Implicit Regularization

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3 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

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Latest papers 35

Oct 6, 2026cs.LG

A Broader Look at Model Merging: Rethinking Implicit Regularization Induced by Task Arithmetic

Model merging aims to build a multi-task model cheaply by combining the weights of individual task-specific models. To perform well across multiple tasks, most existing merging methods use an additional dataset to find the coefficients for the best linear combination of task-specific weight updates. However, we identify an implicit regularization in this standard practice: searching over coefficients restricts the candidate models to a subspace spanned by task-specific weight updates. In this work, we investigate whether this regularization is actually useful. Surprisingly, empirical results show that optimizing merged-model weights without this regularization significantly boosts the performance of common merging methods across multiple architectures, domains, and even in an extremely data-limited scenario where only one instance is available per class. Moreover, directly optimizing the pretrained model weights even outperforms some existing merging methods. Analysis shows that better multi-task weights exist outside the subspace and can be found using multiple methods. We study different strategies for using the additional dataset, discussing their practical use and implications for model merging. Overall, this work calls for revisiting the existing model-merging pipeline, motivating a broader exploration of the weight space and a reconsideration of the implicit regularization induced by task arithmetic.

Sin-Han Yang, Shih-Cheng Huang, Chieh-Yen Lin +3

Appier AI Research · National Taiwan University

Task Vector Merging Multi-Task Optimization Model Merging Implicit Regularization

Oct 5, 2026cs.LG

Mind the Drift: Diagonal Linear Networks Under Large Learning Rates

Large learning rates can qualitatively change the trajectory of neural network training, often pushing optimization into regimes far from classical gradient-flow behavior. The Edge of Stability (EoS) offers a valuable lens on the dynamics such learning rates induce. We study corresponding dynamics in diagonal linear networks, where we uncover a competition between two distinct implicit biases that jointly determine the sparsity of the recovered solution in regression settings. Complementary to the Gain, which captures the average discretization error accumulated by Gradient Descent relative to Gradient Flow, we derive a closely associated but overlooked quantity: the Drift. Under large learning rates, it describes an imbalance between different discretization errors and represents a systematic shift in the optimization trajectory. While the Gain grows monotonically in certain regimes, and can bias towards denser, flatter interpolators, the impact of the Drift depends on its alignment with potential solutions, which can either counteract or reinforce the effect of the Gain. Consequently, its behavior drives model selection, particularly during early training epochs. To validate our theoretical insights, we introduce an intervention that actively steers the Gain to recover sharper, sparser solutions. Thus, our analysis reveals that large learning rates do not universally hinder the recovery of sparse solutions. On the contrary, they can be harnessed to control the implicit bias of training.

Aniket Sanyal, Tom Jacobs, Rebekka Burkholz

CISPA Helmholtz Center for Information Security · Technical University of Munich · Konrad Zuse School of Excellence in Learning and Intelligent Systems (ELIZA)

Edge of Stability Deep Learning Optimization Neural Network Training Dynamics Implicit Regularization

Sep 29, 2026cs.LG

Grokking through the Lens of Minimum-Norm Interpolation

Grokking shows that fitting the training data and learning the underlying signal can occur at very different stages. However, existing theories offer limited quantitative insight into how this delayed generalization depends on inductive bias and signal structure. Our work addresses the gap by developing a statistical theory that characterizes how regularization geometry and signal sparsity govern generalization near interpolation. In particular, we focus on the prototypical setting of high-dimensional regression and identify regimes in which sparsity-promoting regularization makes exact interpolation much more accurate than approximate fitting. In strongly overparameterized noiseless problems, we prove a zero--one generalization law and construct a family of convex norms whose interpolators transition from the trivial risk of the all-zero predictor to exact recovery, while keeping the training error equal to

0

. Furthermore, when feature dimension and sample size are proportional, we provide a precise characterization of training and generalization errors along

\ell_r

-regularization paths. This in turn allows us to quantify the generalization gain that remains near interpolation: we show that this gain increases as the norm becomes more sparsity-promoting and as the target becomes sparser, with a sharp drop in generalization reached for noiseless data and

\ell_1

regularization. Experiments on diagonal linear networks and transformers trained on modular arithmetic demonstrate the generality of our theoretical predictions. Finally, beyond grokking, our work reveals a statistical instability in minimum-norm interpolation: small perturbations in the regularization strength can lead to drastically different generalization, while preserving small training error.

