Quan Xiao

Momin Abbas, Quan Xiao, Lisha Chen et al.

Model-agnostic meta learning (MAML) is currently one of the dominating approaches for few-shot meta-learning. Albeit its effectiveness, the optimization of MAML can be challenging due to the innate bilevel problem structure. Specifically, the loss landscape of MAML is much more complex with possibly more saddle points and local minimizers than its empirical risk minimization counterpart. To address this challenge, we leverage the recently invented sharpness-aware minimization and develop a sharpness-aware MAML approach that we term Sharp-MAML. We empirically demonstrate that Sharp-MAML and its computation-efficient variant can outperform the plain-vanilla MAML baseline (e.g., $+3\%$ accuracy on Mini-Imagenet). We complement the empirical study with the convergence rate analysis and the generalization bound of Sharp-MAML. To the best of our knowledge, this is the first empirical and theoretical study on sharpness-aware minimization in the context of bilevel learning. The code is available at https://github.com/mominabbass/Sharp-MAML.

Quan Xiao, Han Shen, Wotao Yin et al.

Stochastic bilevel optimization, which captures the inherent nested structure of machine learning problems, is gaining popularity in many recent applications. Existing works on bilevel optimization mostly consider either unconstrained problems or constrained upper-level problems. This paper considers the stochastic bilevel optimization problems with equality constraints both in the upper and lower levels. By leveraging the special structure of the equality constraints problem, the paper first presents an alternating implicit projected SGD approach and establishes the $\tilde{\cal O}(ε^{-2})$ sample complexity that matches the state-of-the-art complexity of ALSET \citep{chen2021closing} for unconstrained bilevel problems. To further save the cost of projection, the paper presents two alternating implicit projection-efficient SGD approaches, where one algorithm enjoys the $\tilde{\cal O}(ε^{-2}/T)$ upper-level and $\tilde{\cal O}(ε^{-1.5}/T^{\frac{3}{4}})$ lower-level projection complexity with ${\cal O}(T)$ lower-level batch size, and the other one enjoys $\tilde{\cal O}(ε^{-1.5})$ upper-level and lower-level projection complexity with ${\cal O}(1)$ batch size. Application to federated bilevel optimization has been presented to showcase the empirical performance of our algorithms. Our results demonstrate that equality-constrained bilevel optimization with strongly-convex lower-level problems can be solved as efficiently as stochastic single-level optimization problems.

Meng Zhu, Quan Xiao, Weidong Min

Optimization algorithms are core methods by which machine learning models iteratively minimize loss functions, update parameters, learn from data, and improve performance. Momentum SGD and AdamW represent two important optimization paradigms. AdamW produces stable updates and usually has strong robustness across training scenarios, but its generalization performance is sometimes weaker than that of momentum methods. Momentum SGD can often obtain better generalization after careful tuning, but it is more sensitive to gradient-scale variation and hyperparameter settings. To balance the strengths and weaknesses of the two paradigms, this paper proposes Ada2MS, an optimization algorithm that achieves a smooth transition between AdamW-like behavior and momentum-SGD-like behavior through continuous exponential interpolation between elementwise second-moment estimates and global second-moment estimates. On the visual tasks evaluated in this study, Ada2MS obtains competitive results under a unified optimizer-comparison protocol. The code will be released at https://github.com/mengzhu0308/Ada2MS

Han Shen, Quan Xiao, Tianyi Chen

Bilevel optimization enjoys a wide range of applications in emerging machine learning and signal processing problems such as hyper-parameter optimization, image reconstruction, meta-learning, adversarial training, and reinforcement learning. However, bilevel optimization problems are traditionally known to be difficult to solve. Recent progress on bilevel algorithms mainly focuses on bilevel optimization problems through the lens of the implicit-gradient method, where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle a challenging class of bilevel problems through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the (local) solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem with lower-level constraints yet without lower-level strong convexity. Experiments on synthetic and real datasets showcase the efficiency of the proposed PBGD algorithm.

Quan Xiao, Qing Ling, Tianyi Chen

A major challenge of applying zeroth-order (ZO) methods is the high query complexity, especially when queries are costly. We propose a novel gradient estimation technique for ZO methods based on adaptive lazy queries that we term as LAZO. Different from the classic one-point or two-point gradient estimation methods, LAZO develops two alternative ways to check the usefulness of old queries from previous iterations, and then adaptively reuses them to construct the low-variance gradient estimates. We rigorously establish that through judiciously reusing the old queries, LAZO can reduce the variance of stochastic gradient estimates so that it not only saves queries per iteration but also achieves the regret bound for the symmetric two-point method. We evaluate the numerical performance of LAZO, and demonstrate the low-variance property and the performance gain of LAZO in both regret and query complexity relative to several existing ZO methods. The idea of LAZO is general, and can be applied to other variants of ZO methods.

