Multi-head Transformers Provably Learn Symbolic Multi-step Reasoning via Gradient Descent (2508.08222v1)

Abstract: Transformers have demonstrated remarkable capabilities in multi-step reasoning tasks. However, understandings of the underlying mechanisms by which they acquire these abilities through training remain limited, particularly from a theoretical standpoint. This work investigates how transformers learn to solve symbolic multi-step reasoning problems through chain-of-thought processes, focusing on path-finding in trees. We analyze two intertwined tasks: a backward reasoning task, where the model outputs a path from a goal node to the root, and a more complex forward reasoning task, where the model implements two-stage reasoning by first identifying the goal-to-root path and then reversing it to produce the root-to-goal path. Our theoretical analysis, grounded in the dynamics of gradient descent, shows that trained one-layer transformers can provably solve both tasks with generalization guarantees to unseen trees. In particular, our multi-phase training dynamics for forward reasoning elucidate how different attention heads learn to specialize and coordinate autonomously to solve the two subtasks in a single autoregressive path. These results provide a mechanistic explanation of how trained transformers can implement sequential algorithmic procedures. Moreover, they offer insights into the emergence of reasoning abilities, suggesting that when tasks are structured to take intermediate chain-of-thought steps, even shallow multi-head transformers can effectively solve problems that would otherwise require deeper architectures.

• The paper demonstrates that shallow multi-head transformers, trained via gradient descent, can provably learn complex symbolic multi-step reasoning tasks on path-finding problems.
• The methodology introduces distinct transformer constructions: a single-head model for backward reasoning and a two-head model for forward reasoning with autonomous stage control.
• The paper provides non-asymptotic optimization analyses and generalization guarantees, showing that explicit chain-of-thought traces can substitute depth for symbolic reasoning.

Provable Multi-step Symbolic Reasoning in Shallow Multi-head Transformers

Introduction

This paper rigorously analyzes the learnability and mechanistic implementation of symbolic multi-step reasoning in shallow (one-layer) multi-head transformers trained via gradient descent. The focus is on path-finding tasks in trees, which require chain-of-thought (CoT) reasoning and autonomous subtask control. Two intertwined tasks are considered: backward reasoning (goal-to-root path extraction) and forward reasoning (root-to-goal path extraction, which necessitates reversing the backward path). The work provides explicit transformer constructions, non-asymptotic optimization analyses, and generalization guarantees, demonstrating that even shallow multi-head architectures can provably learn and generalize complex sequential reasoning procedures when equipped with sufficient CoT steps.

Problem Formulation and Task Structure

The symbolic reasoning tasks are defined on trees with randomly assigned node embeddings. The backward reasoning task requires the model to output the path from a leaf (goal node) to the root, traversing parent-child relationships recursively. The forward reasoning task is strictly harder: the model must first solve the backward task, then reverse the resulting path, and crucially, learn to autonomously switch reasoning stages without explicit supervision.

The input to the transformer consists of edge embeddings, root and goal node embeddings, and, for the forward task, stage indicator tokens. Reasoning is performed autoregressively, with the output at each step concatenated to the input for the next step, enabling explicit multi-step CoT traces.

Figure 1: Node ordering in a perfect binary tree of depth $m=3$ used for training and analysis.

Transformer Constructions for Reasoning

Backward Reasoning

A single-head, one-layer transformer suffices for backward reasoning. The construction leverages linearly independent node embeddings and a trainable matrix $B$ such that $A^\top B A = \alpha I_S$ (where $A$ is the embedding matrix and $\alpha$ is a scaling parameter). This yields a sharp attention pattern: the query attends only to itself, enabling exact parent retrieval at each step. The output sequence reconstructs the goal-to-root path via recursive attention and output concatenation.

