1. Bibliographic Information

1.1. Title

The central topic of the paper is a novel approach to improving the reasoning abilities of large language models (LLMs) using Reinforcement Learning (RL) with verifiable rewards. The title "RANDOM Policy Valuation is Enough for LLM REASONING WITH VERIFIABLE REWARDS" suggests a surprising simplification of traditional RL methods for this specific domain.

1.2. Authors

The authors are anonymous, as indicated by "Anonymous authors" and "Paper under double-blind review". This is a common practice for papers submitted for peer review to ensure impartiality.

1.3. Journal/Conference

The paper is "Under review as a conference paper at ICLR 2026". The International Conference on Learning Representations (ICLR) is a highly prestigious and influential conference in the fields of artificial intelligence, machine learning, and deep learning. Publication at ICLR signifies a high level of academic rigor and impact.

1.4. Publication Year

The paper was "Published at (UTC): 2025-10-08T00:00:00.000Z", indicating it was published in 2025.

1.5. Abstract

The paper addresses the limitations of current Reinforcement Learning with Verifiable Rewards (RLVR) methods for Large Language Models (LLMs) in reasoning tasks, such as training instability and diversity collapse, which arise from policy optimization frameworks like PPO and GRPO. The authors observe that RLVR in math reasoning can be modeled as a specialized finite-horizon Markov Decision Process (MDP) with deterministic state transitions, tree-structured dynamics, and binary terminal rewards. Based on this structural insight, they theoretically prove that the optimal action can be recovered from the Q-function of a fixed uniformly random policy, thereby bypassing the complex generalized policy iteration loop. They introduce Random Policy Valuation for Diverse Reasoning (ROVER), a minimalist yet effective RL algorithm that samples actions from a softmax over these uniform-policy Q-values. ROVER is shown to preserve diversity throughout training, allowing sustained exploration. Experimental results across multiple base models and standard math reasoning benchmarks demonstrate ROVER's superior performance in both quality (e.g., $+8.2$ on pass@1, $+16.8$ on pass@256) and diversity ($+20.5$) compared to existing complex methods.

1.6. Original Source Link

The original source link is https://openreview.net/forum?id=ujLgLz6QQa. The PDF link is https://openreview.net/pdf?id=ujLgLz6QQa. The paper is currently "Under review as a conference paper at ICLR 2026".

2. Executive Summary

2.1. Background & Motivation

The paper tackles the challenge of enhancing the reasoning abilities of Large Language Models (LLMs) using Reinforcement Learning with Verifiable Rewards (RLVR). While existing methods, primarily based on policy optimization frameworks like Proximal Policy Optimization (PPO) and Group-Relative Policy Optimization (GRPO), have shown success, they suffer from significant drawbacks:

1. Training Instability: The iterative process of policy evaluation and improvement in Generalized Policy Iteration (GPI) leads to a non-stationary evaluation target, making training unstable.

2. Diversity Collapse / Entropy Collapse: The reward-maximizing nature of these algorithms often causes the policy to converge to a few dominant solution paths, reducing the diversity of generated reasoning steps and limiting exploration. This can hinder LLMs' ability to generalize or find novel solutions.

3. Complexity and Heuristics: To mitigate these issues, current methods rely on complex heuristic tricks (e.g., clipping, KL regularization, data selection) and careful hyperparameter tuning, adding layers of implementation complexity and requiring case-specific adjustments.

   The core problem the paper aims to solve is to overcome these limitations and develop a simpler, more stable, and more diverse RLVR algorithm specifically tailored for LLM reasoning tasks. The paper identifies a crucial gap: PPO and GRPO were developed for general-purpose Reinforcement Learning (RL) settings (e.g., continuous control, stochastic environments, graph-based state spaces), which are fundamentally more complex than the Markov Decision Processes (MDPs) underlying LLM math reasoning. The paper's innovative idea is to exploit the specific structural properties of LLM math reasoning MDPs to simplify the RL problem dramatically.

2.2. Main Contributions / Findings

The paper makes several significant contributions:

1. Theoretical Simplification of RLVR: The authors prove a surprising theoretical result: for finite-horizon episodic MDPs with deterministic transitions, tree-structured state spaces, and binary terminal rewards (which characterize LLM math reasoning), the optimal action can be derived directly from the Q-values evaluated under a fixed, uniformly random policy. This finding fundamentally simplifies RL for this domain by eliminating the need for Generalized Policy Iteration (GPI) and its associated complexities.
2. Introduction of ROVER: They introduce Random Policy Valuation for Diverse Reasoning (ROVER), a practical, minimalist, and scalable RL algorithm based on this theoretical insight. ROVER efficiently parameterizes the Q-function intrinsically within the LLM and uses a softmax over the uniform-policy Q-values for action sampling, balancing quality and diversity. It incorporates group reward centering and reward broadcasting to mitigate reward signal variance and improve credit assignment.
3. Superior Performance and Diversity: Despite its radical simplification, ROVER demonstrates superior performance across diverse math reasoning benchmarks and various LLM scales.

   • Quality: Achieves significant improvements in pass@1 (e.g., $+8.2$ averaged on AIME24, AIME25, and HMMT25 for Qwen3-8B-Base) compared to strong, complicated existing methods.

   • Diversity: Shows substantial improvements in pass@256 (e.g., $+16.8$ averaged on AIME24, AIME25, and HMMT25 for Qwen3-8B-Base) and other diversity metrics ($+20.5$ compared to baselines). ROVER maintains higher policy entropy throughout training, encouraging sustained exploration and diverse reasoning paths.

   • Generalization: ROVER exhibits stronger generalization capabilities on out-of-distribution tasks like GPQA-diamond.

   • Novelty: The method can find novel reasoning strategies that are absent from base models and models trained with standard RL approaches.

     These findings solve the problems of training instability and diversity collapse by offering a simpler, more robust RL framework for LLM reasoning that achieves both high-quality solutions and maintains a rich diversity of reasoning pathways.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To understand this paper, a grasp of several fundamental concepts in Reinforcement Learning (RL) and Large Language Models (LLMs) is essential.

