Original: Dynamic Pricing with Monotonicity Constraint Under Unknown Parametric Demand Model
Revised: Dynamic Pricing with Monotonicity Constraint Under Unknown Parametric Demand Model

Original: Abstract
Revised: Abstract

Original: We consider a Continuum-Armed Bandit problem with an additional monotonicity constraint (or “markdown” constraint) on the actions selected.
Revised: We consider a Continuum-Armed Bandit problem with an additional monotonicity constraint (or “markdown” constraint) on the actions selected.

Original: This problem faithfully models a natural revenue management problem, called “markdown pricing”, where the objective is to adaptively reduce the price over a ﬁnite horizon to maximize  expected revenues.
Revised: This problem faithfully models a natural revenue management problem, called “markdown pricing”, where the objective is to adaptively reduce the price over a ﬁnite horizon to maximize the expected revenues.

Original: Chen ([3]) and Jia et al ([9]) recently showed a tight
Revised: Chen ([3]) and Jia et al ([9]) recently showed a tight

Original: T 3 / 4 regret bound over T rounds under minimal assumptions of unimodality and
Revised: T 3 / 4 regret bound over T rounds under minimal assumptions of unimodality and

Original: Lipschitzness in the reward function.
Revised: Lipschitzness in the reward function.

Original: This bound shows that markdown pricing is strictly harder than unconstrained dynamic pricing (i.e., without the monotonicity constraint), which admits T 2 / 3 regret under the same assumptions ([11]).
Revised: This bound shows that markdown pricing is strictly harder than unconstrained dynamic pricing (i.e., without the monotonicity constraint), which admits T 2 / 3 regret under the same assumptions ([11]).

Original: However, in practice, demand functions are usually assumed to have certain functional forms (e.g. linear or exponential), rendering the demand learning easier and suggesting lower regret bounds.
Revised: However, in practice, demand functions are usually assumed to have certain functional forms (e.g. linear or exponential), rendering the demand learning easier and suggesting better regret bounds.

Original: We introduce a concept, markdown dimension , that measures the complexity of any parametric family, and present optimal regret bounds that improve upon the previous T 3 / 4 bound under this framework.
Revised: In this work we introduce a concept, markdown dimension , that measures the complexity of a parametric family, and present optimal regret bounds that improve upon the previous T 3 / 4 bound under this framework.

Original:  Introduction
Revised:  Introduction

Original: Dynamic pricing under unknown demand arises naturally for the sale of new products, where the demand function is not available in advance.
Revised: Dynamic pricing under unknown demand arises naturally for the sale of new products, where the demand function is not available in advance.

Original: The seller in this case has to learn the demand function over time and  faces a learning-vs-earning trade-off.
Revised: The seller in this case has to learn the demand function over time and hence faces a learning-vs-earning trade-off.

Original: This problem is therefore usually formulated as a Multi-Armed Bandit (MAB) problem.
Revised: This problem is therefore usually formulated as a Multi-Armed Bandit (MAB) model.

Original: Although bandit problems have been well understood theoretically , in practice, however, we rarely see retailers implement such policies.
Revised: Although bandit problems have been well understood theoretically , in practice, however, we rarely see retailers implement such policies.

Original: This is a largely because some practical constraints  are often overlooked by those policies.
Revised: This is a largely because some practical constraints which can potentially cause customer dissatisfaction are overlooked.

Original: For example,  price increases can potentially create a manipulative image of the retailer and negatively impact their ratings.
Revised: For example, a price increase may sometimes create a manipulative image of the retailer and negatively impact their ratings.

Original: For example, for online menu prices, “on average, a 1% price increase leads to 3-5% decrease in online ratings” ([13]).
Revised: For example, [13] analyzed the online menu prices of a set of restaurants and concluded that “on average, a 1% price increase leads to 3-5% decrease in online ratings”.

Original: Therefore, retailersmay sometimes implicitly face a monotonicity constraint (which we call “markdown constraint”), which requires that the prices selected be non-increasing.
Revised: Therefore, retailers sometimes implicitly face a monotonicity constraint (which we call “markdown constraint”), which requires that the prices selected be non-increasing.

Original: A pricing policy that satisﬁes such a constraint is usually referred to as markdown pricing policy.
Revised: A pricing policy that satisﬁes such a constraint is usually referred to as markdown pricing policy.

