Flat Minima

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Latest papers 27

2 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

Oct 8, 2026cs.LG

Do Flatter Minima Drive Better Generalization? An Algorithmic Separation in Grokking

Flat loss landscapes have long been linked to better generalization in neural networks. However, its role as a causal mechanism for generalization is less established. Grokking provides an unique testbed to understand this distinction: models are prone to fit observed data using non-generalizing structure and remain in that regime for prolonged periods, transitioning to generalization only under particular training conditions. In this work, we study whether flat loss landscapes can act as a driving mechanism in this transition. While recent work has argued for flatness as a necessary geometric condition for this transition, we find that biasing training toward flatter solutions using sharpness-aware minimization (SAM) is insufficient to reliably induce this transition, despite producing flatter solutions. However, when SAM is paired with mechanisms that drive generalization such as weight decay, an interesting property emerges: SAM can accelerate the transition to generalizing solutions by up to 4x at the epoch-level. We theoretically untangle this relationship between SAM and weight decay using a minimal interpolating two-layer ReLU model with both memorizing and generalizing solutions. We show that even in this simple setup, flatness alone cannot distinguish a memorizing solution from a generalizing one, while weight decay favors generalizing solutions. However, under a local stability analysis, there exists a window where a memorizing interpolant is locally stable under gradient descent but unstable under SAM in the low-norm regime, which can explain SAM's ability to accelerate this transition. Overall, our results provide a more interpretable account of the role of flatness in driving generalization, especially in settings where models are vulnerable to minimizing loss through learning non-generalizing structure.

Mohnish Harwani

Purdue University

Neural Network Generalization Flat Minima Grokking Sharpness-Aware Minimization

Sep 29, 2026cs.LG

When a Flatness Proxy Is Not a Function: Robustness Certificates and Training Interventions

A valid curvature upper bound need not justify either a robustness certificate or an intervention on an intrinsic predictor property. We demonstrate this distinction for a last-layer relative-flatness proxy used in both settings. First, empirical-risk stationarity does not eliminate pointwise first-order loss terms: at a finite global empirical-risk minimum, the retained certificate expression underestimates a loss increase by over

210\times

. We derive a globally valid, gauge-invariant feature-space repair. Second, common-row softmax shifts preserve predictions and the exact contraction while making the proxy unbounded. Even standard reference-class choices double it on average relative to the centered representation. For a single fixed-feature example with at least three classes, scalar retuning generically cannot align the induced probability updates. Row centering gives the orbit-minimized bound and restores value and full-model gradient invariance under this symmetry. Across 45 paired one-step tests on algorithmic and image models, amplified shifts separate raw-regularized predictors while quotient-regularized predictors remain aligned. Long-horizon CIFAR-10 experiments show substantial, reversible suppression of generalization, while evidence for selective delay after memorization is less consistent. Together, these results show that validity as a curvature upper bound does not by itself justify either inversion into a robustness certificate or differentiation into an intrinsic training intervention.

Vicente Opazo, Jose Calatayud-Mateu, Cristobal Rojas +1

Centro Nacional de Inteligencia Artificial (CENIA) · Barcelona Supercomputing Center (BSC) · Pontificia Universidad Cat´olica de Chile

Flat Minima Certified Robustness

Sep 25, 2026cond-mat.dis-nn

A Flatness-Generalization Relation in the Teacher-Student Tree-Committee Machine

The flatness of the loss landscape at a minimizer is a widely used heuristic for reasoning about neural-network generalization, yet evidence for this relation is mostly empirical and controversial. We study this relation in a teacher-student tree committee machine, where both the ERM estimator and the Hessian spectrum are analytically tractable in the proportional high-dimensional limit. First, we use a zero-temperature Gibbs formulation to obtain predictions for the observables of the typical minimizers of the empirical loss. Secondly, we use Edwards-Jones formalism to derive the limiting Hessian resolvent around these typical minimizers. All predictions agree with finite-size gradient-descent simulations. Finally, we study three measures of flatness, namely the left and right edges and the spectral mean, and check if a decrease in generalization error as the dataset size is increased corresponds to an increase in flatness. We find that the answer strongly depends on the learning task and on the ratio of the number of parameters to the number of data points. In regression, the spectral mean and right edge correlate with the generalization error, while the left edge does so only in the overparametrized regime. In classification this correlation reliably holds only in the highly overparametrized phase, while for underparametrized networks it can even reverse.

