Matrix-Decoupled Concentration for Autoregressive Sequences: Dimension-Free Guarantees for Sparse Long-Context Rewards

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with \emph{any} valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive \emph{reasoning} model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.

Policy gradient methods for Large Language Models optimize a policy $π_θ$ via a surrogate objective computed from samples of a rollout policy $π_{\text{roll}}$. However, modern LLM-RL pipelines suffer from unavoidable implementation divergences -- backend discrepancies, Mixture-of-Experts routing discontinuities, and distributed training staleness -- causing off-policy mismatch ($π_{\text{roll}} \neq π_θ$) and approximation errors between the surrogate and the true objective. We demonstrate that classical trust region bounds on this error scale as $O(T^2)$ with sequence length $T$, rendering them vacuous for long-horizon tasks. To address this, we derive a family of bounds -- both KL-based and TV-based -- including a Pinsker-Marginal bound ($O(T^{3/2})$), a Mixed bound ($O(T)$), and an Adaptive bound that strictly generalizes the Pinsker-Marginal bound via per-position importance-ratio decomposition. Taking the minimum over all bounds yields the tightest known guarantee across all divergence regimes. Crucially, all bounds depend on the maximum token-level divergence $D_{\mathrm{KL}}^{\mathrm{tok,max}}$ (or $D_{\mathrm{TV}}^{\mathrm{tok,max}}$), a sequence-level quantity that cannot be controlled by token-independent methods like PPO clipping. We propose Trust Region Masking (TRM), which masks entire sequences violating the trust region, enabling the first non-vacuous monotonic improvement guarantees for long-horizon LLM-RL.

The alignment of Large Language Models (LLMs) for complex reasoning heavily relies on Reinforcement Learning with Verifiable Rewards (RLVR). However, standard algorithms like GRPO apply sequence-level rewards uniformly to all tokens, creating a severe credit-assignment bottleneck. While on-policy self-distillation attempts to resolve this by conditioning a self-teacher on privileged contexts, direct exposure to raw oracle solutions often induces over-conditioned teacher distributions, implicit answer leakage, and late-stage training collapse. To overcome these limitations, we propose Asymmetric Meta-Reflective Self-Distillation (AMR-SD). Instead of conditioning directly on raw reference traces, AMR-SD inserts a reflection bottleneck: it compresses diagnostic signals -- from verifier outcomes, peer rollouts, or reference feedback -- into concise, self-generated Socratic hints and critiques. Furthermore, we introduce Causal Information Gain (CIG) with an asymmetric, ReLU-gated threshold to translate these reflections into sparse, highly precise token-level advantage modulations. Combined with temporal annealing, this mechanism preserves the base environmental reward while filtering out distributional noise. Experiments across scientific, mathematical, and tool-use benchmarks demonstrate that AMR-SD significantly outperforms existing baselines, achieving robust long-horizon stability and successfully preventing late-stage collapse.