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Social Aspects · Fairness

Linus Bleistein, Mathieu Dagréou, Francisco Andrade, Thomas Boudou, Aurélien Bellet

Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalized OT problem, for which we derive novel finite-sample complexity guarantees. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound on the deviation of fairness when matching unseen data. Finally, we present empirical results illustrating the performance of our approaches and the trade-off between fairness and transport cost.

Optimization · Non-Convex

Ying Chen, Aoxi Li, Javad Lavaei

Sharpness-Aware Minimization (SAM) empirically boosts generalization by seeking parameters that minimize the worst-case loss in a small neighborhood, yet existing theory explains its behavior under either strong convexity or small perturbation radius. We revisit SAM through the bilevel minimax problem $\min_{\theta}\max_{\|\Delta\|\le\rho}l(\theta+\Delta)$ and derive a $(\theta,\Delta)$ gradient flow ODE whose equilibria coincide with the problem’s optimality conditions. A Lyapunov argument—free of convexity assumptions—quantifies how the optimality gap depends on the radius~$\rho$ and local curvature. Discretizing the flow yields a \emph{Multi-step SAM} algorithm that recovers classical SAM as $\rho\!\to\!0$. {Moreover, our analysis and the resulting algorithm remain valid even for large $\rho$, providing principled guidance for aggressive neighborhood exploration.} Experiments on synthetic objectives and CIFAR-10 validate the predicted gains from multiple inner updates, bridging the gap between SAM’s minimax intuition and its practical implementation.

General Machine Learning · Transfer, Multitask and Meta-learning

Zhiyuan Yu, Enneng Yang, Hao Jiang, Guojie Zhu, Feihong He, Peng Wang, Li Shen

Upcycling, a strategy that initializes Mixture-of-Experts (MoE) by replicating pre-trained feed-forward or MoE networks to expand model capacity, has become a popular method in continual learning due to its effectiveness in mitigating catastrophic forgetting. However, existing paradigms rely on indiscriminate expansion prioritize performance at the cost of extreme inefficiency, introducing parameter redundancy without exploiting the structural heterogeneity essential for counteracting forgetting with architectural economy. To address this, we investigate the determinants of forgetting in training dynamics using Centered Kernel Alignment (CKA) and loss landscape flatness to analyze the behavior of pre- and post-expansion MoE layers, uncovering instability in deep-layer representations and heterogeneous expert sensitivity to new tasks, thereby demonstrating the potential of selective upcycling to eliminate redundancy. Consequently, we propose a dynamic bilevel optimization framework to guide adaptive upcycling, featuring an outer loop employing a Gumbel-Softmax differentiable mask to perform Neural Architecture Search (NAS) for adaptive growth, while an inner loop optimizes weight updates via task objectives and CKA-regularized replay. Experiments on TRACE benchmark demonstrate that our proposed method achieves better average accuracy with 80\% forgetting reduction, while effectively eliminating 60\% of redundant parameter expansion that standard upcycling would introduce.

Optimization · Zero-order and Black-box Optimization

Junbin Qiu, Zhaowei Hong, Renzhe Xu, Yao Shu

Accurate Zeroth-Order (ZO) Hessian estimation is a cornerstone of derivative-free methods, essential for tasks such as bilevel optimization, Bayesian inference, and uncertainty quantification. However, obtaining a complete suite of low-variance estimators for the Hessian and its inverse in high-dimensional settings remains a significant challenge. To address this, we propose a unified framework that reinterprets ZO Hessian approximation through the lens of single-step Policy Optimization (PO). This perspective establishes a theoretical equivalence between general ZO Hessian estimators and the Hessian of a smoothed PO objective, unifying distinct classical randomized estimators as specific instances of baseline selection. Building on this foundation, we introduce ZoVH, a comprehensive suite of variance-reduced estimators for the full Hessian matrix, its regularized inverse, and the bias-corrected inverse Hessian-gradient product. ZoVH leverages two key techniques: (1) a unique optimal baseline derived to provably minimize variance, and (2) a query reuse strategy that incorporates historical function queries to enhance sample efficiency without inflating costs. Our rigorous theoretical analysis confirms the unbiasedness of the Hessian estimator, validates the variance optimality of our baseline, provides error bounds for the entire ZoVH suite, and establishes convergence guarantees for the resulting curvature-aware ZO algorithm. Extensive empirical results validate our theoretical findings, demonstrating that ZoVH achieves superior estimation accuracy and convergence performance in real-world applications.

