Non-Personalized Recommendation · Recommender Systems

01 · Motivation

Why bother with non-personalized?

Because personalization needs history, and sometimes there is none.

Personalized algorithms read a user’s row of the URM to tailor recommendations. But that row is often empty: a brand-new user, a logged-out visitor, a freshly added item nobody has touched. This is the cold start problem, and a non-personalized list is the standard fallback. Three reasons these algorithms never go away:

Cold start — with no history, the crowd’s preference is the best available prior.
A strong baseline — “just recommend the popular items” is shockingly hard to beat. Any personalized model that loses to it is broken.
A building block — the bias terms we derive at the end ( $\mu, b_i, b_u$ ) reappear inside matrix factorization (Ch. 8) as the model’s backbone.

why

Mental model

Personalization re-orders items per user. A non-personalized algorithm produces one ordering and hands the same list to everybody. Everything in this chapter is a different way to compute that single ordering of the columns of the URM.

02 · Count, not quality

Top-Popular

Rank items by how many people interacted — the rating value is ignored.

Top-Popular is the simplest recommender that exists: count the non-zero entries in each column of the URM and rank items by that count. The crucial detail — the rating itself is not considered. A column of four 1-star ratings and a column of four 5-star ratings are equally popular.

!

Popular ≠ good

Because only the count matters, Top-Popular may recommend items with lots of negative ratings — it cannot tell a great film from a mediocre one rated equally often. In our toy catalog, Interstellar is tied for the most-rated item yet has nearly the lowest average; Top-Popular treats it as a top pick anyway.

Hands-on

Top-Popular vs Best-Rated

Both columns rank the same 8 movies. Left ranks by how many ratings; right by the average rating. The left bars are nearly equal — counting barely separates them. Click a movie to trace it across both lists.

▼ Top-Popular (count)

▼ Best-Rated (average)

Click a movie to compare its two ranks.

TakeawayCounting interactions and judging quality are different goals. The Martian has the fewest ratings (last by popularity) yet is the 4th best-liked film — the two lists disagree by design.

Formal Definition 2.1 — popularity

Popularity of item $i$ is the number of users who interacted with it — the count of non-zero entries in its URM column:

Top-Popular

\operatorname{pop}(i) = \bigl\lvert \{\, u \in \mathcal{U} : r_{ui} \neq 0 \,\} \bigr\rvert = n_i.

The recommendation is $\arg\max_i \operatorname{pop}(i)$ over unseen items. $r_{ui}$ enters only through $r_{ui} \neq 0$ — its magnitude is discarded.

03 · Quality, with a catch

Best-Rated & the small-sample trap

Averages reward quality — but a single rating can fake it.

Best-Rated fixes Top-Popular’s blind spot by ranking on the average rating instead of the count: for item $i$ with $n_i$ ratings, $\text{avg}_i = \frac{1}{n_i}\sum_u r_{ui}$ . Now quality matters — but a new problem appears.

!

One rating is not evidence

An item rated $\{5,4,3\}$ averages $4.0$ , while an item with a single $5$ averages $5.0$ . Best-Rated ranks the one-rating item first — even though we have almost no evidence about it.

The fix is shrinkage: add a constant $C$ to the denominator. Items with few ratings get pulled toward zero; items with many ratings are barely affected because $C$ is small next to $n_i$ . With $C=1$ , the example becomes $12/4 = 3.0$ vs $5/2 = 2.5$ — the ordering flips back to the well-supported item.

Hands-on

Shrinkage explorer · sum / (n + C)

Best-Rated ranks by average — but a single 5★ shouldn't outrank a long track record. Adding C to the denominator pulls low-support items down. Inject a hyped cold item and drag C.

Shrink term C1.0

1LaLan=43.20

2TopGunn=43.00

3Nottingn=43.00

4MIn=42.80

5LoveActn=42.80

6Avengn=42.80

7Martiann=32.75

8Intern=42.40

TakeawayThe shrink term lives in the denominator — score = sum / (n + C). Items with many ratings (n ≫ C) barely move; one-rating items get discounted hard.

Formal Definition 2.2 — average & shrinkage

Let $n_i$ be the number of users who rated item $i$ . The plain and shrunk averages are

Best-Rated

\text{avg}_i = \frac{\sum_u r_{ui}}{n_i}, \qquad\qquad \widehat{\text{avg}}_i = \frac{\sum_u r_{ui}}{n_i + C}.

Here $r_{ui}$ ranges over non-zero ratings only, and $C \ge 0$ is the shrink term. As $C \to 0$ we recover the plain average; as $C$ grows, low-support items are penalized more heavily.

