{\dagger}$ (denoted simply by $\pi$). To get the best completions from SRPO we first generate completions in the 0-revision (0-rev.) model and then we improve these completions with the self-improvement model. We call the revised outputs 1-revision (1-rev.) completions. We can iterate on the improvement process $N$ times to get $N$-revision ($N$-rev.) completions. We report results from 0-rev. to 5-rev. cases. For IPO and DPO we also report results on 0-rev. and 1-rev. For revising the completions we use IPO and DPO in in-context learning mode with the 0-rev. completions used as contexts. In the case of DPO we use the same loss and hyper-parameters used by (Rafailov et al., ). For IPO since the original paper hasn’t provided the hyper-parameters we used a set of hyper-parameters (i.e., learning rate and regularization constant $\beta$) from the range of hyper-parameters that was working. Furthermore we noticed that the performance of IPO was not affected significantly by the choice of these hyper-parameters. So no significant gain is expected by hyper-parameter tuning.

Datasets. We use the Reddit TL;DR Summarization dataset (Stiennon et al., ) as the main dataset for our experiments(^⋆⋆\star⋆^⋆⋆\star⋆$\star$[https://github.com/openai/summarize-from-feedback](https://github.com/openai/summarize-from-feedback)). For training, there are 116k human-written instruction following examples with reference completions (SFT split) while there are 93k human-annotated preference pairs (Preference split). We also use the XSum dataset test split(^⋆⋆\star⋆^⋆⋆\star⋆$\star$[https://huggingface.co/datasets/csebuetnlp/xlsum](https://huggingface.co/datasets/csebuetnlp/xlsum)) (Narayan et al., ), which contains 11.5k total test examples to measure Out-of-Distribution (OOD) generalization.

Model Setup. We use LLaMA-7B as base model (Touvron et al., ) and a single $8\times\text{NVIDIA H}100$ node to conduct all LLaMA-based experiments. We first supervise fine-tune the model on the SFT split of the TL;DR dataset, before preference training and use the same $\pi_{\text{ref}}$ for all preference training experiments. Below are details on the training recipe for the SFT and preference training stages.

Supervised-fine Tuning. In the SFT stage, we train for 2 epochs, using the AdamW optimizer (Loshchilov & Hutter, ), with $\beta_{1}=0.9$ and $\beta_{2}=0.999$, and $0.1$ weight-decay. We use a cosine decay learning rate (Loshchilov & Hutter, ) with a peak value of $2\times 10^{-5}$ and $3\%$ of all steps being warm-up steps. We use an effective batch-size of 64.

Preference Training. We use our SFT model as $\pi_{\text{ref}}$ and we initialize $\pi$ with $\pi_{\text{ref}}$. All models were trained for 5 epochs on the TL;DR preference split using the same optimization setting of the AdamW optimizer as in the SFT stage with $150$ warmup steps, and an effective batch-size of $128$. To fine-tune the models, we use the default PEFT settings in the TRL library(^⋆⋆\star⋆^⋆⋆\star⋆$\star$[https://github.com/huggingface/trl](https://github.com/huggingface/trl)), using LoRA
(Hu et al., ) with a rank of 16 and an alpha of 32.
For SRPO and IPO, we used $\beta=0.01$ with a learning rate of $2\times 10^{-6}$. For DPO following Rafailov et al. (), we used the common $\beta=0.1$ with a learning rate of $1\times 10^{-6}$ and a constant learning rate schedule.

Evaluation. We use win rates as computed by gpt-4-0613 (OpenAI, ) using the Alpacafarm framework (Dubois et al., ), as the main means for evaluation. We measure performance on both in-distribution and OOD examples at test time in the following manner: For the former, we compute win rates against gold reference completions from the test set of the TL;DR SFT split. For the latter, we measure win rates against gold completions from the test set of the XSum dataset. In both settings, we use the first 1,024 samples from each of the test sets. To estimate the win rate more accurately with confidence intervals, we bootstrap 20 times with replacement from the 1,024 samples, each time using a sample size of 512.
To sample from the self-improvement policy, we first sample $y$ from $\pi(\cdot|x)$. Then, using the same policy, we condition on $y$ to sample from the self-improvement policy, that is $y^{\prime}\sim\pi(\cdot|y,x)$. We refer to generations from $\pi(\cdot|x)$ as 0-revision (0-rev.) and generations from $y^{\prime}\sim\pi(\cdot|y,x)$ as 1-revision (1-rev.) (Bai et al., ). For $N$-revision, we apply the same procedure, conditioning on the sample from the ${N\textrm{-}1}^{\text{th}}$-revision.

