论文检索

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

Kangke Cheng, Jiawei Huang, Jingni Song, Wanlin Zhang, Bangxian Han, Hu Ding

The Gromov-Wasserstein (GW) distance enables comparison across different spaces but remains fragile to structural noise due to its global quadratic coupling. Existing robust extensions primarily rely on node-centric mass relaxation. However, we argue that this strategy is far from sufficient: it only addresses node-induced structural noise (outliers) while neglecting edge-induced distortions where spurious connections exist between valid nodes. To overcome this limitation, we propose the Structurally Robust Gromov-Wasserstein (SRGW) distance, a novel formulation that adaptively filters geometric distortions during optimization. By introducing a structure-aware dual-mask mechanism, our method effectively isolates these stubborn structural outliers while preserving strict marginal constraints for balanced transport. We solve this objective using a Mask-Guided GW Algorithm, which jointly optimizes the transport plan and the structural noise filters. We provide a rigorous theoretical analysis proving that our algorithm converges to a critical point under the Kurdyka-Łojasiewicz framework. Extensive experiments on synthetic geometric matching and real-world subgraph alignment benchmarks demonstrate that Mask-Guided GW achieves superior alignment quality, particularly under severe structural noise.

Optimization · Discrete and Combinatorial Optimization

Cheng-Han Huang, Yongliang Sun, Chaoyan Huang, Ismail Alkhouri, Rongrong Wang

Many combinatorial optimization problems admit quadratic unconstrained binary formulations (QUBO) which can often be relaxed to the box $[0,1]^n$ and optimized using scalable gradient-based methods. However, the resulting non-convex landscape can often contain local optima that are spurious or infeasible. In this paper, we establish sufficient structural conditions on quadratic penalties that rule out these failures, guaranteeing that every local minimizer of the relax problem is both binary and feasible. For each problem we study, we examine existing QUBO formulations when available, identify why they fail when they do, and propose alternative relaxed QUBOs that satisfy our conditions. We show for several common combinatorial problems, including open-pit mining, knapsack, and traveling salesman formulations, that these constructions allow gradient-based methods such as projected gradient descent and Adam to be safely applied to obtain valid binary solutions. Our results clarify when differentiable optimization is a reliable local solver for quadratic combinatorial objectives.

Theory · Deep Learning

Hongkang Li, Hancheng Min, Rene Vidal

Transformer-based diffusion models have demonstrated remarkable performance at generating high-quality samples. However, our theoretical understanding of the reasons for this success remains limited. For instance, existing models are typically trained by minimizing a denoising objective, which is equivalent to fitting the score function of the training data. However, we do not know why transformer-based models can match the score function for denoising, or why gradient-based methods converge to the optimal denoising model despite the non-convex loss landscape. To the best of our knowledge, this paper provides the first convergence analysis for training transformer-based diffusion models. More specifically, we consider the population Denoising Diffusion Probabilistic Model (DDPM) objective for denoising data that follow a \textit{multi-token Gaussian mixture} distribution. We theoretically quantify the required number of tokens per data point and training iterations for the global convergence towards the Bayes optimal risk of the denoising objective, thereby achieving a desired score matching error. A deeper investigation reveals that the self-attention module of the trained transformer implements a \emph{mean denoising} mechanism that enables the trained model to approximate the oracle Minimum Mean Squared Error (MMSE) estimator of the injected noise in the diffusion steps. Numerical experiments validate these findings.

Theory · Optimization

Loucas Pillaud-Vivien, Adrien Schertzer

We consider the problem of jointly learning a one-dimensional projection and a univariate function in high-dimensional Gaussian models. Specifically, we study predictors of the form $f(x)=\varphi^\star(\langle w^\star, x \rangle)$, where both the direction $w^\star \in \mathcal{S}_{d-1}$, the sphere of $\mathbb{R}^d$, and the function $\varphi^\star: \mathbb{R} \to \mathbb{R}$ are learned from Gaussian data. This setting captures a fundamental non-convex problem at the intersection of representation learning and nonlinear regression. We analyze the gradient flow dynamics of a natural alternating scheme and prove convergence, with a rate controlled by the information exponent reflecting the *Gaussian regularity* of the function $\varphi^\star$. Strikingly, our analysis shows that convergence still occurs even when the initial direction is negatively correlated with the target. On the practical side, we demonstrate that such joint learning can be effectively implemented using a Reproducing Kernel Hilbert Space (RKHS) adapted to the structure of the problem, enabling efficient and flexible estimation of the univariate function. Our results offer both theoretical insight and practical methodology for learning low-dimensional structure in high-dimensional settings.

