Learning Mixed Multinomial Logits with Provable Guarantees
Anonymous Author(s)
AffiliationAddress email
Abstract
A mixture of multinomial logits (MMNL) generalizes the single logit model,which is commonly used in predicting the probabilities of different outcomes.
While extensive algorithms have been developed in the literature to learn MMNLmodels, theoretical results are limited.
Built on the Frank-Wolfe (FW) method, wepropose a new algorithm that learns both mixture weights and component-specificlogit parameters with provable convergence guarantees for an arbitrary numberof mixtures.
Our algorithm utilizes historical choice data to generate a set ofcandidate choice probability vectors, each being close to the ground truth with ahigh probability.
We further provide a sample complexity analysis to show that onlya polynomial number of samples is required to secure the performance guarantee ofour algorithm.
Finally, we conduct simulation studies to evaluate the performanceand demonstrate how to apply our algorithm to real-world applications
Introduction
Multinomial logit models (MNL) are widely used in a variety of settings to predict the probabilitiesof different outcomes.
It is termed as multinomial logistic regression in statistics and its formulafor choosing the j -th option,
[Equation]
i exp( v i ) , is well known as the softmax function for multi-classclassification in machine learning.
In choice modeling, it is a type of discrete choice model used toanalyze and predict people’s decisions.
Mixed multinomial logit (MMNL) is a mixture model consisted of multiple individual MNL components and can be adopted when the target data is heterogeneous.
For instance, Yang et al.
[2018]demonstrates that using a single softmax does not have enough capacity to model natural languageand claims this as a bottleneck.
To alleviate this issue, they proposed a new method called Mixtureof Softmaxes (MoS), which has higher expressiveness and can better incorporate the contextualinformation.
Later, new techniques are proposed on top of this architecture to further boost theefficiency, such as the Mixtape from Yang et al.
On the other hand, McFadden and Train[2000] shows that MMNL models can lift the restrictive “independence of irrelevant alternatives”(IIA) assumption present in single MNL models as well as approximate any random utility-baseddiscrete choice models to arbitrary degree of accuracy under appropriate assumptions.
Despite its importance, the study on the learning theory of MMNL is limited.
A common approachin the literature is to assume a parametric family of the mixing distribution (cf.
Train [2009] and
Berry et al.
But this approach suffers from the model misspecification issue, i.e., therecould be systematical errors in the estimation if the assumed parametric family is different fromthe true one.
Expectation-Maximization (EM) algorithm is a widely used non-parametric methodfor mixture models.
But it does not guarantee the estimators’ convergence to the true value of theparameters.
Recent inspiring work by Jagabathula et al.
[2020] creatively applies the Frank-Wolfe
(FW) algorithms to estimate the MMNL model and establish the sub-linear convergence of theproposed method.
However, the major issue with their approach is that the convergence resultonly applies to the aggregate predictors, and cannot recover each individual mixture.
In this paper,we propose a new Stochastic Subregion Frank-Wolfe ( SSRFW ) algorithm, which can generate eachindividual mixture within an ϵ -ball to one of the ground truth vectors with a high probability.
We summarize the main contributions of this paper as follows.
First, we provide a learning algorithmfor MMNL models that comes with provable convergence guarantees of the estimators.
In particular,each generated mixture from our algorithm is shown to be within an ϵ -ball to one of the ground truthwith a high probability.
To the best of our knowledge, this is the first paper to prove these results for
MMNL models with an arbitrary number of mixture types and minimal assumptions on the modelparameters.
Second, we further quantify the sample complexity of the proposed algorithm.
We showthat only a polynomial number of samples is required to achieve a high probability convergenceguarantee.
1.1 Preliminaries
Consider a setting where the decision makers need to choose among a set of [ M ] = { 1 , . .
, M } items,where each alternative has a feature vector z j ∈ R d .
Furthermore, we assume the population consistsof K different components (types), each associated with a unique “preference vector” β k ∈ R d anda mixture weight α k , where (cid:80)
Kk
=
α k = 1 .
In other words, β k ’s are not drawn from a parametricmixing distribution and we consider MMNL in this paper as a type of latent class models, which iswhy we claim our method as non-parametric .
This is consistent with the definitions used in otherliterature such as Train [2009] and Jagabathula et al.
[2020].β k ’s can also be used as user embeddings with high interpretability for decision makers who belong tothe same type, which represent the corresponding type’s taste on different aspects of the alternatives.
Under linear utility assumption, the probability of choosing j -th item for a particular mixture k is
[Equation]
and the aggregated counterpart for the entire population is
[Equation]
To simplify notation, we use q k to denote the logit vector q ( β k ) =
[ q 1 ( β k ) , · · · , q M ( β k )] ⊤ ∈ R M ,and q kj for q j ( β k ) , j ∈ [ M ] , ∀ k .
We will also refer to these logit vectors as choice probabilityvectors .
1.2 Related Work
Learning MMNL models remains an open-research problem.
One common approach in the literatureis to presume a parametric family of distributions on the parameters and apply parametric estimationmethods such as the maximum likelihood estimation (MLE) method (c.f. Train [2009]) or the leastsquare regression model to compute the parameters.
A well-known work in this regime is by Berryet al.
They assume the component-specific parameters are normal distributed and proposed atwo-step estimation method to learn the parameters using an aggregated market share data.
A potentialissue for this stream of methods is model misspecification, which leads to inaccurate predictions.
Inparticular, if the assumed parametric family is different from the true one, there could be systematicalerrors in the estimation.
Non-parametric estimation is also widely recognized when learning MMNL models, with the mostcommonly known and used method being the Expectation-Maximization (EM) algorithm (c.f.
Dempster et al.
Train [2008] studied three different types of EM algorithms using historical choicedata from each individual decision maker.
The two main restrictions for the EM algorithm are 1) thealgorithm can get stuck in local optimum (hence no guarantee on the convergence of the estimators)and 2) the number of mixture types needs to be pre-specified (additional heuristics are needed if suchinformation is not known in advance, which may also introduce undesired noises).
Jagabathula et al.
[2020] recently developed an estimation method for mixture models also from a nonparametric perspective based on the Frank-Wolfe (FW) algorithm (cf.
[Frank and Wolfe, 1956], Jaggi[2013], [Jaggi and Sulovsk`y, 2010], [Bach, 2013], Harchaoui et al.
[2015],Lacoste-Julien and Jaggi[2015]), which is an iterative method originally designed to solve constrained quadratic optimization.
They manage to establish a sublinear convergence rate of the proposed estimation method for MMNLmodels.
But the convergence result only applies to the aggregate choice probabilities.
In contrast, weprovide the convergence guarantee for each individual choice probability (more details about theirapproach and the comparisons will be discussed in Section 2).
There is also another stream of work related to MMNL models from a theoretical learning perspective.
The key idea is to solve a system of equations in terms of all the parameters and prove that there isone and only one solution (identifiability).
Due to the entanglement of the individual logit parametersand mixture weights, as well as the high non-linearity in the system of equations, only 2-MNL models(MMNL with two mixture types) have been studied so far.
