Summary Benign Oscillation of Stochastic Gradient Descent arxiv.org
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The paper investigates how neural networks trained with high learning rates and stochastic gradient descent perform in terms of generalization.
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Key Points
- The paper investigates the generalization properties of neural networks trained using stochastic gradient descent (SGD) with large learning rates.
- The concept of "benign oscillation" is introduced, which refers to the oscillation of the neural network weights caused by large learning rate SGD training that benefits the generalization of the network.
- Theoretical results show that oscillating SGD training with large learning rates effectively learns weak features in the presence of strong features, leading to better generalization performance.
- Numerical experiments demonstrate that a convolutional neural network (CNN) trained with a large learning rate achieves higher test accuracy compared to a CNN trained with a small learning rate.
- The document provides a detailed analysis of SGD in the presence of noise and multiple training data, establishing key properties and proving important results.
- The authors emphasize the importance of hyperparameter selection and weight initialization in achieving stable oscillation and accurate learning in SGD.
- The proof of Theorem 4.3 demonstrates that SGD can effectively learn weak signals in a neural network while avoiding overfitting and memorizing noise.
- The analysis of SGD in the presence of multiple training data points reveals the dynamics of learning the strong signal, learning the weak signal, and memorizing noise.
Summaries
18 word summary
This paper explores the generalization properties of neural networks trained with large learning rates and stochastic gradient descent.
90 word summary
This paper examines the generalization properties of neural networks trained with stochastic gradient descent (SGD) and large learning rates. The authors introduce "benign oscillation," which refers to the oscillation of network weights caused by large learning rate SGD training, and show its positive impact on generalization. Experimental results support these findings. The paper also includes theoretical results, lemmas, and proofs characterizing SGD behavior in the presence of noise and multiple training data. The paper contributes to understanding large learning rate training in deep learning and provides insights for optimization analysis.
214 word summary
This paper investigates the generalization properties of neural networks trained using stochastic gradient descent (SGD) with large learning rates. The authors introduce the concept of "benign oscillation," which refers to the oscillation of neural network weights caused by large learning rate SGD training, and demonstrate its beneficial effect on generalization. The theoretical results show that oscillating SGD training with large learning rates effectively learns weak features alongside strong features, leading to better generalization performance. Numerical experiments using convolutional neural networks (CNNs) support these findings, showing that CNNs trained with large learning rates achieve higher test accuracy compared to those trained with small learning rates. The paper also discusses concentration results, lemmas, and proofs characterizing the behavior of SGD in the presence of noise and multiple training data. The proof of the main result and the conditions required for it are presented. Additionally, proofs for lemmas discussing training dynamics and weak signal learning are provided. The proof of Theorem 4.3 demonstrates the effectiveness of SGD in learning weak signals while avoiding overfitting and memorizing noise. Finally, the proof of Proposition D.5 establishes upper and lower bounds on various terms using lemmas B.5 and D.2. Overall, this paper contributes to the understanding of large learning rate training in deep learning and provides insights for optimization analysis.
568 word summary
This paper explores the generalization properties of neural networks trained using stochastic gradient descent (SGD) with large learning rates. The authors introduce the concept of "benign oscillation" which refers to the oscillation of the neural network weights caused by large learning rate SGD training, which is beneficial for generalization. The paper provides an overview of deep learning and the need for a theoretical understanding of its optimization and generalization properties. It highlights empirical evidence that using a large learning rate improves generalization but lacks theoretical understanding.
The main theoretical results of the paper show that oscillating SGD training with large learning rates effectively learns weak features in the presence of strong features. On the other hand, SGD with small learning rates only learns strong features and makes little progress in learning weak features. This division in feature learning leads to better generalization performance for neural networks trained with large learning rates.
Numerical experiments are provided to support these findings. The authors train convolutional neural networks (CNNs) using large and small learning rates and show that the CNN trained with a large learning rate achieves higher test accuracy than the one trained with a small learning rate. The CNN trained with a small learning rate fails to generalize to testing data without strong features.
In conclusion, this paper provides a theoretical investigation of large learning rate SGD training for neural networks. The concept of benign oscillation is introduced and shown to lead to better generalization performance by effectively learning weak features. The findings are supported by numerical experiments. This work contributes to the understanding of large learning rate training in deep learning and provides insights for optimization analysis.
The document discusses the benign oscillation of SGD in the context of machine learning. It presents lemmas and proofs to characterize the behavior of SGD in the presence of noise and multiple training data. The first section focuses on concentration results and provides lemmas to characterize the concentration properties of random elements involved in the problem.
