Self-Improvement as Coherence Optimization: A Theoretical Account

Mechanisms for continued self-improvement of language models without external supervision remain an open challenge. We propose Peer-Predictive Self-Training (PST), a label-free fine-tuning framework in which multiple language models improve collaboratively by using a cross-model aggregate response as an internal training signal. Given a prompt, models generate responses sequentially; the final aggregated answer, which is often more reliable than individual responses in practice, serves as an internal reference for learning. We measure how informative each intermediate response is about the aggregate using pointwise mutual information (PMI), and use this signal to scale self-training updates: responses already aligned with the aggregate receive smaller updates, while less informative or misaligned responses receive larger ones. On mathematical reasoning benchmarks, including SimulEq, MATH-500-Numeric, and MultiArith, PST improves exact-match accuracy by 2.2--4.3 percentage points across Gemma-2-2B, LLaMA-3.2-1B, and Qwen2.5-1.5B, and reduces the average generator--verifier gap (GV-Gap) by 26--40%, while requiring no external supervision, no teacher--student hierarchy, and only cross-model interactions. These results suggest that peer-predictive feedback from cross-model generations can provide an effective mechanism for self-supervised language-model improvement.

Despite ever-increasing sophistication in language model (LM) pre- and post-training pipelines, many important failures persist: models overcondition on user framing ("sycophancy"), exhibit incomplete logical generalization, and produce confident but incorrect responses. We argue that these failures arise from a modeling assumption permeating all aspects of the pipeline: that behavior can be specified and evaluated independently on single-output pairs. Many model failures are difficult, if not impossible, to detect without reasoning about relationships between a model's responses across inputs. In this position paper, we propose self-consistency as a framework for understanding these failures. We first observe that a wide variety of techniques designed to improve specific aspects of LM behavior-targeting properties as diverse as adversarial robustness and factual coherence-can be understood as special cases of a common "consistency optimization" procedure and addressed with a standard set of optimization tools. We next outline a set of new model properties that could be achieved by optimizing for consistency, and conclude with a discussion of what it would mean to develop generally consistent LMs, including the capabilities they would enable and the objections they raise.

Recent work suggests that large language models (LLMs) can improve through self-evolution (SE), using supervision signals generated by the model itself. In this work, we ask: under a strict closed-loop setup, where the self-evolution algorithm has access only to an unlabeled prompt set and a base model, how close can internally generated supervision come to oracle-supervised training? We analyze four representative strategies in a unified offline self-evolution framework: single-round verification, multi-turn revision with feedback, iterative training, and curriculum learning. Our primary experiments use Knights and Knaves (KK) logical reasoning tasks, which provide deterministic solutions, controlled difficulty levels, and a clean testbed for easy-to-hard generalization. We first show that self-evolution consistently improves over the base model, but plateaus after excessive training compute is invested, and eventually still leaves a non-trivial gap to oracle supervision. We find that multi-turn critic-revision with large models can reach strong self-evolution performance, with Gemma 12B nearly matching oracle-supervised training. Beyond Knights and Knaves, we also evaluate self-evolution on real-world reasoning benchmarks, where gains are also modest. Overall, our results characterize when closed-loop self-evolution can help and show how internally generated supervision remains insufficient under this minimal formulation.