Error-budgeted adaptive rollout allocation

(1) Integration point: modify the stochastic-target data loader rather than the network architecture. For every interior training point $x$, dynamically choose rollout depth $M(x)$ and path count $K(x)$ before constructing its regression target. The same allocation can be used during validation, where computation should concentrate on difficult regions. (2) Pseudocode: begin with a pilot rollout of depth $M_{0}$ and $K_{0}$ paths. Compute returns $Y_i(x)$, mean $\bar Y(x)$, and variance $\widehat v(x)$. Choose a stopping radius $\delta$, an estimated truncation constant $\widehat C_{\mathrm{trunc}}$, and domain diameter $D_{\mathrm{diam}}$. Set $M(x)=\lceil\log(\widehat C_{\mathrm{trunc}}/\varepsilon_{\mathrm{trunc}})/\log(1+\widehat\beta^{2}\delta^{2}/(2D_{\mathrm{diam}}^{2}))\rceil$, clipped to $[M_{0},M_{\max}]$. Set $K(x)=\max(K_{0},\lceil\widehat v(x)/\varepsilon_{\mathrm{MC}}^{2}\rceil)$, clipped to $[K_{0},K_{\max}]$. Reuse pilot paths and draw only missing paths. Optionally weight the squared loss by $1/(\widehat v(x)+\eta)$, with small $\eta>0$, to prevent noisy targets from dominating. (3) Compute from the mathematics: geometric cutoff, sphere times, truncation schedule, and functional dependence on $M$ and $K$. Estimate empirically: $\widehat\beta$, error constants, target variance, and rollout cost. Maintain variance estimates with an exponential moving average over distance-to-boundary bins. (4) First experiment: use a four-dimensional unit-ball Poisson problem with known analytic solution. Compare fixed $(M,K)$ against adaptive allocation at equal total rollout evaluations. Log test error, target variance, average $M$ and $K$, and wall-clock time. Success means the adaptive method reaches the same error with at least 25 percent fewer rollouts, or gives lower error at equal budget without unstable minibatch losses.