Deep Learning · Self-Supervised Learning

Gustav Wagner Zakarias, Lars Kai Hansen, Zheng-Hua Tan

Models initialized from self-supervised pretraining may suffer from poor alignment with downstream tasks, limiting the extent to which subsequent fine-tuning can adapt relevant representations acquired during the pretraining phase. To mitigate this, we introduce BiSSL, a novel bilevel training framework that enhances the alignment of self-supervised pretrained models with downstream tasks by explicitly incorporating both the pretext and downstream tasks into a preparatory training stage prior to fine-tuning. BiSSL solves a bilevel optimization problem in which the lower-level adheres to the self-supervised pretext task, while the upper-level encourages the lower-level backbone to align with the downstream objective. The bilevel structure facilitates enhanced information sharing between the tasks, ultimately yielding a backbone model that is more aligned with the downstream task, providing a better initialization for subsequent fine-tuning. We propose a general training algorithm for BiSSL that is compatible with a broad range of pretext and downstream tasks. We demonstrate that our proposed framework significantly improves accuracy on the vast majority of a broad selection of image-domain downstream tasks, and that these gains are consistently retained across a wide range of experimental settings. In addition, exploratory alignment analyses further underpin that BiSSL enhances downstream alignment of pretrained representations.

Deep Learning · Large Language Models

Hongyeon Yu, Young-Bum Kim, Yoon Kim

LLM workflows, which coordinate structured calls to individual LLMs (each augmented with varying instructions and tools) to achieve a particular goal, offer a promising path towards extending the capabilities of LLMs and building powerful systems that can tackle diverse tasks. However, existing approaches for building such workflows generally rely on human-crafted pipelines and prompts, which presents a substantial bottleneck to widening the scope of their applications. How can automatically induce and optimize such workflows in a data-driven way? And can lessons from optimizing deep learning architectures help the design of workflow induction algorithms? This paper describes a simple approach for automatically inducing LLM workflows. We formulate workflow induction as a bilevel optimization problem: an outer loop which optimizes a high-level sketch of the workflow (in particular how the LLM calls should be structured), and an inner loop which optimizes each individual LLM call one-by one. Both loops are optimized with ``textual gradients'', where for the inner loop we optimize each component in a modular way through ``backpropagating'' textual gradients layer-by-layer. We find that LLM workflows discovered through our WIBOT (\textbf{w}orkflow \textbf{i}nduction through \textbf{b}ilevel \textbf{o}ptimization and \textbf{t}extual gradients) approach performs competitively against strong baselines that automate workflow generation and optimization.

Optimization · Non-Convex

Kaiyi Ji

Although upper bound guarantees for bilevel optimization have been widely studied, progress on lower bounds has been limited due to the complexity of the bilevel structure. In this work, we focus on the smooth nonconvex-strongly-convex setting and develop new hard instances that yield nontrivial lower bounds under deterministic and stochastic first-order oracle models. In the deterministic case, we prove that any first-order zero-respecting algorithm requires at least $\Omega(\kappa^{3/2}\epsilon^{-2})$ oracle calls to find an $\epsilon$-accurate stationary point, improving the optimal lower bounds known for single-level nonconvex optimization and for nonconvex-strongly-convex min-max problems. In the stochastic case, we show that at least $\Omega(\kappa^{5/2}\epsilon^{-4})$ stochastic oracle calls are necessary, again strengthening the best known bounds in related settings. Our results expose substantial gaps between current upper and lower bounds for bilevel optimization and suggest that even simplified regimes, such as those with quadratic lower-level objectives, warrant further investigation toward understanding the optimal complexity of bilevel optimization under standard first-order oracles.

