Ghost Hypergradients: Value-Gap Oracles for Optimistic Bilevel Learning

Optimistic bilevel learning is fragile when the lower-level problem has many minimizers: differentiating an arbitrary selected response can follow a solver-dependent branch rather than the validation-preferred one. We study a value-function alternative based on a vanishing validation tilt. Let $G$ be the lower-level training value and $V_{\tau }$ the value after adding $\tau$ times the validation loss to the lower objective. The scaled value gap ${\Delta }_{\tau }=(V_{\tau }-G)/\tau$ converges, under standard compactness and continuity conditions, to the optimistic objective that minimizes validation loss over the training-optimal set. Building on value-function penalties and equilibrium propagation, we make this perturbation identity algorithmic: by differencing conservative adjoint selections for $V_{\tau }$ and $G$, we obtain ghost hypergradients, exact conservative oracles for the ideal finite-tilt surrogate that avoid differentiating a set-valued argmin map. Read at one shared random outer point, their exact scaled difference is almost surely a surrogate gradient, and a certificate error bound lets solve residuals control the computed direction. We prove local surrogate-error bounds, sharpened to $O({\tau }^{\theta /(1-\theta )})$ for $0<\theta <1$, projected-stationarity convergence, and inexact-solve bounds, exposing the bias–accuracy tradeoff in $\tau$. On an exact flat-valley diagnostic, Ghost preserves the optimistic direction while selected-response gradients are misaligned. In our empirical study, Ghost achieves the best final test accuracy/runtime tradeoff among the compared bilevel oracles. These results support value gaps as practical first-order oracles for nonsmooth optimistic bilevel learning.