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## Contents

**CoFi_RANK: Maximum Margin Matrix Factorization for Collaborative Ranking**

## Problem Statement:

The underlying intelligent tools behind the webshoppers such as Amazon, Netflix, and Apple learn a suggestion function based on the the current user's and the others ratings in order to offer personalized recommendations. To this end, collaborative filtering provided a promising approach in which the rating patterns (of the products) by the current user and the others are used to estimate rates (or ranking) for unrated items. The task is more challenging once the user is unknown for the system (i.e., there is not any rating records from this user). Two different strategies might be incorporated for offering the recommendation list: rating or ranking. Ranking is different from rating in which the set of recommendations is obtaind directly, rather than first finding the rates and then sort them accordingly. For collaborative ratings, Maximum Marging Matrix Factorization (MMMF) had a promising result for estimating the unknown rates. This paper extends the use of MMMF for collaborative ranking. Since the top ranked items (products) are offered to the user, it is more important to predict what a user might like than what he/she dislikes. In other words, the items with higher rank should be ranked more accuratly than the last ones.

### Objectives:

The algorithm should

- directly optimize the ranking scores,
- be adaptable to different scores,
- not need any features extraction besides the actual ratings,
- be scalable and parralizable with large number of items and users.

### Definitions:

##### Polya-Littlewood-Hardy inequality

For any two vectors a, b , their inner product is maximized when a, b are sorted in the same order. That is [math]\lt a,b\gt \lt = \lt sort(a),sort(b)\gt [/math]

##### Discounted Comulative Gain

Consider the rating vector [math]y\in\{1,...,r\}^{n}[/math], and [math]\pi[/math] the permutation of the rating vector. For the truncation threshold [math]k[/math] (the number of recommendations), the Discounted Comulative Gains (DCG) score is defined as: [math] DCGk(y,\pi)=\sum_{i=1}^{k}\frac{2^{y_{\pi_i}-1}}{log(i+2)}[/math] Given the permutation [math]\pi_s[/math] which sorts the [math]y[/math] in decreasing order, the Normalized Discounted Comulative Gains (NDCG) score is given by:[math]NDCGk(y,\pi)=\frac{DCGk(y,\pi)}{DCGk(y,\pi_{s})}[/math]

According to the Polya-Littlewood-Hardy inequality, the DCG has the highest score when the [math]y[/math] is decreasingly ordered. Note that the weighting vector [math]\frac{1}{log(i+2)}[/math] is a monotinically decreasing function.

### Problem Statement

Estimate a matrix [math]F\in R^{m\times u}[/math] and use the values [math]F_{ij}[/math] for ranking the item [math]j[/math] by user [math]i[/math]. Therefore,given a matrix [math]Y[/math] of known ratings (in which row i shows the ratings of a user for item j) we are aiming to maxmize the performance of [math]F[/math] defined as: [math]R(F,Y)=\sum_{i=1}^{u}NDCGk(\Pi^i,Y^i)[/math] in which [math]\Pi^i[/math] is argsort(-F^i), i.e., the permutation that sorts F in decreasing order. The performance function is discritized (piecewise constant) and, thus, non-convex. In this paper, the structured estimation is used to convert this non-convex problem to a convex (upper bound) minimization. This convesion consists of three steps:

- Converting [math]NDCG(\pi,y)[/math] into a loss using the regret
- Linear mapping of rating vector in sorted order
- Using the convex upper-bound technique to combine the regret and linear map into a convex upper-bound minimization.

#### (1) Regret Convesion

Instead of maximizing [math]NDCG(\pi,y)[/math], we minimize the non-negative regret function [math]\Delta(\pi,y):=1-NDCG(\pi,y)[/math] which vanishes at [math]\pi=\pi_{s}[/math] .

### (2) Linear Mapping

Using the Polya-Littlewood-Hardy inequality, we define the linear mapp [math]\psi[/math] to encode the permutation [math]\pi=argsort(f)[/math]. [math]\psi(\pi,f):=\lt c,f_{\pi}\gt [/math]. In this mapping, the [math]c[/math] is a monotonically decreasing function (e.g., [math]c_i=(i+1)^{-0.25}[/math] ).

