NeurIPS 2025 Papers

We study gradient flows for loss landscapes of fully connected feedforward neural networks with commonly used continuously differentiable activation functions such as the logistic, hyperbolic tangent, softplus or GELU function. We prove that the gradient flow either converges to a critical point or diverges to infinity while the loss converges to an asymptotic critical value. Moreover, we prove the existence of a threshold $\varepsilon>0$ such that the loss value of any gradient flow initialized at most $\varepsilon$ above the optimal level converges to it. For polynomial target functions and sufficiently big architecture and data set, we prove that the optimal loss value is zero and can only be realized asymptotically. From this setting, we deduce our main result that any gradient flow with sufficiently good initialization diverges to infinity. Our proof heavily relies on the geometry of o-minimal structures. We confirm these theoretical findings with numerical experiments and extend our investigation to more realistic scenarios, where we observe an analogous behavior.

We develop a Frank-Wolfe algorithm with corrective steps, generalizing previous algorithms including Blended Conditional Gradients, Blended Pairwise Conditional Gradients, and Fully-Corrective Frank-Wolfe. For this, we prove tight convergence guarantees together with an optimal face identification property. Furthermore, we propose two highly efficient corrective steps for convex quadratic objectives based on linear optimization or linear system solving, akin to Wolfe's Minimum-Norm Point algorithm, and prove finite-time convergence under suitable conditions. Beyond optimization problems that are directly quadratic, we revisit two algorithms, Split Conditional Gradient and Second-Order Conditional Gradient Sliding, which can leverage quadratic corrections to accelerate the solution of their quadratic subproblems. We show improved convergence rates for the first and prove broader applicability for the second. Finally, we demonstrate substantial computational speedups for Frank-Wolfe-based algorithms with quadratic corrections across the considered problem classes.

The *balanced loss* is a widely adopted objective for multi-class classification under class imbalance. By assigning equal importance to all classes, regardless of their frequency, it promotes fairness and ensures that minority classes are not overlooked. However, directly minimizing the balanced classification loss is typically intractable, which makes the design of effective surrogate losses a central question. This paper introduces and studies two advanced surrogate loss families: Generalized Logit-Adjusted (GLA) loss functions and Generalized Class-Aware weighted (GCA) losses. GLA losses generalize Logit-Adjusted losses, which shift logits based on class priors, to the broader general cross-entropy loss family. GCA loss functions extend the standard class-weighted losses, which scale losses inversely by class frequency, by incorporating class-dependent confidence margins and extending them to the general cross-entropy family. We present a comprehensive theoretical analysis of consistency for both loss families. We show that GLA losses are Bayes-consistent, but only $H$-consistent for complete (i.e., unbounded) hypothesis sets. Moreover, their $H$-consistency bounds depend inversely on the minimum class probability, scaling at least as $1/\mathsf p _{\min}$. In contrast, GCA losses are $H$-consistent for any hypothesis set that is bounded or complete, with $H$-consistency bounds that scale more favorably as $1/\sqrt{\mathsf p _{\min}}$, offering significantly stronger theoretical guarantees in imbalanced settings. We report the results of experiments demonstrating that, empirically, both the GCA losses with calibrated class-dependent confidence margins and GLA losses can greatly outperform straightforward class-weighted losses as well as the LA losses. GLA generally performs slightly better in common benchmarks, whereas GCA exhibits a slight edge in highly imbalanced settings. Thus, we advocate for both GLA and GCA losses as principled, theoretically sound, and state-of-the-art surrogates for balanced classification under class imbalance.

We consider the following question: given a submodular or supermodular set function $f:2^V \to \mathbb{R}$, how should one minimize or maximize its average value $f(S)/|S|$ over non-empty subsets $S\subseteq V$? This problem generalizes several well-known objectives including Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications [39, 31], we formalize two new broad problems: the Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS) which allow negative and non-monotone functions. Using classical results we observe that DSS, SFM, USSS, UDSS, and MNP are all equivalent under strongly polynomial-time reductions. This equivalence enables algorithmic cross-over: methods designed for one problem can be repurposed to solve others efficiently. In particular we use the perspective of the minimum norm point in the base polyhedron of a sub/supermodular function which, via Fujishige's results, yields the dense decomposition as a byproduct. Via this perspective we show that a recent converging heuristic for DSS, \textsc{SuperGreedy++} [15, 29], and Wolfe’s minimum norm point algorithm are both universal solvers for all of these problems. On the theoretical front, we explain the observation made in recent work [39, 31] that \textsc{SuperGreedy++} appears to work well even in settings beyond DSS. Surprisingly, we also show that this simple algorithm can be used for Submodular Function Minimization, including for example that it can act as an effective minimum $st$ cut algorithm. On the empirical front, we explore the utility of several different algorithms including Fujishige-Wolfe min-norm point algorithm for recent problems. We conduct over 400 experiments across seven problem types and large-scale synthetic and real-world datasets (up to $\approx 100$ million edges). Our results reveal that methods historically considered inefficient, such as convex-programming methods, flow-based solvers, and Fujishige-Wolfe’s algorithm, outperform state-of-the-art task-specific baselines by orders of magnitude on concrete problems like HNSN [39]. These findings challenge prevailing assumptions and demonstrate that with the right framing, general optimization algorithms can be both scalable and state-of-the-art for supermodular and submodular ratio problems.