Brandon Livio Annesi, Davide Straziota, Enrico Maria Malatesta

Department of Quantitative Life Sciences, ICTP Trieste, Italy

Flat Minima Neural Network Generalization Statistical Physics of Learning

Aug 25, 2026cs.LG

Across the Loss Landscape with Progressive Growth

Deep neural networks generalize well despite their highly nonconvex, overparameterized loss landscapes, a phenomenon often associated with the geometry of the minima found by stochastic optimization. We study how incremental grow-and-optimize strategies bias training toward flatter regions by viewing growth as progressive constraint relaxation. Starting from a low-dimensional submodel, we iteratively expand the trainable parameters by unlocking nested random subspaces while freezing the orthogonal complement at the network initialization, re-optimizing after each expansion until the full architecture is reached. Under standard local regularity conditions around non-degenerate minima, we prove that local sublevel sets are well approximated by ellipsoids and that basin accessibility under frozen constraints can be characterized by an explicit effective curvature in the frozen directions. This leads to an explanation of the bias: progressive growth increases the relative weight of wide basins and suppresses sharp ones through a volume effect induced by the frozen constraints. We empirically validate these predictions in controlled toy landscapes and in a realistic ResNet/CIFAR-100 setting and confirm that although progressive subspace growth reliably produces flatter solutions, curvature reductions do not universally translate into improved test performance, highlighting subtleties in the flatness-generalization connection. The code is available at https://github.com/p0lcAi/Across-the-Loss-Landscape.

Paul Caillon, Christophe Cerisara, Alexandre Allauzen

MILES Team, LAMSADE, Université Paris Dauphine-PSL · LORIA CNRS, Campus Scientifique, rue du Jardin Botanique, 54500 Vandoeuvre-les-Nancy · ESPCI-PSL

Deep Learning Optimization Neural Network Training Dynamics Neural Network Generalization Flat Minima

Aug 24, 2026cs.LG

How Much Regularization Survives Averaging? Update Masking in Federated Learning

Federated learning on non-IID data seeks flat minima to generalize across clients, and existing methods borrow sharpness-aware minimization from centralized training. There is a second way to reach flat minima, in which the regularization comes for free from noise added to the parameter updates, and it has never been carried over to the federated setting as an implicit regularizer. We show the reason. Masking charges the optimizer for moving in sharp directions. We prove that when each client draws its own mask, federated averaging weakens that charge by exactly the cohort size, and that giving every client the same mask brings it back by a factor equal to the inverse gradient diversity of the cohort. In our experiment setting on CIFAR-10, that factor is 1.19 out of a possible 10. Turning off minibatch sampling raises it to 8.96, while changing data heterogeneity a thousandfold leaves it between 1.17 and 1.50. The configurations keeping the regularization train far too poorly to use.

Wenhao Yan, Fu Kuroda, Yucheng Jin +1

Faculty of Science & Technology Sophia University Tokyo, Japan · Technical R&D Department Shendian Energy Co., Ltd. Lishui, China

Federated Learning Implicit Regularization Non-IID Federated Learning Federated Learning Aggregation Flat Minima

Aug 4, 2026cs.LG

On the Implicit Flatness Bias of Sharpness-Aware Minimization: A Linear Stability Analysis with Quantitative Hyperparameter Bounds

Sharpness-Aware Minimization (SAM) improves generalization by seeking parameters whose loss is robust to local adversarial perturbations, but the quantitative mechanism underlying its implicit bias toward flat minima remains unclear. In particular, the perturbation radius

ρ

is typically treated as an isolated tuning parameter, despite defining the neighborhood in which SAM measures sharpness. We analyze mini-batch SAM near an interpolating minimum through linear stability. Under local linearization and gradient-noise alignment assumptions, we prove that every linearly stable minimum satisfies

λ_{\max}\leq\sqrt[3]{bΓ/(2ρη^2)}

, where

λ_{\max}

is the largest Hessian eigenvalue,

b

is the batch size,

η

is the learning rate, and

Γ

bounds the gradient norm. The bound quantitatively characterizes SAM's implicit flatness bias: holding the other quantities fixed, a smaller batch size, a larger learning rate, or a larger radius restricts linearly stable SAM to flatter minima. It also exposes a necessary trade-off:

ρ

should be large enough to promote flatness, yet remain local enough to preserve the approximation and stable training. We validate this prediction in a controlled study of 900 models on CIFAR-100 with ResNet-18 and VGG-19, where increasing

ρ

is consistently associated with a smaller largest Hessian eigenvalue across batch-size and learning-rate settings. Finally, we instantiate the analysis in Taylor-Locality Controlled SAM (TLC-SAM), which adjusts

ρ

using the observed Taylor-approximation error and further reduces the top Hessian eigenvalue relative to fixed-radius SAM. Our results provide quantitative hyperparameter bounds and a stability--locality perspective for analyzing and designing SAM variants.