The second section focuses on the single training data case and presents a formal statement and proof for the main result, Theorem 3.2. The conditions and assumptions required for Theorem 3.2 are discussed, including conditions on hyperparameters, weight initialization scale, signal strength, and dimension.
The third section presents proofs for the lemmas in Appendix B.1, which discuss basic properties of training dynamics and the two-layer CNN. These lemmas establish properties of neuron subsets, such as fixed signs of inner products and connections between inner products and model outcomes.
The fourth section presents a fundamental reasoning towards weak signal learning. Lemma B.7 provides a quantitative interpretation of the increasing behavior of inner products, formalizing the role of oscillation in learning weak signals.
The fifth section presents the proof of Theorem B.3, which establishes the benign oscillation of SGD. The proof is done by contradiction and involves bounding the stopping time T(v) and establishing lower bounds on inner products.
The proof of Theorem 4.3 in the document "Benign Oscillation of Stochastic Gradient Descent" is presented. The main goal is to show that the SGD algorithm can effectively learn the weak signal in a neural network while avoiding overfitting and memorizing noise.
The proof of Proposition D.5 begins by expanding the expression for f(tj(Sj,k)) and breaking it down into several terms. The proof utilizes Lemma B.5 and Lemma D.2 to establish upper and lower bounds on various terms.
670 word summary
This paper investigates the generalization properties of neural networks (NN) trained using stochastic gradient descent (SGD) algorithm with large learning rates. The authors introduce the concept of “benign oscillation” which refers to the oscillation of the NN weights caused by the large learning rate SGD training that is beneficial for the generalization of the NN. The authors propose a theoretical framework based on the feature learning perspective of deep learning to explain this phenomenon.
The paper starts by providing an overview of deep learning and the need for a theoretical understanding of its optimization and generalization properties. It highlights the empirical evidence that using a large learning rate in NN training improves generalization but the theoretical understanding of this phenomenon is limited. The paper then introduces the problem settings, including the data generation model and the two-layer convolutional neural network (CNN) used for training.
The main theoretical results of the paper are presented next. The authors show that under certain conditions and assumptions, oscillating SGD training with large learning rates leads to effective learning of weak features in the presence of strong features. On the the other hand, SGD with small learning rates only learns the strong features and makes little progress in learning the weak features. This division in feature learning leads to better generalization performance for NN trained with large learning rates.
The authors provide numerical experiments to demonstrate their findings. They train CNNs using large and small learning rates and show that the CNN trained with a large learning rate achieves higher test accuracy than the one trained with a small learning rate. The CNN trained with a small learning rate fails to generalize to testing data that lacks strong features.
In conclusion, this paper provides a theoretical investigation of large learning rate SGD training for NNs. The authors introduce the concept of benign oscillation and show that it leads to better generalization performance by effectively learning weak features. The findings are supported by numerical experiments. This work contributes to the understanding of large learning rate training in deep learning and provides insights for optimization analysis.
The document discusses the benign oscillation of stochastic gradient descent (SGD) in the context of machine learning. It presents a series of lemmas and proofs to characterize the behavior of SGD in the presence of noise and multiple training data.
The first section focuses on concentration results and provides lemmas to characterize the concentration properties of random elements involved in the problem. These lemmas establish bounds on the number of positive and negative labels in the training data and the weight initialization scale.
The second section focuses on the single training data case and presents a formal statement and proof for the main result, Theorem 3.2. The conditions and assumptions required for Theorem 3.2 are discussed, including conditions on hyperparameters, weight initialization scale, signal strength, and dimension.
The third section presents the proofs for the lemmas in Appendix B.1, which discuss basic properties of training dynamics and the two-layer CNN. These lemmas establish properties of neuron subsets, such as fixed signs of inner products and connections between inner products and model outcomes.
The fourth section presents a fundamental reasoning towards weak signal learning. Lemma B.7 provides a quantitative interpretation of the increasing behavior of inner products, formalizing the role of oscillation in learning weak signals.
The fifth section presents the proof of Theorem B.3, which establishes the benign oscillation of SGD. The proof is done by contradiction and involves bounding the stopping time T(v) and establishing lower bounds on inner products.
The proof of Theorem 4.3 in the document “Benign Oscillation of Stochastic Gradient Descent” is presented. The main goal is to show that the stochastic gradient descent (SGD) algorithm can effectively learn the weak signal in a neural network, while avoiding overfitting and memorizing noise.
