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Deep Learning · Graph Neural Networks

Yongjian Zhong, Liao Zhu, Hieu Vu, Bijaya Adhikari

Subgraph neural networks have recently gained prominence for various subgraph-level predictive tasks. However, existing methods either \emph{1)} apply simple standard pooling over graph convolutional networks, failing to capture essential subgraph properties, or \emph{2)} rely on rigid subgraph definitions, leading to suboptimal performance. Moreover, these approaches fail to model long-range dependencies both between and within subgraphs—a critical limitation, as many real-world networks contain subgraphs of varying sizes and connectivity patterns. In this paper, we propose a novel implicit subgraph neural network, the first of its kind, designed to capture dependencies across subgraphs. Our approach also integrates label-aware subgraph-level information. We formulate implicit subgraph learning as a bilevel optimization problem and develop a provably convergent algorithm that requires fewer gradient estimations than standard bilevel optimization methods. We evaluate our approach on real-world networks against state-of-the-art baselines, demonstrating its effectiveness and superiority.

Optimization · Non-Convex

Siqi Zhang, Xing Huang, Feihu Huang

Bilevel optimization is widely applied in many machine learning tasks such as hyper-parameter learning and meta learning. Recently, many algorithms have been proposed to solve these bilevel optimization problems, which rely on the smoothness condition of objective functions of the bilevel optimization. In fact, some machine learning tasks such as learning language model do not satisfy the smoothness condition of objective functions. More recently, some methods have begun to study generalized smooth bilevel optimization. However, these proposed methods for generalized smooth bilevel optimization only focus on the (strongly) convex lower objective function. Meanwhile, these methods only consider the generalized-smooth upper-level objective, but still require the standard smooth lower-level objective in the bilevel optimization. To fill this gap, in the paper, thus we study the generalized-smooth bilevel optimization with the nonconvex lower-level objective function, where both upper-level and lower-level objectives are generalized-smooth. We propose an efficient single-loop Hessian/Jacobian-free penalty normalized gradient (i.e., PNGBiO) method. Moreover, we prove that our PNGBiO obtains a fast convergence rate of $O(\frac{1}{T^{1/4}})$ for finding a stationary solution, where $T$ denotes the iteration number. Meanwhile, we also propose a stochastic version of our PNGBiO (i.e., S-PNGBiO) method to solve stochastic bilevel problems, and prove that our S-PNGBiO has a fast convergence rate of $O(\frac{1}{T^{1/6}})$. Some experimental results on hyper-parameter learning and meta learning demonstrate efficiency of our proposed methods.

Optimization · Non-Convex

Songtao Lu

Extensive research has shown that a wide range of machine learning problems can be formulated as bilevel optimization, where two levels of learning processes intertwine through distinct sets of optimization variables. However, prevailing approaches often impose stringent assumptions, such as strong convexity of the lower-level loss function or uniqueness of the optimal solution, to enable algorithmic development and convergence analysis. However, these assumptions tend to be overly restrictive in real-world scenarios. In this work, we explore a recently popularized Moreau envelope based reformulation of bilevel optimization problems, accommodating nonconvex objective functions at both levels. We propose a stochastic primal-dual method that incorporates smoothing on both sides, capable of finding Karush-Kuhn-Tucker solutions for this general class of nonconvex bilevel optimization problems. A key feature of our algorithm is its ability to dynamically weigh the lower-level problems, enhancing its performance, particularly in stochastic learning scenarios. Numerical experiments underscore the superiority of our proposed algorithm over existing penalty-based methods in terms of both the convergence rate and the test accuracy.

