Original: Dynamic pricing and assortment under a contextual MNL demand
Revised: Dynamic pricing and assortment under a contextual MNL demand

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: We consider dynamic multi-product pricing and assortment problems under anunknown demand over T periods, where in each period, the seller decides on theprice for each product or the assortment of products to offer to a customer whochooses according to an unknown Multinomial Logit Model (MNL).
Revised: We consider dynamic multi-product pricing and assortment problems under an unknown demand over T periods, where in each period, the seller decides on the price for each product or the assortment of products to offer to a customer who chooses according to an unknown Multinomial Logit Model (MNL).

Original: Such problemsarise in many applications, including online retail and advertising.
Revised: Such problems arise in many applications, including online retail and advertising.

Original: We proposea randomized dynamic pricing policy based on a variant of the Online Newton
Revised: We propose a randomized dynamic pricing policy based on a variant of the Online Newton

Original: Step algorithm (ONS) that achieves a O ( d √ T log( T )) regret guarantee under anadversarial arrival model.
Revised: Step algorithm (ONS) that achieves a O ( d √ T log( T )) regret guarantee under an adversarial arrival model.

Original: We also present a new optimistic algorithm for theadversarial MNL contextual bandits problem, which achieves a better dependencythan the state-of-the-art algorithms in a problem-dependent constant κ (potentiallyexponentially small).
Revised: We also present a new optimistic algorithm for the adversarial MNL contextual bandits problem, which achieves a better dependency than the state-of-the-art algorithms in a problem-dependent constant κ 2 (potentially exponentially small).

Original: Our regret upper bound scales as ˜ O
Revised: Our regret upper bound scales as ˜ O

Original: ( d √ κT + log( T ) /κ  ) ,which gives a stronger bound than the existing ˜ O ( d √ T/κ  ) guarantees.
Revised: ( d √ κ 2 T + log( T ) /κ 2 ) , which gives a stronger bound than the existing ˜ O ( d √ T/κ 2 ) guarantees.

Original:  Introduction
Revised:  Introduction

Original: In this paper, we consider the contextual dynamic pricing and assortment optimization problemsfaced by a seller who sequentially observes a contextual demand under bandit feedback.
Revised: In this paper we consider the contextual dynamic pricing and assortment optimization problems faced by a seller who sequentially observes a contextual demand under bandit feedback.

Original: The goal isto learn the underlying model parameters in order to minimize the seller’s cumulative regret.
Revised: The goal is to learn the underlying model parameters in order to maximize the seller’s profit.

Original: Theseproblems arise in numerous applications, including pricing and product recommendation in onlineretail, as well as click-through rate predictions for web search results.
Revised: These problems arise in numerous applications, including pricing and product recommendation in online retail, as well as click-through rate predictions for web search results.

Original: Sequential learning is especiallyimportant in settings involving short selling seasons or where no historical data is available.
Revised: Sequential learning is especially important in settings involving short selling seasons or where no historical data is available.

Original:  In both the dynamic pricing and assortment problems, we consider a seller who sells a set of Nproducts over a time horizon of T periods.
Revised: Problem formulation. In both the dynamic pricing and assortment problems, we consider a seller who sells a set of N products over a time horizon of T periods.

Original: A new customer arrives in each period t and is offered aset of products or prices.
Revised: A new customer arrives in each period t and is offered a set of products or prices.

Original: In the pricing problem, the customer arrives with a consideration set and theseller has to decide on the prices of the different products in the consideration set.
Revised: In the pricing problem, the customer arrives with a consideration set and the seller has to decide on the prices of the different products in the consideration set.

Original: In the assortmentproblem, the seller selects the subset of products to offer to the customer.
Revised: In the assortment problem, the seller selects the subset of products to offer to the customer.

Original: In both cases we assumethat the customer’s purchase decisions are made according to a multinomial logit model (MNL)([33],[29]) which is widely used in modeling customer preferences.
Revised: In both cases we assume that the customer’s purchase decisions are made according to a multinomial logit model (MNL)([32], [28]) which is widely used in modeling customer preferences.

Revised: In both problems, the objective is to maximize the seller’s expected profit, which is equivalent to minimizing its cumulative expected regret (i.e., the difference between the optimal value and the value obtained by following a given policy).

Revised: In this paper, we study a feature-based model, where the product utilities are a function of both products and customers features.

Revised: In particular, we assume that in each period t , the customer’s utility for each product j ∈ [ N ] can be written under the form x ⊤ t,j θ ∗ , where θ ∗ is an unknown model parameter and x t,j ∈ R d is a feature vector, which can be adversarially chosen (see Sections 2.1 and 3.1 for exact definitions of the problems).

Revised: Furthermore, we allow feature-dependent price sensitivities in our pricing setting and suppose that the price sensitivities can be expressed as x ⊤ t,j α ∗ for some unknown parameter α ∗ .

Revised: Typically, the number of products is significantly larger than the number of features ( d << N ).

Original: Non-contextual dynamic pricing, where the seller sells identical products to the customers over time,was initially investigated by [23] under various assumptions on the demand curve, and has sincebeen extensively studied (see for instance [8], [15], [7] and [16]).
Revised: The objective is to learn across the products and obtain a regret upper bound scaling with d instead of N . Literature review. Non-feature-based dynamic pricing, where the seller sells identical products to the customers over time, was initially investigated by [22] under various assumptions on the demand curve, and has since been extensively studied (see for instance [8], [14], [7] and [15]).

Original: We refer the reader to [14] for anin-depth survey of the area.
Revised: We refer the reader to [13] for an in-depth survey of the area.

Original: On the other hand,  dynamic assortment optimization was first studiedby [9] under the assumption of independent demand for the different products.
Revised: On the other hand, non-feature-based dynamic assortment optimization was first studied by [9] under the assumption of independent demand for the different products.

Original: Dynamic assortmentunder a MNL choice model has been recently considered (see for instance [37], [38] and [40]).
Revised: Dynamic assortment under a MNL choice model has been recently considered (see for instance [36], [37] and [39]).

Original: Inparticular, UCB and Thompson sampling-based policies are proposed in [3] and [4].
Revised: In particular, UCB and Thompson sampling-based policies are proposed in [3] and [4].

Original: These policiesachieve optimal regrets of O ( √ NT ) in the non-feature based setting.
Revised: These policies achieve optimal regrets of O ( √ NT ) in the non-feature based setting.

Original: Another related stream ofliterature is that of combinatorial bandits (see for instance [11], [24] and [35]), and in particular, top kcombinatorial bandits ([36]). In this framework, the agent can pull a subset of arms of cardinality less than k in each round and the total reward obtained is a function of the individual rewards for the armsplayed. However, the reward obtained in our setting for each individual arm depends on the wholeset of arms played in period t , whereas the rewards are supposed independent in the aforementionedworks.
Revised: In this paper, we consider feature-based pricing and assortment problems.

Original: Feature-based dynamic pricing has recently received a lot of attention and been well characterizedwhen only a single product is involved at each step (see for instance  [20],[21], [5] [6], [13], [28],[27], [25] and [34]). The proposed policies achieve the optimal regrets of ˜ O (log( T )) or ˜ O ( √ T ) ,depending on whether a well-separability assumption is satisfied or not.
Revised: Feature-based dynamic pricing has recently received a lot of attention. Most of the existing work study a single product pricing setting under various demand models (linear, binary, generalized linear) and assume that the price sensitivity of each product (i.e., the price coefficient in the demand model) is a known constant (see for instance [19], [20], [5] , [12], [27], [26], [24] and [33]).

Original: We would like to highlight[6], which is the firstwork to incorporate contextual information about the customers under the formof feature-dependent price sensitivities.
Revised: In the single product setting, the most related work is [6], which is the first to incorporate contextual information about the customers under the form of feature-dependent price sensitivities.

Original: The MNL contextual dynamic assortment problem is an extension of the generalized linear banditsproblem, which has been widely studied (see for instance [18], [26] and [2]).
Revised: The benefit of this assumption is illustrated on real datasets. [6] considers a high dimensional setting with a sparsity assumption under a generalized demand model and propose a policy with near-optimal regret.

Original: UCB and TS basedpolicies have recently been proposed for the  MNL contextual dynamic assortment problem (see forinstance [31], [32] and [12]). Problem formulation.
Revised: However, they assume i.i.d. features, whereas we consider a more specific demand model (MNL model), but with adversarial features. In [21], is considered a multiple product version of the problem, with feature-based price sensitivities and under the assumption that the demand follows a MNL model.

Original: In this paper, we consider a setting where multiple products are offeredto the customer in each period.
Revised: In this last work, the features are also assumed to be i.i.d.

