Stochastic Bilevel Optimization

We study nonconvex--strongly-convex bilevel optimization with finite-sum objectives and fully first-order oracles. We propose Sampled Implicit Corrections for Bilevel Optimization (SICBO), which combines a penalty approximation with variance-reduced corrections for the lower-level variables. Under componentwise lower strong convexity, SICBO finds a point $x$ satisfying $\mathbb{E}\|\nablaΦ(x)\|^2\le\varepsilon^2$ within $\tilde{O}(N+\sqrt{N}κ^{7/2}\varepsilon^{-2})$ component-gradient calls, where $N$ is the total number of components and $κ$ is the normalized lower-level condition number. The bound achieves square-root finite-sum dependence using only gradients and preserves the best known deterministic fully first-order condition-number exponent for smooth problems. Our improvement follows from a curvature-dependent bound for nonlinear correction errors, which permits epochs of length $\tildeΘ(\sqrt{N/κ})$ with batches of size $\tildeΘ(\sqrt{Nκ})$. A single-level reduction shows that the dependence on $N$ and $\varepsilon$ is optimal up to logarithmic factors for fixed $κ$.

Recent advances in penalty-based methods for stochastic bilevel optimization (SBO) have eliminated the need for second-order derivative oracles. However, for stochastic nonconvex-strongly convex bilevel problems, existing first-order methods typically rely on nested loops and/or large batch sizes for attaining $O(ε^{-6})$ or $O(ε^{-4})$ sample complexity under standard bounded-variance assumption or mean-square smoothness assumption. Achieving these rates with a single-loop penalty method and a constant batch size remains challenging due to a large penalty value needed for an accurate approximation. To address this challenge, we develop a stochastic SIngle-loop COnstant-Batch first-order penalty method (SICO) that combines two complementary ingredients. First, it performs one stochastic-gradient update per-iteration for both the original lower-level and penalized problems, with a projection that controls the separation between their iterates. Second, it applies an exponential moving average to stabilize the upper-level gradient estimator. We show that this combination achieves $O(ε^{-6})$ sample complexity using only $O(1)$ stochastic-gradient samples per iteration under unbiased, bounded-variance stochastic gradients. Under the additional mean-square smoothness assumption on the lower-level stochastic gradients, the same algorithm improves the complexity to $O(ε^{-4})$ also with $O(1)$ batch size. To the best of our knowledge, this is the first work to match the best-known convergence rate for fully first-order SBO methods using a single loop and a constant batch size. This result addresses an open problem posed in the literature.

In recent years, federated stochastic bilevel optimization has attracted increasing attention due to its wide range of applications in machine learning. To reduce the computational overhead associated with second-order Hessian and Jacobian matrices, several first-order methods have been proposed. However, existing methods typically impose restrictive assumptions on the lower-level function, suffer from a strong dependence on the condition number in their convergence rates, and require different learning-rate scales for variables across the upper- and lower-level problems, limiting their practical applicability and complicating hyperparameter tuning. To address these challenges, we propose a stochastic doubly smoothed gradient method for nonconvex federated stochastic bilevel optimization problems, which decouples the learning rates of upper- and lower-level variables and does not require a strongly-convex lower-level loss function. We establish rigorous theoretical guarantees for the proposed algorithm, demonstrating an improved convergence rate of $O(κ^{15/2}/ε^5)$ and a communication complexity of $O(κ^{4}/ε^3)$, where $κ$ denotes the condition number and $ε$ represents the solution accuracy. Notably, these bounds exhibit significantly better dependence on the condition number $κ$ than those of existing methods. Extensive experiments validate the effectiveness of our algorithm.

We study nonconvex--strongly-convex bilevel optimization under a stochastic first-order oracle. We introduce MRT-FD, a single-loop first-order method that simultaneously tracks the upper-level variable, the lower-level solution, and the auxiliary response arising from implicit differentiation of the hyperobjective. MRT-FD performs one update of each variable per iteration and approximates the second-order derivative actions using order-$p$ finite differences. For any fixed finite smoothness order $p\ge1$ in the lower-level variable, MRT-FD finds an $\varepsilon$-stationary point using $\mathcal{O}(\varepsilon^{-4-2/p})$ stochastic gradient queries. We also prove a matching $Ω(\varepsilon^{-4-2/p})$ oracle lower bound. The lower-bound construction starts from a hard nonconvex minimization chain with a stronger stochastic oracle, and lifts it to a bilevel problem through a sinusoidal coupling with a scalar lower-level variable. Consequently, the dependence on $\varepsilon$ is optimal for every fixed finite $p$, closing the upper--lower complexity gap in this stochastic first-order oracle setting.

