arXiv subjects

Peiran Yu

Publications and source records attributed to Peiran Yu.

13 recordsLinked to original sources

arXiv2026

New Hybrid Fine-Tuning Paradigm for LLMs: Algorithm Design and Convergence Analysis Framework

Fine-tuning Large Language Models (LLMs) typically involves either full fine-tuning, which updates all model parameters, or Parameter-Efficient Fine-Tuning (PEFT), which adjusts a small subset of parameters. However, both approaches have inherent limitations: full fine-tuning is computationally expensive, while PEFT often struggles to learn new knowledge and exhibits suboptimal performance. To overcome these issues, we propose a novel hybrid fine-tuning approach that jointly updates both LLMs and PEFT modules using a combination of zeroth-order and first-order optimization methods. To analyze our new algorithm, we develop a theoretical framework centered on the concept of hybrid smoothness condition, which accounts for the heterogeneous nature of the optimization landscape in joint LLM and PEFT training. We derive a rigorous convergence analysis for the convergence of reshuffling-type SGD algorithm under multiple learning rates and demonstrate its effectiveness through extensive empirical studies across various downstream tasks and model architectures. On the practical side, our results demonstrate consistent performance improvement, making the approach a viable solution for large-scale language model fine-tuning.

cs.AI↗

arXiv2026

Breaking the Correlation Plateau: On the Optimization and Capacity Limits of Attention-Based Regressors

Attention-based regression models are often trained by jointly optimizing Mean Squared Error (MSE) loss and Pearson correlation coefficient (PCC) loss, emphasizing the magnitude of errors and the order or shape of targets, respectively. A common but poorly understood phenomenon during training is the PCC plateau: PCC stops improving early in training, even as MSE continues to decrease. We provide the first rigorous theoretical analysis of this behavior, revealing fundamental limitations in both optimization dynamics and model capacity. First, in regard to the flattened PCC curve, we uncover a critical conflict where lowering MSE (magnitude matching) can paradoxically suppress the PCC gradient (shape matching). This issue is exacerbated by the softmax attention mechanism, particularly when the data to be aggregated is highly homogeneous. Second, we identify a limitation in the model capacity: we derived a PCC improvement limit for any convex aggregator (including the softmax attention), showing that the convex hull of the inputs strictly bounds the achievable PCC gain. We demonstrate that data homogeneity intensifies both limitations. Motivated by these insights, we propose the Extrapolative Correlation Attention (ECA), which incorporates novel, theoretically-motivated mechanisms to improve the PCC optimization and extrapolate beyond the convex hull. Across diverse benchmarks, including challenging homogeneous data setting, ECA consistently breaks the PCC plateau, achieving significant improvements in correlation without compromising MSE performance.

cs.LG↗

arXiv2025

Provably Mitigating Corruption, Overoptimization, and Verbosity Simultaneously in Offline and Online RLHF/DPO Alignment

Reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) are important techniques to align large language models (LLM) with human preference. However, the quality of RLHF and DPO training is seriously compromised by \textit{\textbf{C}orrupted} preference, reward \textit{\textbf{O}veroptimization}, and bias towards \textit{\textbf{V}erbosity}. To our knowledge, most existing works tackle only one of these important issues, and the few other works require much computation to estimate multiple reward models and lack theoretical guarantee of generalization ability. In this work, we propose RLHF-\textbf{COV} and DPO-\textbf{COV} algorithms that can simultaneously mitigate these three issues, in both offline and online settings. This ability is theoretically demonstrated by obtaining length-regularized generalization error rates for our DPO-COV algorithms trained on corrupted data, which match the best-known rates for simpler cases with clean data and without length regularization. Moreover, our DPO-COV algorithm is simple to implement without reward estimation, and is proved to be equivalent to our RLHF-COV algorithm, which directly implies the equivalence between the vanilla RLHF and DPO algorithms. Experiments demonstrate the effectiveness of our DPO-COV algorithms under both offline and online settings.