Original: We study a feature-based model, where the  product utilities are afunction of both products and customers features. In particular, we assume that in each period t ,the customer’s utility for each product j ∈ [ N ] can be written under the form x ⊤ t,j θ ∗ , where θ ∗ isan unknown model parameter and x t,j ∈ R d is a feature vector, which can be adversarially chosen.
Revised: Our multi-product pricing setting directly extends [21], and captures the single product setting with unknown (and feature-based) price sensitivity and adversarial features under a binary demand model with logistic noise, for which, to the best of our knowledge, no algorithm with near-optimal regret is known yet. The MNL feature-based dynamic assortment problem is a variant of the contextual generalized linear bandits problem (see for instance [17], [25] and [2]) with a more complicated state space, in which multiple arms are pulled at the same time.

Original: Furthermore, we allow feature-dependent price sensitivities in our pricing setting and suppose thatthe price sensitivities can be expressed as x ⊤ t,j α ∗ for some unknown parameter α ∗ .
Revised: UCB and TS based policies have recently been proposed for the MNL contextual dynamic assortment problem (see for instance [31], [30] and [11]). However, these work suffer from a dependency in a problem-dependent constant κ 2 (potentially exponentially small), which captures the ’degree of non-linearity’ of the problem.

Original: Typically, thenumber of products is significantly larger than the number of features ( d << N ).
Revised: Another related stream of literature is that of combinatorial bandits (see for instance [10], [23] and [34]), and in particular, top k combinatorial bandits ([35]).

Original: The objective is tolearn across the products and obtain a regret upper bound scaling with d instead of N .
Revised: In this framework, the agent can pull a subset of arms of cardinality less than k in each round and the total reward obtained is a function of the individual rewards for the arms played. However, the reward obtained in our setting for each individual arm depends on the whole set of arms played in period t , whereas the rewards are supposed independent in the aforementioned works.

Original: 1.1 Our contributions.
Revised: 1.1 Our contributions.

Original: In this work, we introducetwo new algorithms for the dynamic contextual pricing and assortmentproblems.
Revised: In this work, we introduce new algorithms for the dynamic contextual pricing and assortment problems.

Original: Our main contribution is the following.
Revised: Our main contribution is the following.

Original: Dynamic pricing .
Revised: Dynamic pricing .

Original: We present a dynamic pricing policy for the multi-product MNL model withadversarial contexts and feature-dependent price sensitivities that achieves a O ( d log( T ) √ T ) regretbound.
Revised: We present a dynamic pricing policy for the multi-product MNL model with adversarial contexts and feature-dependent price sensitivities that achieves a O ( d log( T ) √ T ) regret bound.

Original: This is near-optimal given the Ω( √ T ) lower bound from [22].
Revised: This is near-optimal given the Ω( √ T ) lower bound from [21].

Original: Based on structural propertiesof the MNL model, and more specifically, on the self-concordant-like property of the MNL loglikelihood function, we propose to use a variant of the Online Newton Step method ([19]) to updateour estimators of the model parameters.
Revised: Based on structural properties of the MNL model, and more specifically, on the self-concordant-like property of the MNL log likelihood function, we propose to use a variant of the Online Newton Step method ([18]) to update our estimators of the model parameters.

Original: We combine it with a random price shock policy to forceexploration.
Revised: We combine it with a random price shock policy to force exploration.

Original: Closest to our pricing setting is [22], in which the authors design a multi-product dynamic pricingpolicy under a MNL choice model and feature-dependent price sensitivities.
Revised: Closest to our pricing setting is [21], in which the authors design a multi-product dynamic pricing policy under a MNL choice model and feature-dependent price sensitivities.

Original: The proposed algorithmachieves a ˜ O ( √ T ) regret.
Revised: The proposed algorithm achieves a ˜ O ( √ T ) regret.

Original: However, it relies on a bayesian assumption, namely the feature vectorsare drawn i.i.d.
Revised: However, it relies on a bayesian assumption, namely the feature vectors are drawn i.i.d.

Original: from some unknown distribution.
Revised: from some unknown distribution.

Original: Our work considers an adversarial context and usesan ONS method to update the parameters.
Revised: Our work considers an adversarial context and uses an ONS method to update the parameters.

Original: The connection with the Online Convex Optimizationframework has been exploited in the literature (see for example the stochastic online gradient descentin [20] for the single product setting without price sensitivities). The problem we consider is morechallenging since the presence of feature-dependent price sensitivities implies the existence ofuninformative prices (i.e. prices p such that no pricing policy canhope to learn the true modelparameters when repeatedly pricing at price p ).
Revised: Note that the connection with the Online Convex Optimization framework has been exploited in the literature (see for example the stochastic online gradient descent in [19] for the single product setting without price sensitivities), however, the problem we consider is more challenging since the presence of feature-dependent price sensitivities implies the existence of uninformative prices (i.e. prices p such that no pricing policy can learn the true model parameters when repeatedly pricing at price p ).

Original: Note that all prices are informative in [20].
Revised: Note that all prices are informative in [19].

Original: The linkbetween uninformative prices and the difficulty to design low regret policies was first pointed outby [8], which shows that no algorithm can achieve better regret than Ω( √ T ) in settings involvingsuch prices.
Revised: The link between uninformative prices and the difficulty to design low regret policies was first pointed out by [8], which shows that no algorithm can achieve better regret than Ω( √ T ) in settings involving such prices.

Original: We address this challenge by using appropriate randomized price shocks that forceexploration.
Revised: We address this challenge by using appropriate randomized price shocks that force exploration (we note that adding random shocks was first used in the context of dynamic pricing in [29]; however, our work is the first to simultaneously use random price shocks and an ONS based update, and the analysis of our policy differs from the analysis in [29]). Our results imply a O ( d log( T )) regret in the single product setting with adversarial contexts and without price sensitivity considered in [19], under the extra assumption that the noise follows a logistic distribution. This improves over the O ( √ T ) regret bound of [19] in this case (note that [19] studies more precisely the effect of drifts in the parameters). Note that in the same setting, but with feature-dependent price coefficients, our results also imply a ˜ O ( √ T ) regret.

Original: Adding random shocks is a classical exploration strategy in sequential learning problemsand context pricing see instance [ recent paper [41] obtained a logarithmic regret for the contextual single product pricing problemwithout price sensitivity through a variant of the online newton method.
Revised: We have become aware that a concurrent recent paper [40] also obtained a logarithmic regret for the contextual single product pricing problem without price sensitivity through a variant of the online newton method (for a more general demand model with strictly log-concave noise).

Original: We would like to underscorethat our theoretical results were obtained independently from this work.
Revised: We would like to underscore that our theoretical results were obtained independently from this work.

Original: Furthermore, the algorithmproposed in [41] uses the exp-concavity parameter in the descent step and the regret upper boundscales with this potentially exponentially small constant.
Revised: Furthermore, the algorithm proposed in [40] uses the exp-concavity parameter in the descent step and the regret upper bound provided scales with this potentially exponentially small constant.

Original: By leveraging the self-concordance propertyof the logistic loss, we design an algorithm which does not use such parameter.
Revised: By leveraging the self-concordance property of the logistic loss, we design an algorithm which does not use such parameter, which may be better in practical applications.

Original: We present in
Revised: However, our regret upper bound still depends on this constant.

Original:  Appendix D some numerical comparison between the two algorithms, which confirms that ouralgorithm achieves a significantly better regret than the one in [41] as the exp-concavity parameterdecreases.
Revised: We present in Appendix D some numerical comparison between the two algorithms, which confirms that our algorithm achieves a significantly better regret than the one in [40] as the exp-concavity parameter decreases.

Original: In addition to our main contribution, we present a new algorithm for MNL contextual bandits.
Revised: In addition to our main contribution, we present a new algorithm for MNL contextual bandits.

Original: MNL contextual bandits .
Revised: MNL contextual bandits .

Original: We consider a setting with uniform product revenues and propose a new
Revised: We consider a setting with uniform product revenues and propose a new

Original: UCB-based algorithm for the MNL assortment problem with adversarial contexts.
Revised: UCB-based algorithm for the MNL assortment problem with adversarial contexts.

Original: Our algorithmachieves a regret bound of order ˜ O
Revised: Our algorithm achieves a regret bound of order ˜ O

Original: ( d √ T ) , which is optimal as a function T .
Revised: ( dK √ T ) , which is optimal as a function T .

Original: One major limitationof the state-of-the-art algorithms for the MNL contextual bandits problem ([31], [32] and [12]) isthat the regret bounds scale with a problem dependent constant, which we will refer to as κ  (see
Revised: One major limitation of the state-of-the-art algorithms for the MNL contextual bandits problem ([31], [30] and [11]) is that the regret bounds scale with a problem dependent constant, which we will refer to as κ 2 (see

Original: Section 3 for a formal definition); κ  quantifies the level of non-linearity of the model and can beexponentially small even for moderate size instances ([17]).
Revised: Section 3 for a formal definition); κ 2 quantifies the level of non-linearity of the model and can be exponentially small even for moderate size instances ([16]).

Original: Our regret bound can be expressed as
Revised: Our regret bound can be expressed as

Original: C 1 d √ κT + C 2 d 2 log ( T ) /κ.
Revised: C 1 d (cid:113)(cid:80) T t =1 κ ∗ 2 ,t + C 2 d 2 log( T ) /κ

Revised: 2 (where κ ∗ 2 ,t ≤ 1 are problem dependent constants, whose value decrease as the model is further away from the linear model).