Federated stochastic bilevel optimization has been actively studied in recent years due to its widespread applications in machine learning. However, most existing federated stochastic bilevel optimization algorithms require the computation of second-order Hessian and Jacobian matrices, which leads to longer running times in practice. To address these challenges, we propose a novel federated stochastic variance-reduced bilevel gradient descent algorithm that relies solely on first-order oracles. Specifically, our approach does not require the computation of second-order Hessian and Jacobian matrices, significantly reducing running time. Furthermore, we introduce a novel learning rate mechanism, i.e., a constant single-timescale learning rate, to coordinate the update of different variables. We also present a new strategy to establish the convergence rate of our algorithm. Finally, the extensive experimental results confirm the efficacy of our proposed algorithm.

We investigate stochastic simple bilevel optimization with smooth and possibly nonconvex upper- and lower-level objectives. Existing stochastic extensions of dynamic barrier gradient descent (DBGD) either obtain fast convergence under an unverifiable trajectory-dependent ``rare-visit'' assumption, or remove this assumption at a substantially higher oracle cost. We show that a simple denominator-only regularization of the DBGD multiplier eliminates the need for such an assumption while preserving fast convergence rates. Specifically, our method achieves $(\varepsilon, \varepsilon)$-stationarity in $O(\varepsilon^{-2})$ iterations using $O(\varepsilon^{-4})$ upper-level and $O(\varepsilon^{-7})$ lower-level stochastic gradients, which improves upon the best assumption-free complexities. We additionally derive anytime parameter schedules.

In this work, we study the oracle complexity of finding an $ε$-stationary point for nonconvex-strongly-convex (NC-SC) bilevel optimization using only first-order oracles. Existing methods achieving the best-known complexity guarantees typically rely on double-loop, penalty-based procedures. We propose a novel single-loop algorithm based on a constrained reformulation in which lower-level stationarity is imposed as a constraint. Specifically, we construct a regularized Lagrangian by introducing a quadratic regularizer and restricting the dual variable to a bounded domain, and then apply Smoothed Gradient Descent Ascent [Zhang et al., 2020], with Hessian-vector products approximated via finite differences of gradients. We refer to the resulting deterministic and stochastic algorithms as SGHA and Stoc-SGHA, respectively. In the deterministic setting, SGHA achieves an oracle complexity of $O(\barκ_y^{5}ε^{-2})$, where $\barκ_y$ denotes the relevant condition number. In the stochastic setting, Stoc-SGHA achieves an oracle complexity of $O\left(\barκ_y^{17}ε^{-6}ρ^{-3}\right)$ with probability at least $1-ρ$ for any $ρ\in(0,1)$, and an oracle complexity of $O\left(\barκ_y^{17}ε^{-6}\right)$ in expectation under an additional bounded-iterate assumption. Moreover, under an additional stochastic smoothness assumption imposed only on the lower-level objective, the stochastic oracle complexity of Stoc-SGHA improves to $O\left(\barκ_y^{11}ε^{-4}ρ^{-2}\right)$ with high probability and $O\left(\barκ_y^{11}ε^{-4}\right)$ in expectation, matching the $ε$-dependence of the lower bounds.

Bilevel optimization (BLO) is fundamental to hierarchical decision-making but suffers from critical instability under heavy-tailed stochastic noise. Existing variance-reduction techniques typically rely on myopic magnitude checks, which fail to distinguish informative geometric signals from impulsive outliers. To resolve this, we propose RQ-TTSA (Robust Quantile-guided TTSA), a distribution-aware framework that leverages historical gradient buffers to estimate rolling quantiles for adaptive Huber-style clipping, effectively preserving local optimization geometry while strictly bounding effective variance. Theoretically, we provide a convergence analysis for quantile-guided TTSA under nonconvex-strongly convex assumptions with infinite-variance noise ($p \in (1,2]$), deriving a rate of $\mathcal{O}(T^{-\frac{p-1}{3p-2}})$ that recovers optimal dependence on the heavy-tailed parameter. Empirically, across six diverse tasks, spanning heterogeneous vision benchmarks, dynamic games under momentum poisoning, and offline reinforcement learning, RQ-TTSA consistently outperforms state-of-the-art baselines by eliminating divergence spikes and ensuring stable convergence. Our method demonstrates significant robustness to hyperparameter variations and incurs negligible computational overhead ($\approx 2.7\%$ increase), validating distribution-aware gradient control as a practical and necessary component for reliable bilevel learning.

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.