Jiaxin Deng, Junbiao Pang

Beijing University of Technology

Sharpness-Aware Minimization Flat Minima Hyperparameter Optimization

Jul 24, 2026cs.LG

From Perturbation Correction to Geometry-Aware Sampling: Sharpness-Guided Equilibrium Sampling for Balanced Flat Minima in Long-Tailed Learning

Long-tailed learning couples two sources of poor generalization: head classes dominate training exposure, while under-represented classes often converge to sharper regions of the loss landscape. Conventional re-sampling addresses the former without considering geometry, whereas existing long-tailed sharpness-aware minimization (SAM) methods modify losses or perturbations only after biased mini-batches have been drawn. We introduce Sharpness-Guided Equilibrium Sampling (SGS), which treats the sampling distribution as an active control variable for optimization geometry. SGS dynamically adjusts subsequent mini-batches by increasing the sampling probability of less frequently sampled classes while suppressing classes with large SAM-induced loss changes, using only cumulative class counts and EMA sharpness estimates obtained from the standard SAM update, without class-wise perturbations or additional backward passes. We characterize this sampling process through a continuous-time stochastic differential equation and a sampling-dependent PAC-Bayes analysis, explaining how frequency-sharpness feedback can move training toward a more balanced flatness profile. On CIFAR-100 LT with an imbalance ratio of 100, SGS-SAM improves Focal-SAM by 10.85 points in tail accuracy and 3.56 points overall. On ImageNet-LT, it improves ImbSAM by 6.59 points on tail classes and 1.20 points overall. Its training time is only

1.02\times

that of vanilla SAM. Beyond these gains, SGS establishes a sampling-side route to loss-landscape control, suggesting that future long-tailed methods can jointly regulate data exposure and optimization geometry rather than treating either as fixed.

Jiaxin Deng, Junbiao Pang

School of Information Science and Technology, Beijing University of Technology, Beijing, China

Sharpness-Aware Minimization Long-Tail Learning Adaptive Sampling Flat Minima

Jul 17, 2026cs.LG

Scaling Limits of Constant-Stepsize SGD at Flat Minima

For stochastic gradient descent (SGD) with a constant stepsize

α

, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar

\sqrtα

scaling and a Gaussian limit as

α\downarrow 0

. We show that this behavior changes fundamentally for convex objectives

H

with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize

α

, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an

α

-dependent metric. When the minimizer

x_\star

has local flatness exponent

m\ge2

, meaning that

\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d

as

x\to x_\star

, we obtain a contraction bound with factor

1-cα^{m-1}

, where

c>0

is a constant. This recovers the factor

1-cα

in the quadratic case

m=2

. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale

α^{1/m}

and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation

dY_t=-h_0(Y_t)\,dt+Σ^{1/2}\,dB_t ,

where

h_0

is the limiting drift at the minimizer and

Σ

denotes the asymptotic covariance. This recovers the Gaussian limit when

m=2

and gives generally non-Gaussian stationary limits in the flat case

m>2

. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

Jingyi Zhang, Cheng Mao, Debankur Mukherjee

Georgia Institute of Technology

Stochastic Optimization Convergence Flat Minima Stochastic Gradient Descent

Jul 9, 2026cs.LG

Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima

An important quantity in the theory of gradient descent (GD) is the \emph{sharpness}, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a \emph{single scalar output}, providing a normal form for large-step GD in a neighbourhood of an \emph{isolated} flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with \emph{vector-valued outputs} (including regression with arbitrarily many observations), and (2) to a neighbourhood of a \emph{manifold} of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges, including the solution of a singular partial differential equation via a novel method that may be of independent interest. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

Lachlan Ewen MacDonald, René Vidal

Innovation in Data Engineering and Science (IDEAS) · University of Pennsylvania Pennsylvania, PA 19104

Gradient Descent Flat Minima Gradient Descent Dynamics

Jul 7, 2026cs.LG

Leveraging Extragradient for Effective Sharpness-Aware Minimization in Deep Learning

Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances generalization via the extragradient technique. EISAM uses a two-step update process: a prediction step investigating the geometry of the loss landscape and a perturbation step that refines updates with a base optimizer. This approach achieves better generalization performance than SAM. Crucially, EISAM reduces sensitivity to the perturbation radius, enhancing robustness, and simplifying the tuning across diverse settings. Extensive experiments on benchmark datasets demonstrate that EISAM consistently outperforms SGD, Adaptive Moment Estimation (Adam), and SAM in test accuracy and training efficiency across various architectures. Theoretical analysis further confirms that EISAM tightens the generalization bound by steering parameters toward flatter minima with reduced curvature. Accompanied by a thorough hyperparameter analysis, EISAM offers practical tuning guidance, establishing it as a robust, scalable, and broadly applicable optimization solution that advances both the theory and practice in deep learning.

Yao Fu, Chunxia Zhang, Junmin Liu +3

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China · SGIT AI Lab, State Grid Corporation of China, Xi’an, 710054, China · School of Software, Xi’an Jiaotong University, Xi’an, 710049, China +2

Neural Network Generalization Sharpness-Aware Minimization Flat Minima

Jul 1, 2026cs.CV

Improving Sparse-View 3DGS Generalization via Flat Minima Optimization

Recent advances in neural rendering have established 3D Gaussian Splatting (3DGS) as a highly efficient representation for novel view synthesis, enabling fast training and real-time rendering with strong fidelity. However, when supervision is limited to sparse input views, 3DGS tends to overfit to the observed images and generalize poorly to unseen viewpoints. We address this challenge from the perspective of flat minima (FM) optimization, which seeks solutions that remain stable under small parameter perturbations. Viewing Gaussian parameters as trainable weights, we adapt FM principles to the geometric and dynamic nature of 3DGS with a lightweight training framework. Our method regularizes optimization with controlled Gaussian perturbations that account for each Gaussian's anisotropy and the training progress, preserving fine details while improving robustness to sparse-view overfitting. To further stabilize this flat minima optimization process, we introduce periodic reinitialization, which temporarily returns non-positional parameters to their initial states for a short window. Together, these techniques integrate seamlessly into existing 3DGS pipelines without architectural changes. Experiments on LLFF and Mip-NeRF360 datasets demonstrate improved quantitative metrics and perceptual quality under sparse-view supervision, producing reconstructions that are sharper, more stable, and better generalized to novel viewpoints.

Kangmin Seo, Sangeek Hyun, MinKyu Lee +1

Sungkyunkwan University, South Korea

3D Gaussian Splatting Novel View Synthesis Sparse-View 3D Reconstruction Flat Minima

Jun 27, 2026cs.LG

Closed-Form Steepest Descent Direction toward Flat Minima: Reducing Upper Bounds on the Loss Hessian Eigenspectrum in Neural Networks

The flatness hypothesis suggests that flatness of the loss landscape, as measured by the eigenvalues of the loss Hessian, correlates with better neural network generalization. While various algorithms reduce these eigenvalues, most focus on procedural design, leaving it unclear how data distributions and NN parameters structurally determine directions toward flat minima. Characterizing these directions analytically is generally intractable. To overcome this mathematical difficulty, recent studies derived the Wolkowicz-Styan (WS) upper bound on the maximum eigenvalue of the cross-entropy loss Hessian in three-layer NNs. Although this upper bound is differentiable, its gradient was not derived. Therefore, we analytically derive the gradient of the WS upper bound to characterize directions leading to flat minima. Based on this, we propose Hessian Spectral Range (HSR) Regularization, which updates parameters along the steepest descent direction of the WS bound. Experiments demonstrate that HSR Regularization narrows the Hessian eigenvalue spectrum, avoids sharp minima and saddle points, and promotes convergence to flat minima. Although the applicability of this method is currently limited to cross-entropy loss and three-layer architectures, to the best of the authors' knowledge, this is the first study to report a closed-form gradient that promotes convergence to flat minima without numerical approximations. Therefore, the theoretical analysis of this gradient is expected to contribute to the further development of NNs.