General Machine Learning

Yuanjie Shi, Hooman Shahrokhi, Xuesong Jia, Xiongzhi Chen, Jana Doppa, Yan Yan

Conformal prediction (CP) is a promising uncertainty quantification framework which works as a wrapper around a black-box classifier to construct prediction sets (i.e., subset of candidate classes) with provable guarantees. However, standard calibration methods for CP tend to produce large prediction sets which makes them less useful in practice. This paper considers the problem of integrating conformal principles into the training process of deep classifiers to directly minimize the size of prediction sets. We formulate conformal training as a bilevel optimization problem and propose the {\em Direct Prediction Set Minimization (DPSM)} algorithm to solve it. The key insight behind DPSM is to minimize a measure of the prediction set size (upper level) that is conditioned on the learned quantile of conformity scores (lower level). We analyze that DPSM has a learning bound of $O(1/\sqrt{n})$ (with $n$ training samples),while prior conformal training methods based on stochastic approximation for the quantile has a bound of $\Omega(1/s)$ (with batch size $s$ and typically $s \ll \sqrt{n}$).Experiments on various benchmark datasets and deep models show that DPSM significantly outperforms the best prior conformal training baseline with $20.46\\%\downarrow$ in the prediction set size and validates our theory.

Deep Learning · Algorithms

Quan Xiao, Hui Yuan, A F M Saif, Gaowen Liu, Ramana Kompella, Mengdi Wang, Tianyi Chen

Diffusion models, which iteratively denoise data samples to synthesize high-quality outputs, have achieved empirical success across domains. However, optimizing these models for downstream tasks often involves nested bilevel structures, such as tuning hyperparameters for fine-tuning tasks or noise schedules in training dynamics, where traditional bilevel methods fail due to the infinite-dimensional probability space and prohibitive sampling costs. We formalize this challenge as a generative bilevel optimization problem and address two key scenarios: (1) fine-tuning pre-trained models via an inference-only lower-level solver paired with a sample-efficient gradient estimator for the upper level, and (2) training diffusion model from scratch with noise schedule optimization by reparameterizing the lower-level problem and designing a computationally tractable gradient estimator. Our first-order bilevel framework overcomes the incompatibility of conventional bilevel methods with diffusion processes, offering theoretical grounding and computational practicality. Experiments demonstrate that our method outperforms existing fine-tuning and hyperparameter search baselines.

Probabilistic Methods · Bayesian Models and Methods

Ruth Wan Theng Chew, Quoc Phong Nguyen, Bryan Kian Hsiang Low

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal lower-level solutions poses significant challenges, especially in noisy, constrained, and derivative-free settings, as repeating lower-level optimizations is sample inefficient and predicted lower-level solutions may be suboptimal. We present BILevel Bayesian Optimization (BILBO), a novel Bayesian optimization algorithm for general bilevel problems with blackbox functions, which optimizes both upper- and lower-level problems simultaneously, without the repeated lower-level optimization required by existing methods. BILBO samples from confidence-bounds based trusted sets, which bounds the suboptimality on the lower level. Moreover, BILBO selects only one function query per iteration, where the function query selection strategy incorporates the uncertainty of estimated lower-level solutions and includes a conditional reassignment of the query to encourage exploration of the lower-level objective. The performance of BILBO is theoretically guaranteed with a sublinear regret bound for commonly used kernels and is empirically evaluated on several synthetic and real-world problems.

Optimization

Lisha Chen, Quan Xiao, Ellen Fukuda, Xinyi Chen, Kun Yuan, Tianyi Chen

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.

Optimization

Youran Dong, Junfeng Yang, Wei Yao, Jin Zhang

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.

General Machine Learning · Unsupervised and Semi-supervised Learning

Sagar Shrestha, Xiao Fu

Diversified distribution matching (DDM) finds a unified translation function mapping a diverse collection of conditional source distributions to their target counterparts. DDM was proposed to resolve content misalignment issues in unpaired domain translation, achieving translation identifiability. However, DDM has only been implemented using GANs due to its constraints on the translation function. GANs are often unstable to train and do not provide the transport trajectory information---yet such trajectories are useful in applications such as single-cell evolution analysis and robot route planning. This work introduces *diversified flow matching* (DFM), an ODE-based framework for DDM. Adapting flow matching (FM) to enforce a unified translation function as in DDM is challenging, as FM learns the translation function's velocity rather than the translation function itself. A custom bilevel optimization-based training loss, a nonlinear interpolant, and a structural reformulation are proposed to address these challenges, offering a tangible implementation. To our knowledge, DFM is the first ODE-based approach guaranteeing translation identifiability. Experiments on synthetic and real-world datasets validate the proposed method.

