Generative Recursive Reasoning

Abstract: How should future neural reasoning systems implement extended computation? Recursive Reasoning Models (RRMs) offer a promising alternative to autoregressive sequence extension by performing iterative latent-state refinement with shared transition functions. Yet existing RRMs are largely deterministic, following a single latent trajectory and converging to a single prediction. We introduce \emph{Generative Recursive reAsoning Models (GRAM)}, a framework that turns recursive latent reasoning into probabilistic multi-trajectory computation. GRAM models reasoning as a stochastic latent trajectory, enabling multiple hypotheses, alternative solution strategies, and inference-time scaling through both recursive depth and parallel trajectory sampling. This yields a latent-variable generative model supporting conditional reasoning via $p_θ(y \mid x)$ and, with fixed or absent inputs, unconditional generation via $p_θ(x)$. Trained with amortized variational inference, GRAM improves over deterministic recurrent and recursive baselines on structured reasoning and multi-solution constraint satisfaction tasks, while demonstrating an unconditional generation capability. \href{https://ahn-ml.github.io/gram-website/}{https://ahn-ml.github.io/gram-website}

• The paper introduces GRAM, a novel probabilistic framework that models recursive reasoning as a stochastic process to explore multiple latent trajectories.
• It employs a dual-loop architecture with deterministic proposals and learnable Gaussian perturbations to decouple reasoning depth from output length.
• GRAM achieves superior performance in tasks like Sudoku-Extreme and N-Queens, demonstrating effective multi-solution exploration and structured reasoning.

Generative Recursive Reasoning: A Latent Probabilistic Paradigm for Recursive Neural Reasoning

Motivation and Context

The dominant paradigm in neural reasoning has largely revolved around autoregressive architectures, which extend outputs by sequentially generating tokens or latent embeddings. While successful for many applications, this approach intrinsically conflates computational depth with output length and often fails to efficiently address problems involving multi-solution ambiguity, structured constraints, or the need for extended internal computation. Recursive Reasoning Models (RRMs) offer an alternative by refining a latent state through iterative computation via shared transition functions, decoupling depth from parameter count and output length. However, prior RRMs such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM) are fundamentally deterministic: they always follow a single refinement trajectory, collapsing multiple plausible reasoning paths into a single solution.

To address these limitations, the paper introduces Generative Recursive reAsoning Models (GRAM), a novel framework that generalizes recursive reasoning into a probabilistic latent trajectory setting. GRAM treats recursive computation as a stochastic process, enabling parallel exploration of multiple latent reasoning trajectories, maintaining alternative hypotheses, and supporting both conditional and unconditional generation.

Model Architecture and Training Formulation

GRAM models the conditional probability $p_\theta(y \mid x)$ by marginalizing over stochastic latent reasoning trajectories. For a given input $x$, an embedding is computed and used throughout the recursion. The model operates in two nested loops:

• Inner Loop: For $T$ steps, the latent state is updated with a stochastic transition. Instead of the deterministic updates of earlier RRMs, GRAM computes a deterministic proposal and then adds a learnable, state-dependent Gaussian perturbation.
• Outer Loop: $N_{\text{sup}}$ supervision steps are chained recursively, with the terminal state of one step serving as the initial state for the next.

The model instantiates a hierarchical latent state $(h, l)$, where $h$ is a high-level state updated stochastically at the outer loop, and $l$ is a low-level state refined deterministically in the inner loop. This multi-scale decomposition enables both rapid local refinement and higher-level trajectory exploration.

Figure 1: GRAM samples diverse stochastic latent reasoning trajectories, enabling recovery of multiple solutions in tasks with ambiguous or under-constrained inputs, in contrast to deterministic RRMs which collapse to a single trajectory.

During training, GRAM is formulated as a latent-variable model trained via amortized variational inference, maximizing a trajectory-level evidence lower bound (ELBO). The variational posterior conditions on input and target, while the generative prior conditions solely on the input. Training leverages truncated gradient propagation across supervision steps for scalability.

Figure 2: Both the true ELBO and surrogate training objective monotonically improve during training, validating the effectiveness of truncated deep supervision.

Inference-Time Scaling and Multi-Hypothesis Exploration

A core innovation is GRAM’s dual axis of inference-time scaling:

• Depth: Extending the number of recursive transitions or supervision steps, as in prior RRMs.
• Width: Sampling multiple latent trajectories in parallel, enabling the model to consider many possible solutions or hypotheses simultaneously.

Selection among candidate outputs is achieved via majority voting or a learned Latent Process Reward Model (LPRM) that predicts the final output’s accuracy from its latent state. This parallel exploration capability is unattainable with deterministic RRMs.

Figure 3: GRAM supports superior inference-time scaling in both recursion depth and sample width, with parallel sampling yielding performance improvements unattainable for deterministic predecessors.