Gil Kur, Ileana Rugina, Clémentine Carla Juliette Dominé +1

ETH Zürich · Institute of Science and Technology Austria · Harvard University

Statistical Learning Theory Implicit Regularization Grokking

Sep 23, 2026cs.LG

Minimal-Norm Univariate Two-Layer ReLU Classification: Exact Solutions and Global Optimality with Skip Connections

We study minimal-norm interpolation and

\ell_2

-regularized logistic-loss minimization for binary classification by univariate two-layer ReLU networks. We give complete geometric characterizations of the optimal classifiers in function space, resolving how the solutions depend on whether hidden-layer biases are included in the parameter norm. When biases are unpenalized, the minimal-norm interpolators are exactly the continuous piecewise-affine functions that hug every label switch and have kinks of the appropriate convexity. When biases are penalized, the minimizer is unique in function space, has exactly one kink in each intermediate same-label segment, and is therefore a sparsest positive-margin classifier. We further show that adding a free affine skip connection leaves these function-space solutions unchanged but fundamentally improves the parameter-space landscape: every KKT point of the constrained problem becomes globally optimal, whereas suboptimal KKT points can occur without the skip connection. We establish analogous global-optimality and geometric results for sufficiently weak

\ell_2

-regularization of the logistic loss. In the unpenalized-bias case, we identify an additional sparsity-like restriction, implying that most minimal-norm interpolators cannot arise as small-regularization limits of margin-normalized logistic-loss minimizers. Numerical experiments across varying dataset complexity and network width support the predicted landscape and sparsity phenomena.

Karolina Drabik, Ben Lewis, Antoni Puch +4

Department of Mathematics, Informatics and Mechanics, University of Warsaw, Poland · Department of Computer Science, University of Warwick, UK · INRIA & LMO, Université Paris-Saclay, France

ReLU Neural Networks Neural Network Optimization Binary Classification Implicit Regularization

Sep 9, 2026cs.AI

A Function-Space Approach to the Statistical Mechanics of Learning Dynamics

In the kernel regime, neural-network learning inherits its preferences from a frozen spectrum. During feature learning, this spectrum evolves, yet networks retain systematic biases toward simple, smooth directions. We develop a function-space statistical framework explaining the origin of these preferences, treating functions and their learning operators as macroscopic variables, with parameterization entering through the multiplicity of parameter configurations realizing each function. For mean-squared loss, error relaxes exactly under the evolving learning operator

M=JJ^\ast

. Training stochasticity induces a Gaussian weight over function-space states, while parameter multiplicity contributes an entropic operator

B

, defined by the curvature of its log multiplicity. A local Laplace expansion yields the fluctuation free energy

Φ_{\mathrm{fluc}}(M;B)=\frac{σ_ξ^2}{2}\log\det(M^{-1}+B)+\mathrm{const}

, analogous to an Occam factor. Under mild statistical conditions, this free energy is rotationally stationary exactly when

[M,B]=0

, is minimized by pairing large eigenvalues of

M

with small eigenvalues of

B

, and generates a local restoring force against mismatch. Learning is therefore biased toward faster relaxation along entropically cheaper directions. This preference strengthens with training noise and vanishes in the deterministic limit, beyond gradient-flow accounts of operator alignment. For ReLU networks, we relate entropic curvature to the minimal rearrangement of activation boundaries required for a functional change and bound this structural cost by directional smoothness. Consequently, smooth directions are preferentially learned faster, in a data-adaptive manner, even as the learning operator evolves.

Yizhou Zhang, Weichen Wu, Lun Du +1

Variational AI · The Voleon Group · Independent Researcher +1

Statistical Physics of Learning Representation Learning Neural Network Training Dynamics Statistical Learning Theory Implicit Regularization

Aug 5, 2026cs.LG

Non-asymptotic implicit bias of logistic regression at early-stage gradient descent dynamics

Gradient descent has been of particular interest in modern machine learning beyond sole focus on optimization. Implicit bias emerging from optimization, though not being encoded by the learning objective, often prevents from overfitting to spurious patterns. A typical instance is the max-margin implicit bias of a linear classifier, widely established for exponentially tailed loss functions. Even after having a given dataset separated, the parameter vector continues to evolve towards the max-margin direction asymptotically along the gradient descent dynamics. This phenomenon corroborates a frequent empirical observation of "train longer, generalize better." However, the max-margin convergence is an asymptotic phenomenon, and what is worse, this asymptotic convergence rate is significantly slower than pure convex optimization. Even so, the parameter vector along gradient descent dynamics commonly correlates with the max-margin direction positively (though not exactly) within considerably fewer iterations than the asymptotic rate. By shedding another light on this classical problem, this work aims to understand the mechanism of this early-stage alignment phenomenon. Our theoretical results demonstrate that the parameter vector weakly aligns with the max-margin direction within

O(\exp(\exp(-δ)))

iterations, where

δ>0

is the permissible alignment error, which is shown to be tight. By tracking the radial and tangential flows, our proof operates on the alignment dynamics directly with dataset geometry and gets rid of the asymptotic expansion, which is a key insight to establishing faster weak alignment.

Han Bao

The Institute of Statistical Mathematics · Tohoku University · RIKEN AIP

Logistic Regression Implicit Bias Implicit Regularization Gradient Descent Dynamics

Jul 28, 2026cs.AI

Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks

Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.