cs.LG↗

arXiv2025

Zeroth-Order Methods for Stochastic Nonconvex Nonsmooth Composite Optimization

This work aims to solve a stochastic nonconvex nonsmooth composite optimization problem. Previous works on composite optimization problem requires the major part to satisfy Lipschitz smoothness or some relaxed smoothness conditions, which excludes some machine learning examples such as regularized ReLU network and sparse support matrix machine. In this work, we focus on stochastic nonconvex composite optimization problem without any smoothness assumptions. In particular, we propose two new notions of approximate stationary points for such optimization problem and obtain finite-time convergence results of two zeroth-order algorithms to these two approximate stationary points respectively. Finally, we demonstrate that these algorithms are effective using numerical experiments.

math.OC↗

arXiv2025

Cost-Aware Contrastive Routing for LLMs

We study cost-aware routing for large language models across diverse and dynamic pools of models. Existing approaches often overlook prompt-specific context, rely on expensive model profiling, assume a fixed set of experts, or use inefficient trial-and-error strategies. We introduce Cost-Spectrum Contrastive Routing (CSCR), a lightweight framework that maps both prompts and models into a shared embedding space to enable fast, cost-sensitive selection. CSCR uses compact, fast-to-compute logit footprints for open-source models and perplexity fingerprints for black-box APIs. A contrastive encoder is trained to favor the cheapest accurate expert within adaptive cost bands. At inference time, routing reduces to a single k-NN lookup via a FAISS index, requiring no retraining when the expert pool changes and enabling microsecond latency. Across multiple benchmarks, CSCR consistently outperforms baselines, improving the accuracy-cost tradeoff by up to 25%, while generalizing robustly to unseen LLMs and out-of-distribution prompts.

cs.LG↗

arXiv2025

Revisiting Convergence: Shuffling Complexity Beyond Lipschitz Smoothness

Shuffling-type gradient methods are favored in practice for their simplicity and rapid empirical performance. Despite extensive development of convergence guarantees under various assumptions in recent years, most require the Lipschitz smoothness condition, which is often not met in common machine learning models. We highlight this issue with specific counterexamples. To address this gap, we revisit the convergence rates of shuffling-type gradient methods without assuming Lipschitz smoothness. Using our stepsize strategy, the shuffling-type gradient algorithm not only converges under weaker assumptions but also match the current best-known convergence rates, thereby broadening its applicability. We prove the convergence rates for nonconvex, strongly convex, and non-strongly convex cases, each under both random reshuffling and arbitrary shuffling schemes, under a general bounded variance condition. Numerical experiments further validate the performance of our shuffling-type gradient algorithm, underscoring its practical efficacy.

cs.LG↗

arXiv2025

Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-{\L}ojasiewicz property

We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [12] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Lojasiewicz (KL) properties of suitable potential functions. We also study how the KL exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [12], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [13, Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.

math.OC↗

arXiv2025

Bilevel ZOFO: Efficient LLM Fine-Tuning and Meta-Training

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.

cs.LG↗

arXiv2024

Efficient Fine-Tuning and Concept Suppression for Pruned Diffusion Models

Recent advances in diffusion generative models have yielded remarkable progress. While the quality of generated content continues to improve, these models have grown considerably in size and complexity. This increasing computational burden poses significant challenges, particularly in resource-constrained deployment scenarios such as mobile devices. The combination of model pruning and knowledge distillation has emerged as a promising solution to reduce computational demands while preserving generation quality. However, this technique inadvertently propagates undesirable behaviors, including the generation of copyrighted content and unsafe concepts, even when such instances are absent from the fine-tuning dataset. In this paper, we propose a novel bilevel optimization framework for pruned diffusion models that consolidates the fine-tuning and unlearning processes into a unified phase. Our approach maintains the principal advantages of distillation-namely, efficient convergence and style transfer capabilities-while selectively suppressing the generation of unwanted content. This plug-in framework is compatible with various pruning and concept unlearning methods, facilitating efficient, safe deployment of diffusion models in controlled environments.

