Fine-tuning pre-trained Large Language Models (LLMs) for downstream tasks using First-Order (FO) optimizers presents significant computational challenges. Parameter-Efficient Fine-Tuning (PEFT) methods address these by freezing most model parameters and training only a small subset. However, PEFT often underperforms compared to full fine-tuning when high task-specific accuracy is required. Zeroth-Order (ZO) methods fine-tune the entire pre-trained model without back-propagation, estimating gradients through forward passes only. While memory-efficient, ZO methods suffer from slow convergence and high sensitivity to prompt selection. We bridge these two worlds with Bilevel-ZOFO, a bilevel optimization method that couples fast, local FO-PEFT adaptation at the inner level with stable, memory-efficient ZO updates of the full backbone at the outer level. The FO-PEFT inner loop performs fast, low-memory local adaptation that reduces the variance of ZO estimates and stabilizes the search, guiding the outer ZO updates of the full backbone and reducing prompt sensitivity. In the mean time, the outer ZO provides better generalization ability for PEFT. We provide theoretical convergence guarantees and empirically demonstrate that Bilevel-ZOFO significantly outperforms existing ZO and FO-PEFT methods, achieving 2-4 times faster training while maintaining similar memory efficiency. Additionally, we show by updating the backbone with ZO and adapting only a tiny FO-PEFT block per task, Bilevel-ZOFO combines full-model capacity with few-shot efficiency, making it a very efficient meta-learning algorithm that quickly adapts to new tasks.
Bilevel optimization offers a methodology to learn hyperparameters in imaging inverse problems, yet its integration with automatic differentiation techniques remains challenging. On the one hand, inverse problems are typically solved by iterating arbitrarily many times some elementary scheme which maps any point to the minimizer of an energy functional, known as equilibrium point. On the other hand, introducing parameters to be learned in the energy functional yield architectures very reminiscent of Neural Networks (NN) known as Unrolled NN and thus suggests the use of Automatic Differentiation (AD) techniques. Yet, applying AD requires for the NN to be of relatively small depth, thus making necessary to truncate an unrolled scheme to a finite number of iterations. First, we show that, at the minimizer, the optimal gradient descent step computed in the Deep Equilibrium (DEQ) framework admits an approximation, known as Jacobian Free Backpropagation (JFB), that is much easier to compute and can be made arbitrarily good by controlling Lipschitz properties of the truncated unrolled scheme. Second, we introduce an algorithm that combines a restart strategy with JFB computed by AD and we show that the learned steps can be made arbitrarily close to the optimal DEQ framework. Third, we complement the theoretical analysis by applying the proposed method to a variety of problems in imaging that progressively depart from the theoretical framework. In particular we show that this method is effective for training weights in weighted norms; stepsizes and regularization levels of Plug-and-Play schemes; and a DRUNet denoiser embedded in Forward-Backward iterates.
Parameter-efficient fine-tuning (PEFT) methods, particularly Low-Rank Adaptation (LoRA), offer an efficient way to adapt large language models with reduced computational costs. However, their performance is limited by the small number of trainable parameters. Recent work combines LoRA with the Mixture-of-Experts (MoE), i.e., LoRA-MoE, to enhance capacity, but two limitations remain in hindering the full exploitation of its potential: 1) the influence of downstream tasks when assigning expert numbers, and 2) the uniform rank assignment across all LoRA experts, which restricts representational diversity. To mitigate these gaps, we propose GuiLoMo, a fine-grained layer-wise expert numbers and ranks allocation strategy with GuidedSelection Vectors (GSVs). GSVs are learned via a prior bilevel optimization process to capture both model- and task-specific needs, and are then used to allocate optimal expert numbers and ranks. Experiments on three backbone models across diverse benchmarks show that GuiLoMo consistently achieves superior or comparable performance to all baselines. Further analysis offers key insights into how expert numbers and ranks vary across layers and tasks, highlighting the benefits of adaptive expert configuration. Our code is available at https://github.com/Liar406/Gui-LoMo.git.