• Reinforcement Learning (RL): RL is a machine learning paradigm where an agent learns to make decisions by performing actions in an environment to maximize a cumulative reward. The agent interacts with the environment, observes states, takes actions, and receives rewards, learning an optimal policy that maps states to actions.
• Large Language Models (LLMs): LLMs are deep learning models, typically based on the transformer architecture, that are trained on vast amounts of text data to generate human-like text. They are highly capable of understanding, generating, and reasoning with natural language, often exhibiting "emergent abilities" on complex tasks.
• Markov Decision Process (MDP): An MDP is a mathematical framework for modeling decision-making in situations where outcomes are partly random and partly under the control of a decision-maker. It is defined by a tuple $(\mathcal{S}, \mathcal{A}, \mathcal{P}, \mathcal{R}, \gamma)$.
  • $\mathcal{S}$: A set of possible states the agent can be in.
  • $\mathcal{A}$: A set of possible actions the agent can take.
  • $\mathcal{P}$: A transition function (or probability distribution) $P(s' | s, a)$ that describes the probability of transitioning to state $s'$ from state $s$ after taking action $a$. In a deterministic MDP, $P(s' | s, a)$ is 1 for a single $s'$ and 0 otherwise.
  • $\mathcal{R}$: A reward function R(s, a, s') that specifies the immediate reward received after transitioning from state $s$ to $s'$ via action $a$.
  • $\gamma$: A discount factor ($0 \le \gamma \le 1$) that determines the present value of future rewards. A $\gamma$ of 1 means future rewards are valued equally to immediate rewards.
• Policy ($\pi$): A policy defines the agent's behavior. It's a function $\pi(a | s)$ that maps a state $s$ to a probability distribution over actions $a$, indicating the likelihood of taking action $a$ in state $s$. An optimal policy $\pi^*$ maximizes the expected cumulative reward.
• Q-function (Action-Value Function): The Q-function, denoted $Q^\pi(s, a)$, represents the expected cumulative reward (return) obtained by starting in state $s$, taking action $a$, and then following policy $\pi$ thereafter. The optimal Q-function $Q^*(s, a)$ gives the maximum possible expected return for taking action $a$ in state $s$.
• Value Function ($V$): The value function, denoted $V^\pi(s)$, represents the expected cumulative reward obtained by starting in state $s$ and following policy $\pi$ thereafter.
• Bellman Equation: A fundamental equation in RL that relates the value of a state to the values of its successor states. For a Q-function, the Bellman expectation equation for policy $\pi$ is: $ Q^\pi(s, a) = \mathbb{E}[R_{t+1} + \gamma Q^\pi(S_{t+1}, A_{t+1}) | S_t=s, A_t=a] $ where $R_{t+1}$ is the immediate reward, $S_{t+1}$ is the next state, and $A_{t+1}$ is the action taken in the next state according to $\pi$.
• Generalized Policy Iteration (GPI): GPI is a general concept in RL that describes the interaction between two processes: policy evaluation and policy improvement.
  • Policy Evaluation: Given a policy, estimate its value function (e.g., $V^\pi$ or $Q^\pi$).
  • Policy Improvement: Given a value function, update the policy to be greedy with respect to that value function (i.e., choose actions that lead to higher estimated values). GPI-based algorithms iteratively alternate between these two steps until the policy and value function converge to their optimal forms.
• Reinforcement Learning with Verifiable Rewards (RLVR): A specific application of RL where the reward signal is not learned but is derived from an external, objective verification process (e.g., checking if a math problem solution is correct). This is common in tasks like math reasoning, code generation, or fact verification, where a clear correctness criterion exists.
• Finite-Horizon Episodic MDP: An MDP where the interaction between the agent and environment is divided into distinct episodes, and each episode has a finite, predetermined maximum number of steps (horizon).
• Deterministic Transitions: In this type of MDP, taking a specific action $a$ in state $s$ always leads to the same next state $s'$. There is no randomness in the state transitions.
• Tree-structured Dynamics: This implies that each state has a unique parent, and actions from a state lead to distinct, non-overlapping subtrees. There are no cycles in the state transitions, and no state can be reached from multiple, independent paths from the root, simplifying navigation.
• Binary Terminal Rewards: Rewards are only received at the end of an episode (terminal rewards), and they are binary (e.g., 1 for success/correct solution, 0 for failure/incorrect solution). There are no intermediate rewards during the episode.
• Advantage Function ($A_t$): In policy gradient methods, the advantage function is often used to reduce variance in updates. It is defined as $A^\pi(s, a) = Q^\pi(s, a) - V^\pi(s)$, representing how much better an action $a$ is compared to the average outcome of all actions from state $s$ under policy $\pi$.

3.2. Previous Works

The paper frames its contribution in contrast to several established RL algorithms, particularly those applied to LLM reasoning.

• Proximal Policy Optimization (PPO): PPO (Schulman et al., 2017) is a widely used policy gradient algorithm in deep RL. It aims to keep new policies close to old policies via a clipping mechanism or a KL divergence penalty, preventing large policy updates that could lead to instability. The core objective for PPO is: $ J(\theta)=\mathbb{E}{x \sim \mathcal{X}, y \sim \pi{\theta_{\text{old}}}(x)}\left[\frac{1}{|y|} \sum_{t=0}^{|y|-1}\left(\min \left(\operatorname{IS}{t} A{t}, \operatorname{clip}\left(\operatorname{IS}{t}, 1-\epsilon{\text{low}}, 1+\epsilon_{\text{high}}\right) A_{t}\right)-\beta D_{KL}\left(\pi_{\theta} \mid \pi_{\text{ref}}\right)\right)\right] $ Where:

  • $\pi_{\theta}$ is the current policy, parameterized by $\theta$.
  • $\pi_{\theta_{\text{old}}}$ is the behavior policy used to sample data (the policy from the previous iteration).
  • $\operatorname{IS}_{t} = \pi_{\theta}(a_t | s_t) / \pi_{\theta_{\text{old}}}(a_t | s_t)$ is the importance sampling ratio, which corrects for the difference between the target policy $\pi_{\theta}$ and the behavior policy $\pi_{\theta_{\text{old}}}$.
  • $A_t$ is the advantage function for action $a_t$ in state $s_t$.
  • $\epsilon_{\text{low}}$ and $\epsilon_{\text{high}}$ define the clipping range for the importance sampling ratio, limiting how much the new policy can deviate from the old one.
  • $\beta D_{KL}(\pi_{\theta} \mid \pi_{\text{ref}})$ is a KL regularization term, where $D_{KL}$ is the Kullback-Leibler divergence between the current policy $\pi_{\theta}$ and a reference policy $\pi_{\text{ref}}$ (often the initial LLM or a periodically updated version). This term encourages the policy not to stray too far from a known good policy, preserving exploration and preventing catastrophic forgetting. PPO's design is general-purpose, suitable for various environments, including those with stochastic dynamics and continuous action spaces.

• Group-Relative Policy Optimization (GRPO): GRPO (Shao et al., 2024) is a specialized derivative of PPO tailored for LLM applications. It specifically addresses the high variance of reward signals in LLM generation by sampling multiple responses ($G$ responses) for each prompt and estimating the advantage function $A_t$ within each group using mean and standard deviation normalization: $ A_{t}=\frac{r\left(x, y_{i}\right)-\operatorname{mean}\left(\left{r\left(x, y_{i}\right)\right}{i=1}^{G}\right)}{\operatorname{std}\left(\left{r\left(x, y{i}\right)\right}_{i=1}^{G}\right)} $ This grouping mechanism helps stabilize training compared to using raw rewards.

• REINFORCE++ (Hu et al., 2025a): This is another policy gradient method, likely an enhancement over the basic REINFORCE algorithm. REINFORCE directly optimizes the expected reward by using sample returns as an estimate of the advantage. $REINFORCE++$ likely incorporates techniques to reduce variance or improve stability, similar to how PPO and GRPO refine basic policy gradient ideas.

• DAPO (Diversity-Aware Policy Optimization) (Yu et al., 2025): A policy optimization method designed to address diversity concerns, often by incorporating mechanisms like the clip-higher technique to encourage exploration or prevent premature convergence to a single mode.

  These methods, while effective, often rely on an iterative policy evaluation-policy improvement cycle (Generalized Policy Iteration) and require complex heuristic tricks (clipping, KL regularization, data selection) to manage training instability and diversity collapse.

3.3. Technological Evolution

The evolution of applying RL to LLMs for reasoning tasks has largely followed two phases:

1. Phase 1: Adaptation of General-Purpose RL: Early efforts involved adapting well-established RL algorithms, primarily PPO, which were initially designed for diverse RL benchmarks like Atari games or robotic control. These algorithms, while powerful, inherently carry assumptions and complexities suitable for general MDPs (e.g., stochastic transitions, continuous action spaces, graph-based state spaces with cycles). This adaptation phase demonstrated promising results but also exposed challenges like training instability, sensitivity to hyperparameters, and a tendency for policies to collapse to a few optimal modes, sacrificing diversity. Specialized variants like GRPO emerged to address some LLM-specific issues (e.g., reward variance).

2. Phase 2: Domain-Specific Simplification: This paper represents a shift towards exploiting the unique characteristics of LLM reasoning MDPs. Instead of forcing general RL algorithms onto a specialized problem, it asks whether the problem itself can be simplified. The key insight is that LLM math reasoning tasks, with their deterministic, tree-structured, finite-horizon, and binary-reward nature, form a much simpler MDP than typical RL environments. This allows for a radical simplification of the RL solution, moving away from iterative GPI towards a more direct policy evaluation approach.