Pengcheng Cheng

School of Mathematics, Jilin University Changchun, 130012, China

Riemannian Optimization Implicit Regularization Gradient Descent Dynamics

Jul 24, 2026cs.LG

Beyond Negative-Ridge Endpoints: Mixed-Sign Spectral Regularization via Negative-Shifted Gradient Descent

In overparameterized linear regression, many weak spectral directions act like a ridge penalty on the signal-bearing spectrum; negative ridge is the natural correction, pushing filters above one. The stable negative-ridge endpoint, however, is structurally limited: its pole must stay below the smallest nonzero empirical eigenvalue, and it anti-shrinks smaller eigenvalues more than larger ones. Early-stopped negative-shifted gradient descent escapes this constraint. Its filter is smooth at the would-be pole and mixed-sign-capable: above-ridgeless directions form a leading prefix, with lower directions shrunk or exposure-controlled while stopping sets the crossover. In a Gaussian spike-plus-flat model we discover a Marchenko-Pastur barrier: the shift that cancels the implicit penalty lies a bulk width above the smallest empirical eigenvalue, and the stopped path improves on every admissible endpoint by a polynomial factor in risk under explicit conditions. Our main theorem permits a general high-effective-rank tail: its trace sets the implicit floor, its squared spectrum controls exposure, and the floor-critical path recovers all head scales at once, beyond positive shrinkage and, once scales separate, every uniform rescaling of ridgeless. Handling the noncontractive shifted dynamics is the central technical challenge; localized Duhamel integrals control them. A finite-grid hold-out inequality transfers the separations to the validation-selected algorithm.

Peng Zhao

Department of Applied Economics and Statistics · University of Delaware

Spectral Regularization Early Stopping Gradient Descent Linear Regression Ridge Regression Implicit Regularization

Jul 24, 2026cs.CV

FAIR: Feature-Augmented Implicit Regularization for AI-generated Fake Image Detection

Generalization remains a critical bottleneck in AI-generated image detection. Because many modern generators are proprietary or adversarially modified, existing detectors overfit to the low-level textural patterns of accessible training data, resulting in severe failures on unseen domains. Conventional regularization techniques (e.g.,

L_1

/

L_2

norms, Dropout) apply indiscriminate parametric constraints and fail to provide the domain-invariant structure necessary for cross-generator robustness. To address this, we propose Feature-Augmented Implicit Regularization (FAIR). FAIR introduces an orthogonal, macro-structural prior, specifically, Scene Composition Structure (SCS), during training to geometrically constrain the model's optimization trajectory. By augmenting the primary feature space with domain-invariant SCS features, FAIR explicitly penalizes texture-biased shortcut learning. Crucially, this structural prior is entirely discarded at inference, yielding a smoothed, generalized decision boundary with zero architectural or computational overhead. Extensive evaluations across five massive benchmarks demonstrate that integrating FAIR into state-of-the-art detectors significantly improves cross-generator generalization, boosting accuracy by up to 8.04% and establishing new state-of-the-art robustness in zero-shot transfer scenarios.

Md Redwanul Haque, Manzur Murshed, Manoranjan Paul +1

Deakin University, Burwood, VIC, Australia · Charles Sturt University, Bathurst, NSW, Australia

Domain Generalization AI-Generated Image Detection Implicit Regularization

Jul 22, 2026quant-ph

PN-QNN: Harnessing Physical Noise as a Native Regularizer in Photonic Hybrid Quantum Neural Networks

Physical noise in near-term quantum hardware is usually treated as a nuisance to suppress. We ask whether it can instead act as a hardware-native regularizer for photonic hybrid quantum-classical neural networks (PHQCNNs), analogous to noise-injection regularization in classical deep learning. Using Quandela's Perceval simulator and the MerLin framework, we build PHQCNNs for Iris, Digits, and MNIST and inject Perceval's seven-parameter physical noise model directly into training. A genetic algorithm searches the six continuous noise dimensions and 1 boolean parameter to find, per dataset, the configuration maximizing validation accuracy, compared against a noiseless baseline across five seeds. GA-tuned noise yields modest accuracy gains on Iris (+0.82pp) and Digits (+1.45pp), but a clear degradation on MNIST (-1.21pp). Per-parameter sweeps show that no individual noise parameter is consistently beneficial, motivating the joint search, while a second-order loss expansion shows that physical noise induces a Tikhonov-like regularization term whose effect is dataset-dependent. Physical photonic noise can thus act as a free regularizer, but not universally.

Farah Elnakhal, Alberto Marchisio, Nouhaila Innan +2

Science Division, New York University Abu Dhabi, UAE · eBrain Lab, Division of Engineering, New York University Abu Dhabi, PO Box 129188, Abu Dhabi, UAE · Center for Quantum and Topological Systems, NYUAD Research Institute, New York University Abu Dhabi, UAE +1

Hybrid Quantum-Classical ML Photonic Computing Implicit Regularization Quantum Neural Networks

Jul 16, 2026cs.LG

Adaptive Runge-Kutta Step Control Buys Training Loss, Not Generalization: An Honest Compute-Matched Study of RK-Adam Optimizers