The proof of Proposition D.5 begins by expanding the expression for f(tj(Sj,k)) and breaking it down into several terms. The proof utilizes Lemma B.5 and Lemma D.2 to establish upper and lower bounds on various
1867 word summary
This paper investigates the generalization properties of neural networks (NN) trained using stochastic gradient descent (SGD) algorithm with large learning rates. The authors introduce the concept of "benign oscillation" which refers to the oscillation of the NN weights caused by the large learning rate SGD training that is beneficial for the generalization of the NN. The authors propose a theoretical framework based on the feature learning perspective of deep learning to explain this phenomenon.
The paper starts by providing an overview of deep learning and the need for a theoretical understanding of its optimization and generalization properties. It highlights the empirical evidence that using a large learning rate in NN training improves generalization but the theoretical understanding of this phenomenon is limited. The paper then introduces the problem settings, including the data generation model and the two-layer convolutional neural network (CNN) used for training.
The main theoretical results of the paper are presented next. The authors show that under certain conditions and assumptions, oscillating SGD training with large learning rates leads to effective learning of weak features in the presence of strong features. On the the other hand, SGD with small learning rates only learns the strong features and makes little progress in learning the weak features. This division in feature learning leads to better generalization performance for NN trained with large learning rates.
The authors provide numerical experiments to demonstrate their findings. They train CNNs using large and small learning rates and show that the CNN trained with a large learning rate achieves higher test accuracy than the one trained with a small learning rate. The CNN trained with a small learning rate fails to generalize to testing data that lacks strong features.
In conclusion, this paper provides a theoretical investigation of large learning rate SGD training for NNs. The authors introduce the concept of benign oscillation and show that it leads to better generalization performance by effectively learning weak features. The findings are supported by numerical experiments. This work contributes to the understanding of large learning rate training in deep learning and provides insights for optimization analysis.
The document discusses the benign oscillation of stochastic gradient descent (SGD) in the context of machine learning. It presents a series of lemmas and proofs to characterize the behavior of SGD in the presence of noise and multiple training data.
The first section focuses on concentration results and provides lemmas to characterize the concentration properties of random elements involved in the problem. These lemmas establish bounds on the number of positive and negative labels in the training data and the weight initialization scale. The proofs of these lemmas are referenced to another document.
The second section focuses on the single training data case and presents a formal statement and proof for the main result, Theorem 3.2. The conditions and assumptions required for Theorem 3.2 are discussed, including conditions on hyperparameters, weight initialization scale, signal strength, and dimension. The assumption of oscillations in SGD is also introduced.
The third section presents the proofs for the lemmas in Appendix B.1, which discuss basic properties of training dynamics and the two-layer CNN. These lemmas establish properties of neuron subsets, such as fixed signs of inner products and connections between inner products and model outcomes.
The fourth section presents a fundamental reasoning towards weak signal learning. Lemma B.7 provides a quantitative interpretation of the increasing behavior of inner products, formalizing the role of oscillation in learning weak signals.
The fifth section presents the proof of Theorem B.3, which establishes the benign oscillation of SGD. The proof is done by contradiction and involves bounding the stopping time T(v) and establishing lower bounds on inner products. The proof is organized into several steps, including an analysis of pre-T1 behavior, bounding max inner products, lower bounding inner products, and upper bounding inner products.
Overall, the document provides a detailed analysis of SGD in the presence of noise and multiple training data, establishing key properties and proving important results. The lemmas and proofs contribute to a better understanding of the behavior of SGD and its ability to learn weak signals.
The paper investigates the behavior of stochastic gradient descent (SGD) in the context of a single training data point. The objective function is rearranged to focus on the difference between the predicted output and the true output. The weights are updated according to specific rules based on the inner products between the weights and the signal patches. The authors analyze the conditions on the hyperparameters and the initialization of the weights. They prove concentration results for the initialization and provide necessary conditions for stable oscillation.
The authors introduce Assumption C.2, which includes conditions on the weight initialization scale, signal strength, and dimension. They then present Theorem C.3, which states that under these conditions and with appropriate choice of learning rate, the average loss over iterations decreases and the model does not learn weak signals well enough compared to the initialization.
The paper also includes Lemma C.4, which provides a lower bound on the fitting residual based on the maximal inner product between the weights and the signal vectors. The authors divide the analysis into two stages: exponential growth and stabilization. In the exponential growth stage, they track the maximal inner product between the weights and the signal vectors.
