Reinforcement Learning (RL) with verifiable environments has emerged as a powerful approach for enhancing the reasoning capabilities of Large Language Models (LLMs). While prior research demonstrates that scaling environment quantity improves RL performance, existing manual or individual construction methods suffer from linear scaling limits, thereby hindering scalable reasoning generalization. This paper introduces RACES (\textbf{R}ecursive \textbf{A}utomated \textbf{C}omposition for \textbf{E}nvironment \textbf{S}caling), a framework that conceptualizes verifiable environments as composable building blocks that can be recursively assembled. The key insight is that when the codomain (output type) of one environment matches the domain (input type) of another, they can be automatically fused into a new verifiable environment, enabling recursive composition. RACES is implemented with 300 individual environments and defines a set of composition operators (\textsc{SEQUENTIAL}, \textsc{PARALLEL}, \textsc{SORT}, and \textsc{SELECT}) that induce diverse reasoning patterns. Extensive experiments show that RL training on these composite environments consistently enhances reasoning generalization. Specifically, RACES improves DeepSeek-R1-Distill-Qwen-14B by an average of 3.1 points (from 48.2 to 51.3) and boosts Qwen3-14B performance from 58.8 to 61.1 on six benchmarks, which are unseen during the construction of training environments. Moreover, RACES achieves performance comparable to training on 300 individual environments using only 50 base environments, demonstrating significant efficiency in environment utilization.
We pursue a vision for self-improving language models in which the model does not merely generate problems or traces to imitate, but constructs the environments that train it. In zero-data reasoning RL, this reframes self-improvement from a data-generation loop into an environment-construction loop, where each artifact is a reusable executable object that samples instances, computes references, and scores responses. Whether this vision sustains improvement hinges on a single property: the environments must exhibit stable solve--verify asymmetry, the model must be able to write an oracle once that it cannot reliably execute in natural language on fresh instances. This asymmetry takes two complementary forms. Some tasks are algorithmically hard to reason through but trivial as code: a dynamic program or graph traversal, compiled once, yields unboundedly many calibrated instances. Others are intrinsically hard to solve but easy to verify, like planted subset-sum or constraint satisfaction. Both create a durable gap between proposing and solving that the policy cannot close by gaming the verifier, and it is this gap that keeps reward informative as the learner improves. We instantiate this view in EvoEnv, a single-policy generator, solver method that synthesizes Python environments from ten seeds and admits them only after staged validation, semantic self-review, solver-relative difficulty calibration, and novelty checks. The strongest evidence comes from the already-strong regime: on Qwen3-4B-Thinking, fixed public-data RLVR and fixed hand-crafted environment RLVR reduce the average, while EvoEnv improves it from 72.4 to 74.8, a relative gain of 3.3%. Stable self-improvement, we suggest, depends not on producing more synthetic data, but on models learning to construct worlds whose difficulty stays structurally beyond their own reach.
While reinforcement learning with verifiable rewards (RLVR) is widely used to improve the reasoning capabilities of large language models (LLMs), the generalizability of the resulting models remains poorly understood. In this work, we establish the first non-vacuous generalization bounds for parameter-efficient RLVR fine-tuning at the billion-parameter scale. Our approach adapts PAC-Bayes compression bounds to this setting, and addresses the inherent stochasticity of token generation by applying the Gumbel-max reparameterization trick. To operationalize these bounds, we propose the Progressive RLVR framework, which integrates RLVR with on-policy distillation, TinyLoRA, and model quantization. Progressive RLVR empirically retains 84-97% performance of standard LoRA fine-tuning while producing models that are 14,796x more compressible. We show that this framework yields non-vacuous generalization bounds in four domains: mathematical problem-solving, programming, general-knowledge reasoning, and Text-to-SQL. Our bounds exceed the accuracy of the base model by 9-51% and lie within 6-11% of the accuracy of the fine-tuned models.
Reinforcement learning with verifiable rewards (RLVR) has become a leading paradigm for improving the reasoning ability of large language models through outcome-based supervision. However, verifiable rewards frequently become uninformative at the group level: when all sampled traces of a given prompt receive identical rewards, group-relative advantage estimation provides no gradient signal, even though the traces may differ substantially in reasoning quality. We propose Reasoning Arena, an adaptive training framework that routes such non-diverse reward groups to a judge system instead of discarding them. Beyond examining the final answer, Reasoning Arena constructs trace tournaments, where reasoning traces are compared head-to-head to expose finer-grained preferences within the group, converting reasoning quality into rich relative reward signals. To make reward estimation efficient, rather than exhaustively comparing every pair, each new trace is evaluated against a small, dynamically updated pool of previously generated traces as anchors to efficiently establish a relative ranking. We then fit a Bradley-Terry model on the incomplete comparison graph, enabling scalable RL integration without quadratic pairwise comparisons. Empirical results demonstrate that Reasoning Arena consistently outperforms the RLVR baseline by 7.6% on average in competition mathematics and coding benchmarks. By converting otherwise wasted zero-advantage samples into useful gradient updates, our method accelerates training by 27% to 41%, saving nearly 50% of generation compute, and substantially improves overall reasoning performance.