Quan Xiao, Songtao Lu, Tianyi Chen

Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can match the convergence rate of single-level gradient descent (GD) when addressing bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric, for a nonconvex lower-level objective that satisfies the Polyak-Łojasiewicz (PL) condition. We then propose a Generalized ALternating mEthod for bilevel opTimization (GALET) tailored to BLO with convex PL LL problem and establish that GALET achieves an $ε$-stationary point for the considered problem within $\tilde{\cal O}(ε^{-1})$ iterations, which matches the iteration complexity of GD for single-level smooth nonconvex problems.

Quan Xiao, Tianyi Chen

Bilevel optimization has witnessed a resurgence of interest, driven by its critical role in trustworthy and efficient AI applications. While many recent 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 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 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 the global minimum in both cases.

Quan Xiao, Jindan Li, Zhaoxian Wu et al.

Analog in-memory computing (AIMC) performs computation directly within resistive crossbar arrays, offering an energy-efficient platform to scale large vision and language models. However, non-ideal analog device properties make the training on AIMC devices challenging. In particular, its update asymmetry can induce a systematic drift of weight updates towards a device-specific symmetric point (SP), which typically does not align with the optimum of the training objective. To mitigate this bias, most existing works assume the SP is known and pre-calibrate it to zero before training by setting the reference point as the SP. Nevertheless, calibrating AIMC devices requires costly pulse updates, and residual calibration error can directly degrade training accuracy. In this work, we present the first theoretical characterization of the pulse complexity of SP calibration and the resulting estimation error. We further propose a dynamic SP estimation method that tracks the SP during model training, and establishes its convergence guarantees. In addition, we develop an enhanced variant based on chopping and filtering techniques from digital signal processing. Numerical experiments demonstrate both the efficiency and effectiveness of the proposed method.

Meng Zhu, Quan Xiao, Weidong Min

Since the 21st century, artificial intelligence has been leading a new round of industrial revolution. Under the training framework, the optimization algorithm aims to stably converge high-dimensional optimization to local and even global minima. Entering the era of large language models, although the scale of model parameters and data has increased, Adam remains the mainstream optimization algorithm. However, compared with stochastic gradient descent (SGD) based optimization algorithms, Adam is more likely to converge to non-flat minima. To address this issue, the AdamNX algorithm is proposed. Its core innovation lies in the proposition of a novel type of second-order moment estimation exponential decay rate, which gradually weakens the learning step correction strength as training progresses, and degrades to momentum SGD in the stable training period, thereby improving the stability of training in the stable period and possibly enhancing generalization ability. Experimental results show that our second-order moment estimation exponential decay rate is better than the current second-order moment estimation exponential decay rate, and AdamNX can stably outperform Adam and its variants in terms of performance. Our code is open-sourced at https://github.com/mengzhu0308/AdamNX.

Quan Xiao, Tianyi Chen

Offline data selection and online self-refining generation, which enhance the data quality, are crucial steps in adapting large language models (LLMs) to specific downstream tasks. We tackle offline data selection and online self-refining generations through an optimization perspective. Specifically, bilevel data selection is used for offline data selection with respect to the validation dataset, and we treat online self-refining generation as a model adaptation step of selecting the model trained on current responses that best fits the validation data. Our framework offers a unified understanding of offline data selection and self-refining generation by assigning a learned data weight to each question and response, either explicitly or implicitly. For the first time, we theoretically demonstrate the effectiveness of the bilevel data selection framework and demonstrate its performance gains over unfiltered direct mixing baselines. By combining offline data with validation-weighted online generations, our method enhances fine-tuning performance. Experiments on quality enhancement and safety-aware LLM fine-tuning validate its effectiveness.

Lisha Chen, Quan Xiao, Ellen Hidemi Fukuda et al.

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.

Zhaoxian Wu, Quan Xiao, Tayfun Gokmen et al.

As the economic and environmental costs of training and deploying large vision or language models increase dramatically, analog in-memory computing (AIMC) emerges as a promising energy-efficient solution. However, the training perspective, especially its training dynamic, is underexplored. In AIMC hardware, the trainable weights are represented by the conductance of resistive elements and updated using consecutive electrical pulses. While the conductance changes by a constant in response to each pulse, in reality, the change is scaled by asymmetric and non-linear response functions, leading to a non-ideal training dynamic. This paper provides a theoretical foundation for gradient-based training on AIMC hardware with non-ideal response functions. We demonstrate that asymmetric response functions negatively impact Analog SGD by imposing an implicit penalty on the objective. To overcome the issue, we propose Residual Learning algorithm, which provably converges exactly to a critical point by solving a bilevel optimization problem. We demonstrate that the proposed method can be extended to address other hardware imperfections, such as limited response granularity. As we know, it is the first paper to investigate the impact of a class of generic non-ideal response functions. The conclusion is supported by simulations validating our theoretical insights.