Theory · Optimization

Zhen Qin, Zhuqing Liu, Songtao Lu, Yingbin Liang, Jia (Kevin) Liu

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}\tau})$ for approximate KKT-stationary point convergence under relaxed assumptions, where $p$ and $\tau $ are control parameters for LL learning rate and averaging, respectively.In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity.Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

Han Shen, Pin-Yu Chen, Payel Das, Tianyi Chen

Fine-tuning on task-specific data to boost downstream performance is a crucial step for leveraging Large Language Models (LLMs). However, though fine-tuning enhances the model performance for specialized applications, previous studies have demonstrated that fine-tuning the models on several adversarial samples or even benign data can greatly comprise the model's pre-equipped alignment and safety capabilities. In this work, we propose SEAL, a novel framework to enhance safety in LLM fine-tuning. SEAL learns a data ranker based on the bilevel optimization to up rank the safe and high-quality fine-tuning data and down rank the unsafe or low-quality ones. Models trained with SEAL demonstrate superior quality over multiple baselines, with 8.5\% and 9.7\% win rate increase compared to random selection respectively on Llama-3-8b-Instruct and Merlinite-7b models. Our code is available on github https://github.com/hanshen95/SEAL.

Deep Learning · Graph Neural Networks

Mahalakshmi Sabanayagam, Lukas Gosch, Stephan Günnemann, Debarghya Ghoshdastidar

Machine learning models are highly vulnerable to label flipping, i.e., the adversarial modification (poisoning) of training labels to compromise performance. Thus, deriving robustness certificates is important to guarantee that test predictions remain unaffected and to understand worst-case robustness behavior. However, for Graph Neural Networks (GNNs), the problem of certifying label flipping has so far been unsolved. We change this by introducing an exact certification method, deriving both sample-wise and collective certificates. Our method leverages the Neural Tangent Kernel (NTK) to capture the training dynamics of wide networks enabling us to reformulate the bilevel optimization problem representing label flipping into a Mixed-Integer Linear Program (MILP). We apply our method to certify a broad range of GNN architectures in node classification tasks. Thereby, concerning the worst-case robustness to label flipping: $(i)$ we establish hierarchies of GNNs on different benchmark graphs; $(ii)$ quantify the effect of architectural choices such as activations, depth and skip-connections; and surprisingly, $(iii)$ uncover a novel phenomenon of the robustness plateauing for intermediate perturbation budgets across all investigated datasets and architectures. While we focus on GNNs, our certificates are applicable to sufficiently wide NNs in general through their NTK. Thus, our work presents the first exact certificate to a poisoning attack ever derived for neural networks, which could be of independent interest. The code is available at https://github.com/saper0/qpcert.

Optimization · Non-Convex

Sheng Fang, Yongjin Liu, Wei Yao, Chengming Yu, Jin Zhang

Bilevel optimization, which addresses challenges in hierarchical learning tasks, has gained significant interest in machine learning. Implementing gradient descent for bilevel optimization presents computational hurdles, notably the need to compute the exact lower-level solution and the inverse Hessian of the lower-level objective. While these two aspects are inherently connected, existing methods typically handle them separately by solving the lower-level problem and a linear system for the inverse Hessian-vector product. In this paper, we introduce a general framework to tackle these computational challenges in a coordinated manner. Specifically, we leverage quasi-Newton algorithms to accelerate the solution of the lower-level problem while efficiently approximating the inverse Hessian-vector product. Furthermore, by leveraging the superlinear convergence properties of BFGS, we establish a non-asymptotic convergence analysis for the BFGS adaptation within our framework. Numerical experiments demonstrate the comparable or superior performance of our proposed algorithms in real-world learning tasks, including hyperparameter optimization, data hyper-cleaning, and few-shot meta-learning.

