Version: 1.0.2

# Smart Adaptive Recommendations (SAR) Algorithm

The following document is a subset of the implemented logic. The original can be found here

SAR is a fast scalable adaptive algorithm for personalized recommendations based on user transactions history and items description. It produces easily explainable / interpretable recommendations.

The overall architecture of SAR is shown in the following diagram:

## Input​

The input to SAR consists of:

• transaction (usage) data
• catalog data

Transaction data, also called usage data, contains information on interactions between users and items and has the following schema:

<User Id>,<Item Id>,<Time>

Each row represents a single interaction between a user and an item, which we call a transaction. Here's an example of usage data (displayed as a table for readability, but note that it must be provided in comma-separated format):

User IDItem IDTime
User 1Item 12015/06/20T10:00:00
User 1Item 12015/06/28T11:00:00
User 1Item 22015/08/28T11:01:00
User 1Item 22015/08/28T12:00:01

Note that SAR doesn't require explicit user ratings, which are often noisy and unreliable. It also focuses on implicit events (transactions), which are more telling of user intent.

## Collaborative Filtering​

SAR is based on a simple yet powerful collaborative filtering approach. In the heart of the algorithm is computation of two matrices:

• Item-to-Item similarity matrix
• User-to-Item affinity matrix

Item-to-Item similarity matrix contains for each pair of items a numerical value of similarity between these two items. A simple measure of item similarity is co-occurrence, which is the number of times two items appeared in a same transaction. Let's look at the following example:

Item 1Item 2Item 3Item 4Item 5
Item 153432
Item 234321
Item 343431
Item 432342
Item 521123

Here, for example, cooccur(Item 1, Item 2) = 3, meaning that items 1 and 2 were together in a transaction three times. The Item-to-Item similarity matrix is symmetric. Diagonal elements, occ(Item i), represent the number of occurrences of each item. The advantage of co-occurrence is that it's easy to update. However, it favors predictability, and the most popular items will be recommended most of the time. To alleviate that, two more similarity measures are used: lift and Jaccard. They can be thought of as normalized co-occurrences.

Lift measures how much the co-occurrence of two items is higher than it would be by chance, that is, what is the contribution of interaction of the two items. It is obtained as

lift(Item i, Item j) = cooccur(Item i, Item j) / (occ(Item i) * occ(Item j)) .

Lift favors serendipity / discoverability. For example, items 2 and 5 have the same co-occurrence with item 4, but item 5 in general occurs less frequently than item 2 and will be favored by lift.

Jaccard measure is defined as the number of transactions in which two items appear together divided by the number of transactions in which either of them appears:

Jaccard(Item 1, Item 2) = cooccur(Item1, Item 2) / (occ(Item 1) + occ(Item 2) - cooccur(Item 1, Item 2)) .

Jaccard measure is a tradeoff between co-occurrence and lift and is the default in SAR.

An item is cold if it has no transactions yet or the number of transactions is low, that is, below the SupportThreshold, which is configurable. If one or both items is cold, their item-to-item similarity cannot be estimated from the transactions data and item features must be used. A linear learner is trained using warm items, where the features of the model are (partial) matches on corresponding features of a pair of items. The target is the computed similarity, based on normalized co-occurrences of those items. The model is then used to predict similarities between cold and cold/warm items.

User-to-Item affinity matrix contains for each user-item pair an affinity score of the user towards the item. Affinity score is computed as a weighted number of transactions in which the user and the item appear together, where newer transactions are weighted more than the older transactions. Also, weights are adjusted for the event type. For example, "Purchase" event may be valued four times more than the "Click" event. Therefore, affinity score takes into account frequency, recency and the event type of transactions in which the user appears with the item.

An example User-to-Item affinity matrix is shown in the following table:

Item 1Item 2Item 3Item 4Item 5
User 15.003.002.50
User 22.002.505.002.00
User 32.504.004.50
User 45.003.004.50
User 54.003.002.004.003.50
User 62.00
User 71.00

Here, we can see that User 1 has the highest affinity to Item 1, followed by items 2 and 3. This user didn't have a transaction that included items 4 and 5. On the other hand, User 2 has the highest affinity to Item 3, followed by Item 2 and then items 1 and 4.

## Making Recommendations​

SAR can produce two types of recommendations:

• User recommendations, which recommend items to individual users based on their transaction history, and

### User Recommendations​

Personalized recommendations for a single user are obtained by multiplying the Item-to-Item similarity matrix with a user affinity vector. The user affinity vector is simply a transposed row of the affinity matrix corresponding to that user. For example, User 1 affinity vector is

User 1 aff
Item 15.00
Item 23.00
Item 32.50
Item 4
Item 5

By pre-multiplying this vector with the Item-to-Item similarity matrix, User 1 recommendation scores are obtained:

User 1 rec
Item 144
Item 234.5
Item 339
Item 428.5
Item 515.5

In this case, the recommendation score of an item is purely based on its similarity to Item 5. Assuming that a same item isn't recommended again, items 1 and 4 have the highest score and would be recommended before items 2 and 3.

Now, if this user adds Item 2 to the shopping cart, affinity vector (assuming weight 2 for this transaction) will be

New User aff
Item 10
Item 20
Item 30
Item 40
Item 51

resulting in recommendation scores:

New User rec
Item 12
Item 21
Item 31
Item 42
Item 53

The recommendation score of an item is purely based on its similarity to Item 5 in this case. Assuming that a same item isn't recommended again, items 1 and 4 have the highest score and would be recommended before items 2 and 3. Now, if this user adds Item 2 to the shopping cart, affinity vector (assuming weight 2 for this transaction) will be

New User aff
Item 10
Item 22
Item 30
Item 40
Item 51

resulting in recommendation scores:

New User rec
Item 18
Item 29
Item 37
Item 46
Item 55

Now, Item 1 has the highest score, followed by Item 3 and then Item 4. Note that the advantage of SAR is that the updated scores are obtained without any retraining, that is, without having to recompute Item-to-Item similarities.

#### Interpretability​

SAR provides explainable recommendations. For example, let's look at the reasoning behind the recommendation score for User 1 and Item 4:

rec(User 1, Item 4) *= sim(Item 4, Item 1) * aff(User 1, Item 1)* + sim(Item 4, Item 2) * aff(User 1, Item 2) + sim(Item 4, Item 3) * aff(User 1, Item 3) + sim(Item 4, Item 4) * aff(User 1, Item 4) + sim(Item 4, Item 5) * aff(User 1, Item 5) = 3 * 5 + 2 * 3 + 3 * 2.5 + 4 * 0 + 2 * 0 *= 15 + 6 + 7.5 + 0 + 0 = 28.5*

Clearly, the first term (highlighted) has the highest contribution to the score. We can say that "The algorithm recommends Item 4 to User 1 because it's similar to Item 1, to which User 1 has high affinity". A message like this can be displayed automatically for each recommendation.