Escaping the Variance Trap: Jacobian-Free Dynamics for Root-Finding Bilevel Optimization

• URL: http://arxiv.org/abs/2606.22433v1
• Date: Sun, 21 Jun 2026 10:53:32 GMT
• Title: Escaping the Variance Trap: Jacobian-Free Dynamics for Root-Finding Bilevel Optimization
• Abstract summary: Many central machine learning tasks are fundamentally root-finding problems rather than minimization loss.<n> squared residuals introduce a critical flaw we identify as the Variance Trap.<n>Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians.<n>We propose a Jacobian-free solution using Two-TimeScale Approximation (TTSA) that updates directly along the root error.
• Score: 16.106678611834102
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Many central machine learning tasks, from entropy tuning in reinforcement learning to equilibrating generative adversarial networks, are fundamentally stochastic root-finding problems rather than loss minimization. Yet, they are frequently forced into a minimization framework via squared residuals, introducing a critical flaw we identify as the Variance Trap. Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians; in stochastic settings, these terms act as noise amplifiers, destabilizing convergence. We formalize Root-Finding Bilevel Optimization (RF-BO) as a distinct problem class that bypasses this pathology. We propose a Jacobian-free solution using Two-Time-Scale Stochastic Approximation (TTSA) that updates directly along the root error, structurally avoiding variance amplification. We provide the first non-asymptotic convergence guarantees for TTSA in this setting under Markovian noise. Extensive experiments demonstrate the decisive advantage of this paradigm: compared to squared-residual and implicit-gradient baselines, our framework achieves a 2.6\% top-1 accuracy gain in SimCLR, 17$\times$ faster convergence in non-linear ODE control where baselines fail, significantly improved entropy stability in reinforcement learning, and an 11.1\% quality improvement in generative modeling.

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