Yuto Omae, Kazuki Sakai, Yohei Kakimoto +3

Nihon University (Japan) · National Institute of Technology, Nagaoka College (Japan) · Tokyo City University (Japan)

Gradient Descent Neural Network Optimization Flat Minima

Jun 22, 2026cs.RO

Flatness Preserves Instruction Following in Vision-Language-Action Models

Vision-language-action (VLA) models have the potential for open-world generalization by leveraging pretrained vision-language representations, yet downstream finetuning on limited robot data often degrades these representations, leading to brittle policies that ignore language instructions in favor of visual shortcuts, a failure mode we term instruction blindness. We hypothesize that standard finetuning with limited data applies gradients to a sparse set of points, which manifests as a sharp loss landscape with high-curvature minima. We propose to address this directly through flatness-preserving optimization while finetuning on the exact same data, where learning a flatter landscape results in a model more robust to perturbations in the weight space. Specifically, we demonstrate that simply applying sharpness-aware minimization during VLA finetuning significantly improves instruction following by over 60% across multiple simulation and real-world benchmarks without additional data, architectural modification, or retraining. We further analyze the effect of selective sharpness, quantify its effects, and show that our approach is complementary to existing guidance techniques. Project page can be found at https://haochenz11.github.io/papers/flatness-vla/.

Haochen Zhang, Yonatan Bisk

Carnegie Mellon University

Sharpness-Aware Minimization Vision-Language-Action Models Flat Minima VLMs for Robotics

Jun 18, 2026cs.LG

Fisher-Geometric Sharpness and the Implicit Bias of SGD toward Flat Minima

A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with

η/B

matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.

Md Sakir Ahmed, Kumaresh Sarmah, Hemen Dutta

Department of Electronics and Communication Technology Gauhati University Guwahati, Assam, India · Department of Mathematics Gauhati University Guwahati, Assam, India

Information Geometry Implicit Bias Flat Minima Loss Landscape Geometry Stochastic Gradient Descent

Jun 9, 2026cs.CV

5% > 100%: Flatness Preference is All You Need for Multimodal Parameter-Efficient Fine-Tuning

Parameter-Efficient Fine-Tuning (PEFT) methods provide a streamlined and efficient tool for adapting large models to domain-specific multimodal downstream tasks. Although these methods proved their tangible effects in practice, their principal aspects remain under-explored. Therefore we remain curious about the underlying generalization mechanisms in various PEFT methods and how they can be further enhanced. In this paper, we reveal the flatness preference widely present in various PEFTs, where a small fraction of sharp dimensions dominates the generalization of PEFT. This finding suggests an appealing possibility: we may be satisfied with a better generalization by merely attending to this small fraction of sharp dimensions instead of all of them. Furthermore, we propose Flatness Preference Optimization (FlatPO) to flatten these key sharpness dimensions, leading various PEFTs toward better generalization. Extensive experiments demonstrate the effectiveness of our findings and the proposed method. Code is available at https://github.com/Can-Lin/FlatPO.

Yifan Zhu, Can Lin, Hangjie Yuan +4

School of Computer Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China · College of Computer Science and Technology, Zhejiang University, Zhejiang 310027, China · ETH Zürich, Switzerland

Deep Learning Optimization Flat Minima Parameter-Efficient Fine-Tuning

Jun 3, 2026stat.ML

Flatness and Generalization: Learning Multi-Index Models with Homogeneous Neural Networks

A common heuristic used to explain the generalization of first-order gradient methods on non-convex neural networks is that "flat interpolators generalize well" (Hochreiter and Schmidhuber, 1994; Keskar et al., 2017), where flatness can be measured by the trace of the Hessian of the empirical loss. However, Dinh et al. 2017) showed that, using symmetry of the network that can change flatness while keeping the population and empirical losses unchanged, any interpolator can be made sharper or flatter. This result makes the earlier heuristic statement vacuous. In this paper, we show that for learning an unknown multi-index model with

2

-layer non-convex homogeneous neural networks, there is a connection between flatness and generalization, despite the existence of symmetries. This connection pertains to the "flattest" interpolators, i.e., the interpolators that have orderwise minimum flatness among all interpolators. First, we show that there exists a natural class of non-generalizing interpolators whose flatness cannot be made closer to the flattest possible, even using symmetries. Second, we show that for data generated by a sum of single-index models, if the approximation error and label noise are low, any flattest interpolator achieves small population loss, i.e., the flattest interpolators always generalize. This establishes a direct link between flatness and generalization which applies to a large class of activations and realistic data distributions.

