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Problem 2: Convergence of Gradient Descent in Over-Parameterized Neural Networks Statement: Consider an over-parameterized neural network (i.e., a network with more parameters than necessary to fit the training data) trained using gradient descent on a squared loss function. Prove that, under appropriate initialization and with a sufficiently small learning rate, gradient descent converges to a global minimum of the loss function. Key Points for the Proof: • • • Define the over-parameterization regime and its implications for the loss landscape. Analyze the dynamics of gradient descent in the high-dimensional parameter space. Use tools from optimization theory to show that all local minima are global minima in this setting. Ensure that the initialization is within the basin of attraction for convergence to a global minimum.

Problem 2: Convergence of Gradient Descent in Over-Parameterized Neural Networks Statement: Consider an over-parameterized neural network (i.e., a network with more parameters than necessary to fit the training data) trained using gradient descent on a squared loss function. Prove that, under appropriate initialization and with a sufficiently small learning rate, gradient descent converges to a global minimum of the loss function. Key Points for the Proof: • • • Define the over-parameterization regime and its implications for the loss landscape. Analyze the dynamics of gradient descent in the high-dimensional parameter space. Use tools from optimization theory to show that all local minima are global minima in this setting. Ensure that the initialization is within the basin of attraction for convergence to a global minimum.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 2: Convergence of Gradient Descent in Over-Parameterized Neural Networks Statement: Consider an over-parameterized neural network (i.e., a network with more parameters than necessary to fit the training data) trained using gradient descent on a squared loss function. Prove that, under appropriate initialization and with a sufficiently small learning rate, gradient descent converges to a global minimum of the loss function. Key Points for the Proof: • • • Define the over-parameterization regime and its implications for the loss landscape. Analyze the dynamics of gradient descent in the high-dimensional parameter space. Use tools from optimization theory to show that all local minima are global minima in this setting. Ensure that the initialization is within the basin of attraction for convergence to a global minimum.
Transcribed Image Text:Problem 2: Convergence of Gradient Descent in Over-Parameterized Neural Networks Statement: Consider an over-parameterized neural network (i.e., a network with more parameters than necessary to fit the training data) trained using gradient descent on a squared loss function. Prove that, under appropriate initialization and with a sufficiently small learning rate, gradient descent converges to a global minimum of the loss function. Key Points for the Proof: • • • Define the over-parameterization regime and its implications for the loss landscape. Analyze the dynamics of gradient descent in the high-dimensional parameter space. Use tools from optimization theory to show that all local minima are global minima in this setting. Ensure that the initialization is within the basin of attraction for convergence to a global minimum.
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