Neural Combinatorial Optimisation (NCO) is a promising learning-based approach for solving Vehicle Routing Problems (VRPs) without extensive manual design. While existing constructive NCO methods typically follow an appending-based paradigm that sequentially adds unvisited nodes to partial solutions, this rigid approach often leads to suboptimal results. To overcome this limitation, we explore the idea of the insertion-based paradigm and propose Learning to Construct with Insertion-based Paradigm (L2C-Insert), a novel learning-based method for constructive NCO. Unlike traditional approaches, L2C-Insert builds solutions by strategically inserting unvisited nodes at any valid position in the current partial solution, which can significantly enhance the flexibility and solution quality. The proposed framework introduces three key components: a novel model architecture for precise insertion position prediction, an efficient training scheme for model optimization, and an advanced inference technique that fully exploits the insertion paradigm's flexibility. Extensive experiments on both synthetic and real-world instances of the Travelling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that L2C-Insert consistently achieves superior performance across various problem sizes. The code is available at [https://github.com/CIAM-Group/L2C\_Insert](https://github.com/CIAM-Group/L2C\_Insert).

We develop an analytical framework for understanding how the learned distribution evolves during diffusion model training. Leveraging the Gaussian equivalence principle, we derived exact solutions for the gradient-flow dynamics of weights in one or two layer linear or linear convolutional denoiser settings with arbitrary data, where linear networks converge along principal components, and convolutional networks converge along Fourier modes. Remarkably, these solutions allow us to derive the generated distribution in closed-form and its KL-divergence through training. These analytical results expose a pronounced \emph{spectral bias}, i.e. for both weights and generated distributions, the convergence time of a mode follows an inverse power law of its variance. Empirical experiments on both Gaussian and natural image datasets demonstrate that the power-law spectral bias—remain robust even when using deeper or convolutional architectures. Our results underscore the importance of the data covariance in dictating the order and rate at which diffusion models learn different modes of the data, providing potential explanations of why earlier stopping could lead to incorrect details in image generative model.

Deep neural networks have become foundational in various applications but remain vulnerable to adversarial patch attacks. Crafting effective adversarial patches is inherently challenging due to the combinatorial complexity involved in jointly optimizing critical factors such as patch shape, location, number, and content. Existing approaches often simplify this optimization by addressing each factor independently, which limits their effectiveness. To tackle this significant challenge, we introduce a novel and flexible adversarial attack framework termed IMPACT (Irregular Multi-Patch Adversarial Composition based on Two-phase optimization). IMPACT uniquely enables comprehensive optimization of all essential patch factors using gradient-free methods. Specifically, we propose a novel dimensionality reduction encoding scheme that substantially lowers computational complexity while preserving expressive power. Leveraging this encoding, we further develop a two-phase optimization framework: phase 1 employs differential evolution for joint optimization of patch mask and content, while phase 2 refines patch content using an evolutionary strategy for enhanced precision. Additionally, we introduce a new aggregation algorithm explicitly designed to produce contiguous, irregular patches by merging localized regions, ensuring physical applicability. Extensive experiments demonstrate that our method significantly outperforms several state-of-the-art approaches, highlighting the critical benefit of jointly optimizing all patch factors in adversarial patch attacks.

We study the online multi-class selection problem with group fairness guarantees, where limited resources must be allocated to sequentially arriving agents. Our work addresses two key limitations in the existing literature. First, we introduce a novel lossless rounding scheme that ensures the integral algorithm achieves the same expected performance as any fractional solution. Second, we explicitly address the challenges introduced by agents who belong to multiple classes. To this end, we develop a randomized algorithm based on a relax-and-round framework. The algorithm first computes a fractional solution using a resource reservation approach---referred to as the *set-aside* mechanism---to enforce fairness across classes. The subsequent rounding step preserves these fairness guarantees without degrading performance. Additionally, we propose a learning-augmented variant that incorporates untrusted machine-learned predictions to better balance fairness and efficiency in practical settings.