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.

Lin Wang, Zhichao Wang, Ye Shi, Sai Praneeth Reddy Karimireddy, Xiaoying Tang

Federated Learning (FL) often suffers from a trade-off between global model performance and client-level fairness due to data heterogeneity, which often leads to inconsistent performance of the globally trained models, resulting in unfair outcomes among users. Existing fair FL algorithms face a trade-off: they either sacrifice global model performance to promote fairness or fall short of achieving optimal fairness. In this paper, we propose a novel framework that bridge this trade-off by integrating information-theoretic principles with model alignment. Specifically, we leverage the Maximum Entropy Principle to derive an analytic, closed-form solution for fair aggregation weights, ensuring significant fairness enhancements with minimal computational overhead. To maintain the global model performance, we further employ a step-wise model alignment strategy that synchronizes gradient directions across heterogeneous clients, effectively mitigating the drift induced by local updates. Theoretical analysis proves that our method guarantees convergence even in non-convex settings. Importantly, we push the theoretical frontier of federated fairness by extending performance variance analysis to generalized regression, providing broader guarantees. Extensive experiments on five datasets demonstrate that our approach consistently outperforms state-of-the-art methods, achieving superior fairness without sacrificing global accuracy.

Optimization · Non-Convex

Haosong Zhang, Shenxi Wu, Xingjian Ma, Shirui Bian, Yichi Zhang, Xi Chen, Wei Lin

Deeper modern architectures are costly to train, making hyperparameter transfer preferable to expensive repeated tuning. Maximal Update Parametrization ($\mu$P) helps explain why many hyperparameters transfer across width. Yet depth scaling is less understood for modern architectures, whose computation graphs contain multiple parallel paths and residual aggregation. To unify various non-recurrent multi-path neural networks such as CNNs, ResNets, and Transformers, we introduce a graph-based notion of effective depth. Under stabilizing initializations and a maximal-update criterion, we show that the optimal learning rate decays with effective depth following a universal -3/2 power law. Here, the maximal-update criterion maximizes the typical one-step representation change at initialization without causing instability, and effective depth is the minimal path length from input to output, counting layers and residual additions. Experiments across diverse architectures confirm the predicted slope and enable reliable zero-shot transfer of learning rates across depths and widths, turning depth scaling into a predictable hyperparameter-transfer problem.

General Machine Learning · Transfer, Multitask and Meta-learning

Longtao Xu, Jian Li

Real-world federated systems seldom operate on static data: input distributions drift while privacy rules forbid raw data sharing. We study Federated Domain-Incremental Learning (FDIL), where (i) clients are heterogeneous, (ii) tasks arrive sequentially with shifting domains, and (iii) the label space remains fixed. Two theoretical pillars remain missing for FDIL under partial participation: a guarantee of backward knowledge transfer (BKT) and a convergence rate that holds *uniformly across the task sequence*. We introduce SPECIAL (Server-Proximal Efficient Continual Aggregation for Learning), a simple, memory-free FDIL algorithm that adds a single server-side ``anchor'' to FedAvg: in each round, the server aggregates updates from a uniformly sampled subset of clients and then blends the result with the previous global model via a lightweight proximal step. This anchor curbs cumulative drift without replay buffers, synthetic data, or task-specific heads, leaving communication cost and model size unchanged. Our theory shows that SPECIAL (i) *preserves earlier tasks*: a BKT bound caps any increase in earlier-task loss by a drift-controlled term that shrinks with more rounds, local epochs, and participating clients; and (ii) *achieves task-uniform, communication-efficient convergence* for non-convex FDIL with partial participation: $\mathcal{O}\!\big(\sqrt{E/(NT)}\big)$ in expected gradient norm, with $E$ local epochs, $T$ rounds, and $N$ participating clients, while explicitly separating optimization variance from inter-task drift. Experiments on standard FDIL benchmarks corroborate the theory.