Chierichetti et al.
[2018] considers thesetting where the mixture weights are equal (i.e., each mixture represents 50% of the population)and Tang [2020] recently studied 2-MNL with unknown weights.
This paper enriches the theoreticallearning literature for MMNL models by generalizing the analysis to an arbitrary number of mixturetypes and with minimal assumptions on the model parameters.
1.3 Main Results
The main contribution of this paper is to provide a learning algorithm for MMNL models withprovable convergence guarantees of the estimators.
Specifically we will show that each of thegenerated choice probability vectors from our algorithm is within an ϵ -ball to one of the ground truthvectors with a high probability.
Subsequently, we provide the sample complexity analysis of thealgorithm.
We name our algorithm the Stochastic Subregion Frank-Wolfe ( SSRFW ) algorithm, whileis inspired by the Frank-Wolfe framework by Jagabathula et al.
but has a carefully constructedstochastic feasible region that is learned from the data.
The SSRFW framework assumes to work with repeated choice data, such as consumer panel or surveydata.
This is a reasonable assumption in many real-world scenarios (cf.
Revelt and Train [1998]and Brownstone et al.
We assume the data consists of historical choice decisions from thetarget population of size N .
Specifically, for each time period t = 1 , . .
, T , we assume each decisionmaker
i ’s choice X ( t ) i are i.i.d categorical random variables with support x ∈ [ M ] and probabilitymass function (PMF) specified by q k if he/
she is of type k .
For the MMNL to be learnable, we need to assume that none of the individual MNL is identical to therest.
Let F and G be the cumulative distribution functions (CDF) for two different choice probabilitydistribution q k and q k ′ respectively.
Define the Kolmogorov-Smirnov (KS) distance between the twodistributions as D KS ( q k , q k ′ )
=
sup x | F ( x ) − G ( x ) | .
Definition 1 below is adapted from a similarassumption introduced in Kalai et al.
[2010] to impose some regularity in the MMNL model.
Definition 1.
We call a mixed multinomial logit model (MMNL)
g ϵ -standard if1.
min k α k ≥ ϵ2.
β kd ≤ 1 ϵ , ∀ k, ∀ d3.
D KS ( q k , q k ′ ) ≥ ϵ, ∀ k ̸ = k ′
Definition 1 imposes the following properties on the MMNL model: 1) none of the mixtures has zeroweight; 2) all parameters in the model are bounded; and 3) each individual logit model has somelevel of separation from others.
We consider this as a very mild assumption in our setting, as later wewill see that we will not predetermine the number of mixtures ( K ) and if a mixture/component is notpresent in the observed data (since it has zero weight), the algorithm will simply not learn it.
This isalso one of the advantages of our algorithm over the traditional approaches, such as EM.
The main result is as follows:
Theorem 1.
Let g = (cid:80)
Kk
=1 α k q k be a mixed multinomial logit (MMNL) model over a set of M items.
Assume M ≥ K .
For any ϵ > 0 , 0 < δ < 1 , Algorithm 1 outputs an MMNL ˆ g = (cid:80) K ′ k
=
ˆ α k ˆ q k where K ′ ≥ K such that, with probability ≥ 1 − δ , there exists a many-to-one mapping π : k ′ (cid:55)→ k, k ′ ∈
[ K ′ ] , k ∈
[ K ] such that
[Equation]
and
[Equation]
where ∥ · ∥ is the Euclidean norm.
The number of samples required by Algorithm 1 is O ( 1 ϵ 2 log( 1 δ )) .
The outline of the paper is as follows: Section 2 and Section 3 will introduce the main algorithms,namely the Stochastic Subregion Frank-Wolfe ( SSRFW ) and the Q -construction algorithm, where theoutput of the latter serves as an input to the former.
In Section 4, we will discuss the theoreticalproperties of the SSRFW algorithm.
We conduct simulation studies in Section 5.
All the detailedproofs and parts of simulation studies are included in the appendix.
Stochastic Subregion Frank-Wolfe Algorithm
Recall that we observe each decision maker i ’s choice at time t : x ( t ) i (as realization of X ( t ) i ).
Let
Y ti =
[0 , · · · , 1 , · · · , 0] ⊤ ∈ R M , where Y tij
=
if x ( t ) i =
j .
The observed aggregated choiceprobability vector for period t can be expressed as
[Equation]
where y t ∈ R M .
Define P = { q ( β ) ∈
R M | β ∈ R d } , the set of all valid logit vectors for the givenchoice set and their features.
Denote P as its closure.
Since g = (cid:80) Kk
=1 α k q k , by definition
we haveg ∈
Conv ( P ) .
Our main learning objective is to minimize the difference between the observed (from data) andtheoretical aggregated choice probability, as follows
[Equation]
Note that problem (1) is a non-convex problem in α k and β k and is in general hard to solve.
Weadopt the conditional gradient approach proposed by Jagabathula et al.
[2020], which we will refer toas the original FW algorithm.
Each iteration k is consisted of two steps: 1) the supporting findingstep which searches for a new direction represented by a choice probability vector q ( k ) (as a newlatent class k ) via solving a linear optimization subproblem in the search space Conv ( P ) , and 2)the proportion update step which updates the mixture proportion α i , i = 0 , . .
k , assigned to allchoice probability vectors obtained so far.
The final learning outcome ˆ g is (cid:80) k
i =
α k q ( k ) whereq (0) =
g (0)
∈ P is a random initialization.
This setup corresponds to a fully-corrective variant of thegeneric Frank-Wolfe algorithm, and is guaranteed to find the optimal solution to the optimizationproblem
[Jaggi, 2013] for the decision variable g .
The problem with the original FW approach is that the feasible region Conv ( P ) contains the completeset of all logit vectors and their limiting points.
The former corresponds to non-boundary types ,whose choice model can be characterized by a standard MNL choice model, and the latter correspondsto boundary types , whose standard MNL parameters become unbounded, resulting infinite utilityfor some options and zero for the rest.
Such broad search space Conv ( P ) in FW generates mixturecompositions with a considerable number of the boundary types.
More specifically, this problem isrooted to the linear subproblem solved at each support finding step , whose optimal solution always lieson the extreme point of the feasible region which contains many of these limiting choice probabilityvectors ∈ P \ P as its extreme points.
Often, each limiting choice probability vector corresponds toan unbounded β k and cause the learning outcome unusable for downstream applications.
Thoughsignificant effort is made to justify the legitimacy of these boundary types in the paper, the convergenceresult only applies to the population’s aggregate choice probability, i.e. ˆ g → (cid:80) k
α k q k , which is an Frank-Wolfe property.
In other words, their method fails to recover each of the individual logitmodels ( β k ) and the corresponding mixture weights ( α k ) accurately.
To remedy this issue and seek convergence guarantees for the general MMNL estimation problem,we designed the Stochastic Subregion Frank-Wolfe ( SSRFW ) algorithm, as shown in Algorithm 1.