Original: Therefore, we give a significantly stronger bound than [31], whoseregret term of order √ T scales with 1 /κ  .
Revised: Therefore, we give a significantly stronger bound than [31], whose regret term of order √ T scales with 1 /κ 2 .

Original: Note that our first order term improves for small values ofκ (when the model is further from the linear model).
Revised: Note that our first order term improves for small values of { κ ∗ 2 ,t } .

Original: Moreover, a prior knowledge of the value of κis often presupposed in the existing algorithms, which may be a major hindrance to their practicalimplementation.
Revised: Moreover, a prior knowledge of the value of κis often presupposed in the existing algorithms, which may be a major hindrance to their practical implementation.

Original: The quantity κ  appears for example in the design of the exploration bonuses ofthe UCB-MNL algorithm of [31].
Revised: The quantity κ 2 appears for example in the design of the exploration bonuses of the UCB-MNL algorithm of [31].

Original: Our algorithm does not rely on an a priori knowledge of κ  .
Revised: Our algorithm does not rely on an a priori knowledge of κ 2 .

Original: Ourpolicy is based on optimistic parameter search instead of exploration bonuses, as used in [31].
Revised: Our policy is based on optimistic parameter search instead of exploration bonuses, as used in [31].

Original: Theanalysis relies on a concentration result on the MLE estimators, which uses a generalization of the
Revised: The analysis relies on a concentration result on the MLE estimators, which uses a generalization of the

Original: Bernstein-like tail inequality for self-normalized vectorial martingales in [2] and [17], and leveragesthe self-concordant property of the MNL log loss.
Revised: Bernstein-like tail inequality for self-normalized vectorial martingales in [2] and [16], and leverages the self-concordant property of the MNL log loss.

Original: We would like to mention that a similar result is achieved by [17] and [2] in the logistic banditssetting.
Revised: We would like to mention that a similar result is achieved by [16] and [2] in the logistic bandits setting.

Original: Our results cannot, however, be derived from these two works as the MNL contextual banditsproblem cannot be formulated as a generalized bandits problem [12].
Revised: Our results cannot, however, be derived from these two works since the MNL contextual bandits problem cannot be formulated as a generalized bandits problem [11].

Original: In this work, we show that thetechniques from [17] and [2] can be efficiently extended to the MNL bandit setting. This involvesovercoming a few technical challenges that are specific to the MNL problem.
Revised: In this work, we show that the techniques from [16] and [2] can be efficiently extended to the MNL bandit setting, which involves overcoming a few technical challenges that are specific to the MNL problem.

Original: Notations.
Revised: Notations.

Original: For any vector x ∈ R d and any positive definite matrix M ∈ R d × d , let || x
Revised: For any vector x ∈ R d and any positive definite matrix M ∈ R d × d , let || x

Original: || M = √ x T Mx .
Revised: || M = √ x T Mx .

Original: Also let B p (0 , W ) be the d -dimensional ball of radius W under the norm ℓ p .
Revised: Also let B dp (0 , W ) be the d -dimensional ball of radius W under the norm ℓ p .

Original: In the special case of thenorm ℓ 2 , we will drop the index and refer to the d -dimensional ball as B  (0 , W ) .
Revised: In the special case of the norm ℓ 2 , we will drop the index and refer to the d -dimensional ball as B d (0 , W ) .

Original: For two symmetricmatrices A, B ∈ R d × d , A ⪰ B means that A − B is semi-definite positive.
Revised: For two symmetric matrices A, B ∈ R d × d , A ⪰ B means that A − B is semi-definite positive.

Original: For n ∈ N , we use thenotation [ n ] to denote the set { 1 , ..., n } .
Revised: For n ∈ N , we use the notation [ n ] to denote the set { 1 , ..., n } .

Original: When it is not clear from the context, wewill use a bold fontto denote vectors.
Revised: When it is not clear from the context, we use a bold font to denote vectors.

Original:  Multi-product dynamic pricing 
Revised:  Multi-product dynamic pricing 2.1 Model setting and preliminaries

Original: 2.1 Model setting and preliminaries

Original: We consider a dynamic pricing problem for a seller with N products represented by feature vectorsx 1 , x 2 , ..., x N ∈ R d .
Revised: We consider a dynamic pricing problem for a seller with N products represented by feature vectors x 1 , x 2 , ..., x N ∈ R d .

Original: In each period t , a customer arrives with context z t ∈ R d and a considerationset S t ⊆
Revised: In each period t , a customer arrives with context z t ∈ R d and a consideration set S t ⊆

Original: [ N ] . The sequence of customer features and consideration sets can be chosen adversarially.
Revised: [ N ] , which can be chosen adversarially.

Original: The seller offers prices p t,j for all j ∈ S t and the customer makes a selection based on a MNL modelwhere the attraction parameter depends on product and customer features.

Original: For ease of notation, weconsider a more general setting where customer features z t and consideration set S t are representedby | S t | adversarially chosen feature vectors x t, 1 , ..., x t, | S t | ∈ R d .
Revised: For ease of notation, we consider a more general setting where customer features z t and consideration set S t are represented by | S t | adversarially chosen feature vectors x t, 1 , ..., x t, | S t | ∈ R d .

Original: Notations.

Original: For all t ∈ [ T ] , let k t = | S t | and let K ≤ N be an upper bound on k t for all t .
Revised: For all t ∈ [ T ] , let k t = | S t | and let K ≤ N be an upper bound on k t for all t .

Original: We alsodefine X t as the matrix whose rows are x t,j for j ∈ [ k t ] and we refer to it as the context at time t . 
Revised: We also define X t as the matrix whose rows are x t,j for j ∈ [ k t ] and we refer to it as the context at time t . In each period t , 1.

Revised: The seller observes x t, 1 , ..., x t,k t ∈ R d and offers prices p t,j for all j ∈ [ k t ] .

Original: After seeing the prices { p t,j } j ∈ [ k t ] , the customer  purchases  product j ∈ [ k t ] ∪ { 0 } with probability:
Revised: The customer observes the prices, then purchases one single product j ∈ [ k t ] ∪{ 0 } . The probability to purchase each product j is given by:

Original: [Equation]
Revised: [Equation]

Original: where θ ∗ , α ∗ ∈ R
Revised: where θ ∗ , α ∗ ∈ R

Original: d are two model parameters.
Revised: d are two model parameters, which are unknown to the policy maker.

Original: For all j ∈ [ k t ] , x ⊤ t,j α ∗ represents the price sensitivityof customer t for product j .
Revised: For all j ∈ [ k t ] , x ⊤ t,j α ∗ represents the price sensitivity of customer t for product j .

Original: Note that the customer has always the possibility to leave without makingany purchase (by selecting product j = 0 ).
Revised: Note that the customer has always the possibility to leave without making any purchase (by selecting product j = 0 ).

Revised: The policy maker observes only the customer’s purchase decision.

Original: The binary variable y t,j indicates whether the customerhas purchased product j at time t .
Revised: The binary variable y t,j indicates whether the customer has purchased product j at time t .

Original: For brevity, let γ ∗ =
Revised: For brevity, let γ ∗ =

Original: ( θ ∗ , α ∗ ) and ˜ x t,
Revised: ( θ ∗ , α ∗ ) and ˜ x t,

Original: j = [ x t,j , − p t,j x t,j ] .
Revised: j = [ x t,j , − p t,j x t,j ] .

Original: We can more simply write the utilityu t,j = x ⊤ t,j θ ∗ − x ⊤ t,j α ∗ p t,j associated with each product as u t,j = ˜ x
Revised: We can more simply write the utility u t,j = x ⊤ t,j θ ∗ − x ⊤ t,j α ∗ p t,j associated with each product as u t,j = ˜ x

Original: ⊤ t,j γ ∗ .
Revised: ⊤ t,j γ ∗ .

Revised: Also, for each t and for a given estimator

Revised: γ t of the true parameter at time t , we denote by q t,j ( γ t , ⃗p t ) the estimated purchase probability for product j (obtained by replacing γ ∗ by γ t in (1)).

Original: We make the followingassumptions, which are standard in the dynamic pricing literature:
Revised: We make the following assumptions, which are standard in the dynamic pricing literature:

Original: Assumption 2.1.
Revised: Assumption 2.1.

Original: For all t, j ∈ [ k t ] , || x t,j || 2 ≤ 1 .
Revised: For all t, j ∈ [ k t ] , || x t,j || 2 ≤ 1 .

Original: Assumption 2.2.
Revised: Assumption 2.2.

Original: || ( θ ∗ , α ∗ )
Revised: || ( θ ∗ , α ∗ )

Original: || 2 ≤ W for some known constant W .
Revised: || 2 ≤ W for some known constant W .