TL;DR Results. We test our models on the test set split of the TL;DR dataset in Fig 2 (left panel). For every model, we generate 0-rev. and then use these generations to revise our completions recursively from 1-rev. to 5-rev. using the self-improvement model, and measure the models’ win rate against the human-written gold reference summaries.

We observe that in the case of in-distribution TL;DR SRPO 4-rev. generates high-quality summaries with the highest win rate against the gold summaries, compared to the win-rates of of the baseline methods, as well as other variants of SRPO 0-rev.
Furthermore, we observe that SRPO self-improvement process manages to consistently improve upon SRPO 0-rev. . However, DPO and IPO fail to generate an improved sample through the self-improvement step.

Out-of-distribution (OOD) Results.
To assess robustness in an OOD setting, we test SRPO models trained with TL;DR preference dataset on the XSum test split in Fig. 2 (right panel) (Narayan et al., ). As in the TL;DR case, we observe that self-improvement is effective in improving the performance of SRPO as SRPO 5-rev. generates the highest win rate against the gold summaries, compared to all revisions of the baseline methods, as well as prior revision of SRPO. We also observe that the gap in performance between SRPO and the baselines (in particular compared with DPO) is significantly higher in OOD case.

Figure: Figure 2: We present the win rates of SRPO, IPO, and DPO against human-written summaries (GOLD) as a function of $N$-revisions for both in-distribution (TL;DR) and out-of-distribution (XSum) settings. The curves represent the mean win rates, with shaded areas indicating the st.dev. across 20 bootstrap evaluations, as described in the Evaluation section. Notably, DPO and IPO do not show improvements in their generations, whereas SRPO demonstrates significant improvements with each iteration.
Refer to caption: x3.png

## 8 Discussion and Limitations

In this paper we have developed
Self-Improving Robust Preference Optimization
(SRPO), a brand-new robust offline approach for learning from human preferences. We have proven mathematically and with illustrative examples, that unlike other prominent offline methods like DPO and IPO, the solution of SRPO is completely independent of the behavior policy $\mu$ and thus SRPO is completely robust to changes in $\mu$.

Summary of results. We have tested SRPO on standard summarization tasks both on in-distribution and out-of-distribution (OOD) regimes. We have observed that in the OOD case SRPO outperforms both IPO and particularly the celebrated DPO by a clear margin in terms of win-rate against gold completions, while in the in-distribution case there is less difference between SRPO and the baselines. This is an expected behavior since in-distribution case the robustness aspect of the algorithm matters less. We have observed that although 0 0-revision generation of SRPO performs well, we have observed a boost across the board by revising the generation through the self-improvement model.

Future work and Limitations. In our work we used standard and relatively simple language tasks. In the future we would like to apply SRPO to more challenging multi-task benchmarks in which the existing RLHF methods often specialize to a specific set of tasks more represented in the dataset, whereas SRPO should be more resilient due to its robustness to behavior policy $\mu$.

## Appendix A Ablation: the Effect of Combination Coefficient α 𝛼 \alpha italic_α on SRPO Performance

SRPO loss of Eq. (17) is a convex combination of two losses $\widehat{L}$ and $\widehat{L}_{\dagger}$ via the combination coefficient $\alpha$. To understand how both terms affects the loss we plot the win rates both in in-distribution case and OOD case as a function of $\alpha$ in Fig. 3. We observe that the term that contributes most to the performance of SRPO is $\widehat{L}_{\dagger}$ as in the case of $\alpha=1$ when we only use the loss for improvement model $\widehat{L}_{\dagger}$ we almost match the best performance. On the other hand using only $\widehat{L}$ (i.e., $\alpha=0$) is not enough to achieve top performance. We also observe combining both losses seems to provide some boost in performance especially in OOD case.