cs.LG↗

arXiv2020

Analysis and algorithms for some compressed sensing models based on L1/L2 minimization

Recently, in a series of papers [32,38,39,41], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing $\ell_1$/$\ell_2$ subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [32] satisfies the Kurdyka-Lojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [39, Eq. 11] under mild assumptions. We next extend the $\ell_1$/$\ell_2$ model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [37,44], and extend the algorithm in [39, Eq. 11] by incorporating moving-balls-approximation techniques [4] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions, and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by $ \ell_2 $ norm or Lorentzian norm via solving the corresponding $\ell_1$/$\ell_2$ models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy.

math.OC↗

arXiv2020

Convergence rate analysis of a sequential convex programming method with line search for a class of constrained difference-of-convex optimization problems

In this paper, we study the sequential convex programming method with monotone line search (SCP$_{ls}$) in [46] for a class of difference-of-convex (DC) optimization problems with multiple smooth inequality constraints. The SCP$_{ls}$ is a representative variant of moving-ball-approximation-type algorithms [6,10,13,54] for constrained optimization problems. We analyze the convergence rate of the sequence generated by SCP$_{ls}$ in both nonconvex and convex settings by imposing suitable Kurdyka-Lojasiewicz (KL) assumptions. Specifically, in the nonconvex settings, we assume that a special potential function related to the objective and the constraints is a KL function, while in the convex settings we impose KL assumptions directly on the extended objective function (i.e., sum of the objective and the indicator function of the constraint set). A relationship between these two different KL assumptions is established in the convex settings under additional differentiability assumptions. We also discuss how to deduce the KL exponent of the extended objective function from its Lagrangian in the convex settings, under additional assumptions on the constraint functions. Thanks to this result, the extended objectives of some constrained optimization models such as minimizing $\ell_1$ subject to logistic/Poisson loss are found to be KL functions with exponent $\frac12$ under mild assumptions. To illustrate how our results can be applied, we consider SCP$_{ls}$ for minimizing $\ell_{1-2}$ [60] subject to residual error measured by $\ell_2$ norm/Lorentzian norm [21]. We first discuss how the various conditions required in our analysis can be verified, and then perform numerical experiments to illustrate the convergence behaviors of SCP$_{ls}$.

math.OC↗

arXiv2019

Kurdyka-{\L}ojasiewicz exponent via inf-projection

Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of $\frac12$ for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we show that the KL exponent is $\frac12$ for several important convex optimization models, including some semidefinite-programming-representable functions and some functions that involve $C^2$-cone reducible structures, under conditions such as strict complementarity. Our results are applicable to concrete optimization models such as group fused Lasso and overlapping group Lasso. In addition, for nonconvex models, we show that the KL exponent of many difference-of-convex functions can be derived from that of their natural majorant functions, and the KL exponent of the Bregman envelope of a function is the same as that of the function itself. Finally, we estimate the KL exponent of the sum of the least squares function and the indicator function of the set of matrices of rank at most $k$.

math.OC↗

arXiv2017

Iteratively reweighted $\ell_1$ algorithms with extrapolation

Iteratively reweighted $\ell_1$ algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in [4,5,22,28] can be incorporated to possibly accelerate the iteratively reweighted $\ell_1$ algorithm. We consider three versions of such algorithms. For each version, we exhibit an explicitly checkable condition on the extrapolation parameters so that the sequence generated provably clusters at a stationary point of the optimization problem. We also investigate global convergence under additional Kurdyka-$\L$ojasiewicz assumptions on certain potential functions. Our numerical experiments show that our algorithms usually outperform the general iterative shrinkage and thresholding algorithm in [21] and an adaptation of the iteratively reweighted $\ell_1$ algorithm in [23, Algorithm 7] with nonmonotone line-search for solving random instances of log penalty regularized least squares problems in terms of both CPU time and solution quality.

math.OC↗