Original: Thus motivated, in this work, we consider the markdown pricing problem with unknown demand, under various assumptions.
Revised: Thus motivated, in this work, we consider the markdown pricing problem with unknown demand, under various assumptions.

Original: Although unconstrained dynamic pricing under unknown demand has been extensively studied, little is known about markdown pricing under unknown demand.
Revised: Although unconstrained dynamic pricing under unknown demand has been extensively studied, little is known about markdown pricing under unknown demand.

Original: Recently, [3] and [9] independently showed that unimodality and Lipschitzness in  revenue function (deﬁned as the  price times the mean demand) are necessary to achieve sublinear regret.
Revised: Recently, [3] and [9] independently showed that unimodality and Lipschitzness in the revenue function (deﬁned as the product of the price and the mean demand) are necessary to achieve sublinear regret.

Original: In this setting, any reasonable policy must reduce the price at a moderate rate and stop only when there is sufﬁcient evidence for overshooting the optimal price.
Revised: In this setting, any reasonable policy must reduce the price at a moderate rate and stop only when there is sufﬁcient evidence for overshooting the optimal price.

Original: By selecting suitable parameters, [3, 9] showed that the above policy is indeed optimal, achieving a tight T 3 / 4 regret bound.
Revised: By selecting suitable parameters, [3, 9] showed that the above policy is indeed optimal, achieving a tight T 3 / 4 regret bound.

Original: 36th Conference on Neural Information Processing Systems (NeurIPS 2022).
Revised: [Table]

Original: However, in practice, it is usually assumed that the demand function has a certain parametric form, such as linear, exponential or logit form.
Revised: However, in practice, it is usually assumed that the demand function has a certain parametric form, such as linear, exponential or logit form.

Original: Intuitively, these extra structure enables fasterdemand learning  and suggests better regret bounds.
Revised: Intuitively, this extra structure enables faster learning rate and suggests better regret bounds.

Original: This motivates our ﬁrst question:
Revised: This motivates our ﬁrst question:

Original: Q1)
Revised: Q1)

Original: Can we strengthen the T 3 / 4 regret bound for markdown pricing for parametric demand families?
Revised: Can we improve the T 3 / 4 regret bound for markdown pricing for parametric demand families?

Original: Noticeably, there is a known separation result between under the assumption that the corresponding reward function(i.e. R ( x ) = s · D ( x ) ) is Lipschitz and Unimodal.
Revised: Noticeably, there is a known separation result between under the assumption that the corresponding reward function is Lipschitz and unimodal.

Original: In fact, [3, 9] showed a tight T 3 / 4 regret bound, which is asymptotically higher than T 2 / 3 , the optimal regret bound for unconstrained pricing ([11]) under the same assumptions.
Revised: In fact, [3, 9] showed a tight T 3 / 4 regret bound, which is asymptotically higher than T 2 / 3 , the optimal regret bound for unconstrained pricing ([11]) under the same assumptions.

Original: This highlights the extra complexity caused by  monotonicity constraint, and motivates the next question:
Revised: This highlights the extra complexity caused by the monotonicity constraint, and motivates the next question:

Original: Q2)
Revised: Q2)

Original: Is there still a separation in the regret bound between markdown and unconstrained pricing under parametric assumptions?
Revised: Is there still a provable “separation” between markdown and unconstrained pricing under parametric assumptions?

Original: While one may answer the Q1 and Q2 for speciﬁc parametric families, it is tempting to ﬁnd a uniﬁed framework that captures the hardness of demand learning for a given parametric family.
Revised: While one can answer the Q1 and Q2 for speciﬁc parametric families, we aim at ﬁnding a uniﬁed framework that captures the hardness of demand learning for a given parametric family.

Original: This motivates the following question:
Revised: This motivates the following question:

Original: Q3)
Revised: Q3)

Original: Can we ﬁnd a general framework to unify the regret bounds for different categories of families, rather than speciﬁc results for speciﬁc families?
Revised: Can we ﬁnd a general framework to unify the regret bounds for different categories of families, rather than speciﬁc results for speciﬁc families?

Original: In this work, we propose such a framework by introducing a concept called markdown dimension .
Revised: In this work, we propose such a framework by introducing a concept called markdown dimension .

Original: We provide efﬁcient markdown policies for each markdown dimension, which we also show to be best possible, thereby completely settling the problem.
Revised: We provide efﬁcient markdown policies for each markdown dimension, which we also show to be best possible, thereby completely settling the problem

Original: 1.1 Our Contributions.
Revised: 1.1 Our Contributions.