Figure: Figure 3: We present the win rates of SRPO against human-written summaries (GOLD) as a function of $N$-revision iterations at different $\alpha$ values. We report their mean (curve) $\pm$ st.dev. (shaded area), across 20 bootstrap evaluations, as described in the Evaluation section. We observe that SRPO achieves meaningful iterative improvements capability as the value of $\alpha$ increases.
Refer to caption: x4.png

## Appendix B Experimental Details

We provide the prompt templates used for training and evaluations in section 7.

### B.1 Prompt Templates

#### B.1.1 TL;DR

0-revision:

N-revision:

#### B.1.2 XSum

0-revision:

N-revision:

### B.2 Example Summaries

#### B.2.1 TL;DR

| Post | I have a horrible caffeine addiction, and I don’t like sacrificing any of my daily calories for coffee. I used to drink 5-6 Diet Dr. Peppers a day, but I have switched to almost exclusively drinking only water most days. I do have a Diet Dr. Pepper occasionally, and I know that technically it’s okay to probably have one a day, but I honestly just feel better all around when I drink only water. But if I don’t have some caffeine in the morning, I will be writhing in the floor in pain with a skull-pounding headache by mid-day. I know. Ridiculous, right? So, I went to the vitamin aisle the other day in search of a supplement with a little caffeine and the only thing I found was One A Day Adult Vitacraves Gummies with Energy Support. Here’s the website with [nutritional info] One serving is 20 calories and has 90 mg of caffeine. There’s also some vitamin B6, B12, and folic acid in there that supposedly helps turn food into energy. I have no idea if that’s true. I’ve been taking them a couple days, and so far I do feel more energized and I haven’t had any headaches. The issue is that one bottle is around $9 and there are only 60 in a bottle. You have to take 4 (4!) a day, which is very intriguing considering the brand name and all. So it’ll cost me around $20 a month. Which is admittedly cheaper than what I used to spend on Diet Dr. Peppers, but still. I was just curious if anyone else had a better option for caffeine in a supplement or vitamin? |
| --- | --- |
| GOLD | The vitamin with caffeine I’m taking is on the expensive side, and I’m a self-proclaimed tightwad. Anyone have better options? |
| SRPO 1-rev. | I have a caffeine addiction and need a little bit of caffeine in my day. I’m taking One A Day Adult Vitacraves Gummies with Energy Support, but it’s expensive ($20/month). Any other suggestions for a caffeine supplement or vitamin? |
| SRPO 0-rev. | I have a terrible caffeine addiction and need a little caffeine in my day. I found a vitamin with caffeine in it, but it’s pretty expensive. Any other suggestions for a caffeine supplement or vitamin? |
| IPO 1-rev. | I have a caffeine addiction and need some in my system every day to function, but I don’t like sacrificing calories for coffee. I’ve found a supplement with caffeine, but it’s expensive. Anyone have a better option? |
| IPO 0-rev. | I have a caffeine addiction and need some in my system every day to function, but I don’t like sacrificing calories for coffee. I’ve found a supplement with caffeine, but it’s expensive. Anyone have a better option? |
| DPO 1-rev. | I have a caffeine addiction and I need a supplement or vitamin that has caffeine in it to take a couple times a week so that I don’t have a headache all day. Any recommendations? |
| DPO 0-rev. | I have a caffeine addiction and I need a supplement or vitamin that has caffeine in it to take a couple times a week so that I don’t have a headache all day. Any recommendations? |