Deep Learning · Generative Models and Autoencoders

Zihao Cheng, Binrui Wu, Yuesen Liao, Weizhong Zhang

Diffusion probabilistic models (DPMs)’ sampling process is often inefficient, requiring hundreds to thousands of iterative steps to accurately approximate the diffusion trajectory. This inefficiency limits their practical applicability. Although recent advances in sampling efficiency—such as numerical solvers for diffusion ordinary differential equations (ODEs)—have made progress, significant challenges remain: training-free numerical solvers suffer from the suboptimality of manually designed timestep selection rules and the inherent inconsistency between the forward diffusion process (typically involving thousands of steps) and the reverse denoising process (usually limited to tens of steps). Since timestep selection is inherently a discrete problem and cannot be optimized via gradients, we propose an innovative approach—reparameterizing the timestep scheduling through probabilistic masking, thereby enabling gradient-based optimization of sampling timesteps. To circumvent backpropagation, we employ policy gradient methods. Furthermore, to address the inconsistency between forward diffusion (training) and reverse denoising (inference), we extend this framework into a bilevel optimization paradigm: the inner loop performs additional lightweight training on the model at specific timesteps determined by the outer mask to align forward and reverse processes, while the outer loop optimizes the timestep distribution via probabilistic masking and policy gradient based on generation quality. Under mild assumptions, we theoretically analyze the convergence of the proposed algorithm. Extensive experiments across diverse datasets and samplers demonstrate that this framework effectively enhances sampling efficiency and generation quality while maintaining compatibility with various DPM architectures and advanced ODE solvers.

Deep Learning · Large Language Models

Jie Hao, Rui Yu, Wei Zhang, Huixia Judy Wang, Jie Xu, Mingrui Liu

Effective data selection is essential for pretraining large language models (LLMs), enhancing efficiency and improving generalization to downstream tasks. However, existing approaches often require leveraging external pretrained models, making it difficult to disentangle the effects of data selection from those of the external pretrained models. In addition, they often overlook the long-term impact of selected data if the model is trained for a long period of time, primarily due to the prohibitive cost of full-scale LLM pretraining. In this paper, we introduce BLISS (**B**ileve**L** **I**nfluence **S**coring method for data **S**election): a lightweight data selection method that operates entirely \emph{from scratch}, without relying on any external pretrained oracle models, while explicitly accounting for the long-term impact of selected data. BLISS leverages a small proxy model as a surrogate for the LLM and employs a score model to estimate the long-term influence of training samples if the proxy model is trained to convergence. We formulate data selection as a bilevel optimization problem, where the upper-level objective optimizes the score model to assign importance weights to training samples, ensuring that minimizing the lower-level objective (i.e., training the proxy model over the weighted training loss until convergence) leads to best validation performance. Once optimized, the trained score model predicts influence scores for the dataset, enabling efficient selection of high-quality samples for LLM pretraining. We validate BLISS by pretraining 410M/1B/2.8B Pythia and LLaMA-0.5B models on selected subsets of the C4 dataset. Notably, under the 1B model setting, BLISS achieves $1.7\times$ speedup in reaching the same performance as the state-of-the-art method, demonstrating superior performance across multiple downstream tasks.

Optimization · Non-Convex

Zhiyao Zhang, Myeung Suk Oh, Zhen Qin, Jiaxiang Li, Xin Zhang, Jia (Kevin) Liu

In recent years, bilevel optimization (BLO) has attracted significant attention for its broad applications in machine learning. However, most existing works on BLO remain confined to the single-task setting and rely on the lower-level strong convexity assumption, which significantly restricts their applicability to modern machine learning problems of growing complexity. In this paper, we make the first attempt to extend BLO to the multi-task setting under a relaxed lower-level general convexity (LLGC) assumption. To this end, we reformulate the multi-task bilevel learning (MTBL) problem with LLGC into an equality constrained multi-objective optimization (ECMO) problem. However, ECMO itself is a new problem that has not yet been studied in the literature. To address this gap, we first establish a new Karush–Kuhn–Tucker (KKT)-based Pareto stationarity as the convergence criterion for ECMO algorithm design. Based on this foundation, we propose a weighted Chebyshev (WC)-penalty algorithm that achieves a finite-time convergence rate of $\mathcal{O}(ST^{-\frac{1}{2}})$ to KKT-based Pareto stationarity in both deterministic and stochastic settings, where $S$ denotes the number of objectives, and $T$ is the total iterations. Moreover, by varying the preference vector over the $S$-dimensional simplex, our WC-penalty method systematically explores the Pareto front. Finally, solutions to the ECMO problem translate directly into solutions for the original MTBL problem, thereby closing the loop between these two foundational optimization frameworks.

