Re-ranking Based Diversification: A Unifying View
Shameem A Puthiya Parambath


Secure Client and Server Geolocation Over the Internet
    AbdelRahman Abdou
Carleton University, Ottawa, Canada

ETH Zürich, Switzerland

abdoua@inf.ethz.ch
Paul C. van Oorschot
School of Computer Science

Carleton University, Ottawa, Canada

paulv@scs.carleton.ca

    December 30, 2023
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We analyze different re-ranking algorithms for diversification and show that majority of them are based on maximizing submodular/modular functions from the class of parameterized concave/linear over modular functions.
We study the optimality of such algorithms in terms of the `total curvature'.
We also show that by adjusting the hyperparameter of the concave/linear composition to trade-off relevance and diversity, if any, one is in fact tuning the `total curvature' of the function for relevance-diversity trade-off.




§ INTRODUCTION

Standard personalized recommendation algorithms consider only relevance of the items to a user when making predictions.
This strategy implicitly assumes that relevance of the items are independent of each other.
In practice this draws in similar and redundant items to be recommended.
Moreover, popularity bias in recommender systems makes the situation worse by excluding unpopular but relevant items from recommendations.
In personalized recommendation, diversification is a means to recommend novel, serendipitous items that result in higher user satisfaction <cit.> whereas in group recommendation, diversification is a way to generate consensus recommendations by finding items relevant to a group of diverse users <cit.>.

A common strategy for diversification is re-ranking: given a list of recommendations, re-ranking algorithms reorder the list to account for diversity <cit.>.
Recently many re-ranking based diversification algorithms have been proposed which exploit different aspects of the recommendations like personal popularity tendency <cit.>, genre coverage <cit.>, interest coverage <cit.> etc.
It is already shown that the `diminishing return' property of submodular functions enable the system to trade-off between relevance and diversity <cit.>. 
A function F defined on the subsets of a ground set  is called submodular if for all subsets ,⊆, 

    F() + F() ≥ F(∪) + F(∩)

F is modular if strict equality holds in (<ref>).
Though not very apparent, careful analysis reveals that objective functions corresponding to these algorithms are not only submodular/modular but take a simple form: the composition of a concave/linear function with a modular function.

The purpose of this paper is not to propose a new diversification algorithm; we aim to uncover the exact functional forms of submodular or modular functions used in re-ranking based diversification algorithms and see how this affects the relevance-diversity trade off.
We analyze objective functions for many re-ranking algorithms and show that they belong to the the class of concave/linear over modular functions.
We also study the relevance-diversity trade-off of re-ranking algorithms and establish that relevance-diversity trade-off of these algorithms is tied to the `total curvature' of the objective function.
By changing the hyperparameters associated with the objective function, one is effectively tuning the `total curvature' of the objective.

The remainder of this paper is structured as follows.
We give a brief overview of diversification algorithms in Section <ref>.
In Section <ref>, we show that popular diversification algorithms are based on the class of parameterized concave/linear composition of modular functions.
We also discuss the `total curvature' and the worst case lower bound for submodular maximization in terms of the `total curvature'.
We discuss our experimental results in Section <ref> before concluding the paper in Section <ref>.





§ RELATED WORK


Recommendation diversification problem has been studied extensively in the past <cit.>.
Broadly speaking, diversification algorithms come under two schemes: (i) re-ranking based algorithms (ii) multi-objective optimization based algorithms.
Re-ranking based diversification algorithms or simply re-ranking algorithms are the major point of discussion here.
Given a recommendation list produced by any personalized recommendation algorithms, re-ranking algorithms re-order the items such that relevant and diverse items appear in the top- rankings.
Re-ranking algorithms are easy to implement and popular because they can be readily plugged in to existing personalized recommender systems as a post-processing step.

In the seminal work by <cit.>, the authors introduced MMR, a method to re-rank web search results.
A number of papers followed, especially in specific application settings like web search <cit.> and document summarization <cit.>.
Several re-ranking algorithms have been proposed in recommender systems <cit.>.
A notable characteristic of these algorithms is that the objective function can be represented as a modular/submodular maximization problem with matroid constraints and a solution can be obtained using a simple greedy heuristic.
Moreover, the celebrated result due to <cit.> states that steepest ascent greedy algorithm guarantees a constant approximation worst case lower bound i.e. F(^∗) ≥ (1-1/e)F(^opt) where e is the base of the natural logarithm, ^∗ is the greedy solution and ^opt is the unknown optimal solution.
We analyze these algorithms in detail in the following sections. 