Zhaoxian Wu, Quan Xiao, Tayfun Gokmen et al.

Aiming to accelerate the training of large deep neural models (DNN) in an energy-efficient way, an analog in-memory computing (AIMC) accelerator emerges as a solution with immense potential. In AIMC accelerators, trainable weights are kept in memory without the need to move from memory to processors during the training, reducing a bunch of overhead. However, although the in-memory feature enables efficient computation, it also constrains the use of data parallelism since copying weights from one AIMC to another is expensive. To enable parallel training using AIMC, we propose synchronous and asynchronous pipeline parallelism for AIMC accelerators inspired by the pipeline in digital domains. This paper provides a theoretical convergence guarantee for both synchronous and asynchronous pipelines in terms of both sampling and clock cycle complexity, which is non-trivial since the physical characteristic of AIMC accelerators leads to analog updates that suffer from asymmetric bias. The simulations of training DNN on real datasets verify the efficiency of pipeline training.

Quan Xiao, Hui Yuan, A F M Saif et al.

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.

Quan Xiao, Debarun Bhattacharjya, Balaji Ganesan et al.

Estimating the confidence of large language model (LLM) outputs is essential for real-world applications requiring high user trust. Black-box uncertainty quantification (UQ) methods, relying solely on model API access, have gained popularity due to their practical benefits. In this paper, we examine the implicit assumption behind several UQ methods, which use generation consistency as a proxy for confidence, an idea we formalize as the consistency hypothesis. We introduce three mathematical statements with corresponding statistical tests to capture variations of this hypothesis and metrics to evaluate LLM output conformity across tasks. Our empirical investigation, spanning 8 benchmark datasets and 3 tasks (question answering, text summarization, and text-to-SQL), highlights the prevalence of the hypothesis under different settings. Among the statements, we highlight the `Sim-Any' hypothesis as the most actionable, and demonstrate how it can be leveraged by proposing data-free black-box UQ methods that aggregate similarities between generations for confidence estimation. These approaches can outperform the closest baselines, showcasing the practical value of the empirically observed consistency hypothesis.

Liuyuan Jiang, Quan Xiao, Victor M. Tenorio et al.

Interest in bilevel optimization has grown in recent years, partially due to its applications to tackle challenging machine-learning problems. Several exciting recent works have been centered around developing efficient gradient-based algorithms that can solve bilevel optimization problems with provable guarantees. However, the existing literature mainly focuses on bilevel problems either without constraints, or featuring only simple constraints that do not couple variables across the upper and lower levels, excluding a range of complex applications. Our paper studies this challenging but less explored scenario and develops a (fully) first-order algorithm, which we term BLOCC, to tackle BiLevel Optimization problems with Coupled Constraints. We establish rigorous convergence theory for the proposed algorithm and demonstrate its effectiveness on two well-known real-world applications - hyperparameter selection in support vector machine (SVM) and infrastructure planning in transportation networks using the real data from the city of Seville.

Tianyi Chen, Yuejiao Sun, Quan Xiao et al.

Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends the solution of another optimization problem. Recently, stochastic bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of stochastic optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new optimization method for a class of stochastic bilevel problems that we term Single-Timescale stochAstic BiLevEl optimization (STABLE) method. STABLE runs in a single loop fashion, and uses a single-timescale update with a fixed batch size. To achieve an $ε$-stationary point of the bilevel problem, STABLE requires ${\cal O}(ε^{-2})$ samples in total; and to achieve an $ε$-optimal solution in the strongly convex case, STABLE requires ${\cal O}(ε^{-1})$ samples. To the best of our knowledge, this is the first bilevel optimization algorithm achieving the same order of sample complexity as the stochastic gradient descent method for the single-level stochastic optimization.

Quan Xiao, Canhong Wen, Zirui Yan

K-SVD algorithm has been successfully applied to image denoising tasks dozens of years but the big bottleneck in speed and accuracy still needs attention to break. For the sparse coding stage in K-SVD, which involves $\ell_{0}$ constraint, prevailing methods usually seek approximate solutions greedily but are less effective once the noise level is high. The alternative $\ell_{1}$ optimization is proved to be powerful than $\ell_{0}$, however, the time consumption prevents it from the implementation. In this paper, we propose a new K-SVD framework called K-SVD$_P$ by applying the Primal-dual active set (PDAS) algorithm to it. Different from the greedy algorithms based K-SVD, the K-SVD$_P$ algorithm develops a selection strategy motivated by KKT (Karush-Kuhn-Tucker) condition and yields to an efficient update in the sparse coding stage. Since the K-SVD$_P$ algorithm seeks for an equivalent solution to the dual problem iteratively with simple explicit expression in this denoising problem, speed and quality of denoising can be reached simultaneously. Experiments are carried out and demonstrate the comparable denoising performance of our K-SVD$_P$ with state-of-the-art methods.