Stochastic bilevel optimization (SBO) has become a standard framework for hyperparameter learning, data reweighting, representation learning, and data-mixture optimization in deep learning. Existing exact single-loop SBO methods and memory-efficient surrogate SBO methods either create severe memory pressure for large lower-level neural networks or lack competitive convergence guarantees under standard assumptions. In this paper, we propose BROS, a memory-efficient single-loop SBO method with the same convergence rate order as exact single-loop SBO methods. BROS performs lower and auxiliary updates in randomized subspaces with a Rademacher bi-probe correction that recovers an unbiased Hessian-action estimator. We prove that BROS preserves the $\mathcal O(\varepsilon^{-2})$ sample complexity of MA-SOBA for finding an $\varepsilon$-stationary point under only standard assumptions. Experiments on hyper-data cleaning, data-mixture learning, hyper-representation learning, and ViT sample reweighting show that BROS reduces peak memory by up to 44.9% while closely matching full-space baseline performance.

Bilevel optimization and bilevel minimax optimization have recently emerged as unifying frameworks for a range of machine-learning tasks, including hyperparameter optimization and reinforcement learning. The existing literature focuses on empirical efficiency and convergence guarantees, leaving a critical theoretical gap in understanding how well these algorithms generalize. To bridge this gap, we provide the first systematic generalization analysis for first-order gradient-based bilevel minimax solvers with lower-level minimax problems. Specifically, by leveraging algorithmic stability arguments, we derive fine-grained generalization bounds for three representative algorithms, including single-timescale stochastic gradient descent-ascent, and two variants of two-timescale stochastic gradient descent-ascent. Our results reveal a precise trade-off among algorithmic stability, generalization gaps, and practical settings. Furthermore, extensive empirical evaluations corroborate our theoretical insights on realistic optimization tasks with bilevel minimax structures.

Stochastic bilevel optimization (SBO) has been integrated into many machine learning paradigms recently, including hyperparameter optimization, meta learning, and reinforcement learning. Along with the wide range of applications, there have been numerous studies on the computational behavior of SBO. However, the generalization guarantees of SBO methods are far less understood from the lens of statistical learning theory. In this paper, we provide a systematic generalization analysis of the first-order gradient-based bilevel optimization methods. Firstly, we establish the quantitative connections between the on-average argument stability and the generalization gap of SBO methods. Then, we derive the upper bounds of on-average argument stability for single-timescale stochastic gradient descent (SGD) and two-timescale SGD, where three settings (nonconvex-nonconvex (NC-NC), convex-convex (C-C), and strongly-convex-strongly-convex (SC-SC)) are considered respectively. Experimental analysis validates our theoretical findings. Compared with the previous algorithmic stability analysis, our results do not require reinitializing the inner-level parameters at each iteration and are applicable to more general objective functions.

Stochastic Bilevel Optimization has emerged as a fundamental framework for meta-learning and hyperparameter optimization. Despite the practical prevalence of single-loop algorithms, their theoretical understanding in the stochastic regime remains less developed than that of multi-loop methods. In this paper, we provide a refined convergence analysis of the Single-loop Stochastic Approximate Implicit Differentiation (SSAID) algorithm. Under the squared-gradient stationarity criterion $\|\nablaΦ(x)\|^2\leε$, the corrected proof establishes an oracle complexity of $\mathcal{O}(κ^{14}ε^{-2})$, equivalently an averaged stationarity rate of $\mathcal{O}(κ^7K^{-1/2})$. The result preserves the canonical $\mathcal{O}(ε^{-2})$ dependence on the target accuracy while giving an explicit characterization of the condition-number dependence for stochastic AID-based single-loop methods.

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.

Bilevel optimization has recently attracted significant attention in machine learning due to its wide range of applications and advanced hierarchical optimization capabilities. In this paper, we propose a plug-and-play framework, named PnPBO, for developing and analyzing stochastic bilevel optimization methods. This framework integrates both modern unbiased and biased stochastic estimators into the single-loop bilevel optimization framework introduced in [9], with several improvements. In the implementation of PnPBO, all stochastic estimators for different variables can be independently incorporated, and an additional moving average technique is applied when using an unbiased estimator for the upper-level variable. In the theoretical analysis, we provide a unified convergence and complexity analysis for PnPBO, demonstrating that the adaptation of various stochastic estimators (including PAGE, ZeroSARAH, and mixed strategies) within the PnPBO framework achieves optimal sample complexity, comparable to that of single-level optimization. This resolves the open question of whether the optimal complexity bounds for solving bilevel optimization are identical to those for single-level optimization. Finally, we empirically validate our framework, demonstrating its effectiveness on several benchmark problems and confirming our theoretical findings.