Theory · Learning Theory

SANTANU DAS, Sagnik Chatterjee, jatin batra

In this paper, we tackle a fundamental problem in high-dimensional statistics, namely, learning Single Index Models (SIMs) robustly in the presence of heavy-tailed noise and an adversary that can corrupt a constant fraction of both covariates and responses. Prior research on efficient robust recovery only focuses on monotonic link functions or only limit themselves to Phase Retrieval. Provable efficient robust recovery guarantees for generic nonlinear link functions have remained elusive. In this paper, we obtain the first near-linear time, optimal-sample-complexity robust recovery algorithm for a wide class of nonlinear non-monotonic link functions. Critical to our result is an improved understanding of the squared-loss landscape: we identify a sufficient condition under which a broad class of non linear link functions admit a dimension-independent constant-radius convex basin around the ground truth, establishing statistical identifiability beyond previously known cases. We also leverage second-order Stein's identities to identify a structural condition, that we term Expected Squared Convexity (ESC), that acts as a sufficient condition for spectral initialization techniques to obtain an estimator within the convex basin with error $O(\epsilon^{1/4})$, even under heavy-tailed noise and strong adversarial contamination. This robust initialization technique can be combined with a robust gradient descent phase to break the spectral error barrier, achieving an improved estimation error of $O(\sigma\sqrt{\epsilon})$. Our non-convex optimization framework gives the first efficient sample and time complexity robust recovery results for activation functions such as GeLU and Swish that act as building blocks of modern deep-learning architectures.

General Machine Learning · Clustering

Henri Schmidt, Peter Halmos, Benjamin Raphael

Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix defined on pairs of points. Unlike standard OT, which infers unstructured pointwise mappings, low-rank optimal transport explicitly constrains the rank of the transport plan to infer latent structure. This improves statistical stability and robustness, yields sharper parametric rates for estimating Wasserstein distances adaptive to the intrinsic rank, and generalizes $K$-means to co-clustering. These advantages, however, come at the cost of a non-convex and NP-hard optimization problem. We introduce transport clustering, an algorithm to compute a low-rank OT plan that reduces low-rank OT to a clustering problem on correspondences obtained from a full-rank *transport registration* step. We prove that this reduction yields polynomial-time, constant-factor approximation algorithms for low-rank OT: specifically, a $(1+\gamma)$ approximation for negative-type metrics and a $(1+\gamma+\sqrt{2\gamma})$ approximation for kernel costs, where $\gamma \in [0,1]$ denotes the approximation ratio of the optimal full-rank solution relative to the low-rank optimal. Empirically, transport clustering outperforms existing low-rank OT solvers on synthetic benchmarks and large-scale, high-dimensional datasets.

Theory · Optimization

Zijian Liu

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular $\mathtt{Adam}$ and $\mathtt{AdamW}$, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, $\mathtt{AdaGrad}$, the origin of adaptive gradient methods. We provide the first provable convergence rate for $\mathtt{AdaGrad}$ in non-convex optimization when the tail index $p$ satisfies $\frac{4}{3}<p\leq2$. Notably, this result is achieved without requiring any prior knowledge of $p$ and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by $\mathtt{AdaGrad}$. Lastly, we consider $\mathtt{AdaGrad}\text{-}\mathtt{Norm}$, a popular variant of $\mathtt{AdaGrad}$ in theoretical studies, and show an improved rate that holds for any $1<p\leq2$ under an extra mild assumption.