Algorithm 1: Stochastic Subregion Frank-Wolfe
Input: data y , Q from Algorithm
Initialization: k = 0 ; α (0) =
[1] , a random g (0) =
q (0) chosen from Q 1 while stopping condition not met do k ← k +
Compute q ( k ) =
arg min v ∈ Conv ( Q ) ⟨∇L (cid:16) g ( k
[Equation]
1) (cid:17) ,
[Equation] [Equation]
Compute α ( k ) =
arg min
[Equation]
(cid:32)
[Equation]
s
[Equation]
[Equation] [Equation]
) (cid:33)
Update g ( k ) :
= α ( k )
g (0)
+
[Equation]
(cid:88) s
= α ( k ) s q ( s ) 6 end
Output : choice prob.
q (0) , . .
. , q ( k ) mixture weights.
α ( k ) ∈ ∆ k ⊂ R k +
Note in the above algorithm, after k iterations, we have generated k choice probability vectorsq ( s ) , s = 1 , . . .
, k , all of which will be the output of the learning algorithm.
The mixture weightestimator is slightly different.
α ( k ) ∈
R k +1 represents the mixture weight vector for the k + 1 choiceprobability vectors we have generated so far, where α ( k ) s refers to the s -th element in the vector,corresponding to the mixture associated with q ( s ) .
In other words, the vector length of α increasesby 1 for each iteration and only the final vector α ( k ) is outputted from the algorithm.
The key different in SSRFW is that it takes in an additional input Q from the Q Construction Algorithm(discussed in detail in Section 3), which is the key to eliminate the possibility of producing boundarytypes .
In particular, the new feasible region Conv ( Q ) is a subset of Conv ( P ) , where each elementq ∈ Q will be learned from data and is guaranteed to be within an ϵ -ball of the true choice probabilityvector for some mixture type with high probability.
Subsequently, this also ensures the SSRFWalgorithm to recover the mixture weight for each latent class.
Note that the extreme points of
Conv ( Q ) by definition is a subset of Q .
Since we only care about the extreme points of the feasibleregion, we can safely ignore any points ∈ Conv ( Q ) \ Q and replace Conv ( Q ) with Q in Algorithm 1.
Since Frank-Wolfe will converge to the optimal solution [Jaggi, 2013], and as we will see in Section4 that the optimal solution to problem (1) is an interior point of the feasible region Conv ( Q ) , thestopping condition can simply be set as (cid:13)(cid:13) g ( k ) − y (cid:13) (cid:13) ≤ ϵ .
Q Construction Algorithm
Our primary goal in this section is to learn the input set Q for SSRFW (Algorithm 1).
In particular forthis set Q = { ˆ q ℓ } l =1 ,...,L , we require that ∀ ˆ q ℓ , there exists some mapping π : [ L ] → [ K ] such that(cid:13)(cid:13) ˆ q ℓ − q π ( ℓ ) (cid:13)(cid:13) ≤ ϵ , where q π ( ℓ ) is one of the ground truth logit vectors.
Once this property holds, eachof the SSRFW ’s q outputs will also be ϵ -close to some true choice probability vector as the algorithmalways selects an extreme point of Conv ( Q ) in each iteration.
The two major components for the Q Construction Algorithm is first computing a distance scorematrix (Section 3.1) and then creating subsamples that contain decision makers from only one mixturetype through random seeding and non-uniform sampling (Section 3.2).
Section 3.3 presents thealgorithm and discusses the properties of the learning outcomes.
[Equation]
3.1 Distance Score Matrix
We first define a distance score matrix S , whose element s ij measures the dissimilarity between anytwo decision makers i and j in term of their choice decisions.
Recall that each x ( t ) i are realizations ofi.i.d categorical random variables with PMF q k if i is from mixture k .
Denote its associated empiricalcumulative distribution function (CDF) as F T ( x ; i ) when a total of T decisions are observed.
Inparticular,
[Equation]
Define the distance score function for each pair of decision makers as
[Equation]
where || · || ∞ represents the infinity norm.
3.2 Subsample Construction
We now want to create a number of subsamples such that each subsample of decision makers containsonly one mixture type with high probability.
Each of these subsamples is generated following aprocedure that contains a random seeding step, and a subsampling step.
These steps are repeated Ltimes to create a Q set of size L .
Random Seeding.
This step randomly samples a decision maker i ∈
[ N ] from the population andcan be done with simple uniform sampling technique.
The selected decision maker is called a seed .
We will generate L seeds and later investigate how large L needs to be in order to cover all mixturetypes in Q .
Subsampling Strategy.
This step generates an index set I ( i ) for a seed i such that, with highprobability, decision makers whose indices belonging to this set are of the same type as i .
I ( i ) iscalled the subsample originated from seed i .
Given a selected seed i , we first calculate an accepting probability p
j | i which determines the likelihoodof another decision maker j being accepted to the corresponding subsample
I ( i ) .
It is designed to bea monotonically decreasing function with respect to the distance score we defined in Section 3.1:
[Equation] [Equation]
A simple example can be p j | i = 1 − s
( i, j ) .
Intuitively, when s ( i, j ) is small, seed i and decisionmaker j ’s empirical CDF is close to each other, indicating there is a higher chance that they are of thesame type and sharing the same choice probability q .
Subsequently, p j | i will be larger compared toa large value of s ( i, j ′ ) for another person j ′ , resulting we accept j with higher probability than j ′ .
This is consistent with our objective that we want each subsample to be composed of decision makersof the same type as the seed.
For implementation, we first initiate an empty index set I ( i ) for the selected seed i .
Then we repeatthe following two steps until we reach a desired subsample size n : 1) draw a random sample from set[ N ] \ I ( i ) , 2) accept this sample into the index set with probability p
j | i and reject with 1 − p
j | i .
3.3 Algorithm and Properties
The remaining step is to obtain the set Q whose elements ˆ q ℓ are computed as the average of historicalchoice decisions from decision makers in the subsamples I ℓ , i.e. ˆ q ℓ = 1 nT (cid:88) i ∈ I ℓ
T (cid:88) t =
Y ( t ) i .
Finally, wepresent the Q Construction Algorithm in Algorithm 2.
Discussion We next discuss a few properties of the two algorithms.
SSRFW returns the logit vectors q instead of the parameters β .
We can simply perform maximumlikelihood estimation under the single MNL setting where, if the extreme point corresponding to
Input : score matrix S , number of subsamples L , subsample size n
Initialization: Q = set () 1 for ℓ ← 1 to L do
Choose seed: i ∼ U (0 , N )
Initiate: I ℓ = set () while | I ℓ | ̸ = n do j ← random_sample ([ N ] \ I ℓ )
Generate u ∼ U (0 , 1) if u < p
j | i then
I ℓ .
add ( j ) end end
Compute ˆ q ℓ = 1 nT (cid:88) i ∈ I ℓ
T (cid:88) t =
Y ( t )
i
Q . add (ˆ q ℓ ) 13 end
Output: Q subsample ℓ is selected during an SSRFW iteration, the log-likelihood is
[Equation]
and we have ˆ β ℓ = arg max β l ℓ ( β ) .