Linear Mode Connectivity (LMC) is a notable phenomenon in the loss landscapes of neural networks, wherein independently trained models have been observed to be connected—up to permutation symmetries—by linear paths in parameter space along which the loss remains consistently low. This observation challenges classical views of non-convex optimization and has implications for model ensembling, generalization, and our understanding of neural loss geometry. Inspired by recent studies on LMC in standard neural networks, we systematically investigate this phenomenon within Mixture-of-Experts (MoE) architectures—a class of models known for their scalability and computational efficiency, which combine traditional neural networks—referred to as experts—through a learnable gating mechanism. We begin by conducting a comprehensive analysis of both dense and sparse gating regimes, demonstrating that the symmetries inherent to MoE architectures are fully characterized by permutations acting on both the expert components and the gating function. Building on these foundational findings, we propose a matching algorithm that enables alignment between independently trained MoEs, thereby facilitating the discovery of LMC. Finally, we empirically validate the presence of LMC using our proposed algorithm across diverse MoE configurations—including dense, sparse, and shared-expert variants—under a wide range of model settings and datasets of varying scales and modalities. Our results confirm the existence of LMC in MoE architectures and offer fundamental insights into the functional landscape and optimization dynamics of deep learning models.

Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop a formal framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between "full-data" and stereotypical predictors, (ii) the dominance of a region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.

While energy-based models have recently proven to be a powerful framework for learning to reason with neural networks, their practical application is still limited by computational cost. That is, existing methods for energy-based iterative reasoning suffer from computational bottlenecks by relying on expensive optimization routines during training and especially during inference. Furthermore, these routines may not always converge to minimal energy states, potentially leading to suboptimal reasoning. To address these limitations, we propose a novel and efficient algorithm for energy-based iterative reasoning based on a difference-of-convex (DC) functions approach. Our algorithm achieves a significant speedup compared to prior methods while offering theoretical convergence guarantees ensuring locally minimal energy states. In addition, we achieve state-of-the-art or superior performance on continuous reasoning tasks, as demonstrated by our experiments on multiple benchmark datasets from continuous algorithmic reasoning. As such, our method offers a leap in computational efficiency, enabling faster inference with theoretical guarantees, and hence unlocking the potential of energy-based models for iterative reasoning applications.

The power of foundation models (FMs) lies in their capacity to learn highly expressive representations that can be adapted to a broad spectrum of tasks. However, these pretrained models require additional training stages to become effective for downstream applications. In the multi-task setting, prior works have shown empirically that specific meta-learning approaches for preparing a model for future adaptation through parameter-efficient fine-tuning (PEFT) can outperform standard retraining methods, but the mechanism of the benefits of meta-learning has been largely unexplored. We introduce a framework for generic PEFT-based meta-learning to learn a model that can easily adapt to unseen tasks. For linear models using LoRA, we show that standard retraining is provably suboptimal for finding an adaptable set of parameters and provide strict performance guarantees for our proposed method. We verify these theoretical insights through experiments on synthetic data as well as real-data vision and language tasks. We observe significant performance benefits using a simple implementation of our proposed meta-learning scheme during retraining relative to the conventional approach.

Neural Combinatorial Optimization (NCO) has emerged as a promising learning-based paradigm for addressing Vehicle Routing Problems (VRPs) by minimizing the need for extensive manual engineering. While existing NCO methods, trained on small-scale instances (e.g., 100 nodes), have demonstrated considerable success on problems of similar scale, their performance significantly degrades when applied to large-scale scenarios. This degradation arises from the distributional shift between training and testing data, rendering policies learned on small instances ineffective for larger problems. To overcome this limitation, we introduce a novel learning framework driven by Large Language Models (LLMs). This framework learns a projection between the training and testing distributions, which is then deployed to enhance the scalability of the NCO model. Notably, unlike prevailing techniques that necessitate joint training with the neural network, our approach operates exclusively during the inference phase, obviating the need for model retraining. Extensive experiments demonstrate that our method enables a backbone model (trained on 100-node instances) to achieve superior performance on large-scale Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) of up to 100K nodes from diverse distributions. The source code can be found in https://github.com/CIAM-Group/TTPL.

Previous research has explored the expressivity of Transformer models in simulating Boolean circuits or Turing machines. However, the efficient learnability of Transformers from data has remained an open question. Our study addresses this gap by providing the first polynomial-time learnability results (specifically strong, agnostic PAC learning) for single-layer Transformers with linear attention. We show that learning the optimal multi head linear attention can be recast as finding the optimal kernel predictor in a suitably defined RKHS. Moving to generalization, we construct an algorithm that, given a dataset, checks in polynomial time whether the set of best fit multi head linear attention networks on this data all perform an identical computation--a powerful notion for out of distribution generalization. We empirically validate our theoretical findings on several canonical tasks: learning random linear attention networks, key--value associations, and learning to execute finite automata. Our findings bridge a critical gap between theoretical expressivity and learnability of Transformer models.