Harsh Vardhan, Hossein Taheri, Arya Mazumdar

Neural Network Generalization Flat Minima Statistical Learning Theory Feedforward Neural Networks Multi-Index Models

May 25, 2026cs.LG

Towards the Connection between Activation Sparsity and Flat Minima

The observation that activation sparsity emerges in MLP blocks of standardly trained Transformers offers an opportunity to drastically reduce computation costs without sacrificing performance. To theoretically explain this phenomenon, existing works have shown that activation sparsity does not result from the data properties or data fitting but from the implicit bias of the training process. However, these connections are obtained with strong assumptions, which cannot be applied to deep models standardly trained with a large number of steps. Different from these works, we find that the flatness of loss landscapes is also closely related to the MLP activation sparsity and can serve as a weaker and naturally emerging assumption standard deep networks. Specifically, we find that 1) the MLP activation sparsity equals a ratio between "augmented flatness" (a weighted sum of flatness measures) and the product of the input norm and activation gradient of the MLP. We empirically find that this ratio decreases during training, leading to sparse activations. 2) We also propose the notion of derivative sparsity, which reduces to activation sparsity under ReLU, but further enables pruning in the backward propagation and is more stable than activation sparsity. With the theoretical findings, we can further encourage activation sparsity by decreasing the numerator and increasing the denominator of the ratio using three methods. These plug-and-play modifications can effectively reduce the ratio and produce sparser activations. Experiments on ImageNet-1K and C4 demonstrate relative improvements of at least 36% on inference sparsity and at least 50% on training sparsity over vanilla Transformers, indicating further potential cost reduction in both inference and training

Ze Peng, Jian Zhang, Lei Qi +2

State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210023, China · Institute of Brain-Machine Interface, Nanjing University, Nanjing 210023, China · School of Computer Science and Engineering, Southeast University, Nanjing 211189, China

Efficient Neural Network Inference Activation Sparsity Flat Minima Neural Network Training Dynamics Neural Network Pruning

May 23, 2026cs.LG

Momentum Streams for Optimizer-Inspired Transformers

The residual update of a pre-norm Transformer layer admits an interpretation as one step of a first-order optimizer acting on a surrogate token energy, wherein the attention and MLP sublayers function as gradient oracles. Based on this observation, we build a family of optimizer-inspired Transformers (triple-momentum, Adam/AdamW, Muon, SOAP) and compare them under matched compute. In our main pretraining experiment, the triple-momentum TMMFormer achieves the lowest validation loss, outperforming the vanilla Transformer and prior architectural variants. A controlled ablation and supporting theory show that momentum, not preconditioning, is the main source of the gain. We further show that TMMFormer and other momentum-based designs reach flatter minima than the vanilla Transformer, which leads to less forgetting and better generalization.

Jingchu Gai, Nai-Chieh Huang, Jiayun Wu

Carnegie Mellon University

Deep Learning Optimization Transformer Momentum Methods Neural Network Optimization Flat Minima

May 20, 2026cs.LG

Same Target, Different Basins: Hard vs. Soft Labels for Annotator Distributions

When annotators disagree, that disagreement can reflect epistemic uncertainty rather than simple label noise. We study hard-label delivery as an alternative to the usual choices of collapsing votes to a single label or training directly on the empirical soft-label distribution. We focus on two primary hard-label methods: multipass, which cycles through observed votes while keeping the dataset size fixed, and stochastic label sampling (SLS), which samples one label per example at the start of each epoch. On CIFAR-10H, we find that when only a small number of annotations per example is available, hard-label delivery improves over soft-label training, with larger improvements where the sparse empirical target is farther from the full annotator distribution. When full annotator distributions are available, both hard-label methods match soft-label training. We use deterministic control as an ablation of multipass and shuffled SLS as a control that breaks the example-to-distribution match. We also show that SLS and soft-label cross-entropy optimize the same expected objective. Hard-label delivery also converges to flatter basins, with supporting descriptive evidence from OOD detection on SVHN and CIFAR-100. Overall, these results suggest that multipass is a strong practical default when raw vote counts are available, while SLS offers a lightweight alternative that remains competitive when only a few votes per example are available and matches soft-label training when full annotator distributions are available.

Mirerfan Gheibi, Gashin Ghazizadeh

Independent Researcher.