Optimization · Non-Convex

Zihao Zheng, Irwin King, Songtao Lu

Reinforcement learning (RL) is often hierarchical: an upper-level (UL) learner selects model parameters while a lower-level (LL) decision-making process responds, forming a nested two-level optimization structure captured by bilevel optimization. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and thus miss the competitive structure in applications such as incentive design where multiple policies interact. We study a class of bilevel optimization problems whose LL is a regularized min--max zero-sum Markov game, and whose UL optimizes through the saddle-point equilibrium induced by the LL game. We propose a penalty-based first-order policy-gradient method built on the Nikaido–Isoda function, termed penalty-augmented Nikaido–Isoda descent–ascent (PANDA), which avoids UL hypergradients and requires no second-order information by exploiting the min--max game structure. We prove that PANDA converges to stationary points of this bilevel problem without restrictive assumptions such as convexity on either the UL or LL objectives. Moreover, PANDA reaches an $\epsilon$-stationary point in $\tilde{\mathcal{O}}(\epsilon^{-1})$ iterations with sample complexity $\tilde{\mathcal{O}}(\epsilon^{-3})$, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments further demonstrate superior performance over closely related baselines.

Optimization · Large Scale, Parallel and Distributed

Yihan Zhang, Xinwen Zhang, My T. Thai, Jie Wu, Hongchang Gao

Decentralized stochastic bi-level optimization has been actively studied in recent years. However, existing studies assume that the lower-level loss function is strongly convex, which limits their applicability to many machine learning models. To address this limitation, in this paper, we propose a novel decentralized stochastic first-order optimization algorithm, which does not require second-order Hessian or Jacobian matrices, for the setting where the lower-level loss function is nonconvex but satisfies the Polyak–Łojasiewicz (PL) condition. Additionally, unlike existing single-agent methods that introduce a regularization term to the lower-level loss function to artificially enforce strong convexity, our algorithm does not require such modification. Moreover, our algorithm employs a constant single-timescale learning rate for updating variables, which is different from the time-dependent and two-timescale learning rate schedules used in prior work. To establish the convergence rate, we develop a new convergence analysis framework for the pure PL condition, rather than relying on the artificial strong convexity introduced through regularization in existing single-agent methods. To the best of our knowledge, this is the first algorithm for nonconvex decentralized bi-level optimization that offers theoretical convergence guarantees under mild conditions. Finally, our extensive experimental results on hyperparameter optimization and model pruning applications validate the efficacy of the proposed algorithm.

Optimization · Everything Else

Jonas Ohnemus, Marta Fochesato, Riccardo Zuliani, John Lygeros

Optimal-transport distributionally robust optimization (OT-DRO) robustifies data-driven decision-making under uncertainty by capturing the sampling-induced statistical error via optimal transport ambiguity sets. The standard OT-DRO pipeline consists of a two-step procedure, where the ambiguity set is first designed and subsequently embedded into the downstream OT-DRO problem. However, this separation between uncertainty quantification and optimization may lead to excessive conservatism. We introduce an end-to-end pipeline to automatically learn decision-focused ambiguity sets for OT-DRO problems, where the loss function informs the shape of the ambiguity set, leading to less conservative decisions whose distributional robustness is enforced via data-driven bootstrapping. We formulate the learning problem as a bilevel optimization program and solve it via a hypergradient-based method. By leveraging the recently introduced nonsmooth conservative implicit function theorem, we establish convergence to a critical point of the bilevel problem. We present experiments validating our method on standard portfolio optimization and linear regression tasks.