Optimization · Stochastic

Yuheng Zhao, Yu-Hu Yan, Amit Attia, Tomer Koren, Lijun Zhang, Peng Zhao

Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such methods as "partially parameter-free". In this work, we target achieving "*fully* parameter-free" methods, i.e., the algorithmic inputs do not need to satisfy any *unverifiable* condition related to the true problem parameters. We propose a general and powerful *grid search* framework, named GRASP, with a novel *self-bounding* analysis technique that effectively determines the parameter search ranges, in contrast to previous work. Our method demonstrates generality in: (i) the non-convex case, where we propose a fully parameter-free method that achieves near-optimal convergence rate, up to logarithmic factors; (ii) the convex case, where our parameter-free methods are competitive with strong performance in terms of acceleration and universality. Finally, we contribute a sharper guarantee for the model ensemble, a final step of the grid search framework, under interpolated variance characterization.

Optimization · Non-Convex

Haolin Pan, Lianghong Huang, Dong Jinyuan, Mingjie Xing, Yanjun Wu

Compiler auto-tuning faces a dichotomy between traditional black-box search methods, which lack semantic guidance, and recent Large Language Model (LLM) approaches, which often suffer from superficial pattern matching and causal opacity. In this paper, we introduce ECCO, a framework that bridges interpretable reasoning with combinatorial search. We first propose a reverse engineering methodology to construct a Chain-of-Thought dataset, explicitly mapping static code features to verifiable performance evidence. This enables the model to learn the causal logic governing optimization decisions rather than merely imitating sequences. Leveraging this interpretable prior, we design a collaborative inference mechanism where the LLM functions as a strategist, defining optimization intents that dynamically guide the mutation operations of a genetic algorithm. Experimental results on seven datasets demonstrate that ECCO outperforms the LLVM opt -O3 baseline, achieving an average 24.44% reduction in cycles. Our code is available at https://anonymous.4open.science/r/ECCO-Evidence-Driven-Causal-Reasoning-for-Compiler-Optimization-3AD2.

Optimization · Stochastic

Dimitris Oikonomou, Nicolas Loizou

The stochastic Polyak step size (SPS) has proven to be a promising choice for stochastic gradient descent (SGD), delivering competitive performance relative to state-of-the-art methods on smooth convex and non-convex optimization problems, including deep neural network training. However, extensions of this approach to non-smooth settings remain in their early stages, often relying on interpolation assumptions or requiring knowledge of the optimal solution. In this work, we propose a novel SPS variant, Safeguarded SPS (SPS$_{safe}$), for the stochastic subgradient method, and provide rigorous convergence guarantees for non-smooth convex optimization with no need for strong assumptions. We further incorporate momentum into the update rule, yielding equally tight theoretical results. Comprehensive experiments on convex benchmarks and deep neural networks corroborate our theory: the proposed step size achieves competitive performance to existing adaptive baselines and exhibits stable behavior across a wide range of problem settings. Finally, in the context of deep neural network training, the gradient norms under our step size do not collapse to (near) zero, indicating robustness to vanishing gradients.

Optimization · Non-Convex

Foivos Alimisis, Rustem Islamov, Aurelien Lucchi

Learning rate warm-up -- increasing the learning rate at the beginning of training -- has become a ubiquitous heuristic in modern deep learning, yet its theoretical foundations remain poorly understood. In this work, we provide a principled explanation for why warm-up improves training. We rely on a generalization of the $(L_0, L_1)$-smoothness condition, which bounds local curvature as a linear function of the loss sub-optimality and exhibits desirable closure properties. We show -- both theoretically and empirically -- that this condition is satisfied by common neural architectures and accurately captures the curvature of the optimization landscape early in training. Adapting the learning rate in response to this curvature condition naturally induces a warm-up–like schedule, and we show that this choice yields provably faster convergence guarantees than using a fixed learning rate. Finally, we validate our theoretical insights through experiments on language and vision models, confirming the agreement between our theoretically derived schedule and standard warm-up.

