back_prop#

Series: lecture-python.myst
File: lectures/back_prop.md
Audit date: 2026-08-26
Corpus snapshot: e25fdf2345
Categories audited: writing, math, code, figures, links, admonitions (JAX out of scope)
Overall score: 7.8 / 10
Priority: HIGH

Score breakdown#

Category	Score	One-line note
Writing	3/10	`qe-writing-006` ×5; `qe-writing-003` ×2; `qe-writing-002` ×3, +4 more.
Math	7.5/10	`qe-math-003` ×6.
Code	7.5/10	`qe-code-001` ×6; `qe-code-004` ×1.
JAX	out of scope	JAX rules target `lecture-jax`.
Figures	9/10	`qe-fig-010` ×1.
References	N/A	no citations in this lecture.
Links	10/10	no mechanical violations detected.
Admonitions	10/10	no mechanical violations detected.

Issues#

Critical#

None found.

High severity#

[qe-code-001] (reviewer) — Follow PEP8 unless closer to mathematical notation. Count: 6. Lines: 49, 342, 418, 430, 442, 453. Example: fifteen code lines exceed 79 characters, several badly: 98 at line 49, 100 at 418, 104 at 452, 108 at 430 (jnp.block([dx.reshape((-1, 1)) for dx_tuple in grad(loss)(params, x, y) for dx in dx_tuple ]), which also has a space before the closing bracket, as does 407); the continuation line at 443 is indented to column 10 under a bracket opened at column 11; fifteen lines carry trailing whitespace (357, 359, 364, 373, 379, 381, 386, 453, 455, 457, 458, 460, 507, 508, 519); and the two top-level defs at 340 and 345 are separated by one blank line where PEP8 asks for two.
[qe-math-003] — Use square brackets for matrix notation. Count: 6. Lines: 258, 262, 266, 272, 277, 299. Example: array used as matrix.
[qe-writing-006] — Capitalize lecture titles properly. Count: 5. Lines: 72, 153, 211, 312, 538. Example: H2 Title Case: ‘A Deep (but not Wide) Artificial Neural Network’ (Deep, Wide, Artificial, Neural).
[qe-writing-008] — Remove excessive whitespace between words. Count: 42. Lines: 59, 60, 69, 74, 78, 80, 84, 90, 121, 123, …. Example: 2 spaces.

Medium severity#

[qe-fig-010] — Plotly figures require latex directive. Count: 1. Lines: 1. Example: plotly used with no {only} latex directive.
[qe-writing-001] — Use one sentence per paragraph. Count: 1. Lines: 33. Example: 3 sentences in one paragraph.
[qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 3. Lines: 78, 156, 231. Example: line 78 defines width as “the number of right hand side variables on the right hand side of the function being approximated” - the same phrase twice in one clause; lines 156-162 restate the model that 92-151 has just built (“As mentioned above, for a given input \(x_1\), our approximating function \(\hat f\) evaluated at \(x_1\) equals the ‘output’ \(x_{N+1}\) … computed by iterating on \(x_{i+1}=h_i(w_i x_i + b_i)\)” is {eq}`eq:recursion` again in words); and line 231 is a one-word paragraph (“Here goes.”) followed by five blank lines.
[qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 328, 538. Example: lines 328-470 are fourteen consecutive code cells with no prose between them at all - the reader goes from “The update rule {eq}`eq:sgd` … amounts to a stochastic gradient descent algorithm” (326) straight to jnp.block([[δ * xs[:-1]], [δ]]) (411), L = jnp.diag(δ * ws, k=-1) (415) and D = jax.scipy.linalg.block_diag(...) (418) with nothing saying that these are the \(D\) and \(L\) of the display at 262-282, and the two verification cells at 423-435 have only a comment to explain what is being checked against what. Separately, “How Deep?” (538-545) is placed between the two examples but its content belongs to Example 2, which is the one that actually compares one-, two- and three-layer networks (566-610) - and its third bullet asserts a one-layer identity network “would probably work best” for Example 1 without ever running it.
[qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 3. Lines: 92, 101, 262. Example: the lecture promises to describe “a neuron”, “a network of neurons” and “a neural network as a composition of functions” (65-68) and then does all of it in formulas: the layer definition at 92-98, the composition at 134 and the recursion at 145 have no node-and-arrow diagram anywhere in the file, in a lecture whose subject is a graph; the three activation functions at 101-119 (sigmoid, ReLU, identity) are given as three formulas and never plotted, though three curves on one axes is a two-line cell; and the lower-triangular structure that the whole back-propagation argument turns on (262-282, \(D\) and \(L\)) is shown only as a 25-line array display where a sparsity sketch would carry it. The two figures the lecture does have (528, 606) are approximation plots.

Low severity#

[qe-code-004] — Use quantecon Timer context manager. Count: 1. Lines: 519. Example: %%time.
[qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 1. Lines: 54. Example: line 54 bolds machine learning and artificial intelligence, neither of which the lecture defines - they are field names being emphasised, and the rule reserves bold for definitions; the lecture uses no italic anywhere, so emphasis has nowhere else to go.

Strengths#

The linear-algebra route to back-propagation is verified against automatic differentiation rather than asserted: jnp.max(jnp.abs(dxs_ad_mat - dxs_la)) at 426 checks the derivative matrices agree and 435 checks the loss gradients agree, so the \((I-L)^{-1}D\) identity at 293 is demonstrated numerically.
The {youtube} directive at 227 embeds the source of the triangular-matrix idea directly in the page and attributes it in the preceding line, rather than leaving a bare URL.
The whole derivation is set up so the reader can see why lower-triangularity matters: 213-223 states the two ingredients (chain and product rules, triangular matrices) and the two operations they buy (one triangular inversion plus one multiplication) before any algebra appears.
Example 2 compares one-, two- and three-layer networks on the same axes with distinct trace names (600-610), which is the right way to show the depth question the lecture raises.

Recommended actions#

Convert the six \begin{array} matrix displays at 258, 262, 266, 272, 277 and 299 to bmatrix (qe-math-003, 6 occurrences) and replace the h' transpose-style prime at 246 with explicit derivative notation (qe-math-002).
Fix the four mathematical errors in the derivation, which currently make it unfollowable: w_i x_i + bI at line 96 should be w_i x_i + b_i; the composition at 134 repeats l_1 where l_{N-1} belongs; the differential at 252 has + b_i where it needs + db_i; and e_N at 305 is used with no definition.
Write prose into the fourteen bare code cells at 328-470 tying L (415), D (418) and dxs_la (420) back to the matrices named at 262-282, and say what each of the two verification cells (423, 433) is checking.
Add a diagram of the width-one depth-\(N\) network, and plot the three activation functions at 101-119 - a lecture that introduces neural networks currently shows the reader no network.
Sentence-case the five Title Case H2s at 72, 153, 211, 312 and 538, and collapse the 42 double spaces (qe-writing-008, 42 occurrences, the largest mechanical item here).
Fix the training loop at 501-508: key, _ = random.split(key) discards the new key and line 504 permutes with the fixed random.key(1), so all 300 (or 500) epochs see the identical permutation and the epoch variable is unused.
Wrap the fifteen over-length code lines, strip the fifteen trailing-whitespace lines, replace %%time at 519 with the quantecon Timer context manager (qe-code-004), split the three-sentence paragraph at 33, and unify \mathcal{L} (167, 199) with {\mathcal L} (176, 202).