Theory · Optimization

Jinyi Huang, Jinlong Lei, Guodong Shi

Distributionally Robust Optimization (DRO) is widely used to improve model robustness, with existing methods addressing either geometric perturbations (e.g., input shifts) or statistical contamination (e.g., heavy-tailed noise and outliers) effectively. However, these uncertainty sources often co-exist. Coupling them through a single divergence or optimal transport constraint conflates geometric displacement with loss-based outlierness, which often leads to the discarding of informative high-leverage samples. We introduce nested DRO, a bilevel formulation that combines optimal transport with an outer $\\phi$-divergence constraint to decouple geometric smoothing from statistical robustness. We prove that this structure naturally induces a geometry-invariant, loss-based reweighting mechanism that separates outlier suppression from transport-induced regularization. We derive a tractable strong dual for the resulting non-convex problem and show its equivalence to variance-regularized risk minimization, providing a rigorous theoretical justification for reweighting gains as a natural consequence of dualization. Empirical results on synthetic and real datasets demonstrate that nested DRO consistently outperforms geometry-coupled DRO baselines, particularly under heavy-tailed contamination where preserving high-leverage structure is crucial.

Social Aspects · Privacy

Mohammad Yaghini, Michael Aerni, Junrui Zhang, Nicolas Papernot, Florian Tramer

Privacy auditing has emerged as a practical tool for empirically estimating training data leakage in machine learning models, in contrast to the provable but often overly pessimistic bounds provided by differential privacy analysis. A common strategy is to use membership inference attacks to detect the presence of specific canaries—data points chosen to maximize attack success—in training data. However, existing canary designs are largely heuristic, relying on mislabeled or out-of-distribution samples. We address this gap by formulating canary design as a bilevel optimization problem, where the model is trained in the inner loop and the canary is optimized in the outer loop to maximize its detectability. To solve this problem, we develop OptiFluence, a scalable optimization framework that combines (i) initialization by selecting candidates using influence functions and (ii) unrolled optimization with memory-efficient techniques. Our approach achieves remarkable empirical performance on four datasets. Optimized canaries demonstrate 415$\times$ (CIFAR-10/100) higher detectability than in-distribution baselines, achieving near-perfect detection rates of 99% true positive rate at 0.1% false positive rate. Critically, these canaries transfer effectively across different model architectures without retraining, enabling practical third-party privacy audits. This transferability allows regulators and auditors to assess model privacy without requiring access to proprietary training infrastructure or substantial computational resources.

Optimization · Everything Else

Zihao Zhao, Kai-Chia Mo, Shing-Hei Ho, Brandon Amos, Kai Wang

Differentiable optimization studies how to embed a mathematical program as a differentiable layer in machine learning pipelines. However, existing approaches typically rely on implicit differentiation, involving expensive Hessian computation while differentiating through optimality conditions. To address this challenge, we formulate the differentiable optimization problem as a bilevel optimization instance. We construct a new active-set Lagrangian as a proxy to compute an $\epsilon$-approximate hypergradient using only near-constant $O(\log (1/\epsilon))$ first-order information. We also show that applying this efficient hypergradient oracle to constrained bilevel optimization improves the overall gradient complexity to $\tilde{O}(\delta^{-1}\epsilon^{-3})$ to reach a $(\delta, \epsilon)$-Goldstein stationary point. We implement our method `FFOLayer`, as a drop-in Python library compatible with existing differentiable optimization solvers. Our algorithm shows significantly faster computation with similar convergence compared to other existing solvers. Our code is available [here](https://anonymous.4open.science/anonymize/FFOLayer-B78B).

Optimization · Large Scale, Parallel and Distributed

Yaoshuai Ma, Xiao Wang, Wei Yao, Jin Zhang

Distributed optimization over time-varying directed graphs has shown promising performance in addressing challenges posed by complex communication constraints in real-world scenarios. In many practical settings, however, the direct application of distributed optimization algorithms encounters additional difficulties, most notably hyperparameter tuning, which our empirical observations suggest can be effectively mitigated by integrating bilevel optimization. Motivated by these findings, we study distributed bilevel optimization over time-varying directed networks, a problem that remains largely unexplored due to the compounded challenges arising from consensus bias in dynamic unbalanced communication and the nested optimization structure. In this work, we propose a fully first-order distributed gradient-based algorithm that integrates the Push–Pull (also known as AB) communication strategy with a value function-based penalty method and establish its non-asymptotic convergence properties. Notably, a simplified variant of our analysis framework for nonconvex single-level distributed optimization establishes a convergence rate for the Push–Pull algorithm, thereby resolving an open question concerning its convergence over time-varying directed graphs. Empirical evaluations across diverse tasks, including hyperparameter tuning, data hyper-cleaning, and reinforcement learning, validate the effectiveness and efficiency of the proposed algorithm.