Original: In this work, we make the following contributions.
Revised: In this work, we make the following contributions.

Original: New Complexity Measure for Demand Learning.
Revised: New Complexity Measure for Demand Learning.

Original: We introduce a new concept called markdown dimension , which captures the complexity of performing markdown pricing on a family, answering Q3.
Revised: We introduce a new concept called markdown dimension , which captures the complexity of performing markdown pricing on a family, answering Q3.

Original: Within this framework, we provide a complete settlement of the problem as speciﬁed below.
Revised: Within this framework, we provide a complete settlement of the problem as speciﬁed below.

Original: Markdown Policies with Theoretical Guarantees.
Revised: Markdown Policies with Theoretical Guarantees.

Original: For each ﬁnite markdown dimension d  0 , we present an efﬁcient markdown pricing policy with sublinear regret.
Revised: For each ﬁnite markdown dimension d  0 , we present an efﬁcient markdown pricing policy with sublinear regret.

Original: The policies explore the demand function by proceeding in phases of varying lengths.
Revised: The exploration proceeds in phases of varying lengths.

Original: In each phase j , the policies derive a conﬁdence interval I j ✓ I j  1 for the optimal price p ⇤ based on observations collected at evenly spaced prices, which are lower than the prices used in phase j  1 .
Revised: In each phase j , the policies derive a conﬁdence interval I j ✓ I j  1 for the optimal price p ⇤ based on observations from evenly spaced prices, which are lower than the prices used in phase j  1 .

Original: The explorationthen terminates if there is sufﬁcient evidence that the current price is lower than p ⇤ .
Revised: The exploration terminates if there is sufﬁcient evidence that the current price is lower than p ⇤ .

Original: We show that for d = 0 and d  1 , our policies achieve regret O (log 2 T ) and ˜ O
Revised: We show that for d = 0 and d  1 , our policies achieve regret O (log 2 T ) and ˜ O

Original: ( T dd +1 ) respectively, settling Q1.
Revised: ( T dd +1 ) respectively, settling Q1.

Original: Separation From Unconstrained Pricing: Tight Lower Bounds.
Revised: Separation From Unconstrained Pricing: Tight Lower Bounds.

Original: We complement our upper bounds with a matching lower bound for each integer d .
Revised: We complement our upper bounds with a matching lower bound for each integer d .

Original: More precisely, we show an tight ⌦ (log 2 T ) lower bound for d = 0 , which is lower than the known O (log T ) regret bound without this monotonicity constraint (see [2]).
Revised: More precisely, we show an tight ⌦ (log 2 T ) lower bound for d = 0 , which is lower than the known O (log T ) regret bound without this monotonicity constraint (see [2]).

Original: For ﬁnite d  1 , we show an ⌦ ( T dd +1 ) lower bound, which not only matches our upper bound but is also higher than the tight ˜ ⇥ ( T 1 / 2 ) bound without markdown constraint (see [2, 10]).
Revised: For ﬁnite d  1 , we show an ⌦ ( T dd +1 ) lower bound, which not only matches our upper bound but is also higher than the tight ˜ ⇥ ( T 1 / 2 ) bound without markdown constraint (see [2, 10]).

Original: These lower bound results combined settle Q2.
Revised: These lower bound results combined settle Q2.

Original: Impact of Smoothness: We go further to reﬁne our bounds and investigate the impact of smoothness of reward function around the optimal price.
Revised: Impact of Smoothness: We go further to reﬁne our bounds and investigate the impact of smoothness of reward function around the optimal price.

Original: We consider a sensitivity parameter s , which essentially says that the reward function is only O ( " s ) suboptimal if the price is at an " distance away from the optimal price.
Revised: We consider a sensitivity parameter s , which essentially says that the reward function is only O ( " s ) suboptimal if the price is at an " distance away from the optimal price.

Original: For both ﬁnite and inﬁnite d , we extend our upper bounds to incorporate s .
Revised: For both ﬁnite and inﬁnite d , we extend our upper bounds to incorporate s .

Original: As the most basic case one can verify from Taylor’s Theorem that when the demand function is assumed to admit (i) a continuous second derivative and (ii) an optimal price in the interior of the domain, then s = 2 .
Revised: As the most basic case, one can verify from Taylor’s Theorem that when the demand function is assumed to admit (i) a continuous second derivative, and (ii) an optimal price in the interior of the domain, then s = 2 .