Interpreting optimizers as gradient-flow discretizations has motivated applying higher-order Runge-Kutta (RK) integrators to neural networks. We build a representative Adam variant (Bogacki-Shampine 3(2) RK pair, FSAL reuse, local-error step control) and evaluate it under a strict compute-matched protocol giving every method the same gradient-evaluation budget - an accounting this literature rarely enforces. Under it the RK variant loses to plain Adam on training loss in both minibatch and full-batch (RK's best-case) training. Instrumenting it shows the "adaptivity" is illusory: normalized error stays far below tolerance, the step size pins at its growth cap from step one (98-100 percent of steps), and no rtol x hmax x h0 setting makes it act; tolerances spanning 100x give bit-identical trajectories. The method is exactly fixed-step Adam with an averaged gradient at 3-4x cost. Repairing it (true reject branch; error on the applied map) reverses the full-batch result - about 40x lower training loss than tuned Adam - and a fixed-step control isolates adaptivity (an emergent warmup-and-growth schedule) as the mechanism. But the gain is fragile to the initial step size and does not reach test accuracy. A pre-registered follow-up rules out the obvious explanations: deeper minimization does not overfit, and an explicit temperature knob only hurts - leaving a trajectory effect, the controller selecting a minimum generalizing 1.3-3.4 points below first-order descent at equal depth. An n=10 study confirms one secondary effect: gradient averaging is a genuine implicit regularizer, beating lr-matched Adam and AdamW on 10/10 seeds - yet RMSprop and NAdam match or beat it at a third the per-step cost. Higher-order adaptive integration buys deeper deterministic minimization and a small regularization effect, but nothing a cheaper, well-tuned first-order baseline does not already provide.

Akhilesh Gogikar

1Independent Researcher.

Neural Network Optimization Adaptive Gradient Methods Neural Network Training Dynamics Implicit Regularization

Jul 7, 2026cs.LG

Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data

Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S_t^{-1/2} normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-epsilon crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.

Xiangwu Wang, Chengwei Cao, Yicheng Song +2

1The University of Hong Kong · University of California, San Diego · 3Beijing University of Aeronautics and Astronautics

Sign-Based Optimization Implicit Regularization Gradient Descent Dynamics

Jun 23, 2026cs.CL

Decoherence as Defence and the Magnitude of Noise Regularisation: A Rigorous N -Qubit Theory of Stochastic Quantum Neural Networks for Adversarially Robust Network Intrusion Detection

Stochastic quantum neural networks (SQNNs) encode neuronal activations as qubits, synaptic topology as entanglement, and neural noise through a Lindblad master equation. A recent conference study applied a ring-entangled SQNN to collaborative intrusion detection and reached three conclusions: ring entanglement is \emph{essential} for non-local anomaly detection; an adversarial-resilience bound holds but is \emph{conservative}; and the depolarising channel \emph{fails} to act as a dropout-style regulariser, behaving instead as output noise. It left open whether a per-gate stochastic deactivation (``true quantum dropout'') could regularise where the depolarising channel could not, and whether the loose robustness bound could be replaced by a predictive theory. This paper resolves both and extends the framework to real data and to neutral-atom hardware. We give an

N

-qubit formulation through the stochastic master equation and its vectorised Liouvillian, and prove a \emph{decoherence-contraction theorem}: a depolarising channel of strength

γ

over

L

entangling layers contracts every weight-

w

Pauli read-out by a factor

(1-4γ/3)^{wL}

(for the weight-

1

read-out used here,

(1-4γ/3)^{L}

); building on the general noise-as-defence result of Du et al., we make this quantitative and operational for intrusion detection. On the real NSL-KDD dataset under white-box FGSM and PGD attacks, a depolarising SQNN trained with the channel is, over seven seeds under strong

\ell_\infty

/

\ell_2

attacks, significantly more robust than the noiseless circuit (

\ell_\infty

PGD-

20

,

p=0.04

, large effect) and, critically, never suffers the catastrophic robustness collapse that the noiseless model and gradient-trained classical detectors (which fall from

95\%

to

47\%

) do, cutting robustness variance roughly twofold; we show this robustness arises from a noise-reshaped training boundary rather than from attack-time gradient contraction. For generalisation, we derive an adaptive-penalty formula showing that per-gate dropout implements a curvature-weighted

L_2

penalty

\tfrac{p(1-p)}{2}\sumθ^2\partial^2_θL

in weight space, maximised at

p=1/2

, whereas depolarising noise implements an output-space penalty. A

30

-seed study confirms the formula's quantitative prediction: both mechanisms reduce the train-test gap by a small but statistically significant margin (

\approx\!0.01

;

p<10^{-4}

and

p=0.004

), are statistically indistinguishable from each other, and the effect is concentrated where overfitting is largest; increasing the dropout rate past

1/2

does not help, as the formula predicts. The single-seed dichotomy of prior work does not survive replication. We close with a neutral-atom realisation and a feasibility-by-

N

analysis.

