Richness Is Not Evenness: Ordered Renyi Crossings under Self-Training

Pranjal Srivastava

Recursive self-training of language models, a setting studied in model collapse, can reduce vocabulary while making frequent words, mostly function words, less dominant. We fine-tune English GPT-2 on its own output for nine generations, vary the training-data sampling temperature, and measure every model with one fixed sampler. As training temperature rises, Renyi entropy changes turn from loss to gain at distinct temperatures: min-entropy first, then collision entropy, Shannon entropy and observed richness. Between the outer crossings, word types fall while the most-common word becomes less dominant. Untruncated training at T = 1 lies in this interval: at 13,000 words, generation 9 has 24% fewer types than generation 0 yet collision entropy is 0.91 bits higher. The ordering persists with a finer grid, different measurement samplers, a fixed word budget and five fresh training seeds. Min-entropy again crosses before richness in four further settings (in Hindi only at interpolated roots) and in 12 of 13 distinct measurements under eight text representations. Next-token identities and a categorical simulation isolate decoding and sampling effects; the crossing order remains…