Original: Although the contexts { x t,j } t ∈ [ T ] ,j ∈ [ k t ] can be adversarially chosen, we need to slightly restrict the149 set of feasible contexts to guarantee the positiveness of the price sensitivity for all products.
Revised: Although the contexts { x t,j } t ∈ [ T ] ,j ∈ [ k t ] can be adversarially chosen, we need to slightly restrict the set of feasible contexts to guarantee the positiveness of the price sensitivity for all products.

Original: Following[22], we make the following assumption.
Revised: Following [21], we make the following assumption, which implies that the price-sensitivity of each product is not too close to zero.

Revised: This assumption may be reasonable in practice: when the price of a product goes to infinity, its utility should decrease significantly.

Original: Assumption 2.3.
Revised: Assumption 2.3.

Original: For all t ∈ T, j ∈
Revised: For all t ∈ T, j ∈

Original: [ k t ] , x ⊤ t,j α ∗ ≥ L for some known constant L . 
Revised: [ k t ] , x ⊤ t,j α ∗ ≥ L for some known constant L . Assumption 2.4.

Revised: The upper bound K on the number of products in each set S t is constant.

Original: Pricing policy and Benchmark.
Revised: Pricing policy and Benchmark.

Original: We consider non-anticipating pricing policies π , which depend onlyon the history up to time t − 1 , H t − 1 = ( X 1 , ⃗p 1 , ⃗y 1 , ..., X t − 1 , ⃗p t − 1 , ⃗y t − 1 ) , and the current context
Revised: We consider non-anticipating pricing policies π , which depend only on the history up to time t − 1 , H t − 1 = ( X 1 , ⃗p 1 , ⃗y 1 , ..., X t − 1 , ⃗p t − 1 , ⃗y t − 1 ) , and the current context

Original: X t .
Revised: X t .

Original: The objectiveof the seller is to  minimize its cumulative expected regret:
Revised: The objective is to design a pricing policy so as to minimize the sellers’ cumulative expected regret:

Original: [Equation]
Revised: [Equation]

Original: where ⃗p ∗ t is the optimal vector of prices in period t .
Revised: where ⃗p ∗ t is the optimal vector of prices in period t .

Original: The expectation is taken over the random feedbackand any source of randomization in the policy. 
Revised: The expectation is taken over the random feedback and any source of randomization in the policy. Let p max :

Original: 2.2 Dynamic pricing policy
Revised: = 1+ K max( W, 1) L + 1 K . We show in Lemma ?? that for the algorithm we propose, the prices posted at each time t satisfy p t,j ∈ [0 , p max ] for all j ∈ [ k t ] . Hence we can consider policies π that only post prices in [0 , p max ] .

Revised: Finally, we define the following constant, which provides information about how much a feasible demand curve can deviate from the linear model (a smaller κ 1 implies a larger deviation from the linear model).

Revised: [Equation]

Revised: Note that our pricing policy does not directly use the value of κ 1 .

Revised: However, this constant still appears in our regret bound (see Section 2.2 for more discussion).

Revised: 2.2 Dynamic pricing policy

Original: Our algorithm combines the two following ingredients: a variant of the ONS method and randomprice shocks.
Revised: Our algorithm combines the two following ingredients: a variant of the ONS method and random price shocks.

Original: In particular, we maintain estimators of the parameters which are updated in eachiteration by using a variant of ONS on the log likelihood.
Revised: In particular, we maintain estimators of the parameters which are updated in each iteration by using a variant of ONS on the log likelihood.

Original: In each step, our algorithm selects a myopicvector of prices based on the current estimators and adds random price shocks to force explorationand avoid uninformative prices.
Revised: In each step, our algorithm selects a myopic vector of prices based on the current estimators and adds random price shocks to force exploration and avoid uninformative prices.

Original: We first give the details on the update of the parameters.
Revised: We first give the details on the update of the parameters.

Original: Let ℓ t denote the log loss at time t :
Revised: Given our estimator γ t of the true parameters at time t , we let ℓ t denote the log loss at time t :

Original: [Equation]
Revised: [Equation]

Original: 165 For a time-dependent sequence of positive regularizers { λ t } t ≥ 1 , the estimator obtained beforeprojection after conducting one step of our descent method is:
Revised: For a time-dependent sequence of positive regularizers { λ t } t ≥ 1 (the exact value of the regularizers used in our algorithm is given in Theorem 2.5), the estimator obtained before projection after conducting one step of our descent method is:

Original: [Equation]
Revised: [Equation]

Original: where µ = 1 2(1+ √ 6 KW ) and H t = (cid:80) ts =1 ∇ 2 ℓ s ( γ s ) + λ t I d is the regularized Hessian of the negativelog likelihood.
Revised: where µ = 1 2(1+(1+ p max ) √ 6 KW ) and H t = (cid:80) ts =1 ∇ 2 ℓ s ( γ s ) + λ t I d is the regularized Hessian of the negative log likelihood.

Original: Note that for the MNL model, ℓ t is convex, hence H t ≻ 0 for all t .
Revised: Note that for the MNL model, ℓ t, 1 is convex (as can be deduced from Lemma ?? 1.), hence H t ≻ 0 for all t .

Original: Let B t denote the set B  (0 , W ) ∩ { ( θ, α ) | x ⊤ t,j α ≥ L for j ∈ [ k t ] } .
Revised: Let B t denote the set B 2 d (0 , W ) ∩ { ( θ, α ) | x ⊤ t,j α ≥ L for j ∈ [ k t ] } .

Original: B t represents the set ofparameters which satisfy Assumptions 2.2 and 2.3.
Revised: B t represents the set of parameters which satisfy Assumptions 2.2 and 2.3.

Original: We obtain the new estimator γ t by projecting ˆ γ ton the feasible set of parameters:
Revised: We obtain the new estimator γ t by projecting ˆ γ t on the feasible set of parameters:

Original: [Equation]
Revised: [Equation]

Original: where Π H t − 1 B t ( y ) =
Revised: where Π H t − 1 B t ( y ) =

Original: arg min x ∈ B t ( x − y )
Revised: arg min x ∈ B t ( x − y )

Original: T H t − 1 ( x − y ) is the projection relatively to the norm inducedby H t − 1 .
Revised: T H t − 1 ( x − y ) is the projection relatively to the norm induced by H t − 1 .

Original: As a result, during all the course of the algorithm, our estimator γ t also satisfies the lowerand upper boundedness assumptions.
Revised: As a result, during all the course of the algorithm, our estimator γ t also satisfies the lower and upper boundedness assumptions.

Original: Finally, the seller chooses a perturbation factor δ t = 1 Wt 1 / 4 and computes, independently for eachproduct j ∈ [ k t ] , a random price shock ∆ p t,j which takes value δ t with probability 1 / 2 and − δ t withprobability 1 / 2 .
Revised: Finally, the seller chooses a perturbation factor δ t = 1 Wt 1 / 4 and computes, independently for each product j ∈ [ k t ] , a random price shock ∆ p t,j which takes value δ t with probability 1 / 2 and − δ t with probability 1 / 2 .

Original: The seller posts the vector of prices ⃗p t which is the sum of the random price shocksand the myopic vector of prices g
Revised: The seller posts the vector of prices ⃗p t which is the sum of the random price shocks and the myopic vector of prices g

Original: ( X t α t , X t θ t ) (see Appendix B for a formal definition).
Revised: ( X t α t , X t θ t ) (see Appendix ?? for a formal definition).

Original: The pseudocode of our dynamic pricing policy is presented in Algorithm 1.
Revised: The pseudo code of our dynamic pricing policy is presented in Algorithm 1.

Revised: Running time: The two main computational steps of Algorithm 1 are calculating the inverse of H t and projecting the parameter back in the set of feasible parameters relative to the norm H t .

Revised: The time complexity of the first step is O ( d 3 ) , which is reasonable when d is not too large as in the setting we consider.

Revised: The projection step can also be done efficiently by formulating the problem as a Quadratic

Original:  Input: Upper bound W on || ( θ ∗ , α ∗ )
Revised: Programming problem. Using an online descent method for the parameters estimation allows us to obtain a low regret algorithm despite the presence of adversarial contexts. We would like to note that based on our current analysis, a simpler online method such as a stochastic online gradient descent (as proposed Input: Upper bound W on || ( θ ∗ , α ∗ )

Original: || 2 , lower bound L, sequence of regularizers { λ t } t ≥ 1 .
Revised: || 2 , lower bound L, sequence of regularizers { λ t } t ≥ 1 .