_Length generalization_, the ability of sequence models to generalize to sequences longer than those encountered during training, remains a key challenge for transformers, especially in tasks requiring algorithmic reasoning. Existing theoretical understanding of length generalization is limited, often providing only asymptotic results or focusing on specific problem classes or architectural variants, while empirical approaches frequently rely on ad hoc and often fragile techniques. In this work we introduce a novel framework for analyzing and proving length generalization bounds under specified, verifiable assumptions. A key outcome of the theory is the identification of a natural set of _auxiliary_ tasks, intricately related to the primary task structure, such that strong performance on these auxiliary tasks, alongside the primary task, provably guarantees length generalization within the framework. This motivates a multi-task training procedure that explicitly optimizes performance on both the primary and the identified auxiliary tasks. Empirical evaluations on a variety of synthetic benchmarks known to be challenging for length generalization, including sequence sorting, and reversal, demonstrate that our proposed method yields significant improvements in generalization to substantially longer sequences.

Adafactor is an early memory-efficient optimization algorithm proposed as an alternative to Adam. By eliminating first-order momentum and employing a rank-$1$ matrix factorization to approximate the second-moment matrix, Adafactor achieves near-zero memory overhead compared to traditional gradient descent methods. Despite its practical suitability for large-scale training tasks where memory efficiency is critical, its theoretical convergence analysis remains unexplored, largely due to the challenges posed by its matrix factorization and update clipping mechanisms. In this work, we provide a convergence analysis of Adafactor for non-convex smooth optimization. We establish optimal convergence rates (up to logarithmic factors) for finding stationary points in both deterministic and stochastic settings, the latter under sub-Gaussian noises. Central to our analysis involves viewing Adafactor as an approximation of Adam, and the use of a new proxy step-size to approximate the unique adaptive step-size induced by Adafactor's matrix factorization and update clipping, along with an induction argument to control the gradient magnitude. Our finding may theoretically suggest that involving rank-$1$ matrix approximation of the second-moment matrix in Adam does not fundamentally hinder the convergence.

Solving bilevel optimization (BLO) problems to global optimality is generally intractable. A common surrogate is to compute a hyper-stationary point—a stationary point of the hyper-objective function obtained by minimizing or maximizing the upper-level objective over the lower-level solution set. Existing methods, however, either provide weak notions of stationarity or require restrictive assumptions to guarantee the smoothness of hyper-objective functions. In this paper, we eliminate these impractical assumptions and show that strong (Clarke) hyper-stationarity remains computable even when the hyper-objective is nonsmooth. Our key ingredient is a new structural property, called set smoothness, which captures the variational dependence of the lower-level solution set on the upper-level variable. We prove that this property holds for a broad class of BLO problems and ensures weak convexity (resp. concavity) of pessimistic (resp. optimistic) hyper-objective functions. Building on this foundation, we show that a zeroth-order algorithm that computes approximate Clarke hyper-stationary points with non-asymptotic convergence guarantees. To the best of our knowledge, this is the first computational guarantee for Clarke-type stationarity in nonsmooth BLO. Beyond this specific application, the set smoothness property emerges as a structural concept of independent interest, with potential to inform the analysis of broader classes of optimization and variational problems.

We study the problem of regression with interval targets, where only upper and lower bounds on target values are available in the form of intervals. This problem arises when the exact target label is expensive or impossible to obtain, due to inherent uncertainties. In the absence of exact targets, traditional regression loss functions cannot be used. First, we study the methodology of using a loss functions compatible with interval targets, for which we establish non-asymptotic generalization bounds based on smoothness of the hypothesis class that significantly relaxing prior assumptions of realizability and small ambiguity degree. Second, we propose a novel min-max learning formulation: minimize against the worst-case (maximized) target labels within the provided intervals. The maximization problem in the latter is non-convex, but we show that good performance can be achieved with the incorporation of smoothness constraints. Finally, we perform extensive experiments on real-world datasets and show that our methods achieve state-of-the-art performance.

We propose KOALA++, a scalable Kalman-based optimization algorithm that explicitly models structured gradient uncertainty in neural network training. Unlike second-order methods, which rely on expensive second order gradient calculation, our method directly estimates the parameter covariance matrix by recursively updating compact gradient covariance products. This design improves upon the original KOALA framework that assumed diagonal covariance by implicitly capturing richer uncertainty structure without storing the full covariance matrix and avoiding large matrix inversions. Across diverse tasks, including image classification and language modeling, KOALA++ achieves accuracy on par or better than state-of-the-art second-order optimizers while maintaining the efficiency of first-order methods.