Left branches getting ignored in PyTorch implementation

[Pi]

Bojan and I dug through this work (convo in Yannic's Discord -> DailyPapers channel).

UltraFastBERT/benchmark_pytorch/fff/fff_bmm.py:

    # y = torch.einsum('b i j , b i -> b j', selected_w2s, F.gelu(all_logits))
    y = torch.einsum('b i j , b i -> b j', selected_w2s, all_scores)
    return y

I've removed the .gelu from this line. (Bojan switched to einsum also to improve clarity, but that's not relevant here).

If you're using .gelu then you're discarding information from all negative-scoring nodes, as you're multiplying their y-vector contribution by 0.

Here's a gist with an MNIST example: https://gist.github.com/p-i-/784ea313d21856c286b823f27bf79d90

If you put the .gelu back the accuracy will deteriorate.

Notice:

class Net(nn.Module):
    def __init__(self):
        super(Net, self).__init__()
        self.fc1 = FFF(nIn=28*28, nOut=500)
        self.fc2 = FFF(nIn=500, nOut=10)
        # self.fc1 = FFF(nIn=28*28, nOut=10)
    def forward(self, x):
        x = x.view(-1, 28*28)
        y_hat = self.fc2(torch.relu(self.fc1(x)))
        # y_hat = self.fc1(x)
        return y_hat

So I'm introducing a nonlinearity in between two FFF layers.

I think there's maybe a cleaner way to re-conceptualize the result at which you have arrived.

Each node has a node.x, which points in some direction in INPUT space and a node.y which points in some direction in OUTPUT space.

The node[][].x each represents the normal vector to a region-splitting hyperplane. And for input x, node[p][q].score = dot(x, node[p][q].x) projects the input onto this normal-vector.

If it's positive, it's "sunny-side" of the hyper-plane and we branch up and right, else it's "darkside" and we branch up and left.

Either way we'll reach a new node with a fresh region-splitting hyperplane.

And so, once we've traversed the (depth D) tree (and I'm going to follow the authors in considering a solitary root-node as a tree of depth-0 (D=0)) we have split the input space into 2**D regions, of which we are inside one.

And this to me is the beautiful part.

If we consider the "winning" node sequence node_{1..D}, then node_k.x form a basis for a D-dimensional subspace within our INPUT space. e_1 ... e_D.

And node_k.y form a basis for a D-dimensional subspace within our OUTPUT space. f_1 ... f_D.

And our input x can be written as lambda_1 e_1 + ... + lambda_D e_D + remainderTerm, where lambda_i is just node_i.score

And we're projecting this to lambda_1 f_1 + ... + lambda_D f_D

So the FFF layer is figuring out a "most-useful--D-dimensional-transform" and applying it. It's lerping from a basis over INPUT space to a basis over OUTPUT space.

And this basis-pair depends on where our input x is located in INPUT space. There's 2**D possible basis-pairs.

And the backprop will move the bases around to optimize performance. So it will learn a "most-useful" mapping. It reminds me of LoRA in this way.

There's room for exploring quite a few ideas from here. I've emailed one of the authors.

[pbelcak]

Hi,

Thanks for getting in touch and for the summary. Do you have any action items to suggest with regards to the above?

[Pi]

Bojan and I are actively exploring this area at https://github.com/sap-ient-ai/FFF

There's a Discord linked in the README.md where a group of us are hanging out and trying to get our heads around this rich new conceptual space that your work has broken open.

Fundamental seems the idea that a FF layer is expensive: It is an nIn x nOut matrix, and that compute-power can surely be better put to use.

Fundamental seems the idea of branching conditionally on the input.

Bojan knocked out an experiment yesterday, where we pre-calculate the lambdas (each node we traverse thru we pick up a lambda, so how about just pass the input thru a nIn x depth matrix to pre-gen the lambdas, and then use them to navigate the tree, thus applying them to a chosen set of y-vectors.

Amazingly this benchmarks about the same as a FF layer on CIFAR10.

But there's the challenge of the information-content of the input. Just because a novel FFF-variant performs well on MNIST/CIFAR doesn't give any guarantee it will handle a dataset with higher intrinsic dimensionality (or to put it another way, a more interesting heatmap over the input distribution).

I feel we're bumping into many of the problems that you yourselves encountered in your research.