排序與 BPR：點擊資料怎麼變成推薦目標？

P_u = 使用者 u 的訓練正例集合
I   = 固定項目目錄
D   = {(u, i, j) : i 屬於 P_u，j 屬於 I 但不屬於 P_u}

每個 triple 希望：s(u, i) > s(u, j)

s(u, i) = p_u · q_i + b_i
差值 d  = s(u, i) − s(u, j)

單一比較：ell(d) = −log sigmoid(d) = log(1 + exp(−d))

本例完整目標：
L = 32 個比較的 mean ell(d)
    + 0.02 × 全部 26 個參數的平方和 / 26

d ell / d d = −1 / (1 + exp(d)) = −sigmoid(−d)

d = 0.2：
loss = 0.5981388694
導數 = −0.4501660027

(全局偏置 + 使用者偏置 + p_u·q_i + b_i)
− (全局偏置 + 使用者偏置 + p_u·q_j + b_j)
= p_u·(q_i−q_j) + b_i−b_j

Precision@2 = 前兩名中的相關項目數 / 2
Recall@2    = 前兩名中的相關項目數 / 保留相關項目數

DCG@2  = rel_1 / log2(2) + rel_2 / log2(3)
NDCG@2 = DCG@2 / 最佳可能 DCG@2

RR  = 1 / 第一個相關項目的名次
MRR = 四位使用者 RR 的平均

numpy==1.26.3

python3 -m venv .venv
source .venv/bin/activate
python -m pip install -r requirements.txt
python bpr_and_topk.py > stdout.json

"""Original finite-catalogue BPR-style ranking exercise.

Full-batch training over all train-positive/unobserved pairs, not LearnBPR's
bootstrap SGD. Held-out labels never decide training pairs or hyperparameters.
"""
import json
import math
import platform

import numpy as np


TRAIN_POSITIVES = ({0, 1}, {0, 2}, {3, 4}, {3, 5})
HELD_OUT = ({2}, {1}, {5}, {4})
N_USERS, N_ITEMS, DIMENSION, TOP_K = 4, 6, 2, 2
SEED, UPDATES, LEARNING_RATE, REGULARIZATION = 7, 2000, 0.08, 0.02


def training_triples():
    return np.array([(u, i, j)
        for u, positives in enumerate(TRAIN_POSITIVES)
        for i in sorted(positives)
        for j in range(N_ITEMS) if j not in positives], dtype=int)


def initialize():
    rng = np.random.default_rng(SEED)
    return {"users": rng.normal(0, .1, (N_USERS, DIMENSION)),
            "items": rng.normal(0, .1, (N_ITEMS, DIMENSION)),
            "item_bias": np.zeros(N_ITEMS)}


def all_scores(parameters):
    return parameters["users"] @ parameters["items"].T + parameters["item_bias"]


def loss_and_gradient(parameters, triples):
    users, positive, unobserved = triples.T
    delta_items = parameters["items"][positive] - parameters["items"][unobserved]
    difference = (np.sum(parameters["users"][users] * delta_items, axis=1)
                  + parameters["item_bias"][positive] - parameters["item_bias"][unobserved])
    pair_loss = float(np.logaddexp(0, -difference).mean())
    count = sum(value.size for value in parameters.values())
    squared_norm = sum(float(np.sum(value**2)) for value in parameters.values())
    gradient = {name: 2 * REGULARIZATION * value / count
                for name, value in parameters.items()}
    # d[-log sigmoid(difference)]/d difference = -sigmoid(-difference).
    coefficient = -np.exp(-np.logaddexp(0, difference)) / len(triples)
    np.add.at(gradient["users"], users, coefficient[:, None] * delta_items)
    np.add.at(gradient["items"], positive,
              coefficient[:, None] * parameters["users"][users])
    np.add.at(gradient["items"], unobserved,
              -coefficient[:, None] * parameters["users"][users])
    np.add.at(gradient["item_bias"], positive, coefficient)
    np.add.at(gradient["item_bias"], unobserved, -coefficient)
    return pair_loss + REGULARIZATION * squared_norm / count, pair_loss, difference, gradient