   ROVER fits into this second phase by proposing a minimalist approach that leverages the structural simplicity of LLM reasoning MDPs to achieve superior performance and diversity without the intricate heuristics of its predecessors.

3.4. Differentiation Analysis

The core differences and innovations of ROVER compared to the main methods in related work (like PPO, GRPO, $REINFORCE++$, DAPO) are:

• Bypassing Generalized Policy Iteration (GPI):

  • Existing Methods: All mentioned baselines fundamentally follow the GPI paradigm. They iteratively evaluate the current policy's value function and then improve the policy based on that evaluation. This leads to a non-stationary learning target, causing instability and requiring complex regularization techniques (KL regularization, clipping).
  • ROVER: ROVER completely bypasses the GPI loop. It proves that for the specific MDP structure of LLM reasoning, evaluating the Q-function of a fixed, uniformly random policy is sufficient to derive optimal actions. This eliminates the instability inherent in iterative policy improvement.

• Simplicity vs. Complexity:

  • Existing Methods: These are often complex, requiring multiple components (e.g., separate value networks, target networks, intricate loss functions with clipping and KL divergence terms) and careful tuning of many hyperparameters to maintain stability and diversity.
  • ROVER: ROVER is minimalist. It does not require a separate value network, a target network, or KL regularization. It uses a simplified Q-function parameterization and a straightforward mean-centered reward mechanism. This radical simplification makes it easier to implement and more robust.

• Diversity Preservation:

  • Existing Methods: Tend to suffer from diversity collapse or entropy collapse as policies converge to a single (or few) reward-maximizing modes. Techniques like KL regularization or clip-higher are attempts to mitigate this, but often with mixed success or at the cost of performance.
  • ROVER: By interpreting the Q-values of the uniform policy as probabilities of successful continuations and using a softmax over these Q-values for action sampling, ROVER intrinsically maintains diversity. It explores multiple valid reasoning pathways without needing explicit diversity-promoting rewards or complex sampling techniques.

• Nature of Q-function Evaluation:

  • Existing Methods: Evaluate the Q-function of the current, evolving policy, which means the target for evaluation is constantly changing.

  • ROVER: Evaluates the Q-function of a fixed, uniform random policy. This provides a stable and consistent target for learning, contributing to its stability.

    In essence, ROVER differentiates itself by leveraging a deep theoretical insight into the specific MDP structure of LLM reasoning to drastically simplify the RL process, moving from a complex iterative optimization problem to a more direct policy evaluation problem, while inherently promoting diversity.

4. Methodology

4.1. Principles

The core principle of ROVER is rooted in a fundamental observation about the structure of RLVR problems in LLM math reasoning. These problems can be formalized as a specialized Markov Decision Process (MDP) with the following key properties:

1. Finite-Horizon Episodic: Each problem (episode) has a clear start and end.

2. Deterministic Transitions: Given a state and an action (generating a token), the next state (the sequence with the new token appended) is uniquely determined. There is no randomness in the environment's response to an action.

3. Tree-structured State Space: From the initial prompt, every action branches out, forming a unique path. Each partial sequence has exactly one parent state, and distinct actions lead to disjoint subtrees. This means there are no cycles or convergent paths in the state space.

4. Binary Terminal Rewards: A reward (typically 1 for correct, 0 for incorrect) is only received at the end of the episode, upon completion of the entire generated response. There are no intermediate rewards.

   The paper's surprising theoretical finding is that for an MDP with these specific characteristics, the optimal policy can be derived not by complex iterative policy optimization (like PPO), but by simply evaluating the Q-function of the simplest possible policy: a fixed, uniformly random policy. This Q-function indicates the probability of reaching a correct solution if one were to act randomly from a given state-action pair. By selecting actions greedily based on these Q-values, ROVER can identify optimal paths. Furthermore, to balance optimality with diversity (crucial for LLM reasoning), ROVER leverages these Q-values to construct a softmax policy, allowing exploration of multiple high-value reasoning paths.

4.2. Core Methodology In-depth (Layer by Layer)

4.2.1. Formalizing LLM Math Reasoning as an MDP

The paper first formalizes RLVR for LLMs in math reasoning as an MDP, defined by a tuple $(\mathcal{S}, \mathcal{V}, \mathcal{R}, \mathcal{P}, \gamma, \mathcal{X})$. Let's break down each component:

• State Space ($\mathcal{S}$): This consists of all finite-length strings formed by concatenating elements from the vocabulary. A state $s_t$ at time step $t$ is a sequence of tokens generated so far, starting from the initial prompt $x$.

• Action Space ($\mathcal{V}$): This is the vocabulary set of the LLM. At each step, the LLM chooses a token (action) from this vocabulary.

• Reward Function ($\mathcal{R}$): $\mathcal{R}: \mathcal{S} \times \mathcal{V} \rightarrow \mathbb{R}$ is a binary reward function. The reward is typically 1 if the final generated response $y$ is correct given the prompt $x$, and 0 otherwise. This is a terminal reward, meaning it's only received at the end of the episode.

• Transition Function ($\mathcal{P}$): $\mathcal{P}: \mathcal{S} \times \mathcal{V} \rightarrow \mathcal{S}$ is a deterministic transition function. If the current state is $s_t = \{x, a_0, \ldots, a_{t-1}\}$ and the LLM selects action $a_t$, the next state $s_{t+1}$ becomes the concatenation $\{x, a_0, \ldots, a_t\}$. This clearly illustrates the deterministic and tree-structured nature.

• Discount Factor ($\gamma$): Set to $\gamma=1$. This means future rewards are valued equally to immediate rewards, which is typical for episodic tasks where the total reward is the main concern.

• Initial State Distribution ($\mathcal{X}$): A prompt $x$ is sampled from this distribution at the beginning of each episode.

  The LLM acts as the policy $\pi_{\theta}(\cdot \mid s_t)$, selecting actions (tokens) $a_t$ sequentially until a full response $y = \{a_0, a_1, \ldots, a_{|y|-1}\}$ is formed. The goal is to learn an optimal policy $\pi^* = \arg \max_{\pi} \mathbb{E}_{x \sim \mathcal{X}, y \sim \pi(x)}[r(x, y)]$ that maximizes the expected cumulative reward $r$.

4.2.2. The Uniform Random Policy and its Q-function

The paper introduces the uniform random policy $\pi_u(a \mid s) = \frac{1}{|A|}$, where $|A|$ is the size of the action space (the vocabulary size $|\mathcal{V}|$ in this context). This policy samples every available action with equal probability.

The Q-value for this uniform policy, $Q^{\pi_u}(s, a)$, can be estimated using a simplified Bellman update for deterministic transitions and $\gamma=1$: $ \hat{Q}^{\pi_u}(s, a) \leftarrow r(s, a)+\frac{1}{|A|} \sum_{a' \in A} \hat{Q}^{\pi_u}\left(s', a'\right) $ Where:

• $\hat{Q}^{\pi_u}(s, a)$ is the estimated Q-value of taking action $a$ in state $s$ and then following the uniform policy.

• r(s, a) is the immediate reward for taking action $a$ from state $s$. In this terminal reward setting, r(s, a) would typically be 0 for intermediate steps and the actual terminal reward r(x, y) if s,a leads to a terminal state.

• $s'$ is the next state after taking action $a$ from state $s$.

• $\frac{1}{|A|} \sum_{a' \in A} \hat{Q}^{\pi_u}(s', a')$ represents the average Q-value of all possible actions taken from the next state $s'$ under the uniform policy.

  Traditionally, this mean operator has been considered insufficient for optimal control in general MDPs because it averages across all actions without preference. However, the paper shows this is not the case for their specialized MDP.