Deep Learning · Robustness

Melis Ilayda Bal, Volkan Cevher, Michael Muehlebach

As machine learning models grow in complexity and increasingly rely on publicly sourced data, such as the human-annotated labels used in training large language models, they become more vulnerable to label poisoning attacks.These attacks, in which adversaries subtly alter the labels within a training dataset, can severely degrade model performance, posing significant risks in critical applications.In this paper, we propose $\textbf{Floral}$, a novel adversarial training defense strategy based on support vector machines (SVMs) to counter these threats. Utilizing a bilevel optimization framework, we cast the training process as a non-zero-sum Stackelberg game between an $\textit{attacker}$, who strategically poisons critical training labels, and the $\textit{model}$, which seeks to recover from such attacks. Our approach accommodates various model architectures and employs a projected gradient descent algorithm with kernel SVMs for adversarial training. We provide a theoretical analysis of our algorithm’s convergence properties and empirically evaluate $\textbf{Floral}$'s effectiveness across diverse classification tasks.Compared to robust baselines and foundation models such as RoBERTa, $\textbf{Floral}$ consistently achieves higher robust accuracy under increasing attacker budgets.These results underscore the potential of $\textbf{Floral}$ to enhance the resilience of machine learning models against label poisoning threats, thereby ensuring robust classification in adversarial settings.

Optimization · Non-Convex

Wei Yao, Haian Yin, Shangzhi Zeng, Jin Zhang

Constrained bilevel optimization tackles nested structures present in constrained learning tasks like constrained meta-learning, adversarial learning, and distributed bilevel optimization. However, existing bilevel optimization methods mostly are typically restricted to specific constraint settings, such as linear lower-level constraints. In this work, we overcome this limitation and develop a new single-loop, Hessian-free constrained bilevel algorithm capable of handling more general lower-level constraints. We achieve this by employing a doubly regularized gap function tailored to the constrained lower-level problem, transforming constrained bilevel optimization into an equivalent single-level optimization problem with a single smooth constraint. We rigorously establish the non-asymptotic convergence analysis of the proposed algorithm under the convexity of lower-level problem, avoiding the need for strong convexity assumptions on the lower-level objective or coupling convexity assumptions on lower-level constraints found in existing literature. Additionally, the generality of our method allows for its extension to bilevel optimization with minimax lower-level problem. We evaluate the effectiveness and efficiency of our algorithm on various synthetic problems, typical hyperparameter learning tasks, and generative adversarial network.

Optimization · Global Optimization

Quan Xiao, Tianyi Chen

Bilevel optimization has witnessed a resurgence of interest, driven by its critical role in trustworthy and efficient AI applications. Recent focus has been on finding efficient methods with provable convergence guarantees. However, while many prior works have established convergence to stationary points or local minima, obtaining the global optimum of bilevel optimization remains an important yet open problem. The difficulty lies in the fact that unlike many prior non-convex single-level problems, bilevel problems often do not admit a ``benign" landscape, and may indeed have multiple spurious local solutions. Nevertheless, attaining the global optimality is indispensable for ensuring reliability, safety, and cost-effectiveness, particularly in high-stakes engineering applications that rely on bilevel optimization. In this paper, we first explore the challenges of establishing a global convergence theory for bilevel optimization, and present two sufficient conditions for global convergence. We provide {\em algorithm-dependent} proofs to rigorously substantiate these sufficient conditions on two specific bilevel learning scenarios: representation learning and data hypercleaning (a.k.a. reweighting). Experiments corroborate the theoretical findings, demonstrating convergence to global minimum in both cases.