Bilevel optimization, in which one optimization problem is nested inside another, underlies many machine learning applications with a hierarchical structure -- such as meta-learning and hyperparameter optimization. Such applications often involve sensitive training data, raising pressing concerns about individual privacy. Motivated by this, we study differentially private bilevel optimization. We first focus on settings where the outer-level objective is \textit{convex}, and provide novel upper and lower bounds on the excess risk for both pure and approximate differential privacy, covering both empirical and population-level loss. These bounds are nearly tight and essentially match the optimal rates for standard single-level differentially private ERM and stochastic convex optimization (SCO), up to additional terms that capture the intrinsic complexity of the nested bilevel structure. The bounds are achieved in polynomial time via efficient implementations of the exponential and regularized exponential mechanisms. A key technical contribution is a new method and analysis of log-concave sampling under inexact function evaluations, which may be of independent interest. In the \textit{non-convex} setting, we develop novel algorithms with state-of-the-art rates for privately finding approximate stationary points. Notably, our bounds do not depend on the dimension of the inner problem.
Multi-agent reinforcement learning, despite its popularity and empirical success, faces significant scalability challenges in large-population dynamic games. Graphon mean field games (GMFGs) offer a principled framework for approximating such games while capturing heterogeneity among players. In this paper, we propose and analyze a policy optimization framework for continuous-time, finite-horizon linear-quadratic GMFGs. Exploiting the structural properties of GMFGs, we design an efficient policy parameterization in which each player's policy is represented as an affine function of their private state, with a shared slope function and player-specific intercepts. We develop a bilevel optimization algorithm that alternates between policy gradient updates for best-response computation under a fixed population distribution, and distribution updates using the resulting policies. We prove linear convergence of the policy gradient steps to best-response policies and establish global convergence of the overall algorithm to the Nash equilibrium. The analysis relies on novel landscape characterizations over infinite-dimensional policy spaces. Numerical experiments demonstrate the convergence and robustness of the proposed algorithm under varying graphon structures, noise levels, and action frequencies.
Time-adaptive unit commitment (UC) has recently been investigated to reduce the scheduling costs by flexibly varying the temporal resolution, which is usually determined by clustering the net load patterns. However, there exists a misalignment between cost and net load patterns due to the discrete start-up costs and out-of-merit-order dispatch triggered by ramping and other constraints. The optimal time-adaptive resolution cannot be completely captured by clustering-based method. This paper proposes a cost-oriented method to address this misalignment by a novel bilevel optimization approach that is efficiently solved through a heuristic greedy algorithm. The impact of varying temporal resolution on the final scheduling costs are tested, based on which the temporal resolution is heuristically updated, achieving significant cost reduction without increasing the number of temporal periods. Subsequently, an improved discretized Adam optimization method together with offline warm start and online refinement strategy is proposed to efficiently search for the better temporal resolution configuration. Results show that the proposed cost-oriented UC temporal resolution determination method achieves enhanced cost efficiency.
Model extraction attacks aim to replicate the functionality of a black-box model through query access, threatening the intellectual property (IP) of machine-learning-as-a-service (MLaaS) providers. Defending against such attacks is challenging, as it must balance efficiency, robustness, and utility preservation in the real-world scenario. Despite the recent advances, most existing defenses presume that attacker queries have out-of-distribution (OOD) samples, enabling them to detect and disrupt suspicious inputs. However, this assumption is increasingly unreliable, as modern models are trained on diverse datasets and attackers often operate under limited query budgets. As a result, the effectiveness of these defenses is significantly compromised in realistic deployment scenarios. To address this gap, we propose MISLEADER (enseMbles of dIStiLled modEls Against moDel ExtRaction), a novel defense strategy that does not rely on OOD assumptions. MISLEADER formulates model protection as a bilevel optimization problem that simultaneously preserves predictive fidelity on benign inputs and reduces extractability by potential clone models. Our framework combines data augmentation to simulate attacker queries with an ensemble of heterogeneous distilled models to improve robustness and diversity. We further provide a tractable approximation algorithm and derive theoretical error bounds to characterize defense effectiveness. Extensive experiments across various settings validate the utility-preserving and extraction-resistant properties of our proposed defense strategy. Our code is available at https://github.com/LabRAI/MISLEADER.