With the assumption of linear utility in β , we can obtain theoptimal solution using MLE (c.f. McFadden [1974]).
In general, we prefer learning β k ’s compared to q k ’s.
This makes the algorithm more robust whenitem attributes change over time which results in change in the q k ’s whereas β k values persist.
Toachieve this, we just need to add two additional steps after the index set is created:
• MNL paramter estimation: apply the MLE step outlined above; create B = { ˆ β ℓ } ℓ =1 ,...,L• Map B to Q : fix item attributes of current interest, calculate Q = {
ˆ q | ˆ q = q
( ˆ β ℓ ) , ˆ β ℓ ∈ B}
Next, stochasticity in our algorithm comes from two parts, namely the realization of each decisionmakers’ choices, as well as the randomness in the process of generating subsamples.
Yet we willshow that under appropriate assumptions, SSRFW can still recover all individual mixture parameterswith high probability.
In addition, we do not impose any restriction on the subsamples created during Q construction to bemutually exclusive.
As long as they only contain one single mixture type (i.e. with high purity), itcreates a legitimate estimation of ˆ q or ˆ β with MLE.
Last but not least, this is a more robust strategy than clustering algorithms to segment the population.
Not only do we not require the number of mixtures K as a hyper-parameter, each decision maker’sdata being used more than once for the estimation can be thought of as a special bootstrap mechanismthat has custom weights tailored to our objective in creating homogeneous subsamples.
Theory of the SSRFW Algorithm
Theorem 1 is the main result of this paper.
To prove it, we will break down this section into twoparts, where the first part will examine the provable convergence properties of the estimators, and thesecond part will focus on the sample complexity.
We will outline the proof sketch but all detailedproofs can be found in the Appendix.
Provable Convergence.
Assume K ′ is the total number of iterations that the algorithm has performed before reaching the stopping criteria.
To simplify notation, we denote the generated outcomefrom k -th iteration q ( k ) in SSRFW as ˆ q k to indicate they are the estimators and the final updatedmixture weight corresponding to q ( k ) , α ( K ′ )
k , as ˆ α k .
In other words, we have g SSRFW
=
(cid:80) K ′ k ′
=
1 ˆ α k ′
ˆ q k ′ ,
We use a similar notation g FW for the original FW approach.
[Figure]
Figure 1 describes the necessary components and steps of proving the result.
In particular, Property 1and 2 (purple boxes) are adapted from existing results of the Frank-Wolfe framework [Jaggi, 2013].
In Appendix A.1.1, we prove Property 1 in Lemma 2.
Property 2 follows from the fact that each ofthe ˆ q k ′ , obtained by solving a linear optimization problem, has to be an extreme point (denoted by
E ( · ) ) of the feasible region.
On the other hand, Property 3 and 4 (red boxes) hold as a consequence of the stochastic subregion weconstructed in Algorithm 2.
Property 3 claims that with high probability, the ground truth aggregatedchoice probability vector is still an interior point of the shrunken feasible region Conv ( Q ) (proofin Appendix A.1.2).
Together with Property 1, this tells us that | g SSRFW − (cid:80) k
α k q k | ≤ ϵ .
That isto say the aggregated choice probability vector of SSRFW also converges to the ground truth.
Toachieve Property 4 from the Q construction algorithm, we need to supply enough of data, the detailsof which will be discussed in the sample complexity part.
Combining Property 2 and 4, we have∀ k ′ , ∃ k
= π ( k ′ )
s.t.
| ˆ q k ′ − q k
| ≤ ϵ .
Finally, by doing some algebraic manipulation, we can obtain
[Equation]
Sample Complexity.
Recall we use T as the number of choice data records for each decision maker.
Denote T min as the minimum number required and T min ≥ 1 .
Define in-types to be the decision makers who share the same type as the seed, and out-types to bethe ones of different types.
We focus on the following three properties:
• probability of accepting the in-types,
• probability of rejecting the out-types,
• probability of covering all mixture types.
Consider a seed i ,
with type
k i = k .
Assume there is a total of m k decision makers in
the
populationthat
is of type k
Accepting In-Types.
Denote the empirical choice probability for the seed i as F n and that foranother decision maker j as G m , with n and m representing the number of independent priceexperiments from i and j , respectively.
Recall that
[Equation]
i j ) =
[Equation]
G m (
Theorem 2 (Sample Complexity I) .
Assume i and j are of the same type.
Define p
j | i = 1 − s ( i, j ) .∀
δ > 0 , ϵ > 0 , we can achieve p j | i > 1 − ϵ with probability at least 1 − δ with sample size
T min =
O ( 1 ϵ 2 log (cid:0) 1 δ (cid:1) ) .
Theorem 2 indicates that with sufficient amount of data, there is a high probability to accept ain-type—who shares the same type as the seed—into the subsample if it is chosen from the populationafter the random draw.
Rejecting Out-Types.
Assume a decision maker j is of a different type from the seed i .
We use Fand G to denote their CDFs.
Let F n and G m represent the corresponding empirical CDFs based onsamples drawn from the two different distributions with sample size n and m , respectively.
s ( i, j ) isdefined the same as before.
Theorem 3 (Sample Complexity II) .
Assume F and G correspond to the choice CDFs of two differenttypes and sup x | F ( x ) − G ( x ) | ≥ ξ .
Let F n and G m be independent empirical distribution basedon m and n i.i.d.
samples drawn from F and G , respectively.
Denote T min = min { m, n } .
Definep j | i = 1 − s ( i, j ) .
∀ δ > 0 , ϵ > 0 , we can achieve p j | i < 1 − ξ + ϵ with probability at least 1 − δwith T min =
O ( 1 ϵ 2 log 1 δ ) .
Theorem 3 complements Theorem 2 to rule out the out-types from being selected with a highprobability.
The more separable (larger ξ ) their underlying distributions are, the lower acceptingprobabilities are.
Lemma 1.
For ϵ > 0 and 0 < δ < 1 , with minimum of choice records T required by Theorem2 and Theorem 3, ∀ q ℓ ∈ Q , (cid:13)(cid:13) q ℓ − q π ( ℓ ) (cid:13)(cid:13) ≤ ϵ with probability at least 1 − δ , where ˆ q ℓ =1 nT (cid:80)
[Table]
Lemma 1 states that with enough data samples, the Q -construction algorithm achieves Property 4, asshown in Figure 1, which SSRFW ’s desired provable convergence builds upon.
With Lemma 1, weconclude the proof for Theorem 1.
The complete proof can be found in Appendix A.4.
As a final remark, we discuss the number of subsamples (i.e. L ) needed in the Q -construction process.
We distinguish this from sample complexity as we are not requesting more data points with a larger L .
Instead, we should view L as a computational complexity factor.
Recall that each moving directionderived from the support finding step is viewed as a learned logit vector of one mixture component.
Therefore, we want the Q -construction algorithm to include at least one ˆ q ℓ for each mixture k in orderfor it to be picked in SSRFW .
This impose a requirement on L .
Theorem 4 states that the expectednumber of the number of subsamples is controlled by the smallest mixture weight.
Theorem 4.