4.2.3. Theoretical Foundation: Optimality with Uniform Policy Valuation

The paper's first key theoretical contribution is formalized in Theorem 1.

Theorem 1. Consider a finite-horizon episodic MDP with deterministic transitions, tree-structured state space, and binary terminal rewards $\mathcal{R}(s) \in\{0, R\}$ where $R>0$ ($R$ for a correct solution, 0 otherwise). Let $\pi_u$ be the uniform policy, and $Q^{\pi_u}$ its corresponding $Q$-function. Define the greedy policy with respect to $Q^{\pi_u}$ by $\pi_{\text{greedy}}(s)=\arg \max_{a} Q^{\pi_u}(s, a)$, then $\pi_{\text{greedy}}$ is optimal.

Explanation of Theorem 1: This theorem states that for the specific type of MDP describing LLM math reasoning, one can achieve an optimal policy by a remarkably simple procedure:

1. Calculate the Q-values for every state-action pair assuming a uniformly random policy is followed after the initial action.

2. From any given state $s$, simply choose the action $a$ that has the highest $Q^{\pi_u}(s, a)$ value.

   The intuition behind this is critical: $Q^{\pi_u}(s, a)$ essentially represents the probability that a correct solution can be reached if one takes action $a$ from state $s$ and then proceeds to choose actions randomly. If $Q^{\pi_u}(s, a) = 0$, it means there are no paths starting with (s, a) that lead to a correct solution, even with random exploration. If $Q^{\pi_u}(s, a) > 0$, it means there is at least one path (and thus a probability) of finding a correct solution. By greedily choosing the action with the highest $Q^{\pi_u}(s, a)$, the agent is effectively prioritizing paths that have a higher "density" of successful continuations, leading directly to an optimal strategy for finding any correct solution. This bypasses the need for complex GPI loops that iteratively refine a policy.

4.2.4. Beyond Greedy Selection: Balancing Quality and Diversity

While Theorem 1 guarantees optimality with greedy action selection, such deterministic behavior often leads to mode collapse, where the agent consistently finds only one or a few optimal solutions, sacrificing diversity. For LLM reasoning, diversity (i.e., finding multiple valid ways to solve a problem) is crucial for robustness and pass@k performance.

The paper addresses this by leveraging the probabilistic interpretation of $Q^{\pi_u}(s, a)$: it captures the probability of successful continuations. To balance quality and diversity, ROVER transitions from deterministic greedy selection to a stochastic action selection using a softmax policy: $ \pi_s(a \mid s)=\frac{\exp \left(Q^{\pi_{u}}(s, a) / \rho\right)}{\sum_{a^{\prime}} \exp \left(Q^{\pi_{u}}\left(s, a^{\prime}\right) / \rho\right)} $ Where:

• $\pi_s(a \mid s)$ is the probability of selecting action $a$ in state $s$ under the softmax policy.
• $Q^{\pi_u}(s, a)$ is the Q-value from the uniform random policy.
• $\rho$ is a temperature parameter ($>0$).

  • As $\rho \rightarrow 0$, the softmax approaches greedy selection (the action with the highest $Q^{\pi_u}$ gets probability 1), prioritizing quality over diversity.

  • As $\rho \rightarrow \infty$, the softmax approaches a uniform distribution, prioritizing diversity over quality.

  • Intermediate values of $\rho$ allow sampling actions proportional to their estimated success probability, encouraging exploration of promising but not necessarily singular paths.

    The second key theoretical contribution, Theorem 2, provides a performance guarantee for this softmax policy.

Theorem 2. Consider the same MDP $\mathcal{M}$, and let $Q^{\pi_u}(s, a)$ denote the $Q$-function under the uniform random policy $\pi_u$ from state-action pair (s, a), $N(s)=\left|\left\{a: Q^{\pi_u}(s, a)=0\right\}\right|$ be the number of zero-valued actions at state $s$, A(s) be the number of available actions at state $s$, and $P$ denotes the set of key states where both optimal and suboptimal actions exist, i.e., $P=\{s: 1 \leq N(s) \leq A(s)-1\}$. Given the softmax policy $\pi_s(a \mid s)=\frac{\exp \left(Q^{\pi_{u}}(s, a) / \rho\right)}{\sum_{a^{\prime}} \exp \left(Q^{\pi_{u}}\left(s, a^{\prime}\right) / \rho\right)}$ with temperature $\rho>0$, and $\operatorname{Pr}^{\pi_s}\left(s \mid s_0\right)$ is the probability of reaching $s$ from $s_0$ with the policy $\pi_s$, the value function of the induced policy $\pi_s$ satisfies: $ V^{\pi_s}\left(s_{0}\right) \geq R\left(1-\sum_{s \in P} \operatorname{Pr}^{\pi_s}\left(s \mid s_{0}\right) \frac{N(s)}{N(s)+\exp \left(\max _{a} Q^{\pi_u}(s, a) / \rho\right)}\right) $

Explanation of Theorem 2: This theorem quantifies the trade-off between quality and diversity managed by the temperature parameter $\rho$.

• $V^{\pi_s}(s_0)$ is the expected cumulative reward (value) of starting in state $s_0$ and following the softmax policy $\pi_s$.
• $R$ is the reward for a correct solution.
• The term $\sum_{s \in P} \operatorname{Pr}^{\pi_s}(s \mid s_0) \frac{N(s)}{N(s)+\exp \left(\max _{a} Q^{\pi_u}(s, a) / \rho\right)}$ represents the potential loss in value due to exploring suboptimal paths.
• N(s) is the number of "dead-end" actions (those with $Q^{\pi_u}(s, a)=0$) at state $s$.
• $\max_a Q^{\pi_u}(s, a)$ is the Q-value of the best action at state $s$.
• As $\rho \rightarrow 0$, the $\exp(\cdot/\rho)$ term for the maximum Q-value becomes very large relative to other terms, making the fraction $\frac{N(s)}{N(s)+\exp(\ldots)}$ very small. This causes the entire subtracted term to approach zero, and $V^{\pi_s}(s_0)$ approaches $R$ (the optimal value). This confirms that as temperature decreases, the softmax policy approaches the optimal greedy policy.
• The theorem thus guarantees that even with stochastic sampling for diversity, ROVER maintains a strong performance bound, and this bound tightens as $\rho$ decreases.

4.2.5. Practical Implementation (ROVER Algorithm)

The theoretical insights are translated into a practical and scalable algorithm for LLM math reasoning, ROVER, which is summarized in Algorithm 1.

Algorithm 1: Random Policy Valuation for Diverse Reasoning (ROVER)
Input: pre-trained LLM $\pi_{\theta}$, epochs M, prompt dataset $\mathcal{D}$, group size $n$, lr $\eta$, temperature $\rho$
for epoch $m=\{1, \cdots, M\}$ do
    Set $\pi_{\theta_{\text {old }}} \leftarrow \pi_{\theta}$; Sample a batch of prompts $\mathcal{B} \sim \mathcal{D}$ via $\pi_{\theta_{\text {old }}}$
    for each prompt $x \in \mathcal{B}$ do
        Rollout responses and compute rewards: $\left\{y_{i}\right\}_{i=1}^{n} \sim \pi_{\theta_{\text {old }}}(\cdot \mid x) ; \tilde{r}=r_{i}-\frac{1}{n} \sum_{i=1}^{n} r_{i}$
    for each prompt-response pair $\left\{x, y_{i}\right\}$ in batch do
        for each state $s_{t} \in\left\{x, y_{i}\right\}$ do
            Compute Q-value $Q\left(s_{t}, a_{t}\right) = \rho\left(\log \pi_{\theta}\left(a_{t} \mid s_{t}\right)-\log \pi_{\theta_{\text {old }}}\left(a_{t} \mid s_{t}\right)\right)$
            Obtain target $\hat{Q}\left(a_{t} \mid s_{t}\right) \leftarrow \tilde{r}+\frac{1}{|\mathcal{V}|} \sum_{a_{t+1} \in \mathcal{V}} Q\left(s_{t+1}, a_{t+1}\right)$
            // $\mathcal{V}$: the vocabulary set.
            $\mathcal{L}_{\text {ROVER }}=\frac{1}{\sum_{i=1}^{n}\left|y_{i}\right|} \sum_{i=1}^{n} \sum_{t=0}^{\left|y_{i}\right|-1}\left\|Q\left(s_{t}, a_{t}\right), \operatorname{sg}\left[\hat{Q}\left(s_{t}, a_{t}\right)\right]\right\|^{2}$
            // sg: stop gradient.
            $\theta \leftarrow \theta-\eta \widetilde{\nabla}_{\theta} \mathcal{L}_{\text {ROVER }}$ by an AdamW optimizer

