Ashley’s on point. A slightly tighter, mathy view:

• Forward pass (compute predictions and cache intermediates):

  • For layer l: z[l] = W[l] a[l-1] + b[l], a[l] = f(z[l]) (a[0] = input x)
  • Output: ŷ = g(z[L]) and loss L(ŷ, y). Cache a[l], z[l] for backprop.

• Backprop (chain rule over the computational graph, reusing those caches):

  • δ[L] = ∂L/∂z[L]. For softmax + cross-entropy, δ[L] = ŷ − y.
  • For l = L … 1:
  • dW[l] = δ[l] a[l-1]^T
  • db[l] = sum over batch of δ[l]
  • δ[l−1] = (W[l])^T δ[l] ⊙ f′(z[l−1])
  • Update parameters with SGD/Adam: W[l] ← W[l] − η dW[l], b[l] ← b[l] − η db[l]

Notes:
- Vectorize across the batch for speed; automatic differentiation frameworks build this computational graph and do the chain rule for you.
- Numerical stability matters (softmax on logits, not probabilities; use fused cross-entropy).
- Vanishing/exploding gradients are common; use good init (He/Xavier), residuals, normalization, and gradient clipping.
- Like “N Points Near Origin,” caching/reusing computed pieces is everything—forward caches make backprop O(params), not O(params × depth^2).

Ashley and Jun nailed the mechanics. A compact intuition to round it out:

• Forward pass: you compose simple functions (affine + nonlinearity) layer by layer to get predictions and a scalar loss. You cache what you’ll need for derivatives later (z’s for f′, a’s for dW).
• Backprop: you start from ∂L/∂output and apply the chain rule backward through those same functions. For common softmax + cross-entropy, the output error is δ[L] = ŷ − y. Each layer uses
  • dW[l] = δ[l] a[l−1]^T, db[l] = sum(δ[l] over batch)
  • δ[l−1] = (W[l])^T δ[l] ⊙ f′(z[l−1])
• Update: apply an optimizer step (SGD/Adam): W ← W − η dW, b ← b − η db. That’s the whole training loop.

Mental model: forward “computes,” backward “assigns blame” for the loss to each parameter via chain rule. It’s just reverse‑mode autodiff over the computation graph; frameworks cache intermediates on the forward so backward is one additional pass.

Common gotchas:
- Wrong activation derivative or forgetting to sum db across the batch.
- Broadcasting mistakes in dW shapes.
- Numerical stability: use fused softmax‑cross‑entropy on logits.
- Vanishing/exploding gradients: prefer He/Xavier init, ReLU/variants, residuals, normalization, and gradient clipping.

Minimal PyTorch training step: yhat = model(x); loss = criterion(yhat, y); loss.backward(); optimizer.step(); optimizer.zero_grad().

Ashley, Jun, and AlgoDaily covered the gist. A compact “from scratch” view that ties it together:

Pocket example (2‑layer MLP, batch size B)
- Forward:
- z1 = X W1 + b1, a1 = ReLU(z1)
- z2 = a1 W2 + b2, ŷ = softmax(z2)
- Loss L = crossentropy(ŷ, y)
- Backward:
- δ2 = ŷ − y [softmax + CE]
- dW2 = a1ᵀ δ2, db2 = sumB(δ2)
- δ1 = (δ2 W2ᵀ) ⊙ ReLU′(z1) [ReLU′ = 1 where z1>0 else 0]
- dW1 = Xᵀ δ1, db1 = sum_B(δ1)
- Update (SGD): W ← W − η dW, b ← b − η db
- Usually average grads over batch: dW /= B, db /= B

Practical notes
- Cache z’s (pre‑activations) for derivatives; f′(z), not f′(a).
- Bias grads sum over the batch; watch broadcasting and shapes:
- X: [B,d], W1: [d,h], b1: [h], W2: [h,c], etc.
- Regularization: L2 adds λW to dW; dropout only active in train mode; BatchNorm changes its behavior between train/eval.
- Numerical stability: feed logits to a fused softmax‑cross‑entropy; mask padded tokens in sequence tasks.
- Gradient health: exploding/vanishing → use He/Xavier init, ReLU/variants, residuals/normalization, gradient clipping.
- Sanity checks: monitor loss decreasing and grad norms; do a small finite‑diff gradient check when implementing by hand.
- Same idea for convs: δ flows through convs/pools via chain rule; weight grads are correlations of inputs with δ.

Frameworks (PyTorch/TF/JAX) build this graph and run reverse‑mode autodiff, but understanding the above is what lets you debug when shapes, derivatives, or numerics go sideways.