Escaping the Variance Trap: Jacobian-Free Dynamics for Root-Finding Bilevel Optimization
The authors identify a critical flaw termed the Variance Trap, which arises when stochastic root-finding problems are forced into minimization frameworks via squared residuals. Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians that act as noise amplifiers in stochastic settings. To address this, the paper formalizes Root-Finding Bilevel Optimization (RF-BO) as a distinct problem class to bypass this pathology. A Jacobian-free solution using Two-Time-Scale Stochastic Approximation (TTSA) is proposed to update directly along the root error. The study provides the first non-asymptotic convergence guarantees for TTSA in this setting under Markovian noise. Experiments show a 2.6% top-1 accuracy gain in SimCLR and 17x faster convergence in non-linear ODE control compared to baselines. Additionally, the framework achieves significantly improved entropy stability in reinforcement learning and an 11.1% quality improvement in generative modeling.