LinearBackward

Run the forward pass and, when grad mode is on, register this as the grad_fn of the returned output.

The forward flattens all leading dimensions of x into an M-row batch ($M = \prod_{i<\text{ndim}-1} x.\text{shape}[i]$), dispatches the GEMM through backend::Dispatcher, and reports 2 * M * N * K FLOPs to the profiler scope. Under an active AutocastGuard, the three operands are cast to the effective dtype using astype_op (which preserves autograd) before the kernel call so the matmul takes the native F16 fast path on Metal.

Math

$\mathbf{y} = \mathbf{x}\mathbf{W}^\top + \mathbf{b}$

Shape

• Input x : $(\ast, K)$
• Weight W : $(N, K)$
• Bias b : $(N,)$
• Output : $(\ast, N)$

Compute the three gradients from the incoming grad_out.

Delegates to IBackend::linear_backward which produces $dx = \nabla_{\!x}\ell$, $dW = \nabla_{\!W}\ell$, and $db = \nabla_{\!b}\ell$ in a single backend call using the shapes recorded in input_shapes_. The closed-form is