From 689d80b524c65f0da7ed84a34091b3dfbdb507ae Mon Sep 17 00:00:00 2001 From: "Peter B. Johnson" Date: Wed, 4 Nov 2026 22:58:25 +0000 Subject: [PATCH 1/2] Add embedding-space experiments for Bengio, GPT-2 and SmolLM2 Nearest neighbours, analogies, learned directions and 2D plots, used to pick course examples showing that directions in embedding space carry meaning. report.md has the recommendations; results.md and plots/ are generated by embeddings.py. No production code changes. Co-Authored-By: Claude Opus 5.5 --- educational_material/embeddings/embeddings.py | 396 ++++++++++++++++++ .../embeddings/plots/direction_gender.png | Bin 0 -> 97501 bytes .../embeddings/plots/direction_plural.png | Bin 0 -> 88765 bytes .../plots/direction_view_country_capital.png | Bin 0 -> 93494 bytes .../plots/direction_view_gender.png | Bin 0 -> 92113 bytes .../plots/direction_view_plural.png | Bin 0 -> 92675 bytes .../embeddings/plots/pca_country_capital.png | Bin 0 -> 78823 bytes .../embeddings/plots/pca_gender.png | Bin 0 -> 74044 bytes .../embeddings/plots/pca_plural.png | Bin 0 -> 77018 bytes educational_material/embeddings/report.md | 66 +++ educational_material/embeddings/results.md | 127 ++++++ 11 files changed, 589 insertions(+) create mode 100644 educational_material/embeddings/embeddings.py create mode 100644 educational_material/embeddings/plots/direction_gender.png create mode 100644 educational_material/embeddings/plots/direction_plural.png create mode 100644 educational_material/embeddings/plots/direction_view_country_capital.png create mode 100644 educational_material/embeddings/plots/direction_view_gender.png create mode 100644 educational_material/embeddings/plots/direction_view_plural.png create mode 100644 educational_material/embeddings/plots/pca_country_capital.png create mode 100644 educational_material/embeddings/plots/pca_gender.png create mode 100644 educational_material/embeddings/plots/pca_plural.png create mode 100644 educational_material/embeddings/report.md create mode 100644 educational_material/embeddings/results.md diff --git a/educational_material/embeddings/embeddings.py b/educational_material/embeddings/embeddings.py new file mode 100644 index 0000000..29487e9 --- /dev/null +++ b/educational_material/embeddings/embeddings.py @@ -0,0 +1,396 @@ +"""Do directions in embedding space carry meaning? + +Experiments on the token embeddings of the course's three neural models: +Bengio (Q5), GPT-2 (Q6) and SmolLM2 (Q7). Writes every table to results.md and +the figures to plots/, next to this file. + + poetry run python -m educational_material.embeddings.embeddings + +GPT-2 and SmolLM2 download from Hugging Face on first run (into $HF_HOME, default /tmp/hf). +""" +import itertools +import os +from dataclasses import dataclass, field +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np + +HERE = Path(__file__).resolve().parent +STORAGE = HERE.parent.parent / "evaluation_function" / "models" / "storage" +PLOTS = HERE / "plots" + +FROM_COLOUR, TO_COLOUR, INK, MUTED, SURFACE = "#2a78d6", "#eb6834", "#0b0b0b", "#52514e", "#fcfcfb" + +NEIGHBOUR_WORDS = ["king", "woman", "dog", "house", "red", "happy", "walked", "three", "France", "Monday", "coffee"] + +# (x, y) pairs where y = x + relation. Analogies take every ordered pair of pairs: +# y2 ≈ x2 - x1 + y1, e.g. queen ≈ king - man + woman. +RELATIONS = { + "plural": [("cat", "cats"), ("dog", "dogs"), ("car", "cars"), ("house", "houses"), ("book", "books"), + ("tree", "trees"), ("bird", "birds"), ("hand", "hands"), ("girl", "girls"), ("door", "doors")], + "past