Assume α 1 ≤ α 2 ≤ · · · α K .
The expected number of subsamples L we need to constructis bounded by 1 α 1 log 1 α 1 .
Intuitively, this means if we have one mixture component that is very under-represented, then weneed to create more subsamples to ensure that one of the seeds belongs to this mixture.
Conclusion
In this work, we propose the SSRFW algorithm, which provides an end-to-end solution for learning
MMNL models using historical choice data.
This novel approach utilizes a carefully designedsampling method to construct a meaningful search space.
Not only does it resolve the drawback ofthe original Frank-Wolfe approach with boundary-type issues but also enables us to obtain provableguarantees for the model estimates and sample complexity.
Another major advantages of ouralgorithm is that we do not assume prior knowledge of the number of mixtures K in the model.
It isalso more robust than traditional unsupervised approaches such as clustering method and EM.
In marketing science and operations research, MMNL provides a fundamental tool for demandprediction, which many downstream applications such as pricing and inventory position rely on.
For instance, consider a pricing model that tries to maximize the revenue by solving the followingoptimization problem: max p (cid:80) Mj
=
p j (cid:80) Kk =1 α k q j ( β k ; p ) where p j ≥ 0 , ∀ j ∈ [ M ] .
This requiresan accurate estimation of each individual MNL components.
Due to page limit, we include oursimulation studies and application in real-world setting in the appendix.
We will also briefly discussthe requirements on the datasets in order to apply our algorithm.
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A ProofsA.1 Proofs for Provable ConvergenceA.1.1 Proof of Property 1
Lemma 2.
Denote g ∗ as the optimal solution to the Stochastic Subregion Frank-Wolfe algorithm and g ( k ) denote the k -th iterate generated by Algorithm 1.
Then
[Equation]
for all k ≥ K .
Proof : This Lemma follows directly from the existing results of the original Frank-Wolfe algorithmand its variants [Jaggi, 2013], which states that for an optimization problem min x ∈D f
( x ) where f isa convex and continuously differentiable function and that the domain D is a compact convex set ofany vector space, then for each k ≥ 1 , the iterates x ( k ) of the fully-corrective Frank-Wolfe algorithmsatisfy:
[Equation]
where C f , defined as
[Equation]
is the curvature constant , which measures the “non-linearity” of function f over domain D .
The typeof the Frank-Wolfe we use in Algorithm 1 is precisely the fully-corrective variant in that we optimizefor α ’s in each iteration.
Claim 1.
L ( g ; y )
=
|| g − y
||
2 is a twice differentiable convex function.
Conv ( Q ) is a compactconvex set.
The first statement in Claim 1 is true by definition.
The second statement can be shown by observingthat Q is a finite set, hence compact, followed by the fact that convex hulls of compact set arecompact.
For squared loss function L used in our model, Jagabathula et al.
[2020] proved that C L ≤ 2 .
Theresult of Lemma 2 follows by plugging C L into Equation 2.
□
Note that with enough sample, we also have (cid:13)(cid:13)(cid:13) y − (cid:80)
Kk
=1 α k q k (cid:13)(cid:13)(cid:13) ≤ ϵ by the law of large numbers,which leads to g SSRFW converges to (cid:80) K k
=1 α k q k with high probability.
This shows that Frank-Wolfecan reach any tolerance level ϵ with enough number of iterations by setting appropriate stoppingcriteria.
A.1.2 Proof of Property
Lemma 3 (Wendel [1962]) .
If X 1 , . .
, X n are i.i.d.
random points in R d whose distribution is symmetric with respect to the center O and assigns measure zero to every hyperplane through O , then
[Equation]
Lemma 3 is an interesting result from stochastic geometry, which states that if we randomly sample npoints in a d -dimensional ball, the probability that the convex hull formed using these points containsthe center point can be computed using the above formula.
Corollary 1.
[Equation]
Proof : When n ≥ 2 d − 1 , (cid:0) n − 1 k (cid:1) is a monotonically increasing function of k for k = 0 , . . .
, d − 1 .
We then have
[Equation]
k (cid:19)
[Equation]
d (cid:18)
[Equation]
d
[Equation]
(cid:19) ≤ d ( n − 1) d ( d − 1)!
Therefore, when n ≥ 2 d − 1 ,
P
[Equation] [Equation]
−d − 1 (cid:88) k
= (cid:18) n −k (cid:19) ≥ 1 − d ( d − 1)!
( n − 1) d 2 n −
Since lim n →∞ ( n − 1) d 2 n − 1 = 0 , we get lim
[Equation]
□
Corollary 2.
With high probability, Conv( { q k } 1 ,...,K ) ⊆
Conv( Q )
The proof of Corollary 2 will be included in the proof of Theorem 1 in Appendix A.4.
We willalso quantify what “high probability” it is referring to.
Corollary 2 establishes the fact that g = 470 (cid:80)
Kk =
α k q k ∈ Conv ( { q k } k
=1 ,...,K ) ⊆
Conv ( Q ) with high probability.
We illustrate this idea in
Figure 2 for an intuitive understanding.
[Figure]
For each mixture type k , with enough data points, we have with probability 1 − δ , ∀ ℓ such thatπ ( ℓ ) =
k , ˆ q ℓ is within an ϵ -ball of q k given sufficient number of samples.
In addition, according to
Lemma 3, with high probability, such convex hull (small regions with green firm lines for each k )contains the ground truth choice probability vector q k , i.e. q k ∈ Conv ( { q ℓ ∈ Q| π ( ℓ )
= k } ) .
Sincethe true aggregated choice probability (cid:80)
Kk
=
α k q k is a convex combination of q k ’s (so it is in theblue shaded region) and (cid:83)
k Conv ( { q ℓ ∈ Q| π ( ℓ ) = k } )
⊂
Conv ( Q ) , we have Conv ( Q ) enclosesthe blue region and (cid:80)
Kk
=1 α k q k is an interior point of Conv ( Q ) .
A.2 Proofs for Sample Complexity
We first extend the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality with the following Lemma, whereboth CDFs in the inequality are empirical.
Lemma 4.
Let F n and G m be independent empirical distribution based on m and n i.i.d.
samples drown from a common cumulative distribution F ( · ) .
Denote min { m, n } as T min .
We have
[Equation]
Proof : Lemma 4 differs from the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality in that it investigates the tail probability of the maximum difference between two empirical distributions.
By DKW, we know that
[Equation]
if F n and G m are empirical distributions of samples drawn from their true distribution function Fand G respectively.
We can show
[Equation]
where the first inequality makes use of the fact that sup x | F n − F | + sup x | G m − F | > ϵ impliesthat either sup x | F n − F | > ϵ 2 or sup x | G m − F | > ϵ 2 .
The second inequality comes from theindependent assumption between F n and G m .
On the other hand, we also have
[Equation]
Combining (3) and (4), we can obtain
[Equation]
□
A.2.1 Proof of Theorem 2
Proof : For simpler notation, denote s ( i, j ) = s .
Since i and j are of the same consumer type k , F nand G m are empirical distributions based on n and m samples drawn from the same distribution q k .