def evaluate(score_matrix):
    rows = []
    for user in range(N_USERS):
        candidates = [item for item in range(N_ITEMS) if item not in TRAIN_POSITIVES[user]]
        ranked = sorted(candidates, key=lambda item: (-score_matrix[user, item], item))
        top = ranked[:TOP_K]
        relevant = HELD_OUT[user]
        relevance = [int(item in relevant) for item in ranked]
        hits = sum(relevance[:TOP_K])
        dcg = sum(rel / math.log2(rank + 2) for rank, rel in enumerate(relevance[:TOP_K]))
        ideal = sum(1 / math.log2(rank + 2) for rank in range(min(TOP_K, len(relevant))))
        first_hit = next((rank + 1 for rank, rel in enumerate(relevance) if rel), None)
        rows.append({"user": user, "candidates": candidates, "ranking": ranked,
            "scores_in_ranking_order": [float(score_matrix[user, item]) for item in ranked],
            "held_out_relevant": sorted(relevant), "top_k": top, "hit_count": hits,
            "Precision_at_2": hits / TOP_K, "Recall_at_2": hits / len(relevant),
            "NDCG_at_2": dcg / ideal, "reciprocal_rank_full_candidate_set": 1 / first_hit if first_hit else 0})
    keys = ("Precision_at_2", "Recall_at_2", "NDCG_at_2", "reciprocal_rank_full_candidate_set")
    return {"users": rows, "macro_mean": {key: sum(row[key] for row in rows) / N_USERS for key in keys}}


def main():
    triples = training_triples()
    parameters = initialize()
    initial_loss, initial_pair_loss, initial_differences, _ = loss_and_gradient(parameters, triples)
    initial_scores = all_scores(parameters)
    for _ in range(UPDATES):
        _, _, _, gradient = loss_and_gradient(parameters, triples)
        parameters = {name: value - LEARNING_RATE * gradient[name]
                      for name, value in parameters.items()}
    final_loss, final_pair_loss, final_differences, _ = loss_and_gradient(parameters, triples)
    popularity = np.array([sum(item in positives for positives in TRAIN_POSITIVES)
                           for item in range(N_ITEMS)], dtype=float)
    popularity_scores = np.tile(popularity, (N_USERS, 1))
    scores = all_scores(parameters)
    assert len(triples) == 32 and np.isfinite(scores).all()
    print(json.dumps({"scope": "Hand-set implicit train interactions and held-out relevance; "
        "full finite candidate ranking, no sampled evaluation negatives or online benefit claim.",
        "versions": {"python": platform.python_version(), "numpy": np.__version__},
        "settings": {"users": N_USERS, "items": N_ITEMS, "dimension": DIMENSION, "seed": SEED,
            "updates": UPDATES, "learning_rate": LEARNING_RATE, "regularization": REGULARIZATION,
            "top_k": TOP_K, "parameter_count": sum(value.size for value in parameters.values()),
            "objective": "mean(logaddexp(0,-score_difference)) + .02*sum(all_parameter_squares)/26",
            "optimizer": "simultaneous full-batch gradient descent over all32trainingtriples"},
        "train_positives": [sorted(p) for p in TRAIN_POSITIVES],
        "held_out_relevance": [sorted(p) for p in HELD_OUT], "training_triples": triples.tolist(),
        "training_record": {"initial_objective": initial_loss, "final_objective": final_loss,
            "initial_mean_pair_loss": initial_pair_loss, "final_mean_pair_loss": final_pair_loss,
            "initial_positive_margin_fraction": float(np.mean(initial_differences > 0)),
            "final_positive_margin_fraction": float(np.mean(final_differences > 0))},
        "final_parameters": {key: value.tolist() for key, value in parameters.items()},
        "training_popularity_counts": popularity.tolist(),
        "initial_ranking_evaluation": evaluate(initial_scores),
        "popularity_ranking_evaluation": evaluate(popularity_scores),
        "BPR_style_ranking_evaluation": evaluate(scores),
        "heldout_pairs_also_unobserved_during_training": [
            {"user": u, "item": i, "number_of_training_triples_where_j": int(np.sum((triples[:, 0] == u) & (triples[:, 2] == i)))}
            for u, held in enumerate(HELD_OUT) for i in sorted(held)],
        "scalar_pair_loss_example": {"score_difference": .2, "negative_log_sigmoid": float(np.logaddexp(0, -.2)),
            "derivative_wrt_difference": -1 / (1 + math.exp(.2))},
        "limits": ["Unknown training items are assumed lower-preference comparisons, not proven dislikes.",
            "Triple construction uses TRAIN_POSITIVES and fixed catalogue only, never HELD_OUT.",
            "Held-out relevance can therefore occur as unobserved j in training; not secretly excluded.",
            "All six catalogue items are assumed available; no timestamp/exposure modelling.",
            "No user or item is cold-start; four users and one held-out positive each are a small toy.",
            "Full candidate evaluation filters only training positives; ties use ascending item ID.",
            "Fixed seed/budget/settings were not selected by inspecting the held-out metrics.",
            "Full-batch over32pairs differs from paper LearnBPR bootstrap SGD and its normalization.",
            "Pairwise sigmoid does not establish calibrated click probability or unbiased preference.",
            "One relevance item makes Recall_at_2 a hit indicator; precision maximum here is1/2.",
            "No claim of production convergence, statistically significant lift, or exposure debiasing."]},
        ensure_ascii=False, indent=2))