Let's break down the algorithm step-by-step:

1. Initialization:
   • pre-trained LLM\pi_{\theta}

: The starting point is a pre-trained language model, which provides the initial policy $\pi_{\theta}$.
    *   `epochs M`, `prompt dataset`\mathcal{D}

, group size $n$, lr\eta,temperature $\rho$: These are standard hyperparameters. $n$ responses are rolled out for each prompt, $\eta$ is the learning rate, and $\rho$ is the temperature for the softmax policy.

2. Epoch Loop: The algorithm iterates for $M$ epochs.

3. Policy Snapshot:

   • Set\pi_{\theta_{\text {old }}} \leftarrow \pi_{\theta}

: At the beginning of each epoch, the current policy $\pi_{\theta}$ is saved as $\pi_{\theta_{\text {old}}}$. This `old policy` serves as a stable reference point for calculating the `relative Q-function` and for sampling data.

4.  **Data Sampling (Rollouts)**:
    *   `Sample a batch of prompts`\mathcal{B} \sim \mathcal{D}`via`\pi_{\theta_{\text {old}}}

: A batch of prompts is sampled from the dataset. * for each promptx \in \mathcal{B}do: For each prompt in the batch: * Rollout responses and compute rewards:{y_{i}}{i=1}^{n} \sim \pi{\theta_{\text {old}}}(\cdot \mid x)

: The `LLM` (using the `old policy` $\pi_{\theta_{\text{old}}}$) generates $n$ different responses $y_i$ for each prompt $x$.
        *

\tilde{r}=r_{i}-\frac{1}{n} \sum_{i=1}^{n} r_{i}

: For each generated response $y_i$, a `mean-centered reward` $\tilde{r}$ is computed. $r_i$ is the raw binary reward (0 or 1) for response $y_i$. Subtracting the average reward of the group reduces variance, making the reward signal more stable. This `centered reward` $\tilde{r}$ is then broadcasted to all tokens in the sequence.

5.  **Q-value Computation and Loss Calculation**:
    *   `for each prompt-response pair`\{x, y_{i}\}`in batch do`: The algorithm iterates through each generated `(prompt, response)` pair.
    *   `for each state`s_{t} \in\{x, y_{i}\}`do`: For each token (action $a_t$) in the response $y_i$ at state $s_t$:
        *   **Q-Parameterization**:
            $Q\left(s_{t}, a_{t}\right) = \rho\left(\log \pi_{\theta}\left(a_{t} \mid s_{t}\right)-\log \pi_{\theta_{\text {old }}}\left(a_{t} \mid s_{t}\right)\right)$
            This is a crucial step. Instead of a separate `Q-network`, the `Q-function` is parameterized *intrinsically* through the `LLM`'s parameters $\theta$. It represents the `Q-value` as a relative improvement of the current policy's log-probability over the `old policy`'s log-probability, scaled by `temperature` $\rho$. This formulation centers initial `Q-values` around zero and learns changes relative to a stable baseline, reducing instability.
        *   **Target Q-value Calculation**:
            $\hat{Q}\left(s_{t}, a_{t}\right) \leftarrow \tilde{r}+\frac{1}{|\mathcal{V}|} \sum_{a_{t+1} \in \mathcal{V}} Q\left(s_{t+1}, a_{t+1}\right)$
            This is the `Bellman target` for the `Q-value`. It combines the `mean-centered reward` $\tilde{r}$ (which is broadcasted to all tokens in the sequence) with the average `Q-value` of all possible next actions $a_{t+1}$ from the next state $s_{t+1}$. The summation $\frac{1}{|\mathcal{V}|} \sum_{a_{t+1} \in \mathcal{V}} Q\left(s_{t+1}, a_{t+1}\right)$ represents the expected future value under a `uniform random policy` (as per the theoretical foundation). `Q(`s_{t+1}, a_{t+1}`)` for the next step is computed using the same `relative Q-parameterization`.
        *   **Loss Function ($\mathcal{L}_{\text {ROVER}}$)**:
            $\mathcal{L}_{\text {ROVER }}=\frac{1}{\sum_{i=1}^{n}\left|y_{i}\right|} \sum_{i=1}^{n} \sum_{t=0}^{\left|y_{i}\right|-1}\left\|Q\left(s_{t}, a_{t}\right), \operatorname{sg}\left[\hat{Q}\left(s_{t}, a_{t}\right)\right]\right\|^{2}$
            This is a `Mean Squared Error (MSE)` loss function. It aims to minimize the difference between the current `Q-value` estimate $Q(s_t, a_t)$ and the `target Q-value` $\hat{Q}(s_t, a_t)$. The `sg` operator (`stop gradient`) ensures that gradients are not propagated through the `target Q-value`, treating it as a fixed value during optimization. This stabilizes the training process, similar to how target networks are used in `DQN`. The loss is averaged over all tokens in all responses in the batch.
        *   **Policy Update**:

\theta \leftarrow \theta-\eta \widetilde{\nabla}{\theta} \mathcal{L}{\text {ROVER}}by an AdamW optimizer: The LLM's parameters $\theta$ are updated using the AdamW optimizer to minimize the loss, moving $\theta$ in the direction of the negative gradient $\widetilde{\nabla}_{\theta} \mathcal{L}_{\text {ROVER}}$.

This implementation cleverly combines the theoretical insight of uniform policy valuation with practical techniques for LLM training, resulting in a stable and effective RL algorithm. The intrinsic Q-parameterization leverages the LLM's strong priors, and the mean-centered reward with broadcasting addresses the high variance of LLM rewards.

5. Experimental Setup

5.1. Datasets

The experiments are conducted on both a specific countdown task and a suite of math reasoning benchmarks.

• Countdown Tasks:

  • Dataset: TinyZero (Pan et al., 2025) dataset with 1,024 test problems.
  • Characteristics: Involves finding a sequence of arithmetic operations (+,-,×,÷) on a given array of numbers to reach a target number. It has a restricted search space and often multiple valid solutions, making it suitable for analyzing diversity.
  • Purpose: Used to evaluate the method's ability to handle tasks with multiple correct solutions and to robustly analyze reasoning behavior and diversity.

