In this paper we propose to study generalization of neural networks on small algorithmically generated datasets. In this setting, questions about data efficiency, memorization, generalization, and speed of learning can be studied in great detail. In some situations we show that neural networks learn through a process of "grokking" a pattern in the data, improving generalization performance from random chance level to perfect generalization, and that this improvement in generalization can happen well past the point of overfitting. We also study generalization as a function of dataset size and find that smaller datasets require increasing amounts of optimization for generalization. We argue that these datasets provide a fertile ground for studying a poorly understood aspect of deep learning: generalization of overparametrized neural networks beyond memorization of the finite training dataset.
Grokking, the unusual phenomenon for algorithmic datasets where generalization happens long after overfitting the training data, has remained elusive. We aim to understand grokking by analyzing the loss landscapes of neural networks, identifying the mismatch between training and test losses as the cause for grokking. We refer to this as the "LU mechanism" because training and test losses (against model weight norm) typically resemble "L" and "U", respectively. This simple mechanism can nicely explain many aspects of grokking: data size dependence, weight decay dependence, the emergence of repr…
Faculty of computer and information science, University of Ljubljana（卢布尔雅那大学计算机与信息科学学院）；Faculty of mathematics and physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）
We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and …
To Grok Grokking: Provable Grokking in Ridge Regression
Department of Computer Science, Purdue University, West Lafayette, IN, USA（普渡大学计算机科学系）；Department of Computer Science（计算机科学系）；Applied Mathematics, Weizmann Institute of Science, Israel（科学应用数学系，魏茨曼研究院）；Stein Faculty of Computer（斯坦因计算机科学学院）；Information Science, Ben-Gurion University of the Negev, Israel（信息科学系，本· Gurion 军事大学）
Grokking is a phenomenon where a model trained on an algorithmic task first overfits but, then, after a large amount of additional training, undergoes a phase transition to generalize perfectly. We empirically study the internal structure of networks undergoing grokking on the sparse parity task, and find that the grokking phase transition corresponds to the emergence of a sparse subnetwork that dominates model predictions. On an optimization level, we find that this subnetwork arises when a small subset of neurons undergoes rapid norm growth, whereas the other neurons in the network decay slo…
Beyond Progress Measures: Theoretical Insights into the Mechanism of Grokking
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络空间安全学院）
The grokking phenomenon as reported by Power et al. ( arXiv:2201.02177 ) refers to a regime where a long period of overfitting is followed by a seemingly sudden transition to perfect generalization. In this paper, we attempt to reveal the underpinnings of Grokking via a series of empirical studies. Specifically, we uncover an optimization anomaly plaguing adaptive optimizers at extremely late stages of training, referred to as the Slingshot Mechanism. A prominent artifact of the Slingshot Mechanism can be measured by the cyclic phase transitions between stable and unstable training regimes, an…
Grokking Beyond the Euclidean Norm of Model Parameters
Mila, Quebec AI Institute, Montréal, Quebec, Canada（魁北克AI研究所）；CHU Sainte-Justine Research Center, Montréal, Quebec, Canada（圣-朱斯特研究中心）；CIFAR AI Chair（CIFAR人工智能席位）
University of California, Los Angeles（加州大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full…
MIT（麻省理工学院）；Institute for AI and Fundamental Interactions（人工智能与基础交互研究所）
We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find repr…
Algorithmic approaches endow deep learning systems with implicit bias that helps them generalize even in over-parametrized settings. In this paper, we focus on understanding such a bias induced in learning through dropout, a popular technique to avoid overfitting in deep learning. For single hidden-layer linear neural networks, we show that dropout tends to make the norm of incoming/outgoing weight vectors of all the hidden nodes equal. In addition, we provide a complete characterization of the optimization landscape induced by dropout.
RL Grokking Recipe: How Does RL Unlock and Transfer New Algorithms in LLMs?
University of California, Berkeley（加州大学伯克利分校）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of Washington（华盛顿大学）；Ai2 Main dataset contributor（Ai2 主数据贡献者）
Bocconi University（博科尼大学）；Istituto Nazionale di Fisica Nucleare（意大利国家核物理研究所）；Politecnico di Torino（都灵理工大学）；International Centre for Theoretical Physics（国际理论物理中心）
Learning in Deep Neural Networks (DNN) takes place by minimizing a non-convex high-dimensional loss function, typically by a stochastic gradient descent (SGD) strategy. The learning process is observed to be able to find good minimizers without getting stuck in local critical points, and that such minimizers are often satisfactory at avoiding overfitting. How these two features can be kept under control in nonlinear devices composed of millions of tunable connections is a profound and far reaching open question. In this paper we study basic non-convex one- and two-layer neural network models w…
The University of Sydney（悉尼大学）；Hong Kong Baptist University（香港浸会大学）；JD Explore Academy（京东探索研究院）；University of Science and Technology of China（中国科学技术大学）；Nanjing University of Science and Technology（南京理工大学）；The University of Melbourne（墨尔本大学）
Robust overfitting widely exists in adversarial training of deep networks. The exact underlying reasons for this are still not completely understood. Here, we explore the causes of robust overfitting by comparing the data distribution of \emph{non-overfit} (weak adversary) and \emph{overfitted} (strong adversary) adversarial training, and observe that the distribution of the adversarial data generated by weak adversary mainly contain small-loss data. However, the adversarial data generated by strong adversary is more diversely distributed on the large-loss data and the small-loss data. Given t…
We present an information-theoretic framework for understanding overfitting and underfitting in machine learning and prove the formal undecidability of determining whether an arbitrary classification algorithm will overfit a dataset. Measuring algorithm capacity via the information transferred from datasets to models, we consider mismatches between algorithm capacities and datasets to provide a signature for when a model can overfit or underfit a dataset. We present results upper-bounding algorithm capacity, establish its relationship to quantities in the algorithmic search framework for machi…
In computer vision and natural language processing, innovations in model architecture that increase model capacity have reliably translated into gains in performance. In stark contrast with this trend, state-of-the-art reinforcement learning (RL) algorithms often use small MLPs, and gains in performance typically originate from algorithmic innovations. It is natural to hypothesize that small datasets in RL necessitate simple models to avoid overfitting; however, this hypothesis is untested. In this paper we investigate how RL agents are affected by exchanging the small MLPs with larger modern …
Teaching LLM to Reason: Reinforcement Learning from Algorithmic Problems without Code
University of Science and Technology of China（中国科学技术大学）；Alibaba Group（阿里巴巴集团）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
University of California, San Francisco（加州大学旧金山分校）；U.S. Food and Drug Administration（美国食品药品监督管理局）
After initial release of a machine learning algorithm, the model can be fine-tuned by retraining on subsequently gathered data, adding newly discovered features, or more. Each modification introduces a risk of deteriorating performance and must be validated on a test dataset. It may not always be practical to assemble a new dataset for testing each modification, especially when most modifications are minor or are implemented in rapid succession. Recent works have shown how one can repeatedly test modifications on the same dataset and protect against overfitting by (i) discretizing test results…
