[Short version] Information Loss --> Basin flatness

post by Vivek Hebbar (Vivek) · 2022-05-21T12:59:10.932Z · LW · GW · 0 comments

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This is an overview for advanced readers.  Main post: Information Loss --> Basin flatness [LW · GW]

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" ${\nabla }_{\theta }f(\theta ,x_i)$ for each input. 
2. Let $G$ be the matrix of behavioral gradients.  (The $i^{th}$ column of $G$ is $g_i={\nabla }_{\theta }f(\theta ,x_i)$).^[1]  We can show that $dim\left(manifold\right)\le N-Rank\left(G\right)$.^[2]
3. $Rank\left(Hessian\right)=Rank\left(G\right)$.^[3][4]
4. Flat basin  $\approx$  Low-rank Hessian  $=$  Low-rank $G$  $\approx$  High manifold dimension
5. High manifold dimension  $\approx$  Low-rank $G$  $=$  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 [LW · GW] for details.

1. ^{^}

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

2. ^{^}

   $N$ is the number of parameters in the model.  See claims 1 and 2 here [LW · GW] for a proof sketch.

3. ^{^}

   Proof sketch for $Rank\left(Hessian\right)=Rank\left(G\right)$:

   •  $span(g_1,..,g_k{)}^{\perp }$ is the set of directions in which the output is not first-order sensitive to parameter change.  Its dimensionality is $N-rank\left(G\right)$.
   • At a local minimum, first-order sensitivity of behavior translates to second-order sensitivity of loss.
   • So $span(g_1,..,g_k{)}^{\perp }$ is the null space of the Hessian.
   • So $rank\left(Hessian\right)=N-(N-rank(G\left)\right)=rank\left(G\right)$
4. ^{^}

   There is an alternate proof [LW(p) · GW(p)] going through the result $Hessian=2G{G}^T$.  (The constant 2 depends on MSE loss.)

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