### (3)Convex Upper Bound

The follwoing lemma explains how to find a convex upper-bounds on non-convex optimization.

#### Lemma

Assume that [math]\psi[/math] is the abovementioned linear map, and [math]\pi^{\ast}:=argsort(-f)[/math] is the ranking induced by [math]f[/math]. Then, the follwoing loss function [math]l(f,y)[/math] is convex in [math]f[/math] and it satisfies [math]l(f,y)\gt = \delta(y,\pi^{\ast})[/math]. [math]l(f,y):=\max\underline{\pi}[\Delta (\pi,y)+\lt c,f_{\pi}\gt ][/math].

###### Proof

The argument of the maximization over the permutation [math] \pi [/math] is a linear and therefore convex function in [math]f[/math]. Since [math]\pi^{\ast}[/math] maximizes the [math]\lt c,f_{\pi^{ast}}\gt [/math], we have the upper bound as [math]l(f,y)\geq \Delta (\pi^{\ast},y)+\lt c,f_{\pi^{\ast}}-f\gt \geq \Delta(\pi^{\ast},y)[/math].

## Maximum Margin Matrix Factorization (MMMF)

We replaced the ranking score with a convex upper bound on a regret loss. That is replacing the [math]max R(F,Y)[/math] with the min [math]L(F,Y):=\sum_{i=1}^{u}l(F^i,Y^i)[/math]. For robust estimation, we add a regularization term inspired from MMMF to have a better estimation for the testing. [math]\Omega[F]:=\frac{1}{2} \min_{U,M}[tr MM^{T}+tr UU^{T}][/math]. The mtrix of interest [math]F[/math] is defined as [math]F=UM[/math] in which U is a user matrix and M is an item matrix. In [], the above regularization is a proper norm of [math]F[/math]. The final optimization problem is formalized as [math]\min_{U,M} L(UM,Y_{train})+\frac{\lambda}{2}[tr MM^{T}+tr UU^{T}][/math]. The objective is to learn user matrix [math]U\in R^{u\times d}[/math] and item matrix [math]M\in R^{d\times m}[/math] which store the specific properties of users and items, respectively. For computational concerns, the dimesion d is chosen as [math]d=10[/math] or [math]d=100[/math].

### Algorithm

Even though the obtained objective function may not be jointly convex in M and U, it is convex in M and U individually, once the other term is kept fixed. The alternating subspace descent provides a solution for this problem (although it might not be global minima):

**Repeat**

- For fixed M, minimize the objective function w.r.t. U
- For fixed U, minimize the objective function w.r.t. M

**Until** maximum iteration is reached or the error is small enough.

## Optimization

### Boundle Methods

Boudle method is a general method for solving regularized convex optimization problem. For a fixed element matrix [math]M[/math], the optimization over the user matrix [math]U[/math] is as follows: [math]R(U):=L(UM,Y_{train})+\frac{\lambda}{2} tr UU^{T}[/math]. Solving for the regularizer is simple and fast, however [math]L[/math] requires maximizing [math]l[/math] for all useres and hence, it is so expensive. Boudle method is the cutting plane method with a regularization term added(see in [] for more details on cutting plane method). Cutting plane method approximates [math]L[/math] using Taylor approximation with a lower bound.

##### Cutting Plane Method

Instead of minimizing a convex function directly, we can approximately minimize it by iteratively solving a linear program arising from its lower bound. A convex function [math]g(w)[/math] is bounded from below by its first Taylor approximation. [math]g(w)\geq g(w_0)+\lt w-w_0,\partial_w g(w_0)\gt [/math]

Boudle method is the cutting plane method with a regularization term added to it. Therefore, we would have
[math] L(UM,Y_{train})\geq L(UM',Y_{train})+tr (M-M')^{T} \partial_{M}L(UM',Y) \forall M,M' [/math].
Since this holds for arbitrary [math]M'[/math], we can pick a set of [math]M_i[/math] and use the maximum over the Taylor approximations at locations [math]M_i[/math] to lower-bound [math]L[/math]. Boudle method minimizes this piecewise linear lower bound in combination with the regularization [math]\frac{\lambda}{2} tr UU^{T}[/math] to obtain a new location where to compute the next Taylor approximation and iterate until convergence is achieved.