Neural networks exhibit grokking on small algorithmic datasets, achieving perfect generalization well after overfitting.
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On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima
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The stochastic gradient descent (SGD) method and its variants are algorithms of choice for many Deep Learning tasks. These methods operate in a small-batch regime wherein a fraction of the training data, say $32$-$512$ data points, is sampled to compute an approximation to the gradient. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. We investigate the cause for this generalization drop in the large-batch regime and present numerical evidence that supports the view that large-batch methods tend to converge to sharp minimizers of the training and testing functions - and as is well known, sharp minima lead to poorer generalization. In contrast, small-batch methods consistently converge to flat minimizers, and our experiments support a commonly held view that this is due to the inherent noise in the gradient estimation. We discuss several strategies to attempt to help large-batch methods eliminate this generalization gap.
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Muon outperforms Adam by reducing curvature penalty via lower Normalized Directional Sharpness, as shown via Taylor approximation on LLM training and proven on stylized quadratic problems with heterogeneous curvature.
LE-SAM inverts SAM by fixing the loss budget instead of the parameter-space radius, yielding better generalization across benchmarks.
Online kernel regression equals offline regression with shifted targets; correcting the targets lets online learning match offline performance and outperform true targets in continual image classification.
ConquerNet smooths quantile ReLU networks with convolution for easier training and establishes minimax-optimal nonasymptotic risk bounds over Besov function classes.
Gradient matching empirically recovers implicit regularization effects such as l2 penalties from early stopping and dropout in neural networks.
FP32-converged language models enter a post-convergence phase where INT4 quantization error explodes while FP32 perplexity remains stable, with onset tied to fine convergence rather than learning rate decay.
SGD dynamics in Hilbert spaces are approximated by an SDE with cylindrical noise, with the weak error between discrete and continuous versions shown to be second order in the step size.
Effective noise scale non-monotonically governs model merging success with an optimum, unifying effects of learning rate, weight decay, batch size, and augmentation on the loss landscape.
One training example via RLVR boosts LLM math reasoning from 17.6% to 35.7% average across six benchmarks.
Derives explicit step-size conditions ensuring the metastability behavior of discrete SGD under heavy-tailed noise approximates its continuous SDE limit.
NEB-adapted ravine ensembles for QNNs classifying concentratable entanglement outperform naive methods when local-prediction variability is high and reduce costs, with ravines persisting under depth and qubit scaling.
FLORA is an octree-based deep learning framework with auxiliary data fusion that predicts forest attributes from heterogeneous LiDAR, achieving rRMSE of 12.3% for dominant height and 39% for total volume on 32k French NFI plots.
Hessian eigenvector displacement and inverse participation ratio metrics show SGD stabilizing leading curvature directions while Adam causes more reorganization and parameter localization in MLP training.
Derives closed-form gradient of WS upper bound on Hessian max eigenvalue for 3-layer cross-entropy NNs and proposes HSR regularization to steer toward flat minima.
Model-assisted sampling applies survey sampling variance reduction with auxiliary predictors to stochastic gradients, yielding empirical gains on benchmarks and faster generalization with AdamW.
On-policy self-distillation with sampled demonstrations reduces rollout diversity by amplifying existing probability gaps in the base model, unlike ideal RL which preserves ratios among correct outputs.
Knowledge editing methods redistribute and suppress rather than overwrite facts in LLMs, creating narrow vulnerable regions in representation space that adversarial prompts can exploit.
SFF smooths the non-convex loss landscape of pre-trained LTSMs by linear weight interpolation with a random model, enabling more effective fine-tuning while preserving pre-trained knowledge.
Derives second-order path-kernel interpolation formulas for gradient descent, SGD, and momentum training, adding curvature terms and a concentration estimate around the expected prediction.
For random MPS and Clifford+T circuits, increases in entanglement or T-count correlate with sharper loss minima and worse reconstruction under constrained neural capacity.
MONA integrates Nesterov acceleration into Muon's orthogonalization framework, reporting better convergence than Muon and AdamW on MoE models up to 68B parameters trained on 1T tokens and SOTA fine-tuning results.
Batch normalization amplifies memorization of outlier samples in deep neural networks, directly increasing susceptibility to membership inference attacks.
PAC-Bayes applied to low-sharpness flat minima yields non-vacuous generalization bounds for boolean functions whose Fourier spectra are sparse and low-degree, with parameters estimable by property testing.
citing papers explorer
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Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
Neural networks exhibit grokking on small algorithmic datasets, achieving perfect generalization well after overfitting.
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Why Muon Outperforms Adam: A Curvature Perspective
Muon outperforms Adam by reducing curvature penalty via lower Normalized Directional Sharpness, as shown via Taylor approximation on LLM training and proven on stylized quadratic problems with heterogeneous curvature.
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Fix the Loss, Not the Radius: Rethinking the Adversarial Perturbation of Sharpness-Aware Minimization
LE-SAM inverts SAM by fixing the loss budget instead of the parameter-space radius, yielding better generalization across benchmarks.
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Characterizing and Correcting Effective Target Shift in Online Learning
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ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees
ConquerNet smooths quantile ReLU networks with convolution for easier training and establishes minimax-optimal nonasymptotic risk bounds over Besov function classes.