Optimization · Large Scale, Parallel and Distributed

Feihu Huang, Yuning Luo, Songcan Chen

Large models recently are widely applied in machine learning, so efficient training of large models has received widespread attention. More recently, a useful Muon optimizer is specifically designed for matrix-structured parameters of large models. Although some works have begun to studying the Muon optimizer, the existing Muon and its variants still suffer from high sample complexity or high memory for large models. To fill this gap, we propose a light and fast Muon (LiMuon) optimizer for training large models, which builds on the momentum-based variance reduced technique and randomized Singular Value Decomposition (SVD). In particular, our LiMuon optimizer simultaneously has a lower memory and lower sample complexity than the Muon. Moreover, we prove that our LiMuon has a lower sample complexity of $O(\epsilon^{-3})$ for finding an $\epsilon$-stationary solution of non-convex stochastic optimization under the generalized smooth condition. Numerical experimental results on training Mamba-130M, Qwen2.5-0.5B and ViT models demonstrate effectiveness of our LiMuon optimizer.

Optimization · Non-Convex

Luxuan Li, Xiao Wang, Chunfeng Cui

In this paper, we study a general class of nonconvex constrained stochastic problems with difference-of-convex (DC) regularization, where the feasible set is possibly nonconvex, and the concave part of the DC regularizer is allowed to be nonsmooth. The fundamental challenge lies in maintaining feasibility for nonconvex constraints while achieving favorable oracle complexity. Although single-loop algorithms are efficient in solving unconstrained DC optimization problems, their potential for constrained optimization with DC structure remains largely unexplored. To address this gap, we develop **MoSSP**, a **Mo**mentum-based **S**ingle-loop **S**tochastic **P**enalty method for such problems with provable complexity guarantee. The key idea is to perform a single stochastic proximal-gradient update that approximates the gradient of the Moreau envelope of the composite term, which consists of the penalty function and the convex component of the DC regularizer. Simultaneously, the proximal mapping of its concave component is computed in parallel. We derive two algorithm variants: a Polyak-momentum version with $\mathcal{O}(\varepsilon^{-4})$ oracle complexity for finding stochastic $\varepsilon$-KKT points, and an improved $\mathcal{O}(\varepsilon^{-3})$ version incorporating recursive momentum. Experiment results demonstrate the effectiveness of our proposed algorithms.

Deep Learning · Self-Supervised Learning

Maedeh Zarvandi, Michael Timothy, Theresa Wasserer, Debarghya Ghoshdastidar

Self-supervised learning (SSL) effectively learns representations from massive unlabeled data, yet the resulting models typically operate as black boxes, necessitating domain-specific post-hoc explanations. We introduce KREPES, a unified framework that learns inherently interpretable representations for arbitrary SSL objectives, including SimCLR, BYOL, VICReg. By bridging empirical neural tangent kernel approximations of neural networks with the Representer Theorem for kernels, we express the learned latent space directly via "Representer Landmarks", which are the representations of influential unlabeled training examples. We introduce two novel metrics, "Sample-Specific Influence Score" and "Conceptual Influence Profile", to quantify the transparency of the learned representations. KREPES enables direct audit of the latent space without supervision, for example, revealing an algorithmic bias in the Adult-1M dataset where SSL uses demographic proxies for income. Finally, to ensure scalability to SSL benchmarks with 1M+ samples (ImageNet-1K, Adult-1M), KREPES introduces a novel Nyström approximation-based optimization of any non-convex SSL objective.

Optimization · Everything Else

Guillaume Houry, Jean Feydy, François-Xavier Vialard

A fundamental challenge in data science is to match disparate point sets with each other. While optimal transport efficiently minimizes point *displacements* under a bijectivity constraint, it is inherently sensitive to rotations. Conversely, minimizing *distortions* via the Gromov-Wasserstein (GW) framework addresses this limitation but introduces a non-convex, computationally demanding optimization problem. In this work, we identify a broad class of distortion penalties that reduce to a simple alignment problem within a lifted feature space. Leveraging this insight, we introduce an iterative GW solver with a linear memory footprint and quadratic (rather than cubic) time complexity. Our method is differentiable, comes with strong theoretical guarantees, and scales to hundreds of thousands of points in minutes. This efficiency unlocks a wide range of geometric applications and enables the exploration of the GW energy landscape, whose local minima encode the symmetries of the matching problem.