Forward Reasoning

Forward reasoning requires a two-head, one-layer transformer. The first head specializes in path traversal, while the second head acts as a stage controller, monitoring the current reasoning phase and triggering the transition from backward to forward reasoning upon root detection. The construction uses stage tokens ($s_b$, $s_f$) and block-structured parameter matrices, ensuring that each head autonomously specializes and coordinates. The output sequence first generates the backward path as a scratchpad, then reverses it, with the stage controller head switching the output format at the correct turning point.

Figure 1: Node ordering in a perfect binary tree, illustrating the recursive structure leveraged by the transformer.

Optimization Dynamics and Generalization

Gradient Descent Convergence

The paper provides non-asymptotic convergence analyses for both tasks under orthonormal (e.g., one-hot) embeddings and uniform training distributions over perfect binary trees. For backward reasoning, the diagonal entries of $H = A^\top B A$ grow while off-diagonal entries remain small, provably yielding sharp attention and zero training loss in $\widetilde{O}(1/\epsilon)$ iterations.

Figure 2: Training and test loss curves for backward reasoning, showing rapid convergence and generalization.

Figure 3: Training dynamics of selected entries of $H$, illustrating diagonal dominance and off-diagonal suppression.

For forward reasoning, the multi-phase training dynamics are tracked for all parameter blocks. The two heads' parameters converge to the explicit construction, with specialization and coordination emerging autonomously. The required number of iterations is $\widetilde{O}(1/\epsilon^{3/2})$, reflecting the increased complexity of multi-stage reasoning.

Figure 4: Training and test loss curves for forward reasoning, validating optimization and generalization.

Figure 5: Training dynamics of selected entries of $U_l, V_l$ for $l=1,2,3$, showing specialization and coordination of attention heads.

Generalization Guarantees

Both constructions provably generalize to unseen trees, with test loss bounded by a function of the path length and number of nodes. The transformer learns the underlying algorithmic rule for path-finding, not mere memorization. The generalization bounds scale favorably with tree size and path length, and empirical results confirm zero test loss on diverse tree structures.

Mechanistic Insights and Implications

The analysis reveals that multi-head attention enables autonomous specialization and coordination, even in shallow architectures, when sufficient CoT steps are available. The stage controller head learns to monitor and trigger subtask transitions, while the traversal head implements recursive path extraction. This mechanistic division of labor emerges naturally from gradient-based optimization, without explicit architectural constraints.

The results challenge the necessity of deep architectures for complex reasoning tasks, showing that reasoning trace length (CoT steps) can substitute for depth when intermediate steps are explicitly generated. This aligns with empirical findings in LLMs, where longer CoT traces unlock emergent reasoning abilities.

Practical and Theoretical Implications

• Model Design: Shallow multi-head transformers can be deployed for symbolic reasoning tasks if CoT traces are explicitly generated and multi-head specialization is enabled.
• Training Protocols: Autoregressive, step-by-step supervision is sufficient for learning complex sequential procedures, provided input embeddings are well-structured.
• Generalization: The learned mechanisms are algorithmic and generalize to unseen structures, supporting robust deployment in symbolic domains.
• Scaling: The required number of training iterations scales polynomially with tree size and path length, making the approach practical for moderate-sized symbolic tasks.

Future Directions

• Extension to Arbitrary Graphs: The analysis can be extended to more general graph structures, with attention mechanisms adapting to richer relational dependencies.
• Multi-stage Reasoning in LLMs: Insights from the mechanistic specialization of heads may inform interpretability and controllability in large-scale models.
• Architectural Variants: Investigating the minimal architectural requirements for autonomous subtask control and specialization in other neural architectures.

Conclusion

This work provides a rigorous theoretical and empirical foundation for the learnability and mechanistic implementation of symbolic multi-step reasoning in shallow multi-head transformers. By leveraging explicit CoT traces and multi-head specialization, even one-layer architectures can provably solve and generalize complex sequential reasoning tasks via gradient descent. The findings elucidate the emergence of algorithmic reasoning abilities and inform both model design and interpretability in symbolic domains.