We investigate multi-period mean-risk portfolio optimization for long-horizon Defined Contribution plans, focusing on buffered Probability of Exceedance (bPoE), a more intuitive, dollar-based alternative to Conditional Value-at-Risk (CVaR). We formulate both pre-commitment and time-consistent Mean-bPoE and Mean-CVaR portfolio optimization problems under realistic investment constraints (e.g., no leverage, no short selling) and jump-diffusion dynamics. These formulations are naturally framed as bilevel optimization problems, with an outer search over the shortfall threshold and an inner optimization over rebalancing decisions. We establish an equivalence between the pre-commitment formulations through a one-to-one correspondence of their scalarization optimal sets, while showing that no such equivalence holds in the time-consistent setting. We develop provably convergent numerical schemes for the value functions associated with both pre-commitment and time-consistent formulations of these mean-risk control problems. Using nearly a century of market data, we find that time-consistent Mean-bPoE strategies closely resemble their pre-commitment counterparts. In particular, they maintain alignment with investors' preferences for a minimum acceptable terminal wealth level-unlike time-consistent Mean-CVaR, which often leads to counterintuitive control behavior. We further show that bPoE, as a strictly tail-oriented measure, prioritizes guarding against catastrophic shortfalls while allowing meaningful upside exposure, making it especially appealing for long-horizon wealth security. These findings highlight bPoE's practical advantages for Defined Contribution investment planning.
Machine unlearning aims to remove the influence of specific training samples from a trained model without full retraining. While prior work has largely focused on privacy-motivated settings, we recast unlearning as a general-purpose tool for post-deployment model revision. Specifically, we focus on utilizing unlearning in clinical contexts where data shifts, device deprecation, and policy changes are common. To this end, we propose a bilevel optimization formulation of boundary-based unlearning that can be solved using iterative algorithms. We provide convergence guarantees when first-order algorithms are used to unlearn. Our method introduces tunable loss design for controlling the forgetting-retention tradeoff and supports novel model composition strategies that merge the strengths of distinct unlearning runs. Across benchmark and real-world clinical imaging datasets, our approach outperforms baselines on both forgetting and retention metrics, including scenarios involving imaging devices and anatomical outliers. This work establishes machine unlearning as a modular, practical alternative to retraining for real-world model maintenance in clinical applications.
Out-of-distribution (OOD) generalization remains a fundamental challenge in machine learning. Models trained on one data distribution often experience substantial performance degradation when evaluated on shifted or unseen domains. To address this challenge, the present paper studies the design of training data distributions that maximize average-case OOD performance. First, a theoretical analysis establishes a family of generalization bounds that quantify how the choice of training distribution influences OOD error across a predefined family of target distributions. These insights motivate the introduction of two complementary algorithmic strategies: (i) directly formulating OOD risk minimization as a bilevel optimization problem over the space of probability measures and (ii) minimizing a theoretical upper bound on OOD error. Last, the paper evaluates the two approaches across a range of function approximation and operator learning examples. The proposed methods significantly improve OOD accuracy over standard empirical risk minimization with a fixed distribution. These results highlight the potential of distribution-aware training as a principled and practical framework for robust OOD generalization.
This paper develops a persistently exciting input generating Online Feedback Optimization (OFO) controller that estimates the sensitivity of a process ensuring minimal deviations from the descent direction while converging. This eliminates the need for random perturbations in feedback loop. The proposed controller is formulated as a bilevel optimization program, where a nonconvex full rank constraint is relaxed using linear constraints and penalization. The validation of the method is performed in a simulated scenario where multiple systems share a limited, costly resource for production optimization, simulating an oil and gas resource allocation problem. The method allows for less input perturbations while accurately estimating gradients, allowing faster convergence when the gradients are unknown. In the case study, the proposed method achieved the same profit compared to an OFO controller with random input perturbations, and $1.4\%$ higher profit compared to an OFO controller without input perturbations.
