DPO Optimal Policy & Implicit Reward Proof

The Direct Preference Optimization (DPO) Closed-Form Proof mathematically derives the exact algebraic equivalence between parameterized policies and implicit ground-truth reward functions; Step 1: optimizing the KL-constrained RLHF objective $\max_\pi \mathbb{E}[r(x, y)] - \beta D_{\text{KL}}(\pi \| \pi_{\text{ref}})$ yields the closed-form optimal Boltzmann policy $\pi^*(y|x) = \frac{1}{Z(x)} \pi_{\text{ref}}(y|x) \exp(\frac{1}{\beta} r(x, y))$; Step 2: re-arranging log-terms algebraically expresses the ground-truth reward as $r(x, y) = \beta \ln \frac{\pi^*(y|x)}{\pi_{\text{ref}}(y|x)} + \beta \ln Z(x)$; Step 3: substituting into the Bradley-Terry preference model $P(y_w \succ y_l|x) = \sigma(r(x, y_w) - r(x, y_l))$ cancels out the intractable partition function $Z(x)$ completely, deriving a closed-form binary cross-entropy loss without training explicit reward models.

AI research lab whiteboard theoretical interviews, preference alignment mathematics, and variational loss engineering.

Eliminates unstable two-stage PPO actor-critic training by proving policy log-ratios are isomorphic to ground-truth reward margins under Bradley-Terry preferences.

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