In multi-objective optimization based algorithms, a model optimized for an objective function comprising of both relevance and diversity is used to predict the recommendations.
<cit.> proposed an objective which combines the standard latent factor model with intra-list distance, a measure of diversity.
Similarly, <cit.> proposed diversified collaborative filtering to learn a prediction model for diverse personalized ranking. 



In addition, there has been some work on unifying the performance metrics for diverse recommendations.
<cit.> proposed intra-list distance as a measure of diversity of the recommendation.
In <cit.>, authors argued that accuracy alone does not capture the "fitness of use" of the recommendation and that further metrics are required to capture non-trivial serendipitous recommendations which increases user satisfaction.
A serendipity measure based on average popularity of the items is proposed in <cit.>.



§ DIVERSIFICATION ALGORITHMS



Before we start analyzing different diversification algorithms, we introduce our notation and some basic definitions.
Our discussion lies primarily within the collaborative filtering domain.
We denote the set of items using , set of observed (rated) items using  and  denotes the set of unobserved items.
 denotes a subset of  such that || ≤.
rel(i) indicates the relevance of an item i and rel() indicates the relevance of the set , often defined as the sum of the monotone transformation of relevance of individual items.

Below we discuss some seminal work on re-ranking and show that these algorithms are based on maximizing a submodular/modular objective function from the class of parameterized concave/linear composition of modular functions.
Maximal Marginal Relevance (MMR) <cit.> is one of the earliest re-ranking algorithms with objective function

    F() = ∑_i ∈(λ*sim_1(u,i) - (1-λ) max_j ∈∖{i} sim_2(i,j))

where sim_1(u,i) is a modular function representing the relevance of a user u to the item i and similarly sim_2(i,j) is also a modular function representing the similarity between items i and j.
λ in the above formulation is the hyperparameter that is tuned for relevance-diversity trade-off.
Hence F() is in fact the difference of two modular functions i.e a linear composition of modular functions and is shown to be submodular <cit.>.
Many re-ranking algorithms inspired from MMR were proposed in specific application settings like web search diversification <cit.>.
Those algorithms follow the same functional form as in the case of MMR.
In fact, many re-ranking algorithms like MMR take a simple form as given below

    F() = f(S) + λ g(h(S))

where f is a modular function representing the relevance of recommendation set , g(h) is the diversity term which is the composition of a linear or concave function, g, and a modular function h and λ is the hyperparameter.
Often h is defined as a function of relevance itself, but capturing diversity aspects like coverage, popularity, serendipity etc.

TANGENT algorithm <cit.> proposed an objective function similar to the MMR objective.
The objective function can be written as F() = rel(S) + 1/rel(S).
Here f is simply the sum of relevance of items and g is the concave reciprocal function.
The objective function proposed by <cit.> takes the form F() = rel() +logh()/h().
Here h() is defined as the personal popularity tendency of the set  with respect to a user.
Here diversity term is defined as the composition of the concave function log with the modular function h()/h().
BinomDiv<cit.> also re-ranks the recommendations by defining a diversity term for genre coverage.
Like in the previous case, f corresponds to a modular function indicating the relevance of an item and diversity term is defined as the product of coverage and non-redundancy.
Both coverage and non-redundancy terms are defined in terms of the genre coverage and take the form x^λ with λ∈ [0,1].
The objective function proposed in <cit.> also consists of two terms, the first modular term captures the relevance with respect to neighbouring users and the second term, the neighbour coverage function stands as a surrogate for diversity.
It is defined as the concave composition g(x) = x/1+x of a positive modular function.

It is also very common that only the diversity aspect is considered for re-ranking i.e. f() = 0 in (<ref>).
<cit.> proposed a re-ranking algorithm by considering only the sub-categories associated with items.
The objective function considers only diversity aspect and takes the form F() = ∑_i∈()∑_j ∈(_-i)1/|()||(_-i)| where () represents the categories associated with the set of items  and _-i = ∖{i}.
This is equivalent to the sum of the concave function g(x) = 1/x. 
<cit.> also proposed a re-ranking algorithm considering only the diversity term.
The diversity term takes the form x^λ, λ∈ [0,1] and is defined in-terms of the interest coverage.
In <cit.>, the authors proposed an intent-aware diversification algorithm.
<cit.> showed that intent-aware objective functions are either submodular or modular functions.
The objective function in <cit.> is modular since it is based on re-weighting item based collaborative filtering scores.
Table:<ref> summarizes the essence of the above discussion regarding re-ranking algorithms.