This paper addresses the challenge of graph domain adaptation on evolving, multiple out-of-distribution (OOD) graphs. Conventional graph domain adaptation methods are confined to single-step adaptation, making them ineffective in handling continuous domain shifts and prone to catastrophic forgetting. This paper introduces the Graph Continual Adaptive Learning (GCAL) method, designed to enhance model sustainability and adaptability across various graph domains. GCAL employs a bilevel optimization strategy. The "adapt" phase uses an information maximization approach to fine-tune the model with new graph domains while re-adapting past memories to mitigate forgetting. Concurrently, the "generate memory" phase, guided by a theoretical lower bound derived from information bottleneck theory, involves a variational memory graph generation module to condense original graphs into memories. Extensive experimental evaluations demonstrate that GCAL substantially outperforms existing methods in terms of adaptability and knowledge retention.
Bilevel optimization involves a hierarchical structure where one problem is nested within another, leading to complex interdependencies between levels. We propose a single-loop, tuning-free algorithm that guarantees anytime feasibility, i.e., approximate satisfaction of the lower-level optimality condition, while ensuring descent of the upper-level objective. At each iteration, a convex quadratically-constrained quadratic program (QCQP) with a closed-form solution yields the search direction, followed by a backtracking line search inspired by control barrier functions to ensure safe, uniformly positive step sizes. The resulting method is scalable, requires no hyperparameter tuning, and converges under mild local regularity assumptions. We establish an O(1/k) ergodic convergence rate and demonstrate the algorithm's effectiveness on representative bilevel tasks.
The dependency of the actor on the critic in actor-critic (AC) reinforcement learning means that AC can be characterized as a bilevel optimization (BLO) problem, also called a Stackelberg game. This characterization motivates two modifications to vanilla AC algorithms. First, the critic's update should be nested to learn a best response to the actor's policy. Second, the actor should update according to a hypergradient that takes changes in the critic's behavior into account. Computing this hypergradient involves finding an inverse Hessian vector product, a process that can be numerically unstable. We thus propose a new algorithm, Bilevel Policy Optimization with Nystr\"om Hypergradients (BLPO), which uses nesting to account for the nested structure of BLO, and leverages the Nystr\"om method to compute the hypergradient. Theoretically, we prove BLPO converges to (a point that satisfies the necessary conditions for) a local strong Stackelberg equilibrium in polynomial time with high probability, assuming a linear parametrization of the critic's objective. Empirically, we demonstrate that BLPO performs on par with or better than PPO on a variety of discrete and continuous control tasks.
Fine-tuning large language models (LLMs) on a mixture of diverse datasets poses challenges due to data imbalance and heterogeneity. Existing methods often address these issues across datasets (globally) but overlook the imbalance and heterogeneity within individual datasets (locally), which limits their effectiveness. We introduce Hierarchical Balancing Optimization (HBO), a novel method that enables LLMs to autonomously adjust data allocation during fine-tuning both across datasets (globally) and within each individual dataset (locally). HBO employs a bilevel optimization strategy with two types of actors: a Global Actor, which balances data sampling across different subsets of the training mixture, and several Local Actors, which optimizes data usage within each subset based on difficulty levels. These actors are guided by reward functions derived from the LLM's training state, which measure learning progress and relative performance improvement. We evaluate HBO on three LLM backbones across nine diverse tasks in multilingual and multitask setups. Results show that HBO consistently outperforms existing baselines, achieving significant accuracy gains. Our in-depth analysis further demonstrates that both the global actor and local actors of HBO effectively adjust data usage during fine-tuning. HBO provides a comprehensive solution to the challenges of data imbalance and heterogeneity in LLM fine-tuning, enabling more effective training across diverse datasets.