Empirical Results: Structured and Generative Tasks

Structured Reasoning Benchmarks

GRAM demonstrates consistently superior performance over deterministic baselines (Looped TF, HRM, TRM) on challenging structured reasoning tasks such as Sudoku-Extreme and the ARC-AGI challenge. Key empirical findings include:

• Sudoku-Extreme: GRAM achieves 97.0% accuracy, substantially outperforming all deterministic RRMs, for which the best competitor (TRM) attains 87.4%. Notably, large pretrained LLMs attain 0% in this benchmark, underscoring the unique advantage afforded by recursive computation and multi-trajectory exploration.
• ARC-AGI: The gains persist on abstract visual reasoning tasks, where GRAM again outperforms deterministic recursive baselines.

  Figure 4: GRAM yields higher accuracy on both Sudoku and ARC-AGI, showcasing the practical advantage of stochastic latent transitions in recursive reasoning.

Multi-Solution Constraint Satisfaction

GRAM excels in multi-solution settings such as N-Queens and Graph Coloring, maintaining accuracy and coverage even as the number of valid solutions increases—addressing a failure mode that is structural for deterministic RRMs:

• N-Queens: Deterministic RRMs exhibit catastrophic degradation in both accuracy and coverage as solution multiplicity increases; GRAM sustains 99.7% accuracy and up to 90.3% solution coverage (20 samples, 8x8), outperforming masked diffusion and autoregressive baselines in constraint satisfaction.
• Graph Coloring: GRAM achieves a near-order-of-magnitude reduction in constraint violations relative to autoregressive transformers for 10-node graphs.

(Figure 1, right) (Figure 3, right)

Figure 3: Unlike deterministic recursive models, GRAM maintains robust accuracy as the number of possible solutions increases, demonstrating effective non-collapsed exploration.

Unconditional Generation

GRAM extends its probabilistic recursive computation to unconditional generative modeling. On binarized MNIST:

• Unconditional Generation: GRAM’s Inception Score improves monotonically with increased inference steps (1.85→2.04, 8→256 steps), with FID also dropping accordingly (84.0→73.3).
• Structured Sample Generation: In unconditional Sudoku generation, GRAM achieves 99.05% validity, significantly outperforming discrete diffusion models with fewer parameters and orders-of-magnitude fewer generation steps.

  Figure 5: GRAM refines unconditional image samples over recursive steps, correcting initial errors and improving generative quality via stochastic latent transitions.

  

  Figure 6: GRAM can synthesize highly diverse, fully valid Sudoku boards unconditionally, without any explicit constraint checking.

Ablation Analyses and Qualitative Visualization

Ablation studies confirm that both stochasticity and learned guidance are essential. The absence of either diminishes performance or induces mode collapse, especially on multi-solution tasks. Attempts to introduce naive stochasticity in deterministic RRMs fail to replicate the benefits of GRAM, highlighting the necessity of a variational framework for effective exploration.

Visualization of latent trajectories reveals that GRAM explores and covers larger regions of the solution landscape than deterministic models, directly supporting its empirical superiority on multi-solution and ambiguous tasks.

Figure 7: TRM (deterministic) follows a single trajectory, often failing to escape local minima in the latent loss landscape.

Figure 8: GRAM (with 50 samples) explores diverse regions, with several trajectories reaching optimal low-loss areas.

Implications, Limitations, and Future Directions

GRAM’s generative recursive reasoning paradigm presents significant implications for both practical neural reasoning and theoretical modeling:

• Structured Reasoning: By facilitating robust exploration over solution spaces and supporting parallel hypothesis evaluation, GRAM is well suited for domains requiring constraint satisfaction, combinatorial search, or reasoning under ambiguity.
• Efficient Inference-Time Scaling: The decoupling of reasoning “width” (number of latent samples) from parameter count and “depth” (recursion length) offers a new handle for tuning compute vs. performance without retraining or increasing model size.
• Generative Modeling: The ability to view recursive computation as a generative process enables transfer beyond reasoning to unconditional generation, with promising results in both symbolic and perceptual domains.

However, GRAM’s reliance on sequential deep supervision introduces training inefficiencies compared to fully parallel Transformers, posing challenges for scaling to higher-capacity or longer-horizon problems. Additionally, the risks of generating plausible but invalid structures, and the computational burden of multi-sample inference, warrant further practical study. Future research is likely to investigate memory-efficient training schemes, adaptive sampling methods, scaling laws under varying recursion and width, integration with external search routines, and applications in domains such as scientific discovery, program synthesis, and model-based planning.

Conclusion

Generative Recursive reAsoning Models (GRAM) establish probabilistic multi-trajectory recursion as a powerful design principle for neural reasoning architectures. By introducing stochastic latent transitions into recursive computation and leveraging variational training, GRAM attains superior structured reasoning performance, recovers diverse solutions in multi-solution settings, and enables unconditional generation—all while supporting new axes of computational scaling at inference. The work lays the foundation for future architectures that treat reasoning as a process over distributions of latent trajectories, providing both practical tools for structured tasks and theoretical advances in neural reasoning systems.