RLVR landscapes for iterated multiplications can be benign: Insights from spin-glass theory

2026-09-23

AI Summary

This study addresses the fundamental challenge of reinforcement learning (RL) in acquiring novel reasoning capabilities for algorithmic tasks, investigating whether this difficulty stems from rugged optimization landscapes or inherent training obstacles. By mapping entropy-regularized RL with verifiable rewards (RLVR) to a spin glass model for the first time, and combining tabular policy analysis with end-to-end Transformer training, we rigorously examine the optimization landscape of iterative multiplication tasks. Theoretically, we prove that this landscape is free of spurious local minima traps, revealing that diffusion barriers and gradient errors constitute the primary bottlenecks, and propose corresponding modulators to mitigate them. Empirically, we demonstrate that Transformers trained from scratch can successfully learn iterative chain-of-thought reasoning over non-abelian groups using solely terminal token rewards.

Abstract

Despite the importance of reinforcement learning with verifiable rewards (RLVR), the extent to which it can learn new reasoning capabilities remains debated. Here we study the optimization landscape of RLVR on algorithmic tasks, such as iterated group and quasigroup multiplication. To this end, we map entropy-regularized RLVR over myopic tabular policies onto an energy-based (spin-glass) model over deterministic policies. This mapping upper-bounds what RLVR can achieve, and lets us rigorously characterize the landscape in this tabular setting. We show, both theoretically and experimentally, that for a wide class of models and tasks with uncorrelated inputs, this landscape is benign, containing no local minima that could trap RLVR training. Rather, the practical difficulty of these tasks appears to stem, at least in part, from issues such as diffusive barriers and gradient-estimation error in traversing the landscape. These are genuine obstacles that can prevent a solution from being found, but they are distinct from the landscape itself being rugged. We show that these obstacles can often be mitigated through the choice of entropy regulator. Consistent with this theory, we find that a transformer trained from scratch, using only last-token rewards, successfully learns an algorithmic chain of thought for iterated non-Abelian group multiplications.

Problem

• Reinforcement Learning with Verifiable Rewards
• optimization landscape
• iterated multiplication
• spin-glass theory
• local minima

Innovation

• Reinforcement Learning with Verifiable Rewards
• Spin-Glass Theory
• Optimization Landscape
• Entropy Regularization
• Algorithmic Chain of Thought