• Math Reasoning Benchmarks (Training and Evaluation):

  • Training Dataset: DeepScaler (Luo et al., 2025). This open-source dataset is used for training all models.
  • Evaluation Datasets: A comprehensive suite of widely-acknowledged math reasoning benchmarks are used for evaluation:
    • AIME24 (MAA, 2024) and AIME25 (MAA, 2025): Problems from the American Invitational Mathematics Examination (AIME), a highly challenging competition-level math contest.
    • HMMT25 (Balunović et al., 2025): Problems from the Harvard-MIT Mathematics Tournament (HMMT), another prestigious competition.
    • OlympiadBench (He et al., 2024): A benchmark designed for evaluating LLMs on problems similar to mathematical olympiads, testing advanced reasoning.
    • AMC23 (AI-MO, 2024): Problems from the American Mathematics Competitions (AMC).
    • MATH500 (Hendrycks et al., 2021): A challenging dataset of math problems requiring step-by-step reasoning.
  • Out-of-Distribution (O.O.D) Benchmark:
    • GPQA-diamond (Rein et al., 2024): This benchmark contains 198 graduate-level questions in biology, physics, and chemistry. It is math-unrelated and used to evaluate the generalization capability of the models beyond the training distribution of math problems.
  • Reward Mechanism: A binary reward is assigned by the open-source verification tool math_verify (Kydliček & Face, 2025) upon completion of LLM generation, indicating correctness (1 for correct, 0 for incorrect).

5.2. Evaluation Metrics

The paper employs several metrics to comprehensively evaluate both the quality and diversity of the generated solutions.

1. Pass@1:

   • Conceptual Definition: Pass@1 measures the proportion of problems for which at least one generated solution (out of a single trial) is correct. It is a direct measure of a model's quality or accuracy on a single attempt.
   • Mathematical Formula: $ \text{Pass@1} = \frac{\text{Number of problems with at least one correct solution}}{\text{Total number of problems}} \times 100% $
   • Symbol Explanation:
     • Number of problems with at least one correct solution: The count of unique problems where the model produced at least one correct output.
     • Total number of problems: The total count of problems in the evaluation set.

2. Pass@k:

   • Conceptual Definition: Pass@k extends Pass@1 by measuring the proportion of problems for which at least one correct solution is found within $k$ independent generation attempts. It is a crucial metric for evaluating reasoning breadth and diversity, as a model with higher diversity has a better chance of finding a correct path across multiple trials, even if its single-shot accuracy (Pass@1) is lower.
   • Mathematical Formula: The typical formula for Pass@k (from Chen et al., 2021, "Evaluating Large Language Models Trained on Code") is: $ \text{Pass@k} = \mathbb{E}_{\mathcal{P}} \left[ 1 - \frac{\binom{N-C}{k}}{\binom{N}{k}} \right] $ Where:
     • $\mathcal{P}$: The set of problems.
     • $N$: Total number of samples generated for a problem.
     • $C$: Number of correct samples among $N$ generated samples for a problem.
     • $k$: The number of attempts (trials).
     • $\binom{n}{r}$: The binomial coefficient, representing "n choose r", calculated as $\frac{n!}{r!(n-r)!}$. This formula calculates the probability that at least one out of $k$ samples is correct, given $C$ correct samples out of $N$ total samples for a problem. The expectation is taken over all problems.
   • Symbol Explanation:
     • $\mathbb{E}_{\mathcal{P}}[\cdot]$: Expected value over the set of problems $\mathcal{P}$.
     • $N$: Total number of generated responses per problem (e.g., 256 in the paper).
     • $C$: Number of correct responses for a given problem out of the $N$ generated responses.
     • $k$: The number of independent attempts or trials for which pass@k is being calculated.
     • $\binom{N-C}{k}$: The number of ways to choose $k$ incorrect responses from the N-C incorrect responses.
     • $\binom{N}{k}$: The total number of ways to choose $k$ responses from $N$ responses. The term $\frac{\binom{N-C}{k}}{\binom{N}{k}}$ is the probability of picking $k$ incorrect samples out of $N$ total samples. $1 - \text{this term}$ is the probability of picking at least one correct sample.

3. Number of Distinct Strategies:

   • Conceptual Definition: This metric quantifies reasoning diversity by identifying how many unique problem-solving approaches or strategies a model employs to solve a single problem. A higher number indicates greater diversity.
   • Methodology: The paper uses the method from NoveltyBench (Zhang et al., 2025c). It involves sampling a fixed number of correct responses (e.g., 32) for a problem. An LLM judger (Claude-3.5-Sonnet in this case) is then used to determine if two response pairs exhibit strategically equivalent reasoning.
   • Symbol Explanation: Not a mathematical formula, but a count of distinct clusters of reasoning paths identified by an external LLM judge.

4. Utility (from NoveltyBench):

   • Conceptual Definition: A metric from NoveltyBench that measures the overall usefulness or quality of the diverse solutions found. It might combine correctness with some measure of how distinct or valuable the alternative solutions are. (Specific formula not provided in the paper, but its context implies it's a diversity-related metric).

5. Cosine Distance:

   • Conceptual Definition: A metric used to quantify the semantic difference between reasoning paths or solutions. It measures the cosine of the angle between two vector embeddings of the solutions. A smaller cosine distance (closer to 1) means higher similarity, while a larger cosine distance (closer to 0 for positive vectors, or -1 for opposite vectors) indicates greater dissimilarity. When applied to diversity, a higher average cosine distance between different correct solutions implies greater reasoning diversity.
   • Mathematical Formula: For two non-zero vectors $A$ and $B$, the cosine similarity is: $ \text{cosine similarity}(A, B) = \frac{A \cdot B}{|A| |B|} = \frac{\sum_{i=1}^n A_i B_i}{\sqrt{\sum_{i=1}^n A_i^2} \sqrt{\sum_{i=1}^n B_i^2}} $ The cosine distance is often defined as $1 - \text{cosine similarity}$.
   • Symbol Explanation:
     • A, B: Vector representations (embeddings) of two reasoning paths/solutions.
     • $A \cdot B$: The dot product of vectors $A$ and $B$.
     • $\|A\|$: The Euclidean norm (magnitude) of vector $A$.

6. Maj@k (Majority Voting at k):

   • Conceptual Definition: Maj@k evaluates the performance of a majority voting ensemble. It represents the accuracy if one generates $k$ solutions and takes the answer that appears most frequently (the majority vote). If there's a tie, a random choice is made. This metric assesses test-time scalability and how well a model's diverse outputs converge to the correct answer. A high Maj@k implies that even if individual solutions are not always correct, the correct answer is frequently present among the multiple attempts and can be identified by consensus.
   • Mathematical Formula: Not explicitly provided but implies counting the most frequent answer among $k$ samples and checking its correctness.

5.3. Baselines

The paper compares ROVER against several established RLVR methods.

• GRPO (Group-Relative Policy Optimization) (Shao et al., 2024): A strong baseline specifically designed for LLM RLVR, which addresses reward variance through group-based advantage estimation.
• REINFORCE++ (Hu et al., 2025a): An enhanced policy gradient method, likely incorporating variance reduction techniques over the basic REINFORCE.
• DAPO (Diversity-Aware Policy Optimization) (Yu et al., 2025): A policy optimization variant that aims to improve diversity, often using techniques like clip-higher.
• Base Model: The pre-trained LLM without any RL fine-tuning serves as a crucial baseline to understand the improvement brought by RLVR methods. The paper uses Qwen2.5-3B, Qwen3-8B-Base, Qwen3-4B-Base, and DeepSeek-R1-Distill-Qwen-1.5B as base models.

6. Results & Analysis

6.1. Core Results Analysis

The experimental results consistently show ROVER outperforming all RL baselines across various tasks and model scales, particularly in terms of both solution quality (pass@1) and diversity (pass@k).

