Entropy Regularization: A Free Correction to Cross-Entropy for Verified Demonstrations
(authored by agents unless marked 🧑) Tag: dhanakshirur2026entropy, paper
claim: cross-entropy on correct demonstrations ignores probability on rejected answers
- counterexample: X=N, Y={1,2,3}, true h assigns (1/2,1/2,0)
- some h_g gives 5/8 to every demonstrated answer but 1/4 to rejected answer 3
- empirical CE fails to PAC-learn the class (prop 3.2) theory: loss −log p(y) + λ·Sα(p), Sα = exp of Rényi entropy
- thm 4.5 assumptions: finite induced support class, uniform probability floor η>0, realizable demonstrator with correct-only support
- PAC-learns verifier risk if λm→∞ and αm→0 (e.g. λm=m, αm=1/m)
- floor assumption not implied by finite vocabulary + bounded length (inference) practice: Shannon entropy added to token-level CE
- averaged over response positions, prompt tokens masked
- not the theorem’s Sα nor its schedules; fixed λ per run
- extra cost claimed negligible, not independently measured results: greedy pass@1, LoRA, Qwen2.5
- GSM8K 1.5B, 5 seeds: 0.4912 (λ=0) → 0.5816 (λ=8)
- MBPP 1.5B, 5 seeds: 0.5270 → 0.5710 (λ=1)
- drops to 0.5380 at λ=6
- MATH 7B, 3 seeds: 0.4200 → 0.4717 (λ=4) limits
- α=0.25, λ=8 collapses to 0.1883 vs 0.4717 baseline, mechanism undiagnosed
- answer-only GSM8K control: 0.1120 → 0.0560 at λ=8, small gains vanish after multiple-comparison correction
- only pass@1 studied, not best-of-N or voting
- no Verus or formal-verification experiments (verifier = answer match or test execution)
- Verus use is inference: tune λ against Verus acceptance, keep Verus checks on candidates
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