Theory · Optimization

Emanuele Zangrando, Sara Venturini, Francesco Rinaldi, Francesco Tudisco

Low-rank adaptation methods are a popular approach for parameter-efficient fine-tuning of large-scale neural networks. However, selecting the optimal rank for each layer remains a challenging problem that significantly affects both performance and efficiency. In this paper, we introduce a novel bilevel optimization strategy that simultaneously trains both matrix and tensor low-rank adapters, dynamically selecting the optimal rank for each layer. Our method avoids the use of implicit differentiation in the computation of the hypergradient, and integrates a stochastic away-step variant of the Frank-Wolfe algorithm, eliminating the need for projection and providing identifiability guarantees of the optimal rank structure. This results in a highly efficient and cost-effective training scheme that adaptively allocates the parameter budget across the network layers. On top of a detailed theoretical analysis of the method, we provide different numerical experiments showcasing its effectiveness.

Optimization · Non-Convex

Yan Yang, Bin Gao, Ya-xiang Yuan

Bilevel optimization, with broad applications in machine learning, has an intricate hierarchical structure. Gradient-based methods have emerged as a common approach to large-scale bilevel problems. However, the computation of the hyper-gradient, which involves a Hessian inverse vector product, confines the efficiency and is regarded as a bottleneck. To circumvent the inverse, we construct a sequence of low-dimensional approximate Krylov subspaces with the aid of the Lanczos process. As a result, the constructed subspace is able to dynamically and incrementally approximate the Hessian inverse vector product with less effort and thus leads to a favorable estimate of the hyper-gradient. Moreover, we propose a provable subspace-based framework for bilevel problems where one central step is to solve a small-size tridiagonal linear system. To the best of our knowledge, this is the first time that subspace techniques are incorporated into bilevel optimization. This successful trial not only enjoys $\mathcal{O}(\epsilon^{-1})$ convergence rate but also demonstrates efficiency in a synthetic problem and two deep learning tasks.

Theory · Optimization

Yifan Yang, Hao Ban, Minhui Huang, Shiqian Ma, Kaiyi Ji

Bilevel optimization has recently attracted considerable attention due to its abundant applications in machine learning problems. However, existing methods rely on prior knowledge of problem parameters to determine stepsizes, resulting in significant effort in tuning stepsizes when these parameters are unknown. In this paper, we propose two novel tuning-free algorithms, D-TFBO and S-TFBO. D-TFBO employs a double-loop structure with stepsizes adaptively adjusted by the "inverse of cumulative gradient norms" strategy. S-TFBO features a simpler fully single-loop structure that updates three variables simultaneously with a theory-motivated joint design of adaptive stepsizes for all variables. We provide a comprehensive convergence analysis for both algorithms and show that D-TFBO and S-TFBO respectively require $\mathcal{O}(\frac{1}{\epsilon})$ and $\mathcal{O}(\frac{1}{\epsilon}\log^4(\frac{1}{\epsilon}))$ iterations to find an $\epsilon$-accurate stationary point, (nearly) matching their well-tuned counterparts using the information of problem parameters. Experiments on various problems show that our methods achieve performance comparable to existing well-tuned approaches, while being more robust to the selection of initial stepsizes. To the best of our knowledge, our methods are the first to completely eliminate the need for stepsize tuning, while achieving theoretical guarantees.

Applications · Physics

Kanishk Bhatia, Felix Koehler, Nils Thuerey

The physics solvers employed for neural network training are primarily iterative, and hence, differentiating through them introduces a severe computational burden as iterations grow large. Inspired by works in bilevel optimization, we show that full accuracy of the network is achievable through physics significantly coarser than fully converged solvers. We propose *progressively refined differentiable physics* (PRDP), an approach that identifies the level of physics refinement sufficient for full training accuracy. By beginning with coarse physics, adaptively refining it during training, and stopping refinement at the level adequate for training, it enables significant compute savings without sacrificing network accuracy. Our focus is on differentiating iterative linear solvers for sparsely discretized differential operators, which are fundamental to scientific computing. PRDP is applicable to both unrolled and implicit differentiation. We validate its performance on a variety of learning scenarios involving differentiable physics solvers such as inverse problems, autoregressive neural emulators, and correction-based neural-hybrid solvers. In the challenging example of emulating the Navier-Stokes equations, we reduce training time by 62%.