# Motivation:

• Most ML moderns become statistics once they are trained. Or one has to retrain the model to keep it adaptive to the new environment.

# Main idea:

• Instead of updating weights directly, update the rules (determined by the Hebbian coefficients) that update the weights

# Some details:

• Weights are updated by Hebbian ABCD model, $\Delta w_{i,j} = \eta_w (A_w o_i o_j + B_w o_i +C_w o_j + D_w)$
• Let $bf h$ be the vectorize Hebbian coefficients, ${\bf h}_{t+1} \leftarrow h_t + \frac{\alpha} {n \sigma} \sum_{i=1}^n F_i ({\bf h}_t + \Delta {\bf h}_i)$, where $\Delta {\bf h}_i \sim \mathcal{N} ({\bf 0}, \sigma{\bf I})$ and $F_i$ is a fitnness evalution of ${\bf h}_t + \Delta {\bf h}_i$ (This I am not completely certained how it is evaluated)

## Reformer

Problem statement:

• Transformer is computationally expensive when we have too many keys and queries (need to compute dot product between each pair). It can be memory intensive as well depending on implementations.

Proposed solution:

• Group keys and queries into “buckets” first and only compute dot product of keys and queries in the same bucket.
• Bucket can be implemented as a form of “locality sensitive hashing”. Seriously, I always forget what LSH means. Can they create an even more obscure jargon? (fairly speaking, I think they definitely could)
• I think a simple example to under LSH is the binary case. Assumes two binary vector is close according to the Hamming distance. Then, for a say length-100 binary vector, we can sort them into 4 bins just according to the value of the first two bits. Actually, it is just coset and syndrome coding but with the parity check matrix “degenerated” (care nothing but just for the first two bits). If we assume the bits are independent, this degenerated choice is okay. Otherwise, we can just use a random coset as in Slepian Wolf coding.
• For the reformer case, LSH is implemented as projections of the keys and queries onto random vectors. The signs of the resulting projections will determine the bucket. This is exactly the generalization Slepian Wolf coding to continuous case.
• Since they don’t want to store the bucket projection vectors, they make the step reversible instead. It is basically the same as the lifting scheme in wavelet construction.

## Numpy vs Matlab reshape

Being a Matlab long-term user, I have almost switched to Python completely. But I wasn’t paying attention of different reshape behavior of numpy vs Matlab. It wasted me a night to catch a nasty bug because of that. I always assumed that when I “vectorize” a matrix, it will expand along the row index first as in Matlab. It turns out that numpy’s default behavior is to expand the column index first. This is known as the ‘C’ order vs the Fortran order in Matlab. To override the default behavior, simply set order to ‘F’ in the argument. For example,

z = x.reshape(3,4,order='F')