Original: Initialize θ 1 , α 1 ∈ Bfor all t ≥ 1 do
Revised: Initialize θ 1 , α 1 ∈ Bfor all t ≥ 1 do

Original: For j ∈ [ k t ] , let ∆ p t,j = (cid:26) 1 Wt 1 / 4 w/p 12 −
Revised: For j ∈ [ k t ] , let ∆ p t,j = (cid:26) 1 Wt 1 / 4 w/p 12 −

Original: Wt 1 / 4 w/p
Revised: Wt 1 / 4 w/p

Original: For j ∈ [ k t ] , set p t,j = g ( X t α t , X t θ t )
Revised: For j ∈ [ k t ] , set p t,j = g ( X t α t , X t θ t )

Original: j + ∆ p t,j
Revised: j + ∆ p t,j

Original: Post prices ⃗p t
Revised: Post prices ⃗p t

Revised: Observe feedback ⃗y t

Original: Set γ t +1 =
Revised: Set γ t +1 =

Original: ( θ t +1 , α t +1 ) as follows: 
Revised: ( θ t +1 , α t +1 ) as follows: γ t +1 =

Original: [Equation] end for Using an online descent method for the parameters estimation allows us to obtain a low regretalgorithm despite the presence of adversarial contexts. We would like to note that based on ourcurrent analysis, a simpler online method such as a stochastic online gradient descent (as proposedin [20] for single-product dynamic pricing without price sensitivity) would not allow us to obtainsublinear regret.
Revised: Π H t B t +1 (cid:16) γ t − 1 µ H − 1 t ∇ ℓ t, 1 ( γ t ) (cid:17) end for in [19] for single-product dynamic pricing without price sensitivity) would not allow us to obtain sublinear regret.

Original: We would also like to mention that our method is different from the Online Newton
Revised: We would also like to mention that our method is different from the Online Newton

Original: Step presented in [19], which is the classic online analogue of the Newton method.
Revised: Step presented in [18], which is the classic online analogue of the Newton method.

Original: The ONS methodmoves into the direction of A − 1 t ∇ ℓ t , where A − 1 t is an approximation of the inverse of the Hessian.
Revised: The ONS method moves into the direction of A − 1 t ∇ ℓ t, 1 , where A − 1 t is an approximation of the inverse of the Hessian.

Original: In our case, we move directly into the direction of the inverse of the Hessian multiplied by ∇ ℓ t , andleverage self-concordant-like properties of the negative log likelihood function of the MNL model toshow the convergence of the estimators.
Revised: In our case, we move directly into the direction of the inverse of the Hessian multiplied by ∇ ℓ t, 1 , and leverage self-concordant-like properties of the negative log likelihood function of the MNL model to show the convergence of the estimators.

Original: This allows us to avoid using the exp-concavity parameter1 /κ (often referred to as β in the literature) in the descent step, as is done by the ONS algorithm.
Revised: This allows us to avoid using the parameter 1 /κ 1 (which corresponds to the exp-concavity parameter β in the literature) in the descent step, as is done by the ONS algorithm.

Original: We are ready to present our regret bound.
Revised: We are ready to present our regret bound.

Original: Theorem 2.4.
Revised: Theorem 2.5.

Original: Setting λ 1 = 1 , λ t = d log( t ) for all t ≥ 1 , there is a constant C depending only on
Revised: Setting λ 1 = 1 , λ t = d log( t ) for all t ≥ 1 , there is a constant C depending only on

Original: W, L, K such that the regret of Algorithm 1 is bounded as:
Revised: W, L, K such that the regret of Algorithm 1 is bounded as:

Original: [Equation]
Revised: [Equation]

Original: The proof of Theorem 2.4 is presented in Appendix B. We would like to point out that following [22],we treat the maximum size of a consideration set K as a constant.
Revised: The proof of Theorem 2.5 is presented in Appendix ?? . We would like to point out that, even though our algorithm does not use the parameter 1 /κ 1 , it still appears within the constant C .

Original: Under the assumptions of [20] (single product dynamic pricing with adversarial contexts and constantprice sensitivity), and assuming that the noise has a logistic distribution, we can show that our policyachieves a regret  with a dependency on T of order O (log( T ) + (cid:80)
Revised: Under the assumptions of [19] (single product dynamic pricing with adversarial contexts and constant price sensitivity), and assuming that the noise has a logistic distribution, we can show the following regret bound, which contrasts with the O ( √ T ) upper bound established in [19].

Revised: Corollary 2.6.

Original: Tt =1 tδ t ) where { δ t } Tt =1 representpossible drifts in the ground truth parameter. This contrasts with the O ( √ T + (cid:80) Tt =1 √ tδ t ) upperbound established in [20].
Revised: If k t = 1 for all t , and if the price coefficient is a known constant, then setting λ 1 = 1 , λ t = d log( t ) for all t ≥ 1 , and letting ∆ p t, 1 = 0 at each step, there is a constant C depending only on W such that the regret of Algorithm 1 with regularizers { λ t } and price shocks { ∆ p t, 1 } is bounded as: [Equation]

Original: 2.3 High level ideas and sketch of the proof.
Revised: 2.3 High level ideas and sketch of the proof.

Original: We provide here the main ideas in the proof of Theorem 2.4.
Revised: We provide here the main ideas in the proof of Theorem 2.5.

Original: The technical details are presented in
Revised: The technical details are presented in

Original: Appendix B. We first follow the classical regret analysis for dynamic pricing policies and decomposethe regret between a term due to the error in the estimation of the parameters and a term due to therandom price shocks.
Revised: Appendix ?? . We first follow the classical regret analysis for dynamic pricing policies and decompose the regret between a term due to the error in the estimation of the parameters and a term due to the random price shocks.

Original: In particular, we have the following lemma.
Revised: In particular, we have the following lemma.

Original: Lemma 2.5.
Revised: Lemma 2.7.

Original: There exist constants C 1 , C 2 depending only on W, L, K , such that:
Revised: There exist constants C 1 , C 2 depending only on W, L, K , such that:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Since the variances of the random price shocks are 1 W 2 √ t , the second term is O ( √ T ) .
Revised: Since the variances of the random price shocks are 1 W 2 √ t , the second term is O ( √ T ) .

Original: Therefore,to exhibit a ˜ O ( √ T ) regret bound, it suffices to focus on upper bounding the first term.
Revised: Therefore, to exhibit a ˜ O ( √ T ) regret bound, it suffices to focus on upper bounding the first term.

Original: Note that itfollows from (1) that the regret upper bound does not require the global convergence of the estimatorsto the true parameters.
Revised: Note that it follows from (2) that the regret upper bound does not require the global convergence of the estimators to the true parameters.

Original: We only need to show that they converge sufficiently fast in the directionsgiven by the contexts seen throughout the T periods.
Revised: We only need to show that they converge sufficiently fast in the directions given by the contexts seen throughout the T periods.

Original: For any sequence of prices { ⃗p t } Tt =1 , our online descent method allows us to control the convergenceof the estimated utilities u t,j = x ⊤ t,j θ t − x ⊤ t,j α t p t,j to the true utilities u ∗ t,j = x ⊤ t,j θ ∗ − x ⊤ t,j α ∗ p t,j foreach product j ∈ [ k t ] .
Revised: For any sequence of prices { ⃗p t } Tt =1 , our online descent method allows us to control the convergence of the estimated utilities u t,j = x ⊤ t,j θ t − x ⊤ t,j α t p t,j to the true utilities u ∗ t,j = x ⊤ t,j θ ∗ − x ⊤ t,j α ∗ p t,j for each product j ∈ [ k t ] .

Original: In particular, we have the following lemma.
Revised: In particular, we have the following lemma.

Original: Lemma 2.6.
Revised: Lemma 2.8.

Original: There is a constant ˜ C depending only on W, L, K such that with probability at least 
Revised: There is a constant ˜ C depending only on W, L, K such that with probability at least 1 − log

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: We present the proof in Appendix B. Note that the upper bound in Lemma 2.6 is valid for any sequenceof prices posted by the seller.
Revised: We present the proof in Appendix ?? . Note that the upper bound in Lemma 2.8 is valid for any sequence of prices posted by the seller.

Original: However, it is meaningful only with random price experimentation.
Revised: However, it is not possible, in general, to derive directly Theorem 2.5 from Lemma 2.8 without an appropriate price experimentation scheme.

Original: Suppose at each step we only post the myopic vector of prices ⃗p t = g
Revised: Suppose at each step we only post the myopic vector of prices ⃗p t = g

Original: ( X t α t , X t θ t ) .
Revised: ( X t α t , X t θ t ) .

Original: If the prices{ p t,j } happened to be uninformative (i.e. x ⊤ t,j ( θ t − θ ∗ ) − x ⊤ t,j ( α t − α ∗ ) p t,j = 0 ), then the boundprovided in  the above lemma would  be empty.
Revised: If the prices { p t,j } happened to be uninformative (i.e. x ⊤ t,j ( θ t − θ ∗ ) − x ⊤ t,j ( α t − α ∗ ) p t,j = 0 ), then the left-hand side in Lemma 2.8 would be zero and the bound provided would not be useful.

Original: However, adding random price shocks allows us todeviate from uninformative prices and to derive an upper bound on the first term of (1) based on Lemma 2.6.
Revised: However, adding random price shocks allows us to deviate from uninformative prices and to derive an upper bound on the first term of (2) based on Lemma 2.8.

Original: This concludes the proof of Theorem 2.4.
Revised: This concludes the proof of Theorem 2.5.