Gautier-Edouard Edouard Filardo

CREOGN · Efrei Research Lab, Efrei Paris Panth´eon-Assas Universit´e, Villejuif, France

Network Intrusion Detection Neural Network Robustness Adversarial Robustness Implicit Regularization Quantum Neural Networks

Jun 7, 2026cs.LG

The Spectral Dynamics and Noise Geometry of Muon

Muon replaces a matrix gradient

G=UΣV^\top

by its polar factor

UV^\top

. This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. Under explicit alignment assumptions, we prove that the polar update is the one-step entropy-maximizing choice among bounded updates that use the gradient singular directions and do not adapt to the current weight spectrum. In an underdetermined regression model, we derive exact singular-value dynamics for continuous-time Muon and identify a measurement-dependent condition under which the normalized spectrum moves toward equal nonzero singular values. This geometry also rules out a common low-rank interpretation: at fixed Frobenius norm, Muon's distinguished state has a flat spectrum, whereas nuclear-norm minimization favors spectral concentration. Controlled matrix-sensing experiments separate the effect from simple gradient rescaling, show that norm-matched gradient descent does not reproduce Muon, and recover the predicted flattening trend across broad ablations. In small NanoGPT pretraining, Muon preserves stable rank, has a broad learning-rate plateau, and improves validation loss relative to AdamW; in a matched small-ViT control, the ranking reverses. The resulting picture is regime-dependent: Muon is not universally superior, but its flat-spectrum bias can help when many spectral directions need to remain active.

Pierfrancesco Beneventano, Mahmoud Abdelmoneum, Tomaso Poggio

Massachusetts Institute of Technology

Muon Optimizer Matrix Optimization Spectral Methods Implicit Regularization Gradient Descent Dynamics

Jun 5, 2026cs.LG

Closed-Form Spectral Regularization for Multi-Task Model Merging

Model merging combines several independently fine-tuned experts into a single multi-task model without any training data, reducing the storage, serving, and decentralized-development costs of large foundation models. State-of-the-art merging methods formulate merging as a layer-wise quadratic interference minimization problem. Although this problem admits an exact closed-form pseudoinverse solution, that solution underperforms hundreds of iterations of gradient descent in practice. The iterative loop dominates the cost of the pipeline, yet its effectiveness has remained unexplained. We revisit this regime and show that the iterative solver does not primarily act as an optimizer; rather, it serves as an implicit spectral regularizer for an ill-posed normal equation, where small-eigenvalue directions of the per-layer interference operator amplify proxy noise. Building on this finding, we formalize multi-task model merging as a noisy linear inverse problem and propose a spectral filtering estimator parameterized by a per-direction filter. We instantiate this estimator with SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent, with a hard top-K truncation that suppresses noise-amplifying small-eigenvalue directions. Furthermore, we propose SWUDI-A, an adaptive variant that replaces the global rank hyperparameter with per-layer rank rules, further improving robustness across architectures. Both variants share a single symmetric eigendecomposition per linear layer and require no training data or optimizer state. Across four general benchmarks and a multimodal merging benchmark spanning VQA, Geometry, Chart, OCR, Grounding, and modality merging, our proposed spectral solvers match or outperform state-of-the-art merging methods. Crucially, they reduce wall-clock time by 28-72x and peak GPU memory by up to 50%.

Yongxian Wei, Runxi Cheng, Xingxuan Zhang +4

Shenzhen International Graduate School, Tsinghua University, Shenzhen 518071, China · Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China · Sun Yat-sen University, Shenzhen 510275, China +1

Spectral Regularization Multi-Task Learning Spectral Methods Model Merging Implicit Regularization

Jun 3, 2026cs.CV

Implicit Fuzzification via Bounded Noise Injection for Robust Medical Image Segmentation

Image segmentation remains fundamentally limited by boundary ambiguity arising from sampling-induced information loss and inherent uncertainty in pixel-wise labeling. Although encoder-decoder architectures such as U-Net achieve strong performance, they often produce overconfident predictions that fail to capture transition-region ambiguity. To address this issue, we propose \textbf{NoiseUNet}, a simple yet effective framework that injects bounded perturbations into skip connections to regularize cross-scale feature fusion. This mechanism enforces robustness to local feature variations and promotes boundary-aware representations. Theoretically, the perturbation induces an implicit fuzzification effect, yielding soft, data-driven memberships without requiring explicit fuzzy modeling. We further introduce \textbf{ThyR}, a real-world thyroid ultrasound dataset with inherently ambiguous boundaries. Experiments demonstrate that NoiseUNet consistently improves both segmentation accuracy and boundary fidelity.

Bisheng Tang, Zhangfeng Ma, Chuchu Zhai +4

School of Information Science and Engineering, Provincial Key Laboratory of Informational Service for Rural Area of Southwestern Hunan, Shaoyang University, Shaoyang 422000, China · Xinshao County People’s Hospital, Shaoyang 422900, China

Ultrasound Image Segmentation Medical Image Segmentation Boundary-Aware Segmentation Segmentation Uncertainty Estimation Semantic Segmentation Implicit Regularization

Jun 1, 2026cs.LG

Fast Generalization after Interpolation via Critically Damped Momentum Optimization