04 · The bias baseline

Global Effects

Peel off systematic biases — and discover the ranking was never personal.

Some users rate everything high; some items get rated high by everyone. Global Effects models these systematic offsets explicitly, in six steps — start from the global mean, estimate a shrunk item bias and a shrunk user bias, and add them back up:

Global mean $\mu$ = average of all non-zero ratings.
Centre each rating: $\tilde{r}_{ui} = r_{ui} - \mu$ .
Item bias $b_i = \frac{\sum_u \tilde{r}_{ui}}{n_i + C}$ (how much item $i$ beats the mean).
Centre again: $\tilde{\tilde{r}}_{ui} = \tilde{r}_{ui} - b_i$ .
User bias $b_u = \frac{\sum_i \tilde{\tilde{r}}_{ui}}{n_u + C}$ (how generous user $u$ is).
Predict: $\hat{r}_{ui} = \mu + b_i + b_u$ .

!

Personalized rating, non-personalized ranking

The prediction $\hat{r}_{ui}$ contains a user term $b_u$ , so it looks personalized. But for a fixed user, $\mu$ and $b_u$ are the same for every item — they shift all of that user’s scores by a constant and never change their order. The ranking is driven by $b_i$ alone, so every user gets the same list. It is the exam’s favourite trap.

Hands-on

Global-Effects stepper · μ + b_i + b_u

Add the terms one at a time and watch the order. The user term b_u shifts every score by a constant, so it can never reorder the items — every user gets the same list.

μ = 3.58b_u = +0.43

1TopGun3.58

2MI3.58

3Inter3.58

4Martian3.58

5Notting3.58

6LoveAct3.58

7Aveng3.58

8LaLa3.58

Every item scores μ — a flat tie, no ranking yet.

Takeawayr̂(u,i) = μ + b_i + b_u looks personalized, but μ + b_u is constant across items for a fixed user — the ranking depends on b_i alone. The exam's favourite trap.

Formal Definition 2.3 — Global Effects

With global mean $\mu$ , item counts $n_i$ , user counts $n_u$ , and shrink term $C$ :

Global Effects

\mu = \frac{1}{\lvert\Omega\rvert}\sum_{(u,i)\in\Omega} r_{ui}, \qquad b_i = \frac{\sum_u (r_{ui}-\mu)}{n_i + C}, \qquad b_u = \frac{\sum_i (r_{ui}-\mu-b_i)}{n_u + C}.

Since $\mu + b_u$ is constant across items for a fixed user, $\arg\!\operatorname{sort}_i \hat{r}_{ui} = \arg\!\operatorname{sort}_i b_i$ — independent of $u$ . That is the punchline.

05 · Exam intel

What the exam tests

Define each of the three algorithms; compute an average and a shrunk average by hand; and explain why Global Effects, despite the $b_u$ term, produces a non-personalized ranking. Graders want: Top-Popular ranks by count (the value is ignored); the shrink term in the denominator, never the numerator; and the ranking argument that $\mu + b_u$ shifts scores by a constant.

Q

Worked question — rank by Best-Rated, then by shrunk average (C = 1)

Item A has ratings $\{5, 4, 3\}$ ; item B has a single rating $\{5\}$ .

Plain. $\text{avg}_A = 12/3 = 4.0$ , $\text{avg}_B = 5/1 = 5.0$ → B ranks first (one rating wins).
Shrunk. $\widehat{\text{avg}}_A = 12/(3{+}1) = 3.0$ , $\widehat{\text{avg}}_B = 5/(1{+}1) = 2.5$ → A ranks first. Shrinkage discounts B’s thin evidence and restores the sensible order.

Traps: calling Top-Popular an average (it is a count); putting $C$ in the numerator; calling Global Effects “personalized” because of $b_u$ (the ranking is not).

06 · Self-check

Three questions before you move on

Top-Popular ranks items by:

Why do we shrink the average with a term C in the denominator?

Global Effects predicts r̂(u,i) = μ + b_i + b_u. Why does every user get the same item ranking?

07 · Recap

One-screen summary

Chapter 02 — load-bearing ideas

Top-Popular = count, Best-Rated = average. Popular and good are different goals, and the two rankings disagree.
Shrinkage ( $\div (n_i + C)$ ) protects the average from low-support items — a single rating is not evidence. The $C$ goes in the denominator.
Global Effects gives $\hat{r}_{ui} = \mu + b_i + b_u$ , but the ranking is non-personalized: it depends on $b_i$ alone.
These baselines are what CF (Ch. 5) must beat, and $\mu, b_i, b_u$ become the bias terms of MF (Ch. 8).