REINFORCE Leave-One-Out (RLOO)

REINFORCE Leave-One-Out (RLOO) is a reinforcement learning algorithm based on the classic REINFORCE policy-gradient method. It constructs an unbiased advantage baseline via the Leave-One-Out (LOO) technique.

Algorithm Overview

For clarity, we explain RLOO by contrasting it with GRPO (Group Relative Policy Optimization).

Both GRPO and RLOO estimate advantages via intra-group comparisons to avoid the high variance of a global baseline. Their core differences are mainly in the following aspects:

Difference 1: How the Advantage Baseline Is Constructed

1. GRPO (Group Relative Policy Optimization)

For each prompt, GRPO generates $G$ response samples and normalizes rewards using the group mean and standard deviation:

$$\hat{A}_{i} = \frac{R_i - \text{mean}(\{R_j\}_{j=1}^G)}{\text{std}(\{R_j\}_{j=1}^G)}$$

Where:

• $R_i$ is the reward of the $i$-th sample
• $\text{mean}(\{R_j\}_{j=1}^G) = \frac{1}{G}\sum_{j=1}^G R_j$ is the group mean
• $\text{std}(\{R_j\}_{j=1}^G)$ is the group standard deviation

2. RLOO (REINFORCE Leave-One-Out)

For each prompt, RLOO generates $K$ response samples and constructs the baseline via Leave-One-Out, i.e., for the $i$-th sample, the baseline is the mean of the other $K-1$ samples:

$$\hat{A}_{i} = R_i - \frac{1}{K-1}\sum_{j \neq i} R_j$$

This can be equivalently rewritten as:

$$\hat{A}_{i} = \frac{K}{K-1} \left(R_i - \bar{R}\right)$$

where $\bar{R} = \frac{1}{K}\sum_{j=1}^K R_j$ is the group mean reward.

Note: We use $K$ here to match the notation in the paper. It has the same meaning as $G$ in GRPO and corresponds to the configuration parameter num_generations.

Why Leave-One-Out?

The key advantage is unbiasedness. For the $i$-th sample, its reward $R_i$ is independent of the baseline $\frac{1}{K-1}\sum_{j \neq i} R_j$, hence the advantage estimate is unbiased. In contrast, using the mean including itself as the baseline introduces bias.

Difference 2: How KL Regularization Is Applied

To prevent the policy from drifting too far from the reference policy, both algorithms introduce KL divergence regularization, but in different ways:

GRPO: Adds KL divergence as an independent regularization term to the loss:

$$\mathcal{L}(\theta) = -\mathbb{E}\left[\hat{A}_i \log \pi_\theta(a_i|s_i)\right] + \beta \cdot \text{KL}(\pi_\theta \Vert \pi_{\text{ref}})$$

RLOO: Integrates KL divergence directly into the reward, constructing a modified reward:

$$R'_i = R_i - \beta \cdot \text{KL}(\pi_\theta \Vert \pi_{\text{ref}})$$

where $\beta$ is the KL coefficient (parameter beta), and $\pi_{\text{ref}}$ is the reference policy (typically an SFT model or the initial policy).

Parameter Configuration

RLOO training can be enabled based on GRPOTrainer by setting the following parameters:

# Basic RLOO configuration
--advantage_estimator rloo  # Use RLOO's leave-one-out advantage estimator
--kl_in_reward true         # Integrate KL divergence into the reward (default for RLOO)

You can refer to this script for training.

Important Parameters

• --advantage_estimator: Choose the advantage estimator

  • grpo (default): standardize using group mean and standard deviation
  • rloo: construct the baseline via Leave-One-Out

• --kl_in_reward: Controls where the KL term is applied

  • false: KL as a separate regularization term in the loss (GRPO style)
  • true: subtract KL directly from the reward to form a modified reward (RLOO style)

• --num_generations: Number of samples per prompt, i.e., $K$

• --beta: KL regularization coefficient $\beta$

  • Controls how conservatively the policy updates

Other parameters are consistent with the GRPO arguments.