tense": [("walk", "walked"), ("play", "played"), ("look", "looked"), ("open", "opened"), ("go", "went"), + ("see", "saw"), ("run", "ran"), ("eat", "ate"), ("take", "took"), ("come", "came")], + "comparative": [("big", "bigger"), ("small", "smaller"), ("fast", "faster"), ("old", "older"), + ("young", "younger"), ("tall", "taller"), ("good", "better"), ("bad", "worse")], + "gender": [("man", "woman"), ("king", "queen"), ("uncle", "aunt"), ("actor", "actress"), ("boy", "girl"), + ("brother", "sister"), ("father", "mother"), ("son", "daughter"), ("husband", "wife"), + ("he", "she"), ("prince", "princess"), ("Mr", "Mrs")], + "number → ordinal": [("one", "first"), ("two", "second"), ("three", "third"), ("four", "fourth"), + ("five", "fifth"), ("six", "sixth"), ("seven", "seventh"), ("ten", "tenth")], + "country → capital": [("France", "Paris"), ("Germany", "Berlin"), ("Italy", "Rome"), ("Japan", "Tokyo"), + ("Spain", "Madrid"), ("England", "London"), ("Russia", "Moscow"), ("China", "Beijing"), + ("Egypt", "Cairo"), ("Greece", "Athens")], + "country → nationality": [("France", "French"), ("Germany", "German"), ("Italy", "Italian"), + ("Japan", "Japanese"), ("Spain", "Spanish"), ("China", "Chinese"), + ("Russia", "Russian"), ("Greece", "Greek"), ("England", "English")], + "country → food/drink": [("Italy", "pizza"), ("Japan", "sushi"), ("Germany", "beer"), ("France", "wine"), + ("Mexico", "tacos"), ("India", "curry"), ("Russia", "vodka")], +} + +SHOWCASE = [("king", "man", "woman", "queen"), ("uncle", "man", "woman", "aunt"), + ("actor", "man", "woman", "actress"), ("cats", "cat", "dog", "dogs"), + ("walked", "walk", "play", "played"), ("bigger", "big", "small", "smaller"), + ("Paris", "France", "Germany", "Berlin"), ("French", "France", "Italy", "Italian"), + ("pizza", "Italy", "Japan", "sushi"), ("first", "one", "two", "second")] + +# Directions are averaged over the first pairs and tested on held-out ones. +DIRECTIONS = { + "plural": dict( + train=RELATIONS["plural"][:5], + test=RELATIONS["plural"][5:], + probes=["one", "single", "two", "three", "several", "many", "hundred"], + ), + "gender": dict( + train=[("man", "woman"), ("he", "she"), ("boy", "girl"), ("father", "mother"), ("son", "daughter")], + test=[("king", "queen"), ("uncle", "aunt"), ("actor", "actress"), ("brother", "sister"), + ("husband", "wife")], + probes=["nurse", "doctor", "teacher", "engineer", "secretary", "soldier", "dancer", "programmer"], + ), + "country → capital": dict( + train=RELATIONS["country → capital"][:5], + test=RELATIONS["country → capital"][5:], + probes=[], + ), +} + +PCA_SETS = { + "gender": [("man", "woman"), ("king", "queen"), ("uncle", "aunt"), ("boy", "girl"), ("brother", "sister")], + "plural": [("cat", "cats"), ("dog", "dogs"), ("car", "cars"), ("house", "houses"), ("book", "books")], + "country → capital": [("France", "Paris"), ("Germany", "Berlin"), ("Japan", "Tokyo"), ("Spain", "Madrid"), + ("Russia", "Moscow")], +} + + +def normalise(m): + return m / np.linalg.norm(m, axis=-1, keepdims=True).clip(1e-8) + + +@dataclass +class Space: + """A model's token embedding matrix, restricted to whole-word tokens for lookups.""" + + name: str + vectors: np.ndarray + pieces: list + prefix: str # marks a word-initial token: "▁" (SentencePiece) or "Ġ" (byte-level BPE) + lowercase: bool # Bengio's corpus was lowercased + unit: dict = field(init=False) + + def __post_init__(self): + self.ids = {p: i for i, p in enumerate(self.pieces)} + n = len(self.prefix) + self.word_ids = np.array([i for i, p in enumerate(self.pieces) + if p.startswith(self.prefix) and p[n:].isascii() and p[n:].isalpha()]) + self.unit = {False: normalise(self.vectors), True: normalise(self.vectors - self.vectors.mean(0))} + self.word_unit = {c: m[self.word_ids] for c, m in self.unit.items()} + + def id(self, word): + return self.ids.get(self.prefix + (word.lower() if self.lowercase else word)) + + def has(self, *words): + return all(self.id(w) is not None for w in words) + + def word(self, i): + return self.pieces[i][len(self.prefix):] + + def vec(self, word, centred): + return self.unit[centred][self.id(word)] + + def nearest(self, target, exclude, k, centred): + sims = self.word_unit[centred] @ normalise(target) + skip = {w.lower() for w in exclude} + top = np.argpartition(-sims, k + 30)[:k + 30] + out = [] + for j in top[np.argsort(-sims[top])]: + w = self.word(self.word_ids[j]) + if w.lower() not in skip: + out.append(w) + skip.add(w.lower()) # drop case variants: " King" after " king" + if len(out) == k: + break + return out + + def neighbours(self, word, k=8, centred=True): + return self.nearest(self.vec(word, centred), [word], k, centred) + + def analogy(self, a, b, c, k=5, centred=True): + """a - b + c, e.g. king - man + woman.""" + target = self.vec(a, centred) - self.vec(b, centred) + self.vec(c, centred) + return self.nearest(target, [a, b, c], k, centred) + + def direction(self, pairs, centred=True): + pairs = [p for p in pairs if self.has(*p)] + return normalise(np.mean([self.vec(y, centred) - self.vec(x, centred) for x, y in pairs], axis=0)) + + +def load_bengio(): + import sentencepiece as spm + import torch + + weights = torch.load(STORAGE / "bengio_model.pt", map_location="cpu")["emb.weight"].numpy() + sp = spm.SentencePieceProcessor(model_file=str(STORAGE / "bpe.model")) + pieces = [sp.id_to_piece(i) for i in range(sp.get_piece_size())] + return Space("Bengio", weights, pieces, prefix="▁", lowercase=True) + + +def load_hf(name, model_id): + from transformers import AutoModelForCausalLM, AutoTokenizer + + cache = os.environ.get("HF_HOME", "/tmp/hf") + model = AutoModelForCausalLM.from_pretrained(model_id, cache_dir=cache) + tokenizer = AutoTokenizer.from_pretrained(model_id, cache_dir=cache) + weights = model.get_input_embeddings().weight.detach().float().numpy() + pieces = tokenizer.convert_ids_to_tokens(list(range(weights.shape[0]))) + return Space(name, weights, pieces, prefix="Ġ", lowercase=False) + + +# ---------------------------------------------------------------- experiments + +def vocab_table(spaces): + words = sorted({w for pairs in RELATIONS.values() for p in pairs for w in p} | set(NEIGHBOUR_WORDS), + key=str.lower) + lines = ["| model | vocab | whole-word tokens | test words found | missing |", "|---|---|---|---|---|"] + for s in spaces: + missing = [w for w in words if not s.has(w)] + lines.append(f"| {s.name} | {len(s.pieces)} × {s.vectors.shape[1]} | {len(s.word_ids)} | " + f"{len(words) - len(missing)}/{len(words)} | {', '.join(missing) or '-'} |") + return lines + + +def neighbour_tables(spaces): + lines = [] + for centred in (True, False): + lines += [f"\n### Nearest neighbours ({'centred' if centred else 'not centred'})\n", + "| word | " + " | ".join(s.name for s in spaces) + " |", "|---" * (len(spaces) + 1) + "|"] + for w in NEIGHBOUR_WORDS: + cells = [", ".join(s.neighbours(w, centred=centred)) if s.has(w) else "-" for s in spaces] + lines.append(f"| {w} | " + " | ".join(cells) + " |") + return lines + + +def analogy_scores(space, pairs, centred): + """Top-1 and top-5 hits over all ordered pairs of pairs, plus a baseline that ignores the offset.""" + pairs = [p for p in pairs if space.has(*p)] + top1 = top5 = base5 = n = 0 + for (x1, y1), (x2, y2) in itertools.permutations(pairs, 2): + if len({x1, y1, x2, y2}) < 4: + continue + guesses = space.analogy(x2, x1, y1, centred=centred) + top1 += guesses[0] == y2 + top5 += y2 in guesses + base5 += y2 in space.neighbours(x2, k=5, centred=centred) + n += 1 + return top1, top5, base5, n + + +def analogy_tables(spaces): + lines = [] + for centred in (True, False): + lines += [f"\n### Analogy accuracy ({'centred' if centred else 'not centred'})\n", + "Each cell: top-1 % / top-5 % (baseline top-5 %: nearest neighbours of x2 alone, no offset)" + " [number of analogies].