Ω 9

Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

This is an overview for advanced readers.  Main post: Information Loss --> Basin flatness

Summary:

Inductive bias is related to, among other things:

• Basin flatness
• Which solution manifolds (manifolds of zero loss) are higher dimensional than others.  This is closely related to "basin flatness", since each dimension of the manifold is a direction of zero curvature.

In relation to basin flatness and manifold dimension:

1. It is useful to consider the "behavioral gradients"  for each input.
2. Let  be the matrix of behavioral gradients.  (The  column of  is ).[1]  We can show that .[2]
3. .[3][4]
4. Flat basin    Low-rank Hessian    Low-rank     High manifold dimension
5. High manifold dimension    Low-rank     Linear dependence of behavioral gradients
6. A case study in a very small neural network shows that "information loss" is a good qualitative interpretation of this linear dependence.
7. Models that throw away enough information about the input in early layers are guaranteed to live on particularly high-dimensional manifolds.  Precise bounds seem easily derivable and might be given in a future post.

See the main post for details.

1. ^

In standard terminology,  is the Jacobian of the concatenation of all outputs, w.r.t. the parameters.

2. ^

is the number of parameters in the model.  See claims 1 and 2 here for a proof sketch.

3. ^

Proof sketch for :

•   is the set of directions in which the output is not first-order sensitive to parameter change.  Its dimensionality is .
• At a local minimum, first-order sensitivity of behavior translates to second-order sensitivity of loss.
• So  is the null space of the Hessian.
• So
4. ^

There is an alternate proof going through the result .  (The constant 2 depends on MSE loss.)

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