Optimization · Non-Convex

Yuman Wu, Xiaochuan Gong, Jie Hao, Mingrui Liu

Bilevel optimization is a hierarchical framework where an upper-level optimization problem is constrained by a lower-level problem, commonly used in machine learning applications such as hyperparameter optimization. Existing bilevel optimization methods typically assume strong convexity or Polyak-Łojasiewicz (PL) conditions for the lower-level function to establish non-asymptotic convergence to a solution with a small hypergradient. However, these assumptions may not hold in practice, and recent work (Chen et al. 2024) has shown that bilevel optimization is inherently intractable for general convex lower-level functions with the goal of finding small hypergradients. In this paper, we identify a tractable class of bilevel optimization problems that interpolates between lower-level strong convexity and general convexity via lower-level uniform convexity. For uniformly convex lower-level functions with exponent $p\geq 2$, we establish a novel implicit differentiation theorem characterizing the hyperobjective's smoothness property. Building on this, we design a new stochastic algorithm, termed UniBiO, with provable convergence guarantees, based on an oracle that provides stochastic gradient and Hessian-vector product information for the bilevel problems. Our algorithm achieves $\widetilde{O}(\epsilon^{-5p+6})$ oracle complexity bound for finding $\epsilon$-stationary points. Notably, our complexity bounds match the optimal rates in terms of the $\epsilon$ dependency for strongly convex lower-level functions ($p=2$), up to logarithmic factors. Our theoretical findings are validated through experiments on synthetic tasks and data hyper-cleaning, demonstrating the effectiveness of our proposed algorithm.

Applications · Physics

Ege Demirci, Francesco Bullo, Ananthram Swami, Ambuj K Singh

Recovering missing flows on the edges of a network, while exactly respecting local conservation laws, is a fundamental inverse problem that arises in many systems such as transportation, energy, and mobility. We introduce FlowSymm, a novel architecture that combines (i) a group-action on divergence-free flows, (ii) a graph-attention encoder to learn feature-conditioned weights over these symmetry-preserving actions, and (iii) a lightweight Tikhonov refinement solved via implicit bilevel optimization. The method first anchors the given observation on a minimum-norm divergence-free completion. We then compute an orthonormal basis for all admissible group actions that leave the observed flows invariant and parameterize the valid solution subspace, which shows an Abelian group structure under vector addition. A stack of GATv2 layers then encodes the graph and its edge features into per-edge embeddings, which are pooled over the missing edges and produce per-basis attention weights. This attention-guided process selects a set of physics-aware group actions that preserve the observed flows. Finally, a scalar Tikhonov penalty refines the missing entries via a convex least-squares solver, with gradients propagated implicitly through Cholesky factorization. Across three real-world flow benchmarks (traffic, power, bike), FlowSymm substantially outperforms state-of-the-art baselines in RMSE, MAE and correlation metrics.

Social Aspects · Accountability, Transparency and Interpretability

Chenyang Ren, Huanyi Xie, Shu Yang, Meng Ding, Dongrui Liu, Lijie Hu, Di Wang

As one of the most fundamental models, meta learning aims to effectively address few-shot learning challenges. However, it still faces significant issues related to the training data, such as training inefficiencies due to numerous low-contribution tasks in large datasets and substantial noise from incorrect labels. Thus, training data attribution methods are needed for meta learning. However, the dual-layer structure of meta learning complicates the modeling of training data contributions because of the interdependent influence between meta parameters and task-specific parameters, making existing data influence evaluation tools inapplicable or inaccurate. To address these challenges, based on the influence function, we propose a general data attribution evaluation framework for meta learning within the bilevel optimization framework. Our approach introduces task influence functions (task-IF) and instance influence functions (instance-IF) to accurately assess the impact of specific tasks and individual data points in closed forms. This framework comprehensively models data contributions across both the inner and outer training processes, capturing the direct effects of data points on meta parameters as well as their indirect influence through task-specific parameters. We also provide several strategies to enhance computational efficiency and scalability. Experimental results demonstrate the framework's effectiveness in training data evaluation via several downstream tasks.