Annotator Disagreement Label Distribution Learning Classification Flat Minima

May 19, 2026cs.NE

From Mean-Field Limits to Semiclassical Concentration: Global Convergence of the Canonical Evolutionary Strategy

We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schr{ö}dinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ''survival of the flattest'' phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

Matías Neto, Nicolás Garay, Luis Martí +1

Inria Chile Research Center, Av Apoquindo 2827, Las Condes, RM, Chile · Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

Dynamical Systems Evolutionary Optimization Evolution Strategies Global Optimization Stochastic Optimization Convergence Flat Minima

May 15, 2026cs.LG

Navigating Potholes with Geometry-Aware Sharpness Minimization

Sharpness-aware minimization (SAM) encourages flat minima by perturbing parameters along directions of high loss curvature, but treats all parameter directions uniformly, ignoring the underlying loss geometry. We introduce LLQR+SAM, which combines SAM with a learned preconditioner obtained from the recently proposed LLQR framework, a second-order method that recasts steepest descent as a layerwise linear-quadratic regulator problem. The preconditioner is updated sparsely and maintained as a slow exponential moving average, so it captures a smoothed, low-resolution picture of the loss landscape geometry. The SAM perturbation then operates on top of this learned geometry, probing curvature at a faster timescale. We show that this two-timescale structure is not merely a computational convenience: theoretically, the preconditioner amplifies the SAM escape signal in directions that are flat under the average geometry but locally sharp (potholes). Wide, flat basins, by contrast, remain stable. Empirically, LLQR+SAM gives consistent gains over both SAM and LLQR alone across standard vision and sequence modeling benchmarks, supporting the view that slow learned geometry and fast sharpness correction are genuinely complementary.

Simon Dufort-Labbé, Mehrab Hamidi, Razvan Pascanu +3

Mila, Université de Montréal · Samsung – SAIL Montreal

Sharpness-Aware Minimization Flat Minima Loss Landscape Geometry

May 11, 2026cs.LG

Fix the Loss, Not the Radius: Rethinking the Adversarial Perturbation of Sharpness-Aware Minimization

Sharpness-Aware Minimization (SAM) improves generalization by minimizing the worst-case loss within a fixed parameter-space radius neighborhood. SAM and its variants mainly rely on a first-order linearized surrogate, while flat minima are inherently a second-order (curvature) notion.We revisit this mismatch and propose Loss-Equated SAM (LE-SAM), which inverts the traditional SAM mechanism that fixed perturbation radius with a fixed loss-space budget,effectively removing gradient-norm-dominated learning signals and shifting optimization toward curvature-dominated terms. Extensive experiments across diverse benchmarks and tasks demonstrate the strong generalization ability of LESAM that consistently outperforms SAM and even its variants, achieving the state-of-the-art performance.

Jinping Wang, Qinhan Liu, Zhiwu Xie +1

Wenzhou-Kean University · International Frontier Interdisciplinary Research Institute, Wenzhou-Kean University

Neural Network Generalization Sharpness-Aware Minimization Deep Learning Optimization Nonconvex Optimization Flat Minima

May 9, 2026cs.LG

FedVSSAM: Mitigating Flatness Incompatibility in Sharpness-Aware Federated Learning

Sharpness-aware minimization (SAM) is an effective method for improving the generalization of federated learning (FL) by steering local training toward flat minima. Under data heterogeneity, however, device-side SAM searches for locally flat basins that are incompatible with the flat region preferred by the global objective. We identify this structural failure mode as flatness incompatibility, which explains why improving local flatness alone may provide limited training and generalization improvement for the global model. We reveal that flatness incompatibility arises from data heterogeneity and the friendly adversary phenomenon, and is further amplified by local updates and partial device participation. To mitigate this issue, we propose Federated Learning with variance-suppressed sharpness-aware minimization (FedVSSAM), which constructs a variance-suppressed adjusted direction and uses it consistently in local flatness search, local descent, and global update. FedVSSAM anchors both perturbation and update directions to a more stable global direction, instead of correcting only an isolated local perturbation. We establish non-convex convergence guarantees of FedVSSAM and prove that the mean-square deviation between the adjusted direction and the global gradient is effectively controlled. Experiments demonstrate that FedVSSAM mitigates flatness incompatibility and outperforms the baselines across diverse FL settings.

