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

arXiv: https://arxiv.org/abs/2606.22433 · HTML: https://mlsanity.com/paper/2606.22433/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 37 of its 49 references that are in the map.
- First posted on arXiv 21 Jun 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 30 of 37)

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- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Reward Constrained Policy Optimization](https://mlsanity.com/paper/1805.11074/) (2018)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
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- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping](https://mlsanity.com/paper/2005.10785/) (2020)
- [A framework for bilevel optimization that enables stochastic and global variance reduction algorithms](https://mlsanity.com/paper/2201.13409/) (2022)
- [Revisiting and Advancing Fast Adversarial Training Through The Lens of Bi-Level Optimization](https://mlsanity.com/paper/2112.12376/) (2021)
- [Amortized Implicit Differentiation for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2111.14580/) (2021)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2002.01268/) (2020)
- [A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse](https://mlsanity.com/paper/2112.04660/) (2021)
- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion](https://mlsanity.com/paper/2302.03775/) (2023)
- [On Finding Small Hyper-Gradients in Bilevel Optimization: Hardness Results and Improved Analysis](https://mlsanity.com/paper/2301.00712/) (2023)
- [Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance](https://mlsanity.com/paper/2011.01868/) (2020)
- [Implicit differentiation of Lasso-type models for hyperparameter optimization](https://mlsanity.com/paper/2002.08943/) (2020)

## Abstract (arXiv)

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× 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.