Recent advances in model distillation show that data from advanced reasoning models can effectively train smaller student models. However, standard practices discard incorrect reasoning traces—valuable, yet underutilized data. This paper addresses the critical question: How can both positive and negative distilled reasoning traces be effectively leveraged to maximize LLM reasoning performance in an offline setting? We employ a two-stage training recipe: first, Supervised Fine-Tuning (SFT) on positive traces, followed by a refinement stage using both positive and negative traces. We find that a simple REINFORCE-style objective, which we term the Reinforcement Distillation (REDI) objective, outperforms established preference optimization methods like DPO and SimPO in this distillation context. Our empirical evaluations demonstrate the effectiveness of this approach. Notably, our Qwen-REDI-1.5B model, trained on just 131k traces from the open Open-R1 dataset, achieves an 83.1% score on MATH-500. Its performance matches that of DeepSeek-R1-Distill-Qwen-1.5B, a model trained on 800k proprietary data. This result showcases the remarkable data efficiency of utilizing previously discarded negative traces.

Characterizing the expressive power of the Transformer architecture is critical to understanding its capacity limits and scaling law. Recent works provide the circuit complexity bounds to Transformer-like architecture. On the other hand, position embedding has emerged as a crucial technique in modern large language models, offering superior performance in capturing positional information, which shows great performance for the long context scenario. In this work, we take a circuit complexity perspective and rigorously analyze Transformers augmented with widely adopted positional embeddings. We prove that, under standard complexity assumptions, such models remain incapable of efficiently solving canonical tasks such as arithmetic formula evaluation and Boolean formula value computation. Our results expose a fundamental expressivity limitation that persists despite the remarkable empirical success of positionally-enhanced Transformers. Beyond tightening known complexity bounds, our findings offer new theoretical insights for designing future architectures with provably stronger reasoning and compositional capabilities.