\n", + "| relation | " + " | ".join(s.name for s in spaces) + " |", "|---" * (len(spaces) + 1) + "|"] + for rel, pairs in RELATIONS.items(): + cells = [] + for s in spaces: + t1, t5, b5, n = analogy_scores(s, pairs, centred) + cells.append(f"{100 * t1 / n:.0f} / {100 * t5 / n:.0f} ({100 * b5 / n:.0f}) [{n}]" if n else "-") + lines.append(f"| {rel} | " + " | ".join(cells) + " |") + lines += ["\n### Showcase analogies (centred, top 5; ✓ = expected answer ranked first)\n", + "| analogy | expected | " + " | ".join(s.name for s in spaces) + " |", + "|---" * (len(spaces) + 2) + "|"] + for a, b, c, want in SHOWCASE: + cells = [] + for s in spaces: + if not s.has(a, b, c, want): + cells.append("-") + continue + got = s.analogy(a, b, c) + cells.append(("✓ " if got[0].lower() == want.lower() else "") + ", ".join(got)) + lines.append(f"| {a} - {b} + {c} | {want} | " + " | ".join(cells) + " |") + return lines + + +def direction_tables(spaces): + lines = [] + for name, d in DIRECTIONS.items(): + lines += [f"\n### Direction: {name}\n", + f"Built from {', '.join(f'{y} - {x}' for x, y in d['train'])}. " + "Held-out: share of pairs where the second word projects further along the direction.\n", + "| model | centring | held-out pairs ordered | probe projections (high → low) |", "|---|---|---|---|"] + for s in spaces: + for centred in (True, False): + u = s.direction(d["train"], centred) + test = [p for p in d["test"] if s.has(*p)] + ok = sum(s.vec(y, centred) @ u > s.vec(x, centred) @ u for x, y in test) + probes = sorted(((s.vec(w, centred) @ u, w) for w in d["probes"] if s.has(w)), reverse=True) + lines.append(f"| {s.name} | {'centred' if centred else 'raw'} | {ok}/{len(test)} | " + + (", ".join(f"{w} {v:+.2f}" for v, w in probes) or "-") + " |") + return lines + + +# ---------------------------------------------------------------------- plots + +def style(ax): + ax.set_facecolor(SURFACE) + for side in ("top", "right"): + ax.spines[side].set_visible(False) + for side in ("left", "bottom"): + ax.spines[side].set_color(MUTED) + ax.tick_params(colors=MUTED, labelsize=8) + + +def draw_pairs(ax, space, pairs, P): + for i, (x0, y0) in enumerate(P[0::2]): + x1, y1 = P[2 * i + 1] + ax.annotate("", (x1, y1), (x0, y0), arrowprops=dict(arrowstyle="->", color=MUTED, lw=1.2)) + colours = [FROM_COLOUR, TO_COLOUR] * len(pairs) + for (px, py), word, colour, dy in zip(P, [w for p in pairs for w in p], colours, label_offsets(P)): + ax.plot(px, py, "o", ms=8, color=colour, mec=SURFACE, mew=2) + ax.annotate(space.word(space.id(word)), (px, py), (0, dy), textcoords="offset points", ha="center", + va="center", fontsize=9, color=INK) + + +def label_offsets(P): + """Put a label above its point, or below if that would collide with a label already placed.""" + span = np.ptp(P, axis=0).clip(1e-9) + placed, offsets = [], [] + for point in P / span: + for dy in (10, -11, 21, -22): + spot = point + [0, dy / 300] + if all(abs(spot[0] - q[0]) > 0.12 or abs(spot[1] - q[1]) > 0.05 for q in placed): + break + placed.append(spot) + offsets.append(dy) + return offsets + + +def pair_panels(spaces, title, filename, panel): + """One panel per model, side by side; `panel` returns the 2D points for a model's pairs, or None.""" + fig, axes = plt.subplots(1, len(spaces), figsize=(4.2 * len(spaces), 4), facecolor=SURFACE) + for ax, s in zip(axes, spaces): + style(ax) + ax.set_title(s.name, color=INK, fontsize=11) + ax.set_yticks([]) + drawn = panel(ax, s) + if drawn is None: + ax.set_xticks([]) + ax.text(0.5, 0.5, "too few of these words\nin the vocabulary", transform=ax.transAxes, ha="center", + va="center", color=MUTED, fontsize=9) + else: + draw_pairs(ax, s, *drawn) + ax.margins(0.18) + fig.suptitle(title, color=INK, fontsize=12) + fig.tight_layout() + fig.savefig(PLOTS / filename, dpi=150, facecolor=SURFACE) + plt.close(fig) + + +def plot_pca(spaces, name, pairs): + """Plain 2D PCA of the pairs' embeddings, oriented so the mean arrow points right.""" + def panel(ax, s): + found = [p for p in pairs if s.has(*p)] + if len(found) < 3: + return None + X = np.array([s.vec(w, True) for p in found for w in p]) + X -= X.mean(0) + P = X @ np.linalg.svd(X, full_matrices=False)[2][:2].T + if (P[1::2] - P[0::2]).mean(0)[0] < 0: + P[:, 0] *= -1 + ax.set_xticks([]) + return found, P + + pair_panels(spaces, f"{name}: 2D PCA of the word embeddings", f"pca_{slug(name)}.png", panel) + + +def plot_direction_view(spaces, name, d): + """Held-out pairs placed along a direction learned from other pairs (x) against the main remaining + variation (y, PCA). Arrows pointing right mean the direction generalises to new words.""" + def panel(ax, s): + test = [p for p in d["test"] if s.has(*p)] + if len(test) < 3 or not any(s.has(*p) for p in d["train"]): + return None + u = s.direction(d["train"]) + X = np.array([s.vec(w, True) for p in test for w in p]) + x = X @ u + rest = X - np.outer(x, u) + rest -= rest.mean(0) + y = rest @ np.linalg.svd(rest, full_matrices=False)[2][0] + ax.axvline(0, color=MUTED, lw=0.6, ls=":") + ax.set_xlabel(f"along the {name} direction", color=MUTED, fontsize=9) + return test, np.column_stack([x, y]) + + pair_panels(spaces, f"{name}: held-out pairs along the direction learned from other pairs", + f"direction_view_{slug(name)}.png", panel) + + +def plot_direction(spaces, name, d, labels): + """Each word's position along a direction: held-out pairs on two rows, probe words on a third.""" + fig, axes = plt.subplots(len(spaces), 1, figsize=(9, 2.6 * len(spaces)), facecolor=SURFACE) + rows = [labels[0], labels[1], "probe words"] + for ax, s in zip(axes, spaces): + u = s.direction(d["train"]) + test = [p for p in d["test"] if s.has(*p)] + groups = [[x for x, _ in test], [y for _, y in test], [w for w in d["probes"] if s.has(w)]] + style(ax) + for row, (words, colour) in enumerate(zip(groups, (FROM_COLOUR, TO_COLOUR, MUTED))): + proj = [s.vec(w, True) @ u for w in words] + ax.plot(proj, [row] * len(proj), "o", ms=8, color=colour, mec=SURFACE, mew=2) + for i, (p, w) in enumerate(sorted(zip(proj, words))): + ax.annotate(s.word(s.id(w)), (p, row), (0, (8, -14, 18)[i % 3]), textcoords="offset points", + ha="center", fontsize=8, color=INK) + ax.set_yticks(range(3), rows, fontsize=9, color=INK) + ax.set_ylim(-0.9, 3.0) + ax.set_title(s.name, loc="left", color=INK, fontsize=10) + axes[-1].set_xlabel(f"projection onto the {name} direction (cosine)", color=MUTED, fontsize=9) + fig.tight_layout() + fig.savefig(PLOTS / f"direction_{slug(name)}.png", dpi=150, facecolor=SURFACE) + plt.close(fig) + + +def slug(text): + return "".join(c if c.isalnum() else "_" for c in text.lower()).strip("_").replace("___", "_") + + +def main(): + spaces = [load_bengio(), load_hf("GPT-2", "gpt2"), load_hf("SmolLM2", "HuggingFaceTB/SmolLM2-135M-Instruct")] + + lines = ["# Embedding experiments: raw results", "", + "Generated by `embeddings.py`; see `report.md` for the interpretation. Vectors are unit-normalised; " + "'centred' means the mean embedding was subtracted first. Neighbour searches only consider whole-word " + "tokens and exclude the input words.", "", "### Vocabulary coverage", ""] + lines += vocab_table(spaces) + neighbour_tables(spaces) + analogy_tables(spaces) + direction_tables(spaces) + (HERE / "results.md").write_text("\n".join(lines) + "\n") + + PLOTS.mkdir(exist_ok=True) + for name, pairs in PCA_SETS.items(): + plot_pca(spaces, name, pairs) + plot_direction_view(spaces, name, DIRECTIONS[name]) + plot_direction(spaces, "plural", DIRECTIONS["plural"], ("singular", "plural")) + plot_direction(spaces, "gender", DIRECTIONS["gender"], ("male", "female")) + print(f"Wrote {HERE / 'results.md'} and {PLOTS}/") + + +if __name__ == "__main__": + main() diff --git a/educational_material/embeddings/plots/direction_gender.png b/educational_material/embeddings/plots/direction_gender.png new file mode 100644 index 0000000000000000000000000000000000000000..8998d81ba025ac579daec658664461a1ea260044 GIT binary patch literal 97501 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zMPW7lD*n)hhQ`J^hq7_2vGMqMJ)7-!sXs7HwlgE$e=-Iv|^y%sIEhZ$A`}>+U&iUk0i3t<1VzNuU>((oU4B9BYUqYg>0 zb3%x_&{niBdN*Y3q9RT@mQp@9H`TEB`scT2E1uc++m@7R_t;lBmMD=(Y$h|moremm z{QvNDns^J^|NdHJw*eP}Yxb8fv=3$C_Yk>uZM%1$yjt}V(uuXayaE3HiM35lKebA~ z{(el_XXfs{8~GrG_7)nNnzg5to|PyxGLrVK_R|xrIdRfcIBcE1xQn0hV-&Zi7www3 z8A)O%Rspw$q6uam3dwd>dS{4OW?gdPoGc-{29 zXZ`3X>)CM6+jOLI4dkPOP2~M4lEweryAFZrC8V+Jy_YAmKP28Q5e3=cP3t5*$I^Sts29zGl@=-AV;DF^vJlV$}jfj=Ey%SiVOG>Id|Wy>kR|0DUI}`)6G11N?lKh z%h`d;;TqrDn|{%afws7JrIl$zA+`)ZnML z_><4+{o;#1?UUWWz4(*hZ8p}$pKfRbFf9IbXfxiRg+KrQCRvx4e0CWgzn69_5QZI!h2>i>5c8}cO=p?4|~I>zj|PQ0rIZKgKQqs zz0RX2q)S#azV382tD%q1T0` Date: Thu, 5 Nov 2026 08:05:17 +0000 Subject: [PATCH 2/2] Show that embedding directions are spread across many coordinates Co-Authored-By: Claude Opus 5.5 --- educational_material/embeddings/embeddings.py | 19 ++++++++++++++++++- educational_material/embeddings/report.md | 16 ++++++++++++++++ educational_material/embeddings/results.md | 16 ++++++++++++++++ 3 files changed, 50 insertions(+), 1 deletion(-) diff --git a/educational_material/embeddings/embeddings.py b/educational_material/embeddings/embeddings.py index 29487e9..fa5ce59 100644 --- a/educational_material/embeddings/embeddings.py +++ b/educational_material/embeddings/embeddings.py @@ -250,6 +250,22 @@ def direction_tables(spaces): return lines +def spread_table(spaces): + """How many embedding coordinates each learned direction uses (it is not one axis).""" + lines = ["\n### How spread out the directions are\n", + "Each unit-length direction (centred) over the model's coordinates. If one coordinate carried the concept, " + "it would be near 1 and the rest near 0.\n", + "| direction | model | dims | coords near zero (abs < 0.01) | coords for 90% of length | largest coord " + "| even spread would be |", "|---|---|---|---|---|---|---|"] + for name, d in DIRECTIONS.items(): + for s in spaces: + a = np.sort(np.abs(s.direction(d["train"])))[::-1] + k90 = int(np.searchsorted(np.cumsum(a ** 2), 0.9)) + 1 + lines.append(f"| {name} | {s.name} | {len(a)} | {(a < 0.01).mean():.0%} | {k90} ({k90 / len(a):.0%}) " + f"| {a[0]:.2f} | {1 / np.sqrt(len(a)):.3f} |") + return lines + + # ---------------------------------------------------------------------- plots def style(ax): @@ -380,7 +396,8 @@ def main(): "Generated by `embeddings.py`; see `report.md` for the interpretation. Vectors are unit-normalised; " "'centred' means the mean embedding was subtracted first. Neighbour searches only consider whole-word " "tokens and exclude the input words.", "", "### Vocabulary coverage", ""] - lines += vocab_table(spaces) + neighbour_tables(spaces) + analogy_tables(spaces) + direction_tables(spaces) + for table in (vocab_table, neighbour_tables, analogy_tables, direction_tables, spread_table): + lines += table(spaces) (HERE / "results.md").write_text("\n".join(lines) + "\n") PLOTS.mkdir(exist_ok=True) diff --git a/educational_material/embeddings/report.md b/educational_material/embeddings/report.md index 242e1f5..c4b7fc8 100644 --- a/educational_material/embeddings/report.md +++ b/educational_material/embeddings/report.md @@ -40,6 +40,22 @@ The baseline column is worth knowing about when choosing examples. For plurals, **Directions** (built from five pairs, tested on held-out pairs): GPT-2 and SmolLM2 order every held-out pair correctly for plural, gender and country → capital. Bengio scores 2/5 for plural and 1/4 for gender, which is chance. +## A concept is a direction, not an axis + +Each word is a point in a space with 64 dimensions (Bengio), 768 (GPT-2) or 576 (SmolLM2). The gender direction above is not one of those dimensions. It is the average of woman − man, she − he, girl − boy and so on, scaled to length 1, so it points diagonally through the space and uses nearly every coordinate a little: + +| gender direction | Bengio | GPT-2 | SmolLM2 | +|---|---|---|---| +| dimensions | 64 | 768 | 576 | +| coordinates near zero (abs < 0.01) | 11% | 27% | 17% | +| coordinates needed for 90% of its length | 31 (48%) | 305 (40%) | 259 (45%) | +| largest single coordinate | 0.33 | 0.13 | 0.14 | +| each coordinate if spread perfectly evenly | 0.125 | 0.036 | 0.042 | + +The plural and capital directions look the same (see `results.md`). Training gives no reason for the axes to line up with concepts, so there is no "gender coordinate". Meaning lives in combinations of many coordinates. Measuring a word against a concept is just a dot product: the word's unit vector · the direction gives one number, how far along that concept the word sits. That number is the x-axis in the direction-view plots. A 768-dimensional space has room for very many such directions at once, each nearly at right angles to the others. + +For students, one sentence covers it: a word is a point in 768-D space, and a concept is a direction through that space, not one of its axes. + ## Method notes - **Whole-word tokens only.** Each word is looked up as a single token: "Ġking" (a leading space) in GPT-2 and SmolLM2, which share this convention, and "▁king" in Bengio's SentencePiece vocabulary. Bengio's text was lowercased, so it uses "paris" rather than "Paris". Neighbour searches consider whole-word tokens only, and exclude the input words and their case variants (" King" after " king"). diff --git a/educational_material/embeddings/results.md b/educational_material/embeddings/results.md index effb6c3..23b5845 100644 --- a/educational_material/embeddings/results.md +++ b/educational_material/embeddings/results.md @@ -125,3 +125,19 @@ Built from Paris - France, Berlin - Germany, Rome - Italy, Tokyo - Japan, Madrid | GPT-2 | raw | 5/5 | - | | SmolLM2 | centred | 5/5 | - | | SmolLM2 | raw | 5/5 | - | + +### How spread out the directions are + +Each unit-length direction (centred) over the model's coordinates. If one coordinate carried the concept, it would be near 1 and the rest near 0. + +| direction | model | dims | coords near zero (abs < 0.01) | coords for 90% of length | largest coord | even spread would be | +|---|---|---|---|---|---|---| +| plural | Bengio | 64 | 8% | 29 (45%) | 0.28 | 0.125 | +| plural | GPT-2 | 768 | 24% | 300 (39%) | 0.28 | 0.036 | +| plural | SmolLM2 | 576 | 19% | 251 (44%) | 0.15 | 0.042 | +| gender | Bengio | 64 | 11% | 31 (48%) | 0.33 | 0.125 | +| gender | GPT-2 | 768 | 27% | 305 (40%) | 0.13 | 0.036 | +| gender | SmolLM2 | 576 | 17% | 259 (45%) | 0.14 | 0.042 | +| country → capital | Bengio | 64 | 2% | 29 (45%) | 0.37 | 0.125 | +| country → capital | GPT-2 | 768 | 26% | 321 (42%) | 0.13 | 0.036 | +| country → capital | SmolLM2 | 576 | 20% | 246 (43%) | 0.12 | 0.042 |