Medium

Linear Layer Weight Gradient

Medium

~15 min

code completion

Weight Gradient of a Linear Layer

A linear layer computes:

Y = X W

where $X$ has shape $(m, n)$ , $W$ has shape $(n, k)$ , and $Y$ has shape $(m, k)$ .

Given the upstream gradient $\frac{\partial L}{\partial Y}$ (d_out, shape $(m, k)$ ), the gradient with respect to $W$ is:

\frac{\partial L}{\partial W} = X^{⊤} d_o u t

Why? Each weight $W_{ij}$ affects output column $j$ through every input row. The matrix product $X^{⊤} d_o u t$ sums those contributions over the whole batch.

Check the shapes: $(n \times m) \cdot (m \times k) = (n \times k)$ — this matches $W$ 's shape, which is a good sanity check.

Your task:

Implement linear_weight_grad(X, d_out) that returns $X^{⊤} @ d_o u t$ .

Example Tests

2x2 input, 2x1 upstream gradient

Input: {"X":[[1,2],[3,4]],"d_out":[[1],[1]]}

Expected: [[4],[6]]

Identity input: gradient equals d_out

Input: {"X":[[1,0],[0,1]],"d_out":[[2],[3]]}

Expected: [[2],[3]]

Single sample, 3 features, 1 output

Input: {"X":[[1,2,3]],"d_out":[[1]]}

Expected: [[1],[2],[3]]

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