# Non-Stationary Functional Bilevel Optimization

arXiv: https://arxiv.org/abs/2601.15363 · HTML: https://mlsanity.com/paper/2601.15363/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 49 references that are in the map.
- First posted on arXiv 21 Jan 2026.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [Invariant Risk Minimization Games](https://mlsanity.com/paper/2002.04692/) (2020)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Tracking Slowly Moving Clairvoyant: Optimal Dynamic Regret of Online Learning with True and Noisy Gradient](https://mlsanity.com/paper/1605.04638/) (2016)
- [Amortized Implicit Differentiation for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2111.14580/) (2021)
- [Auxiliary Learning by Implicit Differentiation](https://mlsanity.com/paper/2007.02693/) (2020)
- [Control-Oriented Model-Based Reinforcement Learning with Implicit Differentiation](https://mlsanity.com/paper/2106.03273/) (2021)
- [Efficient Regret Minimization in Non-Convex Games](https://mlsanity.com/paper/1708.00075/) (2017)
- [Value-Function-based Sequential Minimization for Bi-level Optimization](https://mlsanity.com/paper/2110.04974/) (2021)
- [Functional Bilevel Optimization for Machine Learning](https://mlsanity.com/paper/2403.20233/) (2024)
- [Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods](https://mlsanity.com/paper/2207.02829/) (2022)
- [A Bilevel Approach for Parameter Learning in Inverse Problems](https://mlsanity.com/paper/1802.01365/) (2018)
- [Non-Convex Bilevel Optimization with Time-Varying Objective Functions](https://mlsanity.com/paper/2308.03811/) (2023)
- [Trilevel and Multilevel Optimization using Monotone Operator Theory](https://mlsanity.com/paper/2105.09407/) (2021)
- [Online Nonconvex Bilevel Optimization with Bregman Divergences](https://mlsanity.com/paper/2409.10470/) (2024)
- [Time-Varying Multi-Objective Optimization: Tradeoff Regret Bounds](https://mlsanity.com/paper/2211.09774/) (2022)

## Built on by (most cited first, 2 of 2)

- [Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization](https://mlsanity.com/paper/2602.06457/) (2026)
- [Fully First-Order Algorithms for Online Bilevel Optimization](https://mlsanity.com/paper/2602.11665/) (2026)

## Abstract (arXiv)

Functional bilevel optimization (FBO) provides a powerful framework for hierarchical learning in function spaces, yet current methods are limited to static offline settings and perform suboptimally in online, non-stationary scenarios. We propose SmoothFBO, the first algorithm for non-stationary FBO with both theoretical guarantees and practical scalability. SmoothFBO introduces a time-smoothed stochastic hypergradient estimator that reduces variance through a window parameter, enabling stable outer-loop updates with sublinear regret. Importantly, the classical parametric bilevel case is a special reduction of our framework, making SmoothFBO a natural extension to online, non-stationary settings. Empirically, SmoothFBO consistently outperforms existing FBO methods in non-stationary hyperparameter optimization and model-based reinforcement learning, demonstrating its practical effectiveness. Together, these results establish SmoothFBO as a general, theoretically grounded, and practically viable foundation for bilevel optimization in online, non-stationary scenarios.