#### B.2.2 XSum

| Article | A paper detailing its proposals focuses on the need to avoid a hard border. The government stresses there should be no physical infrastructure, such as customs posts, at the border, which has almost 300 crossing points. Critics say the proposals lack credible detail, with Labour deriding the plans for the border as "a fantasy frontier". The government’s paper does not envisage CCTV cameras or number plate recognition technology at the border, or set back from it. Instead, the government is arguing for a wide-ranging exemption under which small and medium-sized businesses will not have to comply with any new customs tariffs. Northern Ireland is the only part of the UK that will share a land border with an EU state post-Brexit. The future management of that border is a highly sensitive issue and is one of three main priorities in UK-EU Brexit negotiations. …What is the customs union? Countries in the customs union do not impose tariffs - taxes on imports - on each other’s goods. Every country inside the union levies the same tariffs on imports from abroad. So, for example, a 10% tariff is imposed on some cars imported from outside the customs union, while 7.5% is imposed on roasted coffee. Other goods - such as soap or slate - have no tariffs. The UK has said it is leaving the EU’s customs union because as a member it is unable to strike trade deals with other countries. |
| --- | --- |
| GOLD | The government has said there must be an "unprecedented solution" for the border between Northern Ireland and the Republic after Brexit. |
| SRPO 1-rev. | The UK wants to avoid a hard border between Northern Ireland and the Republic of Ireland after Brexit by proposing two customs models to the EU. The UK also dismisses the idea of a customs border in the Irish Sea. |
| SRPO 0-rev. | Brexit Secretary David Davis wants a limited transition period to implement new customs arrangements, including consideration of the "unique circumstances" of Northern Ireland and the Republic of Ireland. The UK is trying to prevent the need for physical customs posts along the Irish border. |
| IPO 1-rev. | Theresa May wants a ’seamless’ border between Northern Ireland and the Republic of Ireland after Brexit - but the EU may not accept that as viable. The UK government has published a position paper setting out its ideas on the issue. |
| IPO 0-rev. | Theresa May wants a ’seamless’ border between the Republic of Ireland and Northern Ireland after Brexit - but the EU may not accept that as viable. The UK government has published a position paper setting out its ideas on the issue. |
| DPO 1-rev. | the UK is proposing a "new customs partnership" or a "highly streamlined customs arrangement" to prevent a hard border between Northern Ireland and the Republic of Ireland, in the event of no deal being reached over Brexit. |
| DPO 0-rev. | the UK is proposing a "new customs partnership" or a "highly streamlined customs arrangement" to prevent a hard border between Northern Ireland and the Republic of Ireland, in the event of no deal being reached over Brexit. |

| Article | It follows a row over the removal of personal items from graves in Torfaen which were sent to rubbish tips. Nearly 2,000 people signed a petition calling for legislation on the issue. Public Services Minister Leighton Andrews said he was sympathetic but believed it was a matter for each local authority to "develop and justify their own approaches". Torfaen council said in June 2014 it had organised the clear-up because the over-personalisation of graves in some cemeteries had prompted complaints and made maintenance difficult. The authority apologised for any upset, but said it had made efforts to inform people about the clear-up and had allowed them two months to collect any items they wanted to keep. |
| --- | --- |
| GOLD | Calls for a Wales-wide law stating what tributes can be placed on children’s graves have been rejected by ministers. |
| SRPO {1,..,5}-rev. | Public Services Minister Leighton Andrews has rejected a petition calling for legislation on the removal of personal items from graves, saying it was a matter for each local authority to "develop and justify their own approaches". |
| SRPO 0-rev. | Public Services Minister Leighton Andrews has rejected a petition calling for legislation on the removal of personal items from graves. |
| IPO {0,..,5}-rev. | Vicky Pryce wanted revenge on ex-MP Chris Huhne over him getting points on his licence, so she took the speeding points for him in 2003, a court heard. |
| DPO {0,..,5}-rev. | Vicky Pryce told court she signed speeding points form for her husband Chris Huhne in revenge for him threatening their marriage over his speeding points. |

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. It can be beneficial to provide more explanation of the self-improving policy. This concept is not common in many RLHF literatures. Readers may be curious about how these policies work and how they are implemented. Providing more details can make this paper easier to follow.