With the success that the field of bilevel optimization has seen in recent years, similar methodologies have started being applied to solving more difficult applications that arise in trilevel optimization. At the helm of these applications are new machine learning formulations that have been proposed in the trilevel context and, as a result, efficient and theoretically sound stochastic methods are required. In this work, we propose the first-ever stochastic gradient descent method for solving unconstrained trilevel optimization problems and provide a convergence theory that covers all forms of inexactness of the trilevel adjoint gradient, such as the inexact solutions of the middle-level and lower-level problems, inexact computation of the trilevel adjoint formula, and noisy estimates of the gradients, Hessians, Jacobians, and tensors of third-order derivatives involved. We also demonstrate the promise of our approach by providing numerical results on both synthetic trilevel problems and trilevel formulations for hyperparameter adversarial tuning.
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.
Bilevel optimization is a powerful tool for many machine learning problems, such as hyperparameter optimization and meta-learning. Estimating hypergradients (also known as implicit gradients) is crucial for developing gradient-based methods for bilevel optimization. In this work, we propose a computationally efficient technique for incorporating curvature information into the approximation of hypergradients and present a novel algorithmic framework based on the resulting enhanced hypergradient computation. We provide convergence rate guarantees for the proposed framework in both deterministic and stochastic scenarios, particularly showing improved computational complexity over popular gradient-based methods in the deterministic setting. This improvement in complexity arises from a careful exploitation of the hypergradient structure and the inexact Newton method. In addition to the theoretical speedup, numerical experiments demonstrate the significant practical performance benefits of incorporating curvature information.
Large Language Models (LLMs) have demonstrated remarkable performance across a wide range of applications, e.g., medical question-answering, mathematical sciences, and code generation. However, they also exhibit inherent limitations, such as outdated knowledge and susceptibility to hallucinations. Retrieval-Augmented Generation (RAG) has emerged as a promising paradigm to address these issues, but it also introduces new vulnerabilities. Recent efforts have focused on the security of RAG-based LLMs, yet existing attack methods face three critical challenges: (1) their effectiveness declines sharply when only a limited number of poisoned texts can be injected into the knowledge database, (2) they lack sufficient stealth, as the attacks are often detectable by anomaly detection systems, which compromises their effectiveness, and (3) they rely on heuristic approaches to generate poisoned texts, lacking formal optimization frameworks and theoretic guarantees, which limits their effectiveness and applicability. To address these issues, we propose coordinated Prompt-RAG attack (PR-attack), a novel optimization-driven attack that introduces a small number of poisoned texts into the knowledge database while embedding a backdoor trigger within the prompt. When activated, the trigger causes the LLM to generate pre-designed responses to targeted queries, while maintaining normal behavior in other contexts. This ensures both high effectiveness and stealth. We formulate the attack generation process as a bilevel optimization problem leveraging a principled optimization framework to develop optimal poisoned texts and triggers. Extensive experiments across diverse LLMs and datasets demonstrate the effectiveness of PR-Attack, achieving a high attack success rate even with a limited number of poisoned texts and significantly improved stealth compared to existing methods.
Multi-objective learning under user-specified preference is common in real-world problems such as multi-lingual speech recognition under fairness. In this work, we frame such a problem as a semivectorial bilevel optimization problem, whose goal is to optimize a pre-defined preference function, subject to the constraint that the model parameters are weakly Pareto optimal. To solve this problem, we convert the multi-objective constraints to a single-objective constraint through a merit function with an easy-to-evaluate gradient, and then, we use a penalty-based reformulation of the bilevel optimization problem. We theoretically establish the properties of the merit function, and the relations of solutions for the penalty reformulation and the constrained formulation. Then we propose algorithms to solve the reformulated single-level problem, and establish its convergence guarantees. We test the method on various synthetic and real-world problems. The results demonstrate the effectiveness of the proposed method in finding preference-guided optimal solutions to the multi-objective problem.