Bingnan Xiao, Yuan Gao, Bingcong Li +3

Fudan University · ETH Zurich · Edith Cowan University +1

Sharpness-Aware Minimization Non-IID Federated Learning Nonconvex Optimization Flat Minima Federated Learning Heterogeneous Federated Learning

Feb 2, 2026cs.LG

Decentralized SGD with Controlled Disagreement Finds Flatter Minima

Decentralized training is often regarded as inferior to centralized training because the consensus errors between workers are thought to undermine convergence and generalization. This work challenges this view by introducing decentralized SGD with Adaptive Consensus (DSGD-AC), which uses a time-dependent scaling mechanism to maintain consensus errors throughout the training. We show that adaptive consensus changes the stationary variance of disagreement modes by balancing two effects: it preserves consensus-error magnitude through weaker graph damping while still allowing curvature-dependent damping to shape the disagreement directions. This balance can produce a stronger Hessian-weighted loss-envelope penalty around the deployed model, even when normalized Hessian alignment is weaker than in standard DSGD. Empirical results on image classification show that DSGD-AC reaches flatter solutions and higher test accuracy than standard DSGD and even centralized SGD. Together, these results support consensus errors as a useful implicit regularizer and open a new perspective on the design of decentralized learning algorithms.

Zesen Wang, Mikael Johansson

Division of Decision and Control Systems KTH Royal Institute of Technology

Decentralized Learning Flat Minima Implicit Regularization Decentralized Optimization

Nov 22, 2025cs.CV

Do Flat Minima Improve Sparse Novel View Synthesis?

Despite the success of recent novel view synthesis methods, they tend to struggle in sparse-view settings. This poor generalization to unseen viewpoints is an inherent challenge when training with limited data. To address this, we investigate the relationship between loss sharpness and generalization in novel view synthesis-an underexplored direction. Interestingly, while pursuing flatter minima is widely known to improve generalization in deep learning, reducing loss sharpness is not always beneficial in novel view synthesis. We demonstrate that this difference arises because high-detail regions inherently require a sharp loss landscape for accurate reconstruction, whereas low-detail regions benefit from a flat loss landscape for improving generalization. Based on this insight, we introduce structure-aware sharpness, defined within structure-adaptive neighborhoods, and propose to adaptively adjust the sharpness regularization weight according to the local image structure. This strategy encourages flatter minima for generalization while preserving the loss sharpness necessary to reconstruct fine details. Across various datasets and configurations, our strategy consistently improves a wide range of baselines. Code is available at https://bbangsik13.github.io/FASR.

Youngsik Yun, Dongjun Gu, Youngjung Uh

Yonsei University, Seoul, Korea

Novel View Synthesis Flat Minima

Sep 14, 2025cs.LG

On the Escaping Efficiency of Distributed Adversarial Training Algorithms

Adversarial training has been widely studied in recent years due to its role in improving model robustness against adversarial attacks. This paper focuses on comparing different distributed adversarial training algorithms--including centralized and decentralized strategies--within multi-agent learning environments. Previous studies have highlighted the importance of model flatness in determining robustness. To this end, we develop a general theoretical framework to study the escaping efficiency of these algorithms from local minima, which is closely related to the flatness of the resulting models. We show that when the perturbation bound is sufficiently small (i.e., when the attack strength is relatively mild) and a large batch size is used, decentralized adversarial training algorithms--including consensus and diffusion--are guaranteed to escape faster from local minima than the centralized strategy, thereby favoring flatter minima. However, as the perturbation bound increases, this trend may no longer hold. In the simulation results, we illustrate our theoretical findings and systematically compare the performance of models obtained through decentralized and centralized adversarial training algorithms. The results highlight the potential of decentralized strategies to enhance the robustness of models in distributed settings.

Ying Cao, Kun Yuan, Ali H. Sayed

Institute of Electrical and Micro Engineering in EPFL, Lausanne · Center for Machine Learning Research (CMLR) in Peking University

Adversarial Training Adversarial Robustness Decentralized Learning Flat Minima

Sep 16, 2024cs.LG

Convergence of Sharpness-Aware Minimization Algorithms using Increasing Batch Size and Decaying Learning Rate

The sharpness-aware minimization (SAM) algorithm and its variants, including gap guided SAM (GSAM), have been successful at improving the generalization capability of deep neural network models by finding flat local minima of the empirical loss in training. Meanwhile, it has been shown theoretically and practically that increasing the batch size or decaying the learning rate avoids sharp local minima of the empirical loss. In this paper, we consider the GSAM algorithm with increasing batch sizes or decaying learning rates, such as cosine annealing or linear learning rate, and theoretically show its convergence. Moreover, we numerically compare SAM (GSAM) with and without an increasing batch size and conclude that using an increasing batch size { achieves a lower worst-case

\ell_\infty

adaptive sharpness} than compared with using a constant batch size and learning rate.

Hinata Harada, Hideaki Iiduka

Department of Computer Science Meiji University

Sharpness-Aware Minimization Nonconvex Optimization Stochastic Optimization Convergence Flat Minima