First, clarify the problem: the weight matrix at each layer is the product of all preceding weight matrices, so the network is a deep linear model with tied weights. Then, derive the gradient using the chain rule and matrix calculus, and design a backward pass that caches intermediate products to avoid redundant multiplications.
Pro tip: Emphasize that the backward pass can be computed in O(L) matrix multiplications by reusing cached prefix and suffix products, and mention that this is analogous to efficient backpropagation through time in RNNs.
Restate the setup: for layer l, W_l = W_1 * W_2 * ... * W_{l-1} (or similar), and the loss L is a scalar. Define the input x, the forward pass, and the parameters to differentiate.
Express the output as a product of matrices applied to x, then use the chain rule to write dL/dW_i as a sum of terms involving the gradient of L with respect to intermediate activations and the products of other weight matrices.
Precompute prefix products (from W_1 to W_{i-1}) and suffix products (from W_{i+1} to W_L) to avoid recomputing the full product for each gradient. Use dynamic programming to compute all gradients in O(L) matrix multiplications.
Write pseudocode or actual code for the backward pass, then test with a small network (e.g., 2-3 layers) to ensure gradients match numerical gradients.
Mention memory vs. computation trade-offs (caching vs. recomputation), and note that this approach generalizes to other tied-weight architectures.
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