Original: Note that  Lemma 2.6 immediately implies a
Revised: If there is a single product ( k t = 1 ), and there is no price sensitivity (i.e. for all t , the coefficient in front of p t, 1 is a known constant, that we assume to be 1 without loss of generality), note that from Lemma 2.8, we get (cid:80) t x Tt ( θ t − θ ∗ ) =

Original: O ( d log( T )) regret bound when the model does not involve feature-dependent price sensitivities.
Revised: O ( d log( T )) . Combining this with (2) and the choice of ∆ p t, 1 = 0 for all t , we immediately obtain Corollary 2.6.

Original: In [20], where no price sensitivity is involved and the utility is simply written under the formu t = x ⊤ t θ ∗ , the use of a stochastic gradient descent method allows the author to directly obtain a
Revised: In [19], where no price sensitivity is involved and the utility is simply written under the form u t = x ⊤ t θ ∗ , the use of a stochastic gradient descent method allows the author to directly obtain a

Original: O ( √ T ) bound on the sum (cid:80)
Revised: O ( √ T ) bound on the sum (cid:80)

Original: Tt
Revised: Tt

Original: =1 x ⊤ t ( θ t − θ ∗ ) 2 .
Revised: =1 x ⊤ t ( θ t − θ ∗ ) 2 .

Original: Such a bound, when combined with our randomprice shocks, would only give us a linear regret.
Revised: Such a bound, when combined with our random price shocks, would only give us a linear regret.

Original: By exploiting the special structure of the MNLfunction and using our variant of the ONS instead of an online gradient descent, we obtain thestronger O (log( T )) bound of Lemma 2.6.
Revised: By exploiting the special structure of the MNL function and using our variant of the ONS instead of an online gradient descent, we obtain the stronger O (log( T )) bound of Lemma 2.8.

Original: The proof of Lemma 2.6 is based on a lower and an upper bound on (cid:80)
Revised: The proof of Lemma 2.8 is based on a lower and an upper bound on (cid:80)

Original: Tt =1 ℓ t ( γ t ) − ℓ t ( γ ∗ ) .
Revised: Tt =1 ℓ t, 1 ( γ t ) − ℓ t, 1 ( γ ∗ ) .

Original: Bothinvolve (cid:80) Tt =1 (cid:80) 
Revised: Both involve (cid:80) Tt =1 (cid:80) k

Original: k t j [-=
-] ( x ⊤ t,j ( θ t − θ ∗ ) − x ⊤ t,j ( α t − α ∗ ) p t,j ) 2 .
Revised: t j [-=
-]=1 ( x ⊤ t,j ( θ t − θ ∗ ) − x ⊤ t,j ( α t − α ∗ ) p t,j ) 2 .

Original: The proof of the lower bound exploits theconvexity of ℓ t and is based on a Bernstein inequality for martingales difference sequences.
Revised: The proof of the lower bound exploits the convexity of ℓ t, 1 and is based on a Bernstein inequality for martingales difference sequences.

Original: This issimilar to the inequality used in [20].
Revised: This is similar to the inequality used in [19].

Original: The main technical hindsight lies in the proof of the upperbound.
Revised: The main technical hindsight lies in the proof of the upper bound.

Original: It mimics the analysis of the Online Newton Step method presented in [19], but relies on thespecific structure of the gradient and Hessian of ℓ t for the MNL model.
Revised: It mimics the analysis of the Online Newton Step method presented in [18], but relies on the specific structure of the gradient and Hessian of ℓ t, 1 for the MNL model.

Original: Moreover, it unically exploitsthe self concordant-like property of ℓ t .
Revised: Moreover, it unically exploits the self concordant-like property of ℓ t, 1 .

Original: Let’s first recall the definition of a self-concordant-likefunction.
Revised: Let’s first recall the definition of a self-concordant-like function.

Original: Definition 2.7 (self-concordant-like functions [39]) .
Revised: Definition 2.9 (self-concordant-like functions [38]) .

Original: A convex function f ∈ C 3 ( R n ) is called aself-concordant-like function if:
Revised: A convex function f ∈ C 3 ( R n ) is called a self-concordant-like function if:

Original: [Equation]
Revised: [Equation]

Original: for t ∈ R and M f > 0 , where ϕ ( t ) :
Revised: ≤ M f ϕ ′′ ( t ) || u ||for t ∈ R and M f > 0 , where ϕ ( t ) :

Original: = f ( x + tu ) for any x ∈ dom ( f ) and u ∈ R n .
Revised: = f ( x + tu ) for any x ∈ dom ( f ) and u ∈ R n .

Original: By adapting the proof of Lemma 4 in [39], wecan show that ℓ t satisfies the following property.
Revised: By adapting the proof of Lemma 4 in [38], we show in Appendix ?? the following property.

Original: Proposition 2.8.
Revised: Proposition 2.10.

Original: ℓ t is self-concordant-like with M f = X (cid:112) 6 | S t | .
Revised: ℓ t, 1 is self-concordant-like with M f = (1 + p max ) (cid:112) 6 | S t | .

Original: We  detail in Appendix A some useful properties satisfied by self-concordant-like functions.
Revised: We also detail in Appendix ?? some useful properties satisfied by self-concordant-like functions.

Original:  Improved algorithm for MNL contextual bandits3.1 Problem formulation
Revised:  Improved algorithm for MNL contextual bandits 3.1 Problem formulation

Original: We consider the following MNL dynamic assortment optimization problem, also referred as the MNLcontextual bandits.
Revised: We consider the following MNL dynamic assortment optimization problem, also referred as the MNL contextual bandits.

Original: In each period t , the seller observes feature vectors { x t,j } Nj =1 ∈ R d .
Revised: In each period t , the seller observes feature vectors { x t,j } Nj =1 ∈ R d .

Original: As before,this represents a combination of customer and product features which can be adversarially chosen.
Revised: As before, this represents a combination of customer and product features which can be adversarially chosen.

Original: The seller needs to decide on the set S t ⊆
Revised: The seller needs to decide on the set S t ⊆

Original: [ N ] to offer, with | S t | ≤ K .
Revised: [ N ] to offer, with | S t | ≤ K .

Original: Given the offered assortment,the customer purchases  product j ∈ S t ∪ { 0 }  with probability:
Revised: Given the offered assortment, the customer purchases one single product j ∈ S t ∪ { 0 } . Each product j is purchased with probability:

Original: [Equation]
Revised: [Equation]

Original: where θ ∗ ∈ R
Revised: where θ ∗ ∈ R

Original: d is an underlying model parameter.
Revised: d is an underlying model parameter.

Original: As before, the binary variable y t,j indicateswhether the customer has purchased product j at time t .
Revised: As before, the binary variable y t,j indicates whether the customer has purchased product j at time t .

Original: Note that our model encompasses thecontextual logistic bandit problem with finitely many arms (which corresponds to the case where
Revised: Note that our model encompasses the contextual logistic bandit problem with finitely many arms (which corresponds to the case where

Original: K = 1 ).
Revised: K = 1 ).

Original: The objective is to minimize the cumulative expected regret over the T periods:
Revised: The objective is to minimize the cumulative expected regret over the T periods:

Original: [Equation]
Revised: [Equation]

Original: where S ∗ t denotes the optimal assortment at time t .
Revised: where S ∗ t denotes the optimal assortment at time t .

Original: In [31], the authors study a more general model where a reward r t,j ∈ [0 , 1] is also revealed foreach product at time t .
Revised: In [31], the authors study a more general model where a reward r t,j ∈ [0 , 1] is also revealed for each product at time t .

Original: Wefirst consider the case of uniform rewards ( r t,j = 1 ) for all products.
Revised: We consider the case of uniform rewards ( r t,j = 1 ) for all products.

Original: Maximizing (cid:80) j ∈ S t q t,j ( S t , θ ) over all sets S ⊆
Revised: Maximizing (cid:80) j ∈ S t q t,j ( S t , θ ) over all sets S ⊆

Original: [ N ] of cardinality at most K is now equivalent toselecting the K products which have the highest utility x ⊤ t,j θ .
Revised: [ N ] of cardinality at most K is now equivalent to selecting the K products which have the highest utility x ⊤ t,j θ .

Original: Hence the set S t as well as the optimalset S ∗ t always contain exactly K products.
Revised: Hence the set S t as well as the optimal set S ∗ t always contain exactly K products.

Original: Similarly as in the pricing setting, we make the following two assumptions:
Revised: Similarly as in the pricing setting, we make the following two assumptions:

Original: Assumption 3.1.
Revised: Assumption 3.1.

Original: For all t ∈ [ T ] , j ∈ [ N ] , || x t,j || 2 ≤ 1 .
Revised: For all t ∈ [ T ] , j ∈ [ N ] , || x t,j || 2 ≤ 1 .

Original: Assumption 3.2.
Revised: Assumption 3.2.

Original: ||
Revised: ||

Original: θ ∗ || 2 ≤ W for some known constant W.
Revised: θ ∗ || 2 ≤ W for some known constant W.

Original: Following [31], we also introduce the following constant, which typically appears in connection tothe link function in the generalized linear bandits ([18], [26]).
Revised: Following [31], we also introduce the following constant, which typically appears in connection to the link function in the generalized linear bandits ([17], [25]).

