Linear Layer Backward

Compute the backward pass of a dense layer $Y=XW+bY=XW+b$. Given the layer input X, the upstream gradient dY (same shape as $YY$) and the weights W, return the gradients with respect to the input, the weights and the bias.

Implement solve(X, dY, W).

Returns. Return a tuple (dX, dW, db) where dX = dY @ W.T, dW = X.T @ dY and db sums dY over the batch.

Examples

Example 1

Input:

solve([[1.0, 2.0]], [[1.0, -1.0]], [[1.0, 0.0], [0.0, 1.0]])

Output:

([[1.0, -1.0]], [[1.0, -1.0], [2.0, -2.0]], [1.0, -1.0])

Example 2

Input:

solve([[1.0], [2.0]], [[3.0], [4.0]], [[2.0]])

Output:

([[6.0], [8.0]], [[11.0]], [7.0])