if __name__ == "__main__":
    main()

Image source and reuse notice — DS65 selected illustrations

These files are selected teaching illustrations. They are not claims that the referenced source authors ran this article's prototype. Each image is separately attributed below.

1. fig-01-recommendation-pipeline.png and mobile-fig-01-recommendation-pipeline.png
Source: Google Developers, “Recommendation systems overview,” https://developers.google.com/machine-learning/recommendation/overview/types . The source page states that its content is available under Creative Commons Attribution 4.0 International (CC BY 4.0), https://creativecommons.org/licenses/by/4.0/ . The page's code samples are separately Apache 2.0; this image is sourced from the page's own diagram, not its code samples. The SVG's simple database-stack icons were retained. Change: the source SVG was rendered on white; mobile includes the complete figure and enlarged source-region crops. No new labels or data were added.
Portions of this page are modifications based on work created and shared by Google and used according to terms described in the Creative Commons 4.0 Attribution License.

2. fig-02-observed-to-pairwise.png, mobile-fig-02-observed-to-pairwise.png, and cover.png
Source: Aston Zhang, Zachary C. Lipton, Mu Li, and Alexander J. Smola, Dive into Deep Learning, “Personalized Ranking,” https://d2l.ai/chapter_recommender-systems/ranking.html . The book is licensed under Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0), https://creativecommons.org/licenses/by-sa/4.0/ . Changes: mobile repeats the whole diagram and crops its original left/right panels for reading; the cover fits the complete diagram on a white canvas. No cells, labels, arrows, or values were added. These adapted assets are provided under CC BY-SA 4.0.

3. fig-03-bpr-objective.png and mobile-fig-03-bpr-objective.png
Source and license: the same D2L chapter and four authors listed above, CC BY-SA 4.0. The image is a browser capture of the chapter's final displayed BPR objective row (Eq. 21.5.2), not the original Rendle paper figure. Change: source whitespace was cropped and the original equation row was proportionally scaled; the preceding derivation is omitted. Mobile repeats and enlarges the same row. No equation glyphs, symbols, or values were added or changed. This adapted asset is provided under CC BY-SA 4.0.

4. fig-04-pairwise-hinge.png and mobile-fig-04-pairwise-hinge.png
Source and license: the same D2L chapter and four authors listed above, CC BY-SA 4.0. The image is a browser capture of the chapter's pairwise hinge objective, Eq. 21.5.3. Change: mobile repeats the complete formula and equation number, then enlarges the original formula body; the equation number remains in the whole image. No equation glyphs, symbols, or values were added or changed. This adapted asset is provided under CC BY-SA 4.0.

5. fig-05-evaluation-selection-flow.png and mobile-fig-05-evaluation-selection-flow.png
Source: Diego Carraro and Derek Bridge, “A sampling approach to Debiasing the offline evaluation of recommender systems,” Journal of Intelligent Information Systems 58, 311–336 (2022), https://doi.org/10.1007/s10844-021-00651-y . Fig. 2 is covered by the article-specific Creative Commons Attribution 4.0 International (CC BY 4.0) statement, https://creativecommons.org/licenses/by/4.0/ ; the figure has no separate third-party exclusion credit. Changes: the complete publisher figure was proportionally enlarged to 1600 px wide. Mobile keeps the integrated original figure whole, near its source width. The flow is the paper's explicit-rating evaluation setup; it is not a BPR or click-log experiment.

License scope is limited to the named source content and does not extend to unrelated content on those sites or to original-paper figures not listed here.