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

A new arXiv paper (2609.28625v1) maps entropy-regularized reinforcement learning with verifiable rewards (RLVR) over myopic tabular policies onto a spin-glass energy model and shows that, for a wide class of models and tasks with uncorrelated inputs, the optimization landscape is benign and contains no local minima that could trap RLVR training. The authors report that practical difficulty on tasks such as iterated group and quasigroup multiplication instead stems from diffusive barriers and gradient-estimation error, obstacles they say can often be mitigated through the choice of entropy regulator. Consistent with the theory, a transformer trained from scratch using only last-token rewards learned an algorithmic chain of thought for iterated non-Abelian group multiplications.

arXiv:2609.28625v1 Announce Type: new 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.