The following are the results from Table 1 of the original paper:

Pass@1 Performance:

• Consistent Superiority: ROVER consistently achieves the highest pass@1 scores across all mathematical benchmarks and for both Qwen3-4B-Base and Qwen3-8B-Base models. This indicates its strong ability to produce correct solutions on a single attempt.
• Significant Gains on Challenging Tasks: The improvements are particularly pronounced on the more challenging competition-level tasks like AIME24, AIME25, and HMMT25. For the Qwen3-8B-Base model, ROVER shows relative improvements of $+47.1$ on AIME24 (30.6 vs 20.8 for DAPO), $+35.9$ on AIME25 (22.7 vs 16.7 for $REINFORCE++$), and nearly doubles the performance on HMMT25 (14.6 vs 7.1 for $REINFORCE++$) compared to the best-performing baselines. This suggests ROVER is highly effective in tackling complex reasoning problems.
• Overall Average Improvement: For Qwen3-8B-Base, ROVER achieves an average pass@1 of 48.4, representing a $+19.6$ absolute improvement over the base model's 28.8, and a substantial lead over the best baseline (DAPO at 41.1).
• Generalization on O.O.D. tasks: ROVER also achieves the best performance on GPQA-diamond, a math-unrelated out-of-distribution benchmark. This indicates that the reasoning skills learned by ROVER are more robust and generalizable, not merely overfitting to the math domain.

6.2. Diversity Analysis

ROVER's strength lies not just in its quality but significantly in its ability to maintain and leverage diversity.

The following figure (Figure 9 from the original paper) shows the pass@k results of ROVER and baselines on Qwen3-8B-Base:

该图像是一个图表，展示了不同算法在不同 $k$ 值下的通过率（pass@$k$）。图中包含ROVER、DAPO、REINFORCE++、GRPO和Qwen3-BB-Base五种算法的性能对比，横轴为 $k$ 值，纵轴为通过率。各算法的性能随 $k$ 值增大而提升，ROVER算法的表现最佳。

(a) AIME 2024

该图像是一个图表，展示了不同算法在不同 k 值下的表现。其中 ROVER 算法用红色星形标记，DAPO 用紫色方形标记，REINFORCE++ 用蓝色三角形标记，GRPO 用绿色细线标记，Qwen3-BB-Base 用灰色菱形标记。图表显示随着 k 值增加，各算法的表现趋于提高，特别是 ROEVER 和 DAPO 算法的效果显著。

(b) AIME 2025

该图像是一个示意图，展示了多种强化学习方法在不同参数 $k$ 下的多样性表现。图中包含了ROVER、DAPO、REINFORCE++、GRPO和Qwen3-88-Base的曲线，显示了它们在多样性指标上的变化趋势。

(c) HMMT 2025

Pass@k Performance:

• Pass@k measures the probability of solving a problem within $k$ attempts, reflecting reasoning breadth and diversity.

• As shown in Figure 9, ROVER demonstrates sustained and significant performance gains as $k$ increases from 1 to 256. This is in stark contrast to standard RL baseline methods (GRPO, DAPO, REINFORCE++), which, while improving pass@1, often quickly saturate or even underperform the base model at large $k$ values. For example, DAPO's performance on AIME25 and HMMT25 drops below the base model for higher $k$ values, indicating mode collapse.

• ROVER achieves $+16.8$ improvement over the best baseline on pass@256 averaged across AIME24, AIME25, and HMMT25.

• This sustained improvement is particularly evident on the most challenging HMMT25 task (Figure 9c), where ROVER's pass@k score continues to accelerate, while baselines have plateaued. This highlights ROVER's ability to consistently explore and discover diverse correct reasoning paths.

• The paper attributes this to ROVER's higher policy entropy during training (see Figure 20 in Appendix), which enables sustained exploration of different reasoning strategies.

  The following figure (Figure 10 from the original paper) shows the Quality-Diversity tradeoff:

  该图像是一个散点图，展示了不同方法在质量（Pass@1）上的表现。ROVER（标记为红星）在图中位于较高的质量值，而其他方法如GRPO、REINFORCE++和DAPO分布在较低的质量区域。

  Number of Distinct Strategies and Quality-Diversity Tradeoff:

• Figure 10 (main body, with more detailed plots in Appendix) quantifies reasoning diversity using the "number of distinct strategies" metric from NoveltyBench. This metric relies on an LLM judger (Claude-3.5-Sonnet) to classify distinct problem-solving approaches.

• ROVER achieves a relative diversity improvement of $+6.8$ over GRPO and $+20.5$ over the average of all three baselines.

• Crucially, ROVER consistently improves the Pareto front between quality (e.g., pass@1 or overall success rate) and diversity. This means it can achieve higher quality while simultaneously maintaining greater diversity, a critical advantage over conventional RL approaches that often struggle to achieve both.

  The following figure (Figure 7 from the original paper) provides a visualization example to demonstrate the solution diversity of ROVER:

  该图像是一个散点图，展示了不同策略在测试分数和多样性（离散解数量）上的表现。图中标注了多种策略，包括GRPO w/o KL、GRPO w/ KL-coef=0.002和GRPO w/ Clip_higher，ROVER（我们的方法）在图中用红星标出，显示出相对较高的质量和多样性。

(c) Diversity&Quality

• Figure 7c (Diversity&Quality for Countdown task) visually reinforces this. ROVER is positioned in the top-right quadrant, indicating both high test score (quality) and high diversity (number of distinct solution equations). For instance, it successfully finds 17 diverse solution equations, while other methods find only 3 distinct ones for the countdown task.

6.3. Ablation Studies / Parameter Analysis

The paper includes an ablation study on the temperature parameter (\rho) for the softmax action selection.

The following figure (Figure 8 from the original paper) shows the performance under different $\rho$ values:

该图像是图表，展示了不同参数下 ROVER 算法的熵和测试得分在训练步骤中的变化。左侧图表显示了熵的下降趋势，右侧图表展示了平均测试得分随步骤的变化。不同颜色的线条代表不同参数设置，清晰地展示了这些因素对结果的影响。

(a) Entropy

该图像是图表，展示了不同参数下ROVER算法的训练效果。纵轴为算法性能，横轴为训练步骤。不同颜色的曲线代表了不同的时间参数 $t = 1, 0.1, 0.001, 3$，可见随着训练步骤的增加，性能逐渐下降。

(b) Test Score

Ablation on Temperature $\rho$ (Countdown Task):

• Role of $\rho$: The temperature $\rho$ in the softmax policy controls the exploration-exploitation trade-off. Lower $\rho$ values lead to more greedy, deterministic behavior (exploitation), while higher values promote diverse sampling (exploration).
• Optimal $\rho=1$: The study (Figure 8) confirms that $\rho=1$ achieves a robust and desirable performance. This choice aligns with common LLM sampling practices and requires no task-specific tuning.
• Impact of High $\rho$: A higher temperature causes under-exploitation and slower convergence, as the policy becomes too random and doesn't sufficiently prioritize promising paths.
• Impact of Low $\rho$: A lower temperature (e.g., $\rho=0.001$) triggers premature exploitation, leading to an accelerated collapse in policy entropy (Figure 8a) and a constrained exploration space. This results in severe training instability and poor test scores (Figure 8b). The near-deterministic policy fails to explore sufficiently.
• Entropy Maintenance: Figure 8a clearly shows that ROVER with $\rho=1$ maintains significantly higher entropy compared to lower $\rho$ values, allowing for sustained exploration. This directly correlates with better performance.

6.4. Behavioral Analysis

The following figure (Figure 11 from the original paper) shows Maj@k performance of ROVER and baselines on Qwen3-8B-Base:

该图像是图表，展示了不同算法在数学推理任务中的表现。左侧图表显示了多种算法（如ROVER、DAPO、REINFORCE++和GRPO）在不同参数k下的性能变化，右侧图表呈现了另一个指标maJ@k的评估结果。每个算法的表现通过不同颜色和标记进行区分，便于比较。

Test-time Scaling with Maj@k:

• Maj@k (majority voting at $k$ samples) evaluates how well a model scales at test time by aggregating multiple generations.

• Figure 11 demonstrates that ROVER's Maj@k performance scales robustly, consistently improving upon the base model across all $k$ values, even on the challenging HMMT25 task.