2. Can you provide more details about the evaluation? For example, what are the parameters, such as the accuracy compared to humans, of the AI evaluation platform? Exactly how do we measure the winning rate between any two models? Revealing such details can make this work much easier to reproduce.
soundness: 3
strengths: 1. This paper studies the critical problem of finding a more efficient RLHF algorithm. It could have great potential in many important real-world applications.

2. The idea of using a self-improving policy in solving the RLHF problem is novel.

3. The mathematical explanation of how to solve the min-max learning problem they propose is thorough.

3. The SRPO algorithm aims at training a robust policy against the quality of the dataset, making it more trustworthy than many other RLHF algorithms.

4. The experiment design is reasonable in general, and the results look positive.
summary: This paper studies the problem of reinforcement learning from human feedback (RLHF). It designs a novel RLHF algorithm called SRPO based on training a robust self-improving policy by solving a min-max problem. It explains the math in detail on how to find the solution to this min-max problem. The experiment results show that SRPO achieves a higher winning rate against the gold dataset than existing state-of-the-art RLHF algorithms like DPO and IPO.
weaknesses: 1. There is no theoretical guarantee of the learning outcome. This makes the whole theoretical part weak. Is there a chance to provide any theoretical guarantee on the performance of the policy learned by SRPO under some assumptions?

2. The design of SRPO algorithm is novel, but the description of its motivation can be improved. Currently, the motivation is that 'Instead, it is more natural to learn that given a query x and a completion y what would be the improved completion upon y'. However, in general, you can also say that to some other learning algorithms as long as they improve the quality of their policy to generate better completion after each training iteration. It is better to find some more unique motivation for the design of the SRPO algorithm.

3. The SRPO algorithm's winning rate against other algorithms is not provided. This paper only measures the winning rate against a gold-standard dataset. A policy that has a higher winning rate against a fixed dataset than that of another policy does not necessarily mean this policy is better than that policy. To verify that the policy learned by SRPO is better than the policies learned by other algorithms, it is necessary to compare these policies against each other directly.

---

confidence: 4
contribution: 2
initial_rating: 5
presentation: 3
questions: Please see above.
soundness: 3
strengths: The paper is clearly written and easy to follow, with each step of the SRPO algorithm systematically derived.

The derivations provided in the paper appear rigorous, demonstrating a well-founded approach to preference optimization. The use of a min-max objective with a focus on self-improvement mechanisms is an interesting contribution.
summary: This paper introduces a novel method, Self-Improving Robust Preference Optimization (SRPO), which aims to address two primary limitations in current preference optimization techniques. First, existing methods generally lack an inherent self-improvement mechanism at inference time, which limits adaptability. Second, they often rely heavily on the training task and distribution used to generate preference data, which affects the robustness of solutions. To address these issues, SRPO is designed as an offline preference optimization method that leverages a min-max formulation to learn a robust generative policy. This policy generates completions that require minimal adjustment when optimized through a self-improvement policy. The authors apply a DPO/IPO-inspired derivation to show that this min-max objective can be effectively optimized through standard supervised learning techniques. Experimental results on the TL;DR Summarization dataset indicate that SRPO achieves better in-distribution (ID) and out-of-distribution (OOD) performance compared to DPO and IPO baselines.
weaknesses: The evaluation is currently limited to a single dataset (TL;DR Summarization) and only compares SRPO against two baselines, DPO and IPO. Conducting experiments on additional datasets would strengthen the empirical claims related to robustness.

The paper lacks (empirical) comparisons to more recent preference optimization methods, such as SimPO (Meng et al., 2024) and RPO (Liu et al., 2024), which integrate recent advancements over DPO. 

In Figure 3, SRPO achieves improved performance with revisions, likely due to its self-improvement mechanism. However, unlike DPO and IPO, which lack this self-correction feature, SRPO appears to incur additional inference costs to achieve its performance benefits.