The authors conclude that their findings provide insights into the behavior of SGD in the small learning rate regime for single training data points. They emphasize the importance of hyperparameter selection and initialization in achieving stable oscillation and accurate learning.
Overall, this paper contributes to the understanding of SGD in the context of single training data points and provides necessary conditions for stable oscillation. The results have implications for hyperparameter selection and weight initialization in machine learning algorithms.
The proof of Theorem 4.3 in the document "Benign Oscillation of Stochastic Gradient Descent" is presented. The main goal is to show that the stochastic gradient descent (SGD) algorithm can effectively learn the weak signal in a neural network, while avoiding overfitting and memorizing noise. The proof is organized into several steps.
First, the boundedness and sign stability of the SGD algorithm are established. It is shown that the inner products associated with the strong signal are bounded and remain stable throughout the training process. The proof also demonstrates that the inner products associated with the weak signal and noise are negligible compared to the strong signal.
Next, the exponential learning rate of the weak signal is proven. It is shown that the SGD algorithm can effectively learn the weak signal with an exponential rate of convergence. This is achieved by analyzing the dynamics of the inner products and showing that they increase linearly with time.
Finally, the control of noise memorization is addressed. It is proven that the SGD algorithm can effectively avoid memorizing noise and focus on learning the relevant features of the data. This is achieved by bounding the maximum absolute value of the noise inner products over time.
Overall, the proof demonstrates that the SGD algorithm can effectively learn the weak signal in a neural network while avoiding overfitting and memorizing noise. The results provide insight into the behavior of SGD and its ability to generalize well to unseen data.
The proof of Proposition D.5 begins by expanding the expression for f(tj(Sj,k)) and breaking it down into several terms. The proof utilizes Lemma B.5 and Lemma D.2 to establish upper and lower bounds on various terms in the expression. The proof also makes use of the local sign stability assumption to ensure that the inner products grow proportionally. By applying these bounds and using induction, the proposition is proven for all steps t+1.
The proof of Proposition D.4 is also done by induction. The base case is trivial, and it is assumed that the proposition holds for each step t1,...,t. Using Equation (D.4), the proof shows that the weight update at step t+1 can be bounded by the sum of the weight update at step t and a term involving the data labels and the inner products. By applying Lemma D.2 and Lemma A.3, upper bounds on the weight updates are established, and the final result is obtained by combining these bounds.
The proof of Proposition D.1 utilizes Equation (D.2) to derive an expression for the weight update at step t+1. By analyzing the expression and using the local sign stability assumption, it is shown that the sign of the inner products does not change during the weight update. This result holds for all inner products, and therefore, the proposition is proven.
The proof of Lemma B.5 in the multiple data setting begins by defining a quantity that measures the change in the inner products. By applying Lemma D.2 and Lemma A.3, upper and lower bounds on this quantity are established, which depend on the number of data points and certain constants. By comparing these bounds, it is shown that if the lower bound is sufficiently large, then the upper bound must also be large. This condition ensures that the sign stability holds, which completes the proof.
Overall, these proofs establish important results related to the weight updates and sign stability in stochastic gradient descent. These results provide insights into the behavior of the algorithm and can be used to analyze its convergence properties.
The document discusses the behavior of stochastic gradient descent (SGD) in the context of machine learning. The authors analyze the dynamics of SGD and its ability to learn from training data. They focus on the case of multiple training data points and a small learning rate.
In the first stage of the analysis, the authors track the maximal inner product between the weight vectors and the signal vectors, as well as the inner product between the weight vectors and the noise vectors. They establish lower bounds on the fitting residual and derive upper bounds on the growth of these inner products.
In the second stage, the authors show that before the model learns the weak signal or memorizes any noise vectors, it already fits a proportion of the data points by exploiting the strong signal. They provide bounds on the average loss over iterations and show that the model does not learn the weak signal as well as it learns the strong signal.
In the third stage, the authors analyze the memorization of noise vectors by the model. They show that after fitting the strong signal, the model interpolates the entire dataset by memorizing noise vectors. They provide a reference point for this stage and analyze the evolution dynamics during this interval.
The authors conclude that SGD can effectively learn from multiple training data points with a small learning rate. They highlight the importance of differentiating between strong and weak signals and show that SGD can exploit strong signals to fit data points. They also discuss the memorization of noise vectors and how it leads to interpolation of the entire dataset.
Overall, this analysis provides insights into the behavior of SGD in machine learning and highlights its ability to learn from training data. The authors present detailed mathematical proofs and establish important bounds on various parameters. This research contributes to our understanding of SGD and its applications in training machine learning models.