A central problem in machine learning is that models can achieve near-perfect training performance while generalizing substantially less well to unseen examples. This gap is especially acute in high-dimensional, low-sample regimes, where many interpolating solutions exist and optimization must implicitly select among minima with different generalization properties. Following recent theoretical advances on optimization dynamics near the interpolation threshold, we note that the two-regime structure of risk minimization, with loss minimization followed by complexity minimization, motivates a biphasic optimization schedule. We thus theoretically demonstrate that GROKtimizer, a biphasic strategy that combines rapid convergence to interpolation with Critically Damped Momentum (CDM)-based post-interpolation norm minimization, offers a natural solution for selecting low-norm interpolating solutions. Under a local quadratic model of the post-interpolation basin, GROKtimizer provides a quadratic speedup over classical gradient descent, with provable optimality among first-order optimizers. To showcase the applicability of our method, we evaluate GROKtimizer on several synthetic benchmarks common in the classical grokking literature and on various real-world datasets. Finally, we reconcile our findings with the flat-minima hypothesis, highlighting the importance of post-interpolation dynamics in the construction of high-quality, generalizing models.

Luca Muscarnera, Silas Ruhrberg Estévez, Yuanzhang Xiao +1

Neural Network Generalization Momentum Methods Neural Network Optimization Implicit Regularization

May 29, 2026math.NA

Stochastic Rounding Increases Small Singular Values

Over the past half-dozen years, stochastic rounding (SR) has regained significant attention as a quantization scheme for low-precision floating-point arithmetic, with applications spanning numerical analysis and modern machine learning systems. Recent work has shown that SR acts as an implicit regularizer by increasing the smallest singular value of extremely tall-and-thin (or, symmetrically, short-and-fat) matrices. In this work, we substantially sharpen and extend this understanding in two directions. First, we show that the regularization effect of SR is not restricted to extreme aspect ratio regimes: it persists for matrices with constant aspect ratio. Second, we demonstrate that SR does not merely regularize the smallest singular value, but instead lifts entire clusters of singular values at the tail of the spectrum. Together, these results provide a more general characterization of stochastic rounding as a spectral regularizer, revealing that its effects extend beyond extremal aspect ratios and act on a broader portion of the singular value spectrum.

Linkai Ma, Tingzhou Yu, Petros Drineas

Department of Mathematics University of Alberta

Spectral Regularization Implicit Regularization

May 28, 2026cs.LG

Improving Adversarial Robustness of Attribution via Implicit Regularization

The adversarial robustness of attributions is a fundamental requirement for reliable explainability in deep learning, yet existing approaches typically rely on computationally expensive explicit regularization. In this work, we show that attribution robustness can arise implicitly from the learning dynamics of standard stochastic gradient descent. We theoretically motivate this effect through connections between parameter-space and input-space curvature, and validate it across architectures, datasets, and attribution methods, with negligible computational overhead. In contrast, we prove that such robustness gains often does not transfer to attention-based attribution under softmax normalization, due to inherent entropy constraints, and we validate this limitation experimentally. Finally, we show that replacing softmax attention with kernel-based attention restores the robustness gains in transformer models. Our results highlight learning dynamics as a principled and practical mechanism for robust explainability, and reveal fundamental limitations of attention-based attribution under normalization.

Amir Mehrpanah, Matteo Gamba, Hossein Azizpour

Department of Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden · Department of Computer Science, Brown University, USA · Science for Life Laboratory, Stockholm, Sweden

Transformer Interpretability Neural Network Robustness Adversarial Robustness Attention Mechanisms Implicit Regularization

May 27, 2026math.OC

Implicit Regularization in Perturbed Deep Matrix Factorization: Spectral Conditions and Stability

This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings.

Jingzhe Wang, Hung-Hsu Chou

Department of Informatics and Networked Systems, University of Pittsburgh · Department of Mathematics, University of Pittsburgh

Algorithmic Stability Low-Rank Matrix Decomposition Low-Rank Approximation Matrix Optimization Implicit Regularization Matrix Factorization+1

May 26, 2026cs.LG

Mildly Overparameterized ReLU Networks on Orthogonal Data: Incremental Learning and Implicit Bias

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as

m \gtrsim \log(n)

, where

m

is the network width and

n

is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared

\ell_2

-norm scaling as

\sqrt{n}

, which is within a constant factor of the minimal

\ell_2

-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.

James Town, Etienne Boursier, Ben Lewis +2

University of Warwick · INRIA LMO, Université Paris-Saclay

ReLU Neural Networks Neural Network Optimization Incremental Learning Neural Network Training Dynamics Implicit Regularization

May 25, 2026cs.LG

From Latent Space to Training Data: Explainable Specialization in Minimal MLPs

We here study whether training biases can make hidden neurons specialize in minimal one-hidden-layer MLPs, and whether such specialization improves prototype-based reconstruction of the training dataset from the learned weights. We consider Gaussianactivation MLPs of width equal to dataset size and compare three structural losses that respectively encourage coverage of the training samples, separation between neuron-induced prototypes, and low overlap of hidden responses, against the standard fitting baseline. Experiments on uniformly sampled one-dimensional datasets show a stable pattern from N = 3 to N = 100 across 480 controlled runs. Coverage regularization gives the lowest mean reconstruction error at every tested size and raises the prototype-usage specialization ratio relative to the standard baseline, while separation has mixed effects and overlap penalties are systematically harmful. We show that the harm is not an optimization failure: overlap-active approaches fit the data as well as overlap-free ones but route the optimizer to a degenerate equilibrium in which prototype centers are pushed outside the convex hull of the training inputs. Coverage cannot reward this expulsion and acts as an attractor: separation admits it only at large temperature and overlap admits it at the nominal hyperparameter choice. A direct τ-sweep on the separation-only mask and a prototype-position visualization at N = 100 confirm the mechanism. The findings yield a simple design principle for prototype-recoverability-aware training: every repulsive structural loss must be compensated by a compatible attractor, or it will collapse the latent geometry it was meant to refine.