According to Lemma 4,
[Equation]
Let δ = 4 exp( − 12 T min ϵ 2 ) , we have T min =
O ( 1 ϵ 2 log 1 δ ) .
□
A.2.2 Proof of Theorem
Proof : By definition,
[Equation]
Therefore,
[Equation]
where the first inequality is based on (5) and the second inequality makes use of the fact thatsup x | F n − F | +sup x | G m − F | > ξ − ϵ implies that either sup x | F n − F | > ξ − ϵ 2 or sup x | G m − F | >ξ − ϵ 2 .
The equality comes from the independent assumption between F n and G m .
The last inequalityis from the DWK inequality.
We want to restrain the sampling probability such that p j | i = 1 − s is within ϵ -radius of the smallestpossible sampling probability, which is 1 − ξ , i.e.
[Equation]
as a result of Equation 6.
Let δ = 4 exp( − 12 T min ϵ 2 ) .
Then we have T min =
O ( 1 ϵ 2 log 1 δ ) .
□
Additional discussion.
We further discuss the impact of M on the sample complexity.
As discussed above, we mainly relate the number of time periods we need to collect the data for, i.e. T , and ϵ and δ is through the DKW inequality.
To put things simple, consider the case where we want to bound the probability that the supremum of the discrepancy between the empirical CDF and the true CDF larger than ϵ to be no bigger than δ , as follows:
[Equation]
This gives T = 12 ϵ 2 log 2 δ .
Define a random variable U
= sup x ∈
[ M ] | F T ( x ) − F ( x ) | .
For a larger
M ′ , similarly define V = sup x ∈
[ M ′ ] | F T ( x ) − F ( x ) | .
Since the set that the supremum is taken islarger for V , we have V ≥ U .
Subsequently, we have P ( V > ϵ ) ≥ P ( U > ϵ ) .
Assume P ( V > ϵ ) =
δ + ξ for some ξ ≥ 0 .
The relationship between T and ϵ and δ is thus
T =
2 ϵ 2 log 2 δ + ξ for the M ′ case.
Now if we want to bound P ( V > ϵ ) by δ , instead of δ + ξ , weneed to further increase the value of T , hence increase the sample complexity.
This analysis can be generalized directly to the two-sample DKW inequality for both in-type andout-type bounds and suggests an impact of M on the sample complexity.
This agrees with theintuition that for a larger choice set, we need to collect more data since otherwise the empiricalchoice distribution over the set will be very sparse, making the score function computed based on theempirical CDF less credible.
However, we think such impact, compared to ϵ and δ , is rather implicitas it is enclosed in the sup function.
A.3 Proof of Theorem 4
Proof : As in Algorithm 1, we use L as the number of subsamples we need to construct.
Let L k bethe number of subsamples needed to hit the k -th mixture type after k − 1 types of seeds have beenselected.
We have L =
L 1 + · · · + L K .
We first construct a simple and fake scenario where we have K ′ mixture types with each mixtureweight equal to α 1 , i.e. K ′ =
1 α 1 .
Similarly, we can define L ′ and L ′ k as above and also have
L ′ =
L ′ 1 + · · · + L ′ K ′ .
Think of L ′ and L ′ k , k = 1 , . .
, K ′ as random variables
and we know theprobability of selecting a seed from a new type k is p k
= K ′ − k +1 K ′
since in the fake scenario, eachtype has the same probability α 1 of being chosen.
This tells us that L ′ k has a geometric distributionwith expectation 1 p i =
K ′ K ′ − k +1 .
By the linearity of expectations we have
[Equation]
where H ′ K is the K ′ -th
harmonic number.
Using the asymptotics of the harmonic numbers, we get
[Equation]
Since α 1 ≤ · · · ≤ α K , we know K ′ ≥ K .
On the other hand, we know the E [ L k ] ≤ E
[ L ′ k ] sincethere is a higher probability of choosing any mixture type k ≥ 2 , due to the same reason, i.e. α k ≥ α 1 .
Therefore, we have E [ L ] ≤ 1 α 1 log( 1 α 1 )
We can further characterize the probability of event H , which describes the event that all mixturetypes are included in the constructed set Q by creating L subsamples.
Claim 2.
For any δ > 0 , we have P ( H L ) ≥
−
δ with L chosen according to the criteria describedbelow.
Proof : Denote Z Lk as the event that k -th mixture type is not being chosen as seed in the L trials.
Similarly, we can define Z ′
Lk for the fake scenario as described above.
We then have
[Equation]
Denote W k as the event that the convex hull formed by the set { ˆ q ℓ = 1 nT (cid:80) i ∈ I ℓ (cid:80)
Tt =1 Y ( t ) i
| π ( i ) =
k }for mixture type k covers the true choice probability vector.
Note that each 1 T (cid:80) Tt =1 Y ( t ) i can beviewed as a sample mean of q k and by central limit theorem, it is symmetric with respect to q k , henceso are the ˆ q ℓ ’s.
According to Lemma 3 and Corollary 1,
[Equation]
when L ≥ 2 d −
Putting everything together, we have
[Equation]
For any δ > 0 , we can then choose L such that 1 − δ ≤ RHS of Equation (11) and L ≥ 2 d − 1 .
□
A.4 Proof of Theorem 1
We first discuss Corollary 2 and what “high probability” refers to.
Denote W k as the event that the convex hull formed by the subsamples for a mixture type k coversthe true choice probability vector, as in Appendix A.3.
If we have subsampled all mixture typesand for each type k , event W k occurs, we can obtain q k ∈ Conv ( Q ) .
Subsequently, we have
Conv ( { q k } 1 ,...,K ) ⊆
Conv ( Q ) .
On the other hand, we have already analyzed the probability for event that subsampled all mixturetypes and Y k occurs ∀ k to occur, which is precisely H L as defined above.
Specifically, it happens withprobability ≥
[Equation]
As L increases, this number quicklyincreases to 1.
This completes the proof of Corollary 2.
Finally, we combine all the results above and prove the provable convergence part in Theorem 1.
Proof : As illustrated in Figure 1, we want to show with probability ≥ 1 − δ
we have
[Equation]
First, S.1 is proved by Lemma 2 (Property 1: | g FW − (cid:80) k
α k q k | ≤ ϵ ) and Corollary 2 (Property 3:(cid:80) k
α k q k ∈ Conv ( Q ) ), together with the fact that P (cid:16)(cid:12)(cid:12)(cid:12)
[Equation]
(cid:80) Tt =
[Equation]
y
[Equation]
−
[Equation]
k
[Equation]
≤ ϵ (cid:17) > 1 − δ bycentral limit theorem.
Since √ T ( 1 T (cid:80) Tt =1 y t − (cid:80) k
α k q k ) d → N (0 , σ 2 ) , number of samples requiredis also in the order of 1 ϵ 2 log( 1 δ ) .
Second, S.2 is also a combined result by Frank-Wolfe’s solving a linear program as an intermediatestep (Property 2: ∀ k ′ , ˆ q k ′
∈ E ( Conv ( P )) , E ( · ) denoting the extreme point set of the input region)and by construction using Algorithm 2 (Property 4: ∀ q ∈ E ( Conv ( Q )) , ∃ k s.t.