 §.§ Diversity-Relevance Trade-off

We analyze the relevance-diversity trade-off in re-ranking algorithms to get a deeper theoretical understanding.
In majority of the re-ranking algorithms, the trade-off in relevance and diversity is obtained by tuning the associated hyperparameter.
For example, in case of MMR <cit.> based algorithms λ is the hyperparameter controlling the trade-off.
Similarly in <cit.>, λ parameter controls the relevance-diversity trade-off.
In further discussions, we consider the re-ranking algorithm proposed in <cit.> as a use case.

The total curvature of a non-decreasing submodular set function with respect to a set  is defined as


    α = max_j∈F(∖{j}) + F(j) - F()/F(j)

Intuitively, total curvature measures how far F is from being modular and (<ref>) represents the distance of a monotone submodular function to modularity.
Total curvature can take values between 0 and 1, and it is zero in case of modular functions and one in case of matroid rank function.
<cit.> extended the result in <cit.> and gave a tighter lower bound for the submodular maximization problem  in terms of total curvature.
According to <cit.>, F(^∗) ≥1/α(1-e^-α)F(^opt).

The curvature can be computed very easily for any submodular function, assuming a value oracle model.

When the curvature is 1, this gives the standard approximation bound given in <cit.> and for any other values it strengthens the approximation bound in <cit.>.
For example, if the curvature value α = 0.1, we are guaranteed that F(S^∗) ≥ 0.95F(S^opt) compared to the standard result of F(S^opt) ≥ 0.63F(S^∗).

It should be noted that curvature does not rely on specific functional form of F but only on the marginal gains.

We argue that by changing the hyperparameter, one effectively changes the total curvature of the re-ranking objective.
One can adjust the parameter such that the re-ranking objective is simply a modular function.
For example, by setting λ = 1 in <cit.> or setting λ = 1 in <cit.>,
total curvature of the resulting modular function can be made 0, and the recommendation list will be least diverse containing mostly relevant items.
Similarly, by adjusting the hyperparameter such that total curvature is 1, one can obtain the most diverse set of recommendations.



§ EXPERIMENTS


In this section, we experimentally validate our claims on a movie recommendation task. 
We followed the same experimental protocol as given in <cit.>.
We used the benchmark MovieLens 1M dataset and carried out holdout validation by splitting the data into training and test set such that 5% of the original data goes into testing and the remaining goes into training.
The reported results are the average over the five splits.


We used regularized weighted non-negative matrix factorization to extract the user and item features <cit.>.
The extracted user/item features are used to estimate the user-user/item-item similarity matrix.
We used the diversity objective proposed in <cit.>. 






 §.§ Evaluation Metrics

The performance of diverse ranking tasks are evaluated using three metrics: a serendipity metric and a dissimilarity metric to measure the distinctiveness of the recommendations and Discounted Cumulative Gain (DCG) as the ranking metric.

DCG is a binary ranking metric and it is calculated by discretizing the observed rating values such that rating values of 4 and 5 are deemed as relevant and as irrelevant otherwise.

We define serendipity score (SS) as the inverse of the average popularity of the recommended items which are not rated by the user.
    SS is a good indicator of the 'non-obvious' recommendations <cit.>.
We used feature distance (FD) as a measure of dissimilarity between the recommended items, and is defined as the average euclidean distance between the item feature vectors in the recommended set and it is has been used as a diversity objective in the past <cit.>.




 §.§ Results & Discussion

The results of our experiments are plotted in Fig <ref>.

For any value of α less than 1, greedy algorithm returns a mix of diverse and relevant items with the lower bound guarantee provided by the corresponding α value.

As the total curvature (α) value increases from 0 to 1, DCG value decreases.
For α = 0, the re-ranking objective becomes a pure modular function and the greedy algorithm returns recommendations containing the most relevant items as indicated by the highest DCG value.
As α value approaches 1, the objective becomes `easy' to `difficult', and many diverse items are added to the recommendation list.
A modular objective is `easy' as the solution obtained using greedy heuristic is optimal whereas submodular objective is `difficult' as the greedy solution is sub-optimal.
Consequently, SS and FD values increases as the value of α increases from 0 to 1, sacrificing DCG values.

As argued by <cit.>, diversification approach bears particularly high impact in recommendation with sequential consumption schemes like online video or music streaming.
There should be a right mix of items which are diverse yet relevant to the user.
By adjusting the hyperparameter of the re-ranking algorithms such that total curvature value is closer to 0.5, one can hopefully get the right mix of items for consumption.


§ CONCLUSION



We analyzed different diversification algorithms and have shown that majority of these algorithms are based on maximizing a submodular/modular objective function from the class of parameterized concave/linear over modular functions.
We also showed that by varying the parameter of the concave/linear composition, one effectively tunes the `total curvature' of the objective for relevance-diversity trade-off.

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