How Neural Reward Models Learn Features for Policy Optimization: A Single-Index Analysis

Reward modeling is not only a prediction problem: in KL-regularized policy optimization, the learned reward is exponentiated to define the deployed policy, so downstream value depends on errors in reward-tilted regions. We study this feedback in a Gaussian single-index model with $ r^* (x)=\sigma^* (\langle\theta^* ,x\rangle)$ and $x\sim\mathcal N(0,I_d)$. We analyze a two-stage neural reward model that first learns the hidden direction $\theta^*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression. Exponential reward weighting changes the Hermite signal available to the first layer; for any feature-learning temperature $\beta_1$ above a dimension-free $O(1)$ threshold, a constant fraction of neurons recover the hidden direction, with weak-recovery complexity governed by the generative exponent. After feature recovery, we derive tilted-policy value-gap bounds for an idealized label-weighted fit with weights $e^{y/\beta_2}$ and a more practical surrogate-weighted fit with weights $e^{r_{a_0}(x)/\beta_2}$. Keeping the $\beta_2$-dependence explicit yields an admissible set of deployment temperatures, balancing the gain from lowering $\beta_2$ against the learning cost amplified by exponential weighting; in the surrogate-weighted case, proxy-dependent factors shrink this admissible set.