∥ q − q k ∥ ≤ ϵ ).
Subsequently, we can show S.3 .
Denote K ′ as the number of mixtures output by the SSRFW algorithm.
Using S.1 , we first write
[Equation]
According to S.2 , ∃ π such that π ( k ′ )
=
k and we can write ˆ q k ′ =
q π ( k ′ )
+ ϵ ′ where | ϵ ′ | ≤ ϵ ′ .
Rearranging Eqn.
(10) gives(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)
K (cid:88) k
= q k   (cid:88) k
[Equation]
(cid:12)(cid:12)(cid:12)
[Equation]
By triangle inequality, we get
[Equation]
Since q k is some arbitrary non-zero vector, we must have (cid:16)(cid:80) k ′ :
π ( k ′ )
=
k ˆ α k ′ − α k (cid:17) ≤ 2 ϵ ′ , ∀ k , whichcompletes S.3 by letting ϵ = 2 ϵ ′ .
Note the above result holds assuming K ′ ≥ K .
To see why this is always the case, consider thelinear system Q x =
[ q 1 q 2 . . .
q K ][ x 1 , x 2 , . .
, x K ] ⊤ = g , where Q ∈ R M × K and g = (cid:80)
Kk
=
α k q k .
According to Definition 1, D KS ( q k , q k ′ )
≥ ϵ , we know that all q k ’s are linearly independent.
Since
M ≥ K , rank ( Q ) =
rank ( Q | g ) =
K .
The linear system has a unique solution that x = α , whereall x k ’s are non-zero.
On the other hand, we have g SSRFW
=
(cid:80) K ′ k ′
=
1 ˆ α k ′
ˆ q k ′ and ∥ g SSRFW − g ∥
≤ ϵ .
Assume K ′ <
K , then upto a difference of ϵ , the linear system ˆ Q x = g , where ˆ Q =
[Equation]
∈
R M × K ′ , is inconsistent.
In other words, we will not be able to obtain a g SSRFW that is ϵ -close
tog , making it impossible to reach the stopping condition in the SSRFW algorithm.
Therefore, thealgorithm will keep going for more iterations, until we have at least K ′ =
K .
Finally, according to Theorem 2, Theorem 3, the sample complexity is O ( 1 ϵ 2 log( 1 δ )) .
□
Additional discussion.
In the proof, we showed that number of mixtures returned by SSRFW , K ′ , isat least the ground truth number of mixtures, K .
A natural question to ask is that how the misalignednumber of mixtures affect the learning result, if K ′ ̸
= K .
In many situations, this would not be aproblem.
Consider the case that ∃ k 1 , k 2 , such that π ( k 1 ) =
π ( k 2 )
=
k while the rest are all one-to-onemapping.
According to Theorem 1, we have P ( | q k i − q k | < ϵ ) ≥ 1 − δ , for i
= 1 , 2 and 588 P ( | α k 1 + α k 2 − α k | < ϵ ) ≥ 1 − δ .
We can view the ground-truth MMNL model as an ( M +1) -MNLmodel, where the original k -th mixture is now divided into two MNL components which share theidentical logit parameters, where one of them has mixture weight α k 1 and the other one α k − α k 1 .
Itis not hard to see that for the first component, we learned the correct mixture weight with an ϵ -closelogit vector
ˆ q k 1 while for the second, the mixture weight is off by at most ϵ with an ϵ -close logitvector
ˆ q k 2 .
Finally, we give some comment on the mapping function π .
Note that we do not need this informationother than using it as a tool in the proofs, though we can design heuristics to learn the mapping.
In real world applications, we do not know the ground truth parameters, so we cannot derive suchmapping anyways.
On the other hand, if we run the algorithm multiple times, we will get differentresults due to the stochasticity embedded in the algorithm.
In general, the lexicographic order of themixtures is not important and can be reordered arbitrarily.
B Numerical Experiments
In this section, we demonstrate the performance of our algorithm using numerical simulationand on the Nielsen Consumer Panel Data ( https://www.chicagobooth.edu/research/kilts/datasets/nielsenIQ-nielsen ).
B.1 Simulation Study
Consider in a setting where there are N consumers (i.e. decision makers) who make purchasedecisions among a set of M products for T periods.
Working with synthetic data allows us tomeasure the model performance regarding consumer type recovery since we have the knowledge ofthe underlying ground truth.
Data Generation As an example experiment, we use following parameters: N = 2000 , K =5 , M = 10 , d = 10 .
We vary the total number of experiment epochs from T = 5 to 300 .
Mixturetypes are indexed by capital letters, i.e., “A", “B", “C", “D", “E", and products are indexed by numbersfrom 1 to 10 .
In the experiment, we also include an offset option, allowing the consumer to choose tonot buy anything.
Features of different options and preference vector β k ’s are randomly generated in the interval[ − 1 , 1] .
α k ’s are randomly generated such that (cid:80) k
α k = 1 and the minimum mixture proportionmin k
α k ≥ 1 K +3 to ensure that not a particular type is under-represented.
We set L = 75 .
Subsample Purity We first examine the purity of each subsample obtained.
The reason we areinterested in this quantity is that if a constructed subsample has high purity, it means that the empiricalˆ q we obtained from the subsample contains only one mixture type and thus can provide an estimatefor the logit vector for that type with high accuracy.
Figure 3 shows the average subsample purity withrespect to the total number of experiment epochs.
We can see that the constructed subsamples canachieve 90% purity with as few as 30 experiment epochs, and quickly reaches 99% around T = 150 .
Quality of Q Next we evaluate the quality of the constructed set Q after applying Algorithm 2 tothe simulated data.
According to SSRFW , the estimated choice probability outcome is essentially asubset of these candidate vectors in Q —the ones emitted by SSRFW at each iteration.
Therefore, thehigher the quality of Q —in the sense it is concentrated near the ground truth—the better mixtureestimation we can obtain using the learning algorithm SSRFW .
We claim each subsample is a representation of a particular mixture type in the ground truth if themajority of the selected consumers in that subsample are of that type.
For instance, if a subsamplecontains 47 of type A, 2 of type B, and 1 of type E, we will evaluate the estimates ˆ q from thissubsample against ˆ q A .
In Figure 4, we plot the distribution of choice probability values with respect to each product in theoption set for all mixture types, with Figure 4a showing the result with 50 experiment epochs and
Figure 4b with 300 experiment epochs.
[Figure]
These figures show that the estimated choice probability values are very concentrated near the groundtruth when T = 300 and are also reasonably good even with a small T = 50 , which we will proceedwith for the following evaluation.
Mixture Recovery After feeding the constructed Q to Algorithm 1, SSRFW generates 8 mixturetypes in terms of logit vectors (as expected, larger than the ground truth number of mixtures).
Wenow want to compare these generated logit vectors against the ground truth.
We refer to the mappingπ ( j ) =
i as the “closest type” between a generated type j and an ground truth (GT) type i againusing the majority rule mentioned above.