(Meng et al., 2024) SimPO: Simple Preference Optimization with a Reference-Free Reward

(Liu et al., 2024) Provably Mitigating Overoptimization in RLHF: Your SFT Loss is Implicitly an Adversarial Regularizer

---

confidence: 3
contribution: 3
initial_rating: 5
presentation: 3
questions: - DPO and IPO both have "revisions"? Does this mean they received the same in context prompt as SRPO?
- The primary thesis of this work is that current preference optimization methods should not expect the ideal completion to be written and PPO/DPO/IPO are solving a fundamentally different task. In that vein, how does PPO compare under these same conditions? Is the lack of robustness also observed?
- Overoptimization has been observed in DPO and other direct alignment algorithms. Is this observed with SRPO as well? Why/Why not? [0]


[0] https://arxiv.org/abs/2406.02900
soundness: 4
strengths: The observations made by this paper are well supported empirically. In particular, they use a nice synthetic bandit example to show robustness. Further, they experimentally verify on the TL;DR dataset and XSum on out of distribution examples. This paper further does a good job elucidating the derivation of the objective.
summary: This paper proposes Self Improving Robust Preference Optimization (SRPO), a new method of LLM fine tuning. The authors propose using in-context learning to iteratively generate higher quality completions while being robust to the underlying offline dataset. The paper proposes a min-max objective which aims to maximize the quality of the improvement (of the improvement model) and minimize the amount by which you can be improved (for another language model). The paper then analyzes optimizations for each of the models independently and derive a convex combination of these losses. Finally, the authors experimentally verify their theory with experiments on TL;DR and XSum.
weaknesses: - The proposed method empirically performs similar to other methods under the same amount of inference compute.
- The paper does not provide much theoretical justification of the combination loss.

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 2
questions: Is the problem studied by the authors related to continual learning under specific trade-offs [1]?

---

[1] https://arxiv.org/abs/2305.14782
soundness: 3
strengths: Although this is not my field of research, the paper was sufficiently clear. Its motivations are well-stated, and the results easy to grasp. They seem relevant to the RLHF literature.
summary: The authors propose Self-Improving Robust Preference Optimization (SRPO) to address two of the main drawbacks of existing RLHF methods. That is, (a) models trained with RLHF can learn from mistakes or negative examples through RL mechanism or contrastive loss at the time of training. However at the time of inference they are not equipped with an innate mechanism to correct mistakes by self-improvement. (b) The optimal solution of existing methods is highly task-dependent and thus it is difficult for them to generalize to new tasks. SRPO overcomes these shortcomings, and its effectiveness is shown also empirically.
weaknesses: Probably due to my unfamiliarity with this field, I am not able to find major weaknesses to the paper. Some minor considerations follow.

Line 62: "at the time of inference It would be very" should be changed to "at the time of inference, it would be very".

What is $\mathbb{R}^*_+$ in line 162?

Line 195: "Eq. equation 2" should be "equation 2". This typo is repeated throughout the manuscript.

Meta-review: This work proposes a new preference optimization algorithm, self-improving robust preference optimization (SIRPO).  This DPO variant is capable of performing in-context learning to self-improve during execution of the policy.  Importantly, the authors also propose a robust training method which improves out-of-distribution generalization, a problem of considerable interest to the community.

Reviewers generally found the paper to be well-written and well-motivated.  Initially there were some concerns about the generalizability of the results and theoretical underpinnings, but this was addressed during the rebuttal period, in which the authors added an additional benchmark task (post-training llama 3.1 using the Ultra Feedback dataset, and evaluating on Arena-Hard prompts).  SRPO was evaluated by comparing win rates against RPO and SIMPO baselines, and appears to robustly outperform these methods under a variety of conditions and hyper parameter configurations.  Theoretical justifications of the loss function were addressed via Theorem 1 and revisions.

Overall, reviewers and the AC found this work compelling enough for publication, and I commend the authors on their compelling rebuttal and revision.  Please work to thoroughly address any remaining issues and analyses suggested by the reviewers.  For the CR, I would recommend not including the first figure (especially on the first page, since the reader will have no context on how to interpret it!).  The other variants of this figure are more informative, and an illustration is not needed to make the rather intuitive idea behind this work.

Decision: Accept (Poster)