• This superior scalability is attributed to ROVER's ability to maintain a diverse distribution of valid reasoning paths. In contrast, baseline methods suffer from mode collapse, where they confidently converge on similar incorrect solutions. This prevents Maj@k from yielding significant performance gains, as multiple incorrect solutions are voted for. ROVER provides a richer set of diverse, correct solutions, making majority voting more effective.

  Enhanced Reflection Behaviors (Forking Tokens):

• The paper analyzes forking tokens (tokens associated with rethinking, self-correction, or exploring alternative paths) in the generated outputs.

• ROVER-trained models generate a significantly higher proportion of these forking tokens (e.g., 'wait', 'however') compared to baselines. This is detailed in Figure 12 (in Appendix).

• This suggests that ROVER encourages the model to actively reflect, verify, and pivot between different reasoning strategies, rather than rigidly committing to a single path. This behavioral characteristic directly supports its observed diversity and ability to find novel solutions.

• Figure 15 (in Appendix) provides an example from AIME24 where ROVER discovers two additional novel strategies not found by the base model or GRPO-trained models, showcasing its potential to push the reasoning boundary.

6.5. Additional Results on Countdown Task

The following figure (Figure 6 from the original paper) shows test scores, entropy, and diversity&quality on the countdown task:

该图像是一个图表，展示了不同政策优化方法在训练过程中的平均测试得分（Avg@64 Test Score）随步骤变化的趋势。图中包含ROVER方法及不同参数设置下GRPO的表现，显示了在训练步骤增加时，测试得分的提升情况。

(a) Test Score

该图像是一个图表，展示了不同算法在训练过程中的熵变化。曲线分别表示ROVER（红色）、GRPO w/o KL（蓝色）和不同KL系数的GRPO（绿色和棕色）的表现，横轴为步数，纵轴为熵值，反映了算法的稳定性和多样性。

(b) Entropy

该图像是一个散点图，展示了不同策略在测试分数和多样性（离散解数量）上的表现。图中标注了多种策略，包括GRPO w/o KL、GRPO w/ KL-coef=0.002和GRPO w/ Clip_higher，ROVER（我们的方法）在图中用红星标出，显示出相对较高的质量和多样性。

(c) Diversity&Quality

• Figure 6(a) shows ROVER reaching the highest ceiling performance in terms of test score on the Countdown task, surpassing all baselines after 400 training steps.
• Figure 6(b) illustrates that ROVER's policy entropy decays gracefully while remaining significantly higher than baselines, which either collapse or fluctuate erratically. This high entropy is key to ROVER's sustained exploration and diversity.
• Figure 6(c) combines Diversity and Quality for the Countdown task, showing ROVER leading in both aspects.

7. Conclusion & Reflections

7.1. Conclusion Summary

The paper introduces ROVER (Random Policy Valuation for Diverse Reasoning), a novel and minimalist Reinforcement Learning with Verifiable Rewards (RLVR) algorithm specifically designed for LLM reasoning tasks. By identifying and exploiting the unique structural properties of LLM math reasoning MDPs (finite-horizon, deterministic, tree-structured, binary terminal rewards), the authors provide a theoretical proof that the optimal action can be derived from the Q-function of a fixed uniformly random policy. This allows ROVER to bypass the traditional, complex, and often unstable Generalized Policy Iteration (GPI) loop used by methods like PPO and GRPO.

ROVER implements this principle by intrinsically parameterizing the Q-function within the LLM and using a softmax over these uniform-policy Q-values for action sampling, which effectively balances solution quality with diversity. Practical enhancements like group reward centering and reward broadcasting further stabilize training.

Empirical results across diverse math reasoning benchmarks (e.g., AIME, HMMT) and model scales consistently demonstrate ROVER's superior performance in both pass@1 (quality) and pass@k (diversity), as well as improved generalization on O.O.D tasks. Its ability to maintain high policy entropy, encourage diverse reasoning strategies, and scale effectively at test time (Maj@k) highlights its robustness and effectiveness.

7.2. Limitations & Future Work

The authors acknowledge the following limitations of their work:

• Computational Resources: The experiments were limited to math reasoning tasks and models up to 8B parameters due to restricted computational resources. This implies that the scalability and performance on much larger LLMs (e.g., 70B, 100B+) or more complex, open-ended reasoning tasks (beyond structured math problems) are yet to be fully explored.

• Task Domain: The theoretical foundation relies on specific MDP properties (deterministic transitions, tree-structured state space, binary terminal rewards). While this holds for many structured reasoning tasks, it might not directly apply to RLVR problems with stochastic transitions (e.g., environments with external uncertainty), continuous rewards, or graph-structured state spaces (e.g., complex interactive environments where states can be reached via multiple paths or cycles).

  Potential future work could involve:

• Investigating ROVER's applicability to a broader range of LLM tasks, especially those with more complex MDP characteristics.

• Scaling ROVER to larger LLMs to confirm its benefits in higher-capacity models.

• Exploring mechanisms to extend ROVER's principles to MDPs with non-binary rewards or some level of stochasticity, potentially through approximations or hybrid approaches.

7.3. Personal Insights & Critique

This paper presents a truly elegant and impactful idea. The core insight – that a complex RL problem can be dramatically simplified by carefully analyzing the underlying MDP structure – is a powerful reminder of the importance of theoretical foundations in algorithm design. The move from Generalized Policy Iteration to mere Random Policy Valuation is a paradigm shift for this specific domain, offering a more stable and robust alternative to existing PPO-based methods.

One of the most appealing aspects of ROVER is its simplicity. In a field increasingly characterized by intricate architectures, complex loss functions, and endless heuristic tuning, ROVER demonstrates that a minimalist approach, grounded in sound theory, can yield superior results. This simplification not only reduces implementation complexity but also likely contributes to its observed stability and generalization capabilities.

The emphasis on diversity is also highly valuable. For LLMs, finding multiple correct ways to solve a problem is often more important than just finding one, as it enhances robustness, interpretability, and the ability to generalize or adapt to new, subtly different problems. ROVER's intrinsic mechanism for diversity, stemming from the probabilistic interpretation of Q-values of a uniform policy, is more principled than many ad-hoc diversity-promoting techniques.

Potential areas for further exploration or critique:

• The Universality of MDP Assumptions: While the paper clearly states the MDP assumptions, it would be beneficial to have a more detailed discussion on which LLM tasks precisely fit these assumptions and which do not. For instance, tasks requiring external tool use or interaction with dynamic, unpredictable environments might break the deterministic or tree-structured assumptions.

• Computational Cost of Summation: The target Q-value calculation involves a summation over the entire vocabulary set $|\mathcal{V}|$ for each action $a_t$ ($$\sum_{a_{t+1} \in \mathcal{V}} Q\left(s_{t+1}, a_{t+1}\right)$$). For very large vocabularies, this could be computationally intensive. While the paper states ROVER is scalable, further insights into how this summation is efficiently handled (e.g., sampling-based approximation, or if $|\mathcal{V}|$ is not excessively large in practice for sub-token LLM generation) would be helpful.

• Temperature Sensitivity: Although the ablation shows $\rho=1$ is robust for the countdown task, the optimal temperature might still vary for different LLM base models or vastly different task complexities. A deeper analysis of how to dynamically or adaptively set $\rho$ could be interesting.

• Comparison to Beam Search: The softmax sampling with temperature can be seen as a form of diverse beam search. A more direct comparison of ROVER's sampling strategy against advanced diverse beam search methods (which are common in LLM decoding for diversity) might provide further insights into its unique advantages beyond just the RL training loop.

  Overall, ROVER represents a significant step towards developing more efficient, stable, and diverse RL methods for LLM reasoning by grounding the approach in the specific structural properties of the problem. Its simplicity and strong empirical performance make it a compelling contribution to the field.