[Figure]
Figure 5 compares j and π ( j ) using the cumulative distribution (CDF) of the choice probabilityvectors, with orange lines representing learning outcome and blue lines the ground truth.
We cansee that the results from SSRFW are very close to the true mixture CDFs (Figure 5a).
In contrast, theoriginal FW algorithm [Jagabathula et al., 2020] does not recover the true choice probabilities (Figure5b), which are mostly boundary-types that correspond to the extreme points of the feasible region tothe individual LP subproblem in each FW iteration.
We further compare the mixture weight estimates α ’s in Table 1.
From the table, we can see that if wehave a one-to-one mapping (such as A and E), the α estimates from the algorithm are close to the truevalues.
If we have many-to-one mappings, i.e., the algorithm outputs multiple mixtures for the samelatent class, the sum of the estimated mixture weights is also close to the true values of each mixture.
A Further Comparison with Original FW and EM We define another performance measure asthe weighted average distance between the predicted choice probabilities and its closest ground truth,i.e. ∥ (cid:80) k
[Equation]
q
[Equation]
k
[Equation]
i )=
k ˆ q Alg i ∥ , where α k and q k are ground truth and Alg is one of FW ,
SSRFW , and EM.
We vary the number of experiment epochs T to examine how additional data canhelp improve the performance of MMNL learning algorithms.
[Table]
In Figure 6a, we can see that the total discrepancy is consistently smaller using the SSRFW compared tothe original FW , which further justifies the good performance of our algorithm in predicting individualmixtures.
In Figure 6b, we can see that both SSRFW and EM benefit from having more repetitivechoice data, especially when T ≤ 30 , which agrees with the observations we learned from Sample
Purity and Quality of Q part.
[Figure]
Since EM requests the number of mixtures as an additional hyperparameter, we compare the performance of SSRFW to EM with various different hyperparameter ˜ K values, as shown in Figure 6b.
Wecan see from the plot that when ˜ K < K , SSRFW usually outperforms the EM algorithm.
On the otherhand, when ˜ K > = K , EM achieves a marginally better performance.
However, in general we donot have this information, therefore we adopted the conventional strategy (c.f. Train [2008]) to use
AIC/BIC to determine the best ˜ K , which corresponds to the lowest of these two criteria respectively.
Both AIC/BIC measure the relative quality of statistical models for a given dataset, by balancing thetrade-off between goodness of fit and the simplicity of the model and are commonly used in modelselection.
In our experiment, K = 3 gives the lowest value for both AIC and BIC.
This indicates weshould choose K = 3 , yet it will result in worse estimation quality as shown in Figure 6b.
Last but not least, we can see that the major issue of EM is that it suffers from its instability.
In reality,there is no way to tell whether the EM method is stuck at local optimum vs global optimum.
Incontrast, our algorithm is designed to give theoretical guarantees on the estimators.
In summary, thecombination of SSRFW ’s theoretical guarantees on the learning result, sample complexity analysis, aswell as the capability to recover individual level of mixture parameters makes the unique contributionof this work.
One limitation in our framework is related to data collection.
For each decision maker, we require aseries of historical choice data over a fixed set of options in order to get a reliable empirical cumulativedistribution of their choice probability vector.
However, this problem is not unique to our frameworkand EM also requires the same type of data.
Thanks to the high digitalization level of many industries, we can see that such data are widely available in a variety of settings, ranging from training NLPmodels with mixture of topics of text corpus to learning consumer preferences via data that recordtheir purchases over some product sets, which we will discuss next.
B.2 Experiments on Nielsen Consumer Panel Data
In this section, we demonstrate how we have applied SSRFW to the Nielsen Consumer Panel data.
Thiscomprehensive dataset is provided by the Kilts Center for Marketing at the University of Chicago
Booth School of Business, NielsenIQ, and Nielsen.
It contains panelists (i.e., households) purchasedecisions on grocery items included in the NielsenIQ food and nonfood departments (roughly 1.4M
UPC codes) dated back to 2004 with regular annual updates The panel size varies from 40K to 60Kand the characteristics include product description, brand, multipack, size, etc.
This panel data iswidely used for longitudinal studies in marketing science.
B.2.1 Data Curation
We consider applying the algorithm to a substitute set of products under a particular category.
This isa realistic setting as consumers usually choose one item from the substitution set.
We curated data forsix different categories, including yogurt, cereal, snack, candy, soft drinks and pet food and providesome summary statistics in Table 2.
[Table]
.2.2 Experiment Setup
We cannot evaluate the model performance the same as in simulation studies since we no longerhave the ground truth knowledge.
Instead, we split the data into a training and test set, with theformer used for learning the model parameters and the latter for evaluation.
Specifically, we apply ouralgorithm to the training set, and use the learned parameters to compute the theoretical aggregatedmarket share (cid:80) k
ˆ α k ˆ q k .
Then we compute its distance to the aggregated market share of the test set,i.e. ∥ (cid:80) k
ˆ α k ˆ q k − y test ∥ .
The assumption is, if we have similar mixture composition in the trainingand test set, then the estimated parameter values from the training set should yield aggregated choiceprobability values close to that of the test set.
To avoid randomness in the data split, we used a 10-foldcross validation, with the entire process repeated for five times.
B.2.3 Results
Figure 7 plots the distribution of ∥ (cid:88) k
[Equation]
y test ∥ from the repeated runs of both algorithms.
Wecan see that SSRFW in general outperforms the original FW algorithm in that the discrepancy isdistributed close 0.
[Figure]
Figure 8 plots the deviation of product-level choice probability values from the test set,|
[Equation]
y test j , for j = 1 , 2 , 3 , 4 , 5 .
The light orange horizontal line indicates a zero deviation and we observe that the predicted aggregated choice probability per product level from SSRFWis more concentrated around zero than the original FW.
In addition, there is a smaller variance withrespect to different runs.
[Figure] [Figure]
Next, we examine the number of iterations required until convergence.
We also report the percentageof active directions for the given number of iterations.
We think these two metrics measures theeffectiveness of each direction being chosen during the learning process.
From Figure 9, we can see the FW algorithm usually requires more iterations, yet the percentageof active directions is lower than SSRFW .
This suggests that more computation is wasted during thelearning process.
The last metric we look at is the percentage of boundary types in the learning result.
Figure 10 showsthat the original FW still exhibits the same problem of generating boundary-type logit vectors while
SSRFW is much less likely to suffer from this problem.
Finally, as requested by Kilts Center for Marketing at the University of Chicago School of Business,we make the following disclaimers:
• Researcher(s)’ own analyses calculated (or derived) based in part on data from NielsenConsumer LLC and marketing databases provided through the NielsenIQ Datasets at theKilts Center for Marketing Data Center at The University of Chicago Booth School ofBusiness.•
The conclusions drawn from the NielsenIQ data are those of the researcher(s) and do notreflect the views of NielsenIQ.
NielsenIQ is not responsible for, had no role in, and was notinvolved in analyzing and preparing the results reported herein.