Original: [Equation]
Revised: [Equation]

Original: 268 A smaller value of κ  can be interpreted as a bigger deviation from the linear model.
Revised: A smaller value of κ 2 can be interpreted as a bigger deviation from the linear model.

Original: As mentionedbefore, the regret bound of the dynamic assortment policies of [31] and [12] exhibit a harmfuldependency in κ  .
Revised: As mentioned before, the regret bound of the dynamic assortment policies of [31] and [11] exhibit a harmful dependency in κ 2 .

Original: Besides, an a priori knowledge of the value of κ  is presupposed.
Revised: Besides, an a priori knowledge of the value of κ 2 is presupposed.

Original: Our goal is todesign a dynamic assortment policy which does not require prior knowledge of κ  and achieves aregret with a better dependency in κ  .
Revised: Our goal is to design a dynamic assortment policy which does not require prior knowledge of κ 2 and achieves a regret with a better dependency in κ 2 .

Original: For all t ∈ [ T ] , let 
Revised: For all t ∈ [ T ] , let κ ∗ 2 ,t

Original: [Equation]
Revised: = (cid:80) j ∈ S ∗ t q t,j ( S ∗ t , θ ∗ ) q t, 0 ( S ∗ t , θ ∗ ) .

Original: 274 κ ∗ t represents the degree of non-linearity for the optimal set S ∗ t and depends on the unknown parameterθ ∗ as well as on the feature vectors present at time t .
Revised: κ ∗ 2 ,t represents the degree of non-linearity for the optimal set S ∗ t and depends on the unknown parameter θ ∗ as well as on the feature vectors present at time t .

Original: We show that the ˜ O ( √ T ) term of our regretbound can be replaced by a ˜ O (cid:18)(cid:113)(cid:80) T t =1 κ ∗ t (cid:19) term.
Revised: We show that the ˜ O ( √ T ) term of our regret bound can be replaced by a ˜ O (cid:18) (cid:113) (cid:80) Tt = 1 κ ∗ 2 ,t (cid:19) term.

Original: Note that we always have κ ∗ t ≤ 1 hence this isa strict improvement.
Revised: Note that we always have κ ∗ 2 ,t ≤ 1 hence this is a strict improvement.

Original: As a result, a high level of non-linearity at time t induces a smaller κ ∗ t andpositively impacts the regret.
Revised: As a result, a high level of non-linearity at time t induces a smaller κ ∗ 2 ,t and positively impacts the regret.

Original: 3.2 Dynamic assortment policy
Revised: 3.2 Dynamic assortment policy

Original: We design a tight confidence set for the true parameter θ ∗ and use it to construct upper confidencebounds on the utility of each product at time t .
Revised: We design a tight confidence set for the true parameter θ ∗ and use it to construct upper confidence bounds on the utility of each product at time t .

Original: Our algorithm relies on optimistic parameter searchover the confidence interval, as used by [2] in the logistic bandits setting.
Revised: Our algorithm relies on optimistic parameter search over the confidence interval, as used by [2] in the logistic bandits setting.

Original: However, in our setting, theseller’s decision at time t involves the choice of multiple products.
Revised: However, in our setting, the seller’s decision at time t involves the choice of multiple products.

Original: Hence we cannot build a uniqueoptimistic estimator θ t as in [2].
Revised: Hence we cannot build a unique optimistic estimator θ t as in [2].

Original: The key idea is to do the optimistic parameter search independentlyfor each product, generating a set of parameters { ˜ θ t,j } N j =1 such that with high probability, x ⊤ t,j ˜ θ t,j isan upper bound on the utility x ⊤ t,j θ ∗ of product j .
Revised: The key idea is to do the optimistic parameter search independently for each product, generating a set of parameters { ˜ θ t,j } N j =1 such that with high probability, x ⊤ t,j ˜ θ t,j is an upper bound on the utility x ⊤ t,j θ ∗ of product j .

Original: Confidence set.
Revised: Confidence set.

Original: The main ingredient is the design of a confidence set for θ ∗ .
Revised: The main ingredient is the design of a confidence set for θ ∗ .

Original: We classically start bycomputing the maximum likelihood estimator of θ ∗ .
Revised: We classically start by computing the maximum likelihood estimator of θ ∗ .

Original: Let ˆ θ t be the unique minimizer of the followingfunction, for a sequence of time-dependent regularizers { λ t } Tt =1 : [Equation] j [Equation]{+=
+}
Revised: Let ˆ θ t be the unique minimizer of the following function, for a sequence of time-dependent regularizers { λ t } Tt {+=
+}

Original: s
Revised: (the exact values are given in Theorem 3.4):

Original: [Equation]
Revised: [Equation]

Original: j [Equation] ˆ θ t satisfies the equation ∇L λ t t ( θ ) = 0 , where the gradient of L λ t t ( θ ) is given by: [Equation]
Revised: ˆ θ t satisfies the equation ∇L λ t t ( θ ) = 0 , where the gradient of L λ t t ( θ ) is given by: ∇L λ t t ( θ ) = (cid:80) t −s =1 (cid:80) j ∈ S s ( q s,j ( S s , θ ) − y s,j ) x s,j + λ t ˆ θ t .

Original: 293 Following [2], for all δ ∈ [0 , 1) , we now define a confidence set C t ( δ ) for θ ∗ as follows:
Revised: Following [2], for all δ ∈ [0 , 1) , we now define a confidence set C t ( δ ) for θ ∗ as follows:

Original: [Equation]
Revised: [Equation]

Original: δ ) } , where g t ( θ ) := [Equation] (cid:88)
Revised: where g t ( θ ) := (cid:80) t − 1 s =1 (cid:80) j ∈ S

Original: [Equation] 295 H t ( θ ) is the Hessian of the regularized negative log likelihood evaluated at θ , [Equation]
Revised: s q s,j ( S s , θ ) x s,j + λ t θ , where H t ( θ ) is the Hessian of the regularized negative log likelihood evaluated at θ , i.e.,

Original: =
Revised: H

Original: [Equation]
Revised: [Equation]

Original: S
Revised: θ ) :=

Original: [Equation]
Revised: [Equation]

Original: and 
Revised: and where

Original: [Equation]
Revised: [Equation]

Original:  The following proposition is the analogue, in the multi-product setting, of Proposition 1 in [2] andestablishes that C t ( δ ) is a confidence set for θ ∗ .
Revised: dλ t (cid:19)(cid:19) . The following proposition is the analogue, in the multi-product setting, of Proposition 1 in [2] and establishes that C t ( δ ) is a confidence set for θ ∗ .

Original: The details are given in Appendix C.3.
Revised: The details are given in Appendix ?? .

Original: Proposition 3.3.
Revised: Proposition 3.3.

Original: Let δ ∈ (0 , 1] .
Revised: Let δ ∈ (0 , 1] .

Original: Then P ( ∀ t, θ ∗ ∈ C t ( δ )) ≥ 1 − δ .
Revised: Then P ( ∀ t, θ ∗ ∈ C t ( δ )) ≥ 1 − δ .

Original: The proof of Proposition 3.3 builds upon the new Bernstein-like tail inequality for self-normalizedvectorial martingales presented in [17].
Revised: The proof of Proposition 3.3 builds upon the new Bernstein-like tail inequality for self-normalized vectorial martingales presented in [16].

Original: However, Theorem 1 in [17] does not directly apply to oursetting because of the correlation between the variables { ε t,j :=
Revised: However, Theorem 1 in [16] does not directly apply to our setting because of the correlation between the variables { ε t,j :=

Original: y t,j − q t,j ( S t , θ ∗ ) } j ∈ S t induced bythe presence of multiple purchase options in period t .
Revised: y t,j − q t,j ( S t , θ ∗ ) } j ∈ S t induced by the presence of multiple purchase options in period t .

Original: We thus present a generalization of Theorem 1in [17] for the multiple products setting that handles such correlation.
Revised: We thus present a generalization of Theoremin [16] for the multiple products setting that handles such correlation.

Original: Algorithm.
Revised: Algorithm.

Original: Before describing our algorithm, let’s introduce the following notation, which generalizesthe choice probabilities given by (1) to the case where the model parameters corresponding to eachproduct are uncorrelated.
Revised: Before describing our algorithm, let’s introduce the following notation, which generalizes the choice probabilities given by (3) to the case where the model parameters corresponding to each product are uncorrelated.

Original: More precisely, we now consider  ˜ θ ∈ R d × N .
Revised: More precisely, we now consider some estimator ˜ θ ∈ R d × N of the true parameters.

Original: For all j ∈ [ N ] , ˜ θ j representsthe model parameter associated with product j .
Revised: For all j ∈ [ N ] , ˜ θ j represents the parameter associated with product j .