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Estimating Implicit Regularization in Deep Learning
Gradient matching empirically recovers implicit regularization effects such as l2 penalties from early stopping and dropout in neural networks.
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When Flat Minima Fail: Characterizing INT4 Quantization Collapse After FP32 Convergence
FP32-converged language models enter a post-convergence phase where INT4 quantization error explodes while FP32 perplexity remains stable, with onset tied to fine convergence rather than learning rate decay.
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Stochastic Modified Equations for Stochastic Gradient Descent in Infinite-Dimensional Hilbert Spaces
SGD dynamics in Hilbert spaces are approximated by an SDE with cylindrical noise, with the weak error between discrete and continuous versions shown to be second order in the step size.
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How does the optimizer implicitly bias the model merging loss landscape?
Effective noise scale non-monotonically governs model merging success with an optimum, unifying effects of learning rate, weight decay, batch size, and augmentation on the loss landscape.
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Reinforcement Learning for Reasoning in Large Language Models with One Training Example
One training example via RLVR boosts LLM math reasoning from 17.6% to 35.7% average across six benchmarks.
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First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise
Derives explicit step-size conditions ensuring the metastability behavior of discrete SGD under heavy-tailed noise approximates its continuous SDE limit.
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Ravines in quantum cost landscapes: opportunities for improved VQA predictions
NEB-adapted ravine ensembles for QNNs classifying concentratable entanglement outperform naive methods when local-prediction variability is high and reduce costs, with ravines persisting under depth and qubit scaling.
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FLORA: A deep learning approach to predict forest attributes from heterogeneous LiDAR data
FLORA is an octree-based deep learning framework with auxiliary data fusion that predicts forest attributes from heterogeneous LiDAR, achieving rRMSE of 12.3% for dominant height and 39% for total volume on 32k French NFI plots.
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Characterizing Optimizer-Dependent Training Dynamics Through Hessian Eigenvector Displacement and Localization
Hessian eigenvector displacement and inverse participation ratio metrics show SGD stabilizing leading curvature directions while Adam causes more reorganization and parameter localization in MLP training.
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Closed-Form Steepest Descent Direction toward Flat Minima: Reducing Upper Bounds on the Loss Hessian Eigenspectrum in Neural Networks
Derives closed-form gradient of WS upper bound on Hessian max eigenvalue for 3-layer cross-entropy NNs and proposes HSR regularization to steer toward flat minima.
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Stochastic Gradient Optimization with Model-Assisted Sampling
Model-assisted sampling applies survey sampling variance reduction with auxiliary predictors to stochastic gradients, yielding empirical gains on benchmarks and faster generalization with AdamW.
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On-Policy Self-Distillation with Sampled Demonstrations Reduces Output Diversity
On-policy self-distillation with sampled demonstrations reduces rollout diversity by amplifying existing probability gaps in the base model, unlike ideal RL which preserves ratios among correct outputs.
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Exposing the Illusion of Erasure in Knowledge Editing for LLMs
Knowledge editing methods redistribute and suppress rather than overwrite facts in LLMs, creating narrow vulnerable regions in representation space that adversarial prompts can exploit.
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Lost in the Non-convex Loss Landscape: How to Fine-tune the Large Time Series Model?
SFF smooths the non-convex loss landscape of pre-trained LTSMs by linear weight interpolation with a random model, enabling more effective fine-tuning while preserving pre-trained knowledge.
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Second-Order Path Kernel Interpolation Formulas in Machine Learning
Derives second-order path-kernel interpolation formulas for gradient descent, SGD, and momentum training, adding curvature terms and a concentration estimate around the expected prediction.
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Comparing Classical Simulation and Sample-Based Learning of Quantum Systems
For random MPS and Clifford+T circuits, increases in entanglement or T-count correlate with sharper loss minima and worse reconstruction under constrained neural capacity.
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MONA: Muon Optimizer with Nesterov Acceleration for Scalable Language Model Training
MONA integrates Nesterov acceleration into Muon's orthogonalization framework, reporting better convergence than Muon and AdamW on MoE models up to 68B parameters trained on 1T tokens and SOTA fine-tuning results.
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Batch Normalization Amplifies Memorization and Privacy Risks
Batch normalization amplifies memorization of outlier samples in deep neural networks, directly increasing susceptibility to membership inference attacks.
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A Sharper Picture of Generalization in Transformers
PAC-Bayes applied to low-sharpness flat minima yields non-vacuous generalization bounds for boolean functions whose Fourier spectra are sparse and low-degree, with parameters estimable by property testing.
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Spectral Lens: Activation and Gradient Spectra as Diagnostics of LLM Optimization
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Sharpness-Aware Pretraining Mitigates Catastrophic Forgetting
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Generalization at the Edge of Stability
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Language Models (Mostly) Know What They Know
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Scaling Laws for Transfer
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Sharpness-Aware Minimization for Efficiently Improving Generalization
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Large Batch Optimization for Deep Learning: Training BERT in 76 minutes
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Flatness Preserves Instruction Following in Vision-Language-Action Models
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