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Danica
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The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

For a simple way to see this:

  • If we're at a (local) optimum, the full gradient (across the entire training dataset) will be zero. In the interpolating regime common to modern neural networks, the individual gradient for each training point may even be exactly zero; depending on the loss function and how much you've trained / etc, it might instead be mean zero but approximately normal, etc.

  • At initialization, when we're very far from a solution, the gradient may be extremely similar for different datapoints (and very far from zero).

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (When using square loss, this means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

For a simple way to see this:

  • If we're at a (local) optimum, the full gradient (across the entire training dataset) will be zero. In the interpolating regime common to modern neural networks, the individual gradient for each training point may even be exactly zero; depending on the loss function and how much you've trained / etc, it might instead be mean zero but approximately normal, etc.

  • At initialization, when we're very far from a solution, the gradient may be extremely similar for different datapoints (and very far from zero).

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (When using square loss, this means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

For a simple way to see this:

  • If we're at a (local) optimum, the full gradient (across the entire training dataset) will be zero. In the interpolating regime common to modern neural networks, the individual gradient for each training point may even be exactly zero; depending on the loss function and how much you've trained / etc, it might instead be mean zero but approximately normal, etc.

  • At initialization, when we're very far from a solution, the gradient may be extremely similar for different datapoints (and very far from zero).

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (When using square loss, this means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

added 576 characters in body
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Danica
  • 25.2k
  • 2
  • 76
  • 140

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

For a simple way to see this:

  • If we're at a (local) optimum, the full gradient (across the entire training dataset) will be zero. In the interpolating regime common to modern neural networks, the individual gradient for each training point may even be exactly zero; depending on the loss function and how much you've trained / etc, it might instead be mean zero but approximately normal, etc.

  • At initialization, when we're very far from a solution, the gradient may be extremely similar for different datapoints (and very far from zero).

In some particular limits (network becoming infinitely wide, initialized near zero, trained via via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (ThisWhen using square loss, this means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (This means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

For a simple way to see this:

  • If we're at a (local) optimum, the full gradient (across the entire training dataset) will be zero. In the interpolating regime common to modern neural networks, the individual gradient for each training point may even be exactly zero; depending on the loss function and how much you've trained / etc, it might instead be mean zero but approximately normal, etc.

  • At initialization, when we're very far from a solution, the gradient may be extremely similar for different datapoints (and very far from zero).

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (When using square loss, this means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.

Source Link
Danica
  • 25.2k
  • 2
  • 76
  • 140

The exact answer is going to depend greatly on the type of network, the inputs, how it's trained....

In some particular limits (network becoming infinitely wide, initialized near zero, trained via SGD, no batch normalization, ...), the "neural tangent kernel" regime has an answer here: activations are distributed according to a particular Gaussian process, and their derivatives are too. (This means the final result corresponds to kernel ridge regression / Gaussian process regression with a particular kernel.) See e.g.