Original: The  probability that item j is purchased at time t ifassortment S ⊆
Revised: The estimated probability that item j is purchased at time t if assortment S ⊆

Original: [ N ] is offered is then given by:
Revised: [ N ] is offered is computed as:

Original: [Equation]
Revised: [Equation]

Original: Now, at time t , our algorithm uses the previous contexts and observations to compute the maximumlikelihood estimator ˆ θ t as defined above and constructs the confidence set C t ( δ ) .
Revised: Now, at time t , our algorithm uses the previous contexts and observations to compute the maximum likelihood estimator ˆ θ t as defined above and constructs the confidence set C t ( δ ) .

Original: Then, for eachproduct j ∈ [ N ] , the algorithm finds an optimistic parameter ˜ θ t,j = arg max θ ∈ C t ( δ ) x
Revised: Then, for each product j ∈ [ N ] , the algorithm finds an optimistic parameter ˜ θ t,j = arg max θ ∈ C t ( δ ) x

Original: ⊤ t,j θ .
Revised: ⊤ t,j θ .

Original: Let z t,jdenote the resulting upper bound on the utility of product j, i.e., z t,j = x ⊤ t,j ˜ θ t,j .

Original: We offer the set S tof the K items maximizing the optimistic expected revenue ˜ r t ( S, ˜ θ ) :
Revised: We offer the set S t of the K items maximizing the optimistic expected revenue ˜ r t [Equation] S [Equation]

Revised: ˜

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Revised: Since we assumed all prices to be unit, this is equivalent to offering the K products with highest x ⊤ t,j ˜ θ t,j .

Revised: In the case of non-uniform revenues, our algorithm is still valid; however, an extra factor 1 /κ 2 would appear in the regret upper bound with our current analysis.

Revised: We also note that Algorithm 2 is mainly of theoretical interest, since computing each ˜ θ t,j remains computationally expensive.

Original: We  present the following upper bound on the regret of ourdynamic assortment policy.
Revised: We now present the our upper bound on the regret of our policy.

Original: Theorem 3.4.
Revised: Theorem 3.4.

Original: For δ = 1 K 2 T 2 , λ 1 = 1 , and λ t = d log( tK ) for all t ≥ 1 , the regret of Algorithm 2satisfies, for someuniversal constants ˜ C 1 and ˜ C 2  :
Revised: For δ = 1 K 2 T 2 , λ 1 = 1 , and λ t = d log( tK ) for all t ≥ 1 , the regret of Algorithm 2 satisfies, for some constants ˜ C 1 and ˜ C 2 that do not depend on d, K, T and that depend only polynomially on W :

Original: [Equation]
Revised: [Equation]

Original: Algorithm2 OFU-
Revised: Algorithm

Original: Input: Upper bound K on the size of an assortment, δ , sequence { λ t } Tt =1 . for all t ≥ 1 do
Revised: Input: Upper bound K on the size of an assortment, δ , sequence { λ t } Tt = for all t ≥ 1 do

Original: Observe feature vectors { x t, 1 , ..., x t,N }.
Revised: Observe feature vectors { x t, 1 , ..., x t,N }

Original: Set
Revised: Set

Original: ˆ θ t = arg min L λ t t ( θ ).
Revised: ˆ θ t = arg min L λ t t ( θ )

Original: For j = 1 , ..., N , set [Equation]
Revised: For j = 1 , ..., N , set ˜ θ t,j = arg max θ ∈ C t ( δ ) x ⊤ t,j θ

Original: Offer set S t = arg max | S |≤ K ˜ r t ( S, ˜ θ ) .
Revised: Offer set S t = arg max | S |≤ K ˜ r t ( S, ˜ θ ) .

Original: Receive feedback ⃗y t. end for
Revised: Receive feedback ⃗y t end for

Original: 3.3 High level ideas and sketch of the proof
Revised: 3.3 High level ideas and sketch of the proof

Original: We first condition on the event that θ ∗ ∈ C t ( δ ) , for all t ≥ 1 .
Revised: We first condition on the event that θ ∗ ∈ C t ( δ ) , for all t ≥ 1 .

Original: Proposition 3.3 shows that this happenswith high probability.
Revised: Proposition 3.3 shows that this happens with high probability.

Original: From Lemma C.3 (Appendix C.2), we have the following concentration resulton the optimistic parameter ˜ θ t,j associated with each product j ∈ [ N ] :
Revised: From Lemma ?? (Appendix ?? ), we have the following concentration result on the optimistic parameter ˜ θ t,j associated with each product j ∈ [ N ] :

Original: [Equation]
Revised: [Equation]

Original: Now, using the optimistic choice of assortment made by the algorithm as well as the fact that x ⊤ t,j ˜ θ t,jis an upper bound on the true utility for each product, we can first bound the regret as follows:R ( T [Equation] =
Revised: Now, using the optimistic choice of assortment made by the algorithm as well as the fact that x ⊤ t,j ˜ θ t,j is an upper bound on the true utility for each product, we can first bound the regret as follows:

Original: [Equation]

Original: 

Original: [Equation]
Revised: [Equation]

Original: (cid:88)

Original: [Equation]

Original:  (cid:88)

Original: j

Original: [Equation]

Original: t

Original: [Equation]

Original: (cid:88)

Original: j

Original: [Equation]

Original: t ˜

Original: [Equation]

Original: [Equation]

Original: ˜ θ

Original: [Equation]

Original:   ≤

Original: T (cid:88) t = (cid:88) j ∈ S t q t

Original: [Equation]

Original: j ( S t

Original: [Equation]

Original: θ ∗ ) q t

Original: [Equation]

Original: 0 ( S t

Original: [Equation] θ ∗ [Equation] where R 2 ( T ) is a second order term which we will prove to be of order
Revised: where R 2 ( T ) is a second order term which we will prove to be of order

Original: O ( d 2 K 4 log( KT ) 2 /κ  ) .
Revised: O ( d 2 K 4 log( KT ) 2 /κ 2 ) .

Original: Thesecond inequality follows from the fact that ˜ r t ( S t , ˜ θ t ) ≥ ˜ r t ( S ∗ t , ˜ θ ∗ ) and the last one results from a

Original: Taylor expansion.

Original: To show that the first term is of order O(cid:18)
Revised: To show that the first term is of order O

Original: [Equation]
Revised: [Equation]

Original: Tt =1 κ ∗ t (cid:19) , we firstuse the concentration result stated in (2), which implies the following upper bound, for some C > 0 :
Revised: , we first use the concentration result stated in (4), which implies the following upper bound, for some C > 0 :

Original: [Equation]
Revised: [Equation]

Original: Using that H t ( θ ∗ ) 
Revised: Using that H t ( θ ∗ ) ⪰

Original: ⪰ κ  (cid:80) T s =1 (cid:80) j ∈ S s x s,j x ⊤ s,j , we could already show that this term is of order˜ O ( √ T ) by applying the Elliptical Potential Lemma from [1].
Revised: κ 2 (cid:80) T s =1 (cid:80) j ∈ S s x s,j x ⊤ s,j , we could already show that this term is of order ˜ O ( √ T ) by applying the Elliptical Potential Lemma from [1].

Original: However, we would then obtaina term with linear dependency in 1 /κ  .
Revised: However, we would then obtain a term with linear dependency in 1 /κ 2 .

Original: We show that the local information given by the terms{ q t,j ( S t , θ ∗ ) q t, 0 ( S t , θ ∗ ) } and the self-concordance-like property of the log loss can be used to derivea tighter bound on the above sum.
Revised: We show that the local information given by the terms { q t,j ( S t , θ ∗ ) q t, 0 ( S t , θ ∗ ) } and the self-concordance-like property of the log loss can be used to derive a tighter bound on the above sum.

Original: The complete version of the proof is provided in Appendix C.2.
Revised: The complete version of the proof is provided in Appendix ?? .

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we study contextual dynamic pricing and assortment optimization problems under a
Revised: In this paper, we study contextual dynamic pricing and assortment optimization problems under a

Original: MNL choice model.
Revised: MNL choice model.

Original: We present a dynamic pricing policy based on a variant of the Online Newton
Revised: We present a dynamic pricing policy based on a variant of the Online Newton

Original: Step method combined with random price shocks that achieves near-optimal regret for the MNL modelwith adversarial contexts and feature-dependent price sensitivities.
Revised: Step method combined with random price shocks that achieves near-optimal regret for the MNL model with adversarial contexts and feature-dependent price sensitivities.

Original: We also propose a new optimisticalgorithm for the adversarial MNL contextual bandits problem.
Revised: We also propose a new optimistic algorithm for the adversarial MNL contextual bandits problem.

Original: Both our algorithms leverage the selfconcordant property of the MNL log likelihood function to achieve better dependency on potentiallyexponentially small parameters than existing algorithms.
Revised: Both our algorithms leverage the selfconcordant property of the MNL log likelihood function to achieve better dependency on potentially exponentially small parameters than existing algorithms.

Original: An interesting research direction would beto extend our results to other choice models, such as the nested logit model, which is another widelyused model in the Revenue Management literature.
Revised: An interesting research direction would be to extend our results to other choice models, such as the nested logit model, which is another widely used model in the Revenue Management literature.

Original: Checklist
Revised: Checklist

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