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:

- It is useful to consider the
**"behavioral gradients"**for each input. - Let be the matrix of behavioral gradients. (The column of is ).
^{[1]}We can show that .^{[2]} - .
^{[3]}^{[4]} - Flat basin Low-rank Hessian Low-rank High manifold dimension
- High manifold dimension Low-rank Linear dependence of behavioral gradients
- A case study in a very small neural network shows that "information loss" is a good qualitative interpretation of this linear dependence.
- 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.

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

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

^{^}**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

- is the set of directions in which the output is
^{^}There is an alternate proof going through the result . (The constant 2 depends on MSE loss.)