Enrique Alba, Ezequiel Lopez-Rubio

aITIS Software, University of Malaga, Spain

Multilayer Perceptrons Neural Network Interpretability Implicit Regularization

May 23, 2026cond-mat.stat-mech

Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

Khen Cohen, Mark Glass, Meir Feder +1

School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel · School of Electrical and Computer Engineering, Tel Aviv University, Tel Aviv, Israel

Combinatorial Optimization Quadratic Unconstrained Binary Optimization Sparse Coding Implicit Regularization

May 21, 2026cs.LG

Implicit Regularization of Mini-Batch Training in Graph Neural Networks

Mini-batch training of Graph Neural Networks (GNNs) is fundamentally different from training on i.i.d. data: sampling a subgraph alters the topology and introduces boundary effects, leading prior work to develop structure-aware samplers that preserve local connectivity and reduce embedding variance. Surprisingly, we demonstrate that the simplest possible scheme, Random Node Sampling (RNS), training on the induced subgraph of uniformly sampled nodes, matches or outperforms full-graph training on 8 of 10 datasets at a fraction of the wall-clock time and memory. To explain this, we apply backward error analysis to graph mini-batch Stochastic Gradient Descent (SGD) and show that it implicitly minimizes the sampled loss plus a regularizer proportional to the mini-batch gradient variance, a quantity directly shaped by the sampler. Although RNS discards local structure, it produces mini-batches whose expected loss is closer to the full-graph loss, and whose per-batch gradients have lower variance, yielding a better implicit objective. Our analysis reframes the choice of graph sampler as a form of implicit regularization, and identifies RNS as a strong, theoretically grounded method for scalable GNN training.

Clement Wang, Antoine Vialle, Robin Vaysse +1

Institut Polytechnique de Paris · Mirakl

Graph Neural Networks Implicit Regularization Stochastic Gradient Descent

May 19, 2026cs.LG

Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach

The remarkable success of the Adam in training neural networks has naturally led to the widespread use of its descent-ascent counterpart, Adam-DA, for solving zero-sum games. Despite its popularity in practice, a rigorous theoretical understanding of Adam-DA still lags behind. In this paper, we derive ordinary differential equations (ODEs) that serve as continuous-time limits of the Adam-DA. These ODEs closely approximate the discrete-time dynamics of Adam-DA, providing a tractable analytical framework for understanding its behavior in zero-sum games. Using this ODE approach, we investigate two fundamental aspects of Adam-DA: local convergence and implicit gradient regularization. Our analysis reveals that the roles of the first- and second-order momentum parameters in zero-sum games are exactly the opposite of their well-documented effects in minimization problems. We validate these predictions through GAN experiments across multiple architectures and datasets, demonstrating the practical implications of this reversed momentum effect.

Yi Feng, Weiming Ou, Xiao Wang

Aarhus University, Aarhus, Denmark · Shanghai University of Finance and Economics, Shanghai, China · MoE Key Laboratory of Interdisciplinary Research of Computation and Economics, Shanghai University of Finance and Economics, Shanghai, China

Zero-Sum Games Min-Max Optimization Adaptive Gradient Methods Neural Network Training Dynamics Implicit Regularization Gradient Descent Dynamics

May 18, 2026stat.ML

Canonical Regularisation of Wide Feature-Learning Neural Networks

Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

George Whittle, Pranav Vaidhyanathan, Juliusz Ziomek +2

Department of Engineering Science University of Oxford

Kernel Ridge Regression Representation Learning Neural Network Training Dynamics Ridge Regression Implicit Regularization

May 14, 2026cs.LG

Beyond What to Select: A Plug-and-play Oscillatory Data-Volume Scheduling for Efficient Model Training

Data selection accelerates training by identifying representative training data while preserving model performance. However, existing methods mainly focus on designing sample-importance criteria, i.e., deciding what to select, while typically fixing the selected data volume as the target ratio throughout training. Thus, they are often dynamic in sample identity but static in data volume. In this work, we revisit data selection from an optimization perspective and show that selected-data training induces an implicit regularization effect modulated by the instantaneous selection ratio. This reveals a key trade-off: lower ratios amplify selection-induced regularization, whereas higher ratios preserve data coverage and optimization fidelity. Motivated by this insight, we propose PODS, a Plug-and-play Oscillatory Data-volume Scheduling framework. Rather than introducing another sample-scoring metric, PODS serves as a lightweight module that dynamically schedules how much data to select over training. Under the target selection ratio, PODS alternates between low-ratio regularization phases and high-ratio recovery phases to exploit selection-induced regularization without sacrificing optimization stability. With its lightweight, ratio-level, and task-agnostic design, PODS is compatible with existing static and dynamic selection methods and broadly applicable across training paradigms. Experiments across various datasets, architectures, and tasks show that PODS consistently improves the efficiency-generalization trade-off, e.g., reducing ImageNet-1k training cost by 50% with improved accuracy and accelerating LLM instruction tuning by over 2x without performance degradation.

Suorong Yang, Hanqi Zhu, Hai Gan +4

1National University of Singapore · 2Nanjing University · 3Hokkaido University +1

Deep Learning Optimization Data Selection Training Data Selection Implicit Regularization

May 11, 2026cs.AI

Optimizer-Induced Mode Connectivity: From AdamW to Muon

Mode connectivity has been widely studied, yet the role of the optimizer remains underexplored. We revisit it through optimizer-induced implicit regularization, asking how connectivity behaves when restricted to solutions constrained by a given optimizer. For two-layer ReLU networks, we show that solutions from a single optimizer -- AdamW, Muon, or others in the Lion-

\mathcal{K}

family -- form a connected set at sufficiently large width, a result not implied by prior work. We then characterize how optimizer-induced regions interact: at large width two different regions can be disjoint or overlap depending on regularization, while in our small-width example AdamW and Muon converge to disconnected zero-loss components separated by a provable loss barrier. Empirically, in GPT-2 pretraining, we observe same-optimizer paths preserve each model's spectrum while cross-optimizer paths traverse a smooth transition. Our results reveal optimizer-dependent structure beyond classical mode connectivity literature.

Fangzhao Zhang, Sungyoon Kim, Erica Zhang +2

Stanford University

Mode Connectivity Neural Network Optimization Implicit Regularization

May 6, 2026stat.ML

Estimating Implicit Regularization in Deep Learning

Deep learning systems are known to exhibit implicit regularization (alt. implicit bias), favoring simple solutions instead of merely minimizing the loss function. In some cases, we can analytically derive the implicit regularization -- connecting it to an equivalent penalty that augments the learning objective. However, modern deep learning systems are complex, carrying modifications to the training procedure and architecture (e.g. early stopping, minibatching, dropout) whose effects are not always directly interpretable. Although estimating the resulting implicit regularization could aid theorists in algorithm design and practitioners in interpreting their hyperparameter choices, this problem has received little direct attention. It is also tractable: regularization makes weight updates deviate from loss gradients, promising a signal for identifying implicit bias. Here we provide gradient matching methods that can be used to empirically estimate the implicit regularization. Our method works on networks with known regularization, recovering popular explicit penalties like

\ell_1

and

\ell_2

. It also replicates known implicit effects, like the quadratic weight penalty induced by early stopping in gradient descent, demonstrating that it can be used to test theories of implicit regularization. Crucially, because our method is empirical, it can handle implicit regularization in arbitrary networks. We demonstrate this use by characterizing the effects of dropout in deep networks, showing implicit

\ell_2

effects in this popular method. Our work shows that practitioners can use gradient matching to understand regularization in networks with implicit biases that are too complicated to derive analytically.

Joseph H. Rudoler, Kevin Tan, Giles Hooker +1

University of Pennsylvania

Neural Network Optimization Implicit Bias Implicit Regularization

Apr 25, 2026cs.LG

When Does Removing LayerNorm Help? Activation Bounding as a Regime-Dependent Implicit Regularizer

Dynamic Tanh (DyT) removes LayerNorm by bounding activations with a learned tanh(alpha x). We show that this bounding is a regime-dependent implicit regularizer, not a uniformly beneficial replacement. Across GPT-2-family models spanning 64M to 3.78B parameters and 1M to 118M tokens, with Llama and ViT cross-checks, DyT improves validation loss by 27.3% at 64M/1M but worsens it by 18.8% at 64M/118M; the 1M benefit vanishes with capacity (+1.7% at 3.78B), while the 118M penalty reaches +27.9%. The mechanism is measurable: 49% of DyT activations saturate at 1M versus 23% at 118M, and a 500-step saturation heuristic classifies DyT's sign with 75% raw in-sample accuracy on the 12-cell GPT-2 calibration set (AUC 0.75; 64% when adding Scale 5 stress cells), correctly labels 3/3 Llama checks, but only reaches 50% raw leave-one-scale-out accuracy. Three interventions support the bounding explanation: HardTanh reproduces the regime pattern, increasing alpha at 118M monotonically reduces DyT's penalty, and vanilla+dropout(p=0.5) matches DyT's data-rich loss. We also localize Llama-DyT collapse to SwiGLU gating, where saturation separates collapse from convergence in a 3-seed component ablation (r=0.94). Scope: all experiments are compute-limited (T/P < 1.84), below Chinchilla-optimal training.

Lucky Verma

Independent Researcher

Neural Network Activation Functions Layer Normalization Neural Network Training Dynamics Implicit Regularization