Original: We highlight our results for s = 2 in red in Table  where ⇥ denotes tight upper and lower bounds.
Revised: We highlight our results for s = 2 in red in Table 1, where ⇥ denotes tight upper and lower bounds.

Original: Markdown Dimension
Revised: Markdown Dimension

Original: Markdown Pricing
Revised: Markdown Pricing

Original: Unconstr.
Revised: Unconstr.

Original: Pricing d =⇥ (log 2 T ) ⇥ (log T ) [2] 1  d <˜ ⇥
Revised: Pricing d =⇥ (log 2 T ) ⇥ (log T ) [2] 1  d <˜ ⇥

Original: ( T d/ ( d +1) )
Revised: ( T d/ ( d +1) )

Original: ˜ ⇥ ( p T ) [2] d =˜ ⇥
Revised: ˜ ⇥ ( p T ) [2] d =˜ ⇥

Original: ( T 3 / 4 ) [3, 9] ˜ ⇥ ( T 2 / 3 ) [11]
Revised: ( T 3 / 4 ) [3, 9] ˜ ⇥ ( T 2 / 3 ) [11]

Original: 1.2 Related Work
Revised: 1.2 Related Work

Original: In the Multi-Armed Bandit (MAB) problem, the player is offered a ﬁnite set of arms, with each arm providing a random revenue from an unknown probability distribution speciﬁc to that arm.
Revised: In the Multi-Armed Bandit (MAB) problem, the player is offered a ﬁnite set of arms, with each arm providing a random revenue from an unknown probability distribution speciﬁc to that arm.

Original: The objective of the player is to maximize the total revenue earned by pulling a sequence of arms (e.g. [12]).
Revised: The objective of the player is to maximize the total revenue earned by pulling a sequence of arms (e.g. [12]).

Original: Our pricing problem generalizes this framework by using an inﬁnite action space [0 , 1] with each price p corresponding to an action whose revenue is drawn from an unknown distribution with mean R ( p ) .
Revised: Our pricing problem generalizes this framework by using an inﬁnite action space [0 , 1] with each price p corresponding to an action whose revenue is drawn from an unknown distribution with mean R ( p ) .

Original: In the Lipschitz Bandit problem (see, e.g., [1]), it is assumed that each x 2 [0 , 1] corresponds to an arm with mean reward µ ( x ) , where µ is an unknown L -Lipschitz function.
Revised: In the Lipschitz Bandit problem (see, e.g., [1]), it is assumed that each x 2 [0 , 1] corresponds to an arm with mean reward µ ( x ) , where µ is an unknown L -Lipschitz function.

Original: For the one-dimensional case, [11] showed a tight ˜ ⇥ ( T 2 / 3 ) regret bound.
Revised: For the one-dimensional case, [11] showed a tight ˜ ⇥ ( T 2 / 3 ) regret bound.

Original: Recently there is an emerging line of work on bandits with monotonicity constraint.
Revised: Recently there is an emerging line of work on bandits with monotonicity constraint.

Original: [3] and [9] recently independently considered the dynamic pricing problem with monotonicity constraint under unknown demand function, and proved nearly-optimal T 3 / 4 regret bound assuming the reward function is Lipschitz and unimodal.
Revised: [3] and [9] recently independently considered the dynamic pricing problem with monotonicity constraint under unknown demand function, and proved nearly-optimal T 3 / 4 regret bound assuming the reward function is Lipschitz and unimodal.

Original: Moreover, they also showed that these assumptions are indeed minimal, in the sense that no markdown policy achieves o ( T ) regret without any one of these two assumptions.
Revised: Moreover, they also showed that these assumptions are indeed minimal, in the sense that no markdown policy achieves o ( T ) regret without any one of these two assumptions.

Original: Motivated by fairness constraints, [8] and [15] considered a general online convex optimization problem where the action sequence is required to be monotone.
Revised: Motivated by fairness constraints, [8] and [15] considered a general online convex optimization problem where the action sequence is required to be monotone.

Original: Other constraints on the arm sequence motivated by practical problems are also considered in the literature.
Revised: Other constraints on the arm sequence motivated by practical problems are also considered in the literature.

Original: For example, motivated by the concern that customers are hostile to frequent price changes (see e.g. [5]), [4, 14] and [16] considered pricing policies with a given budget in the number of price changes .
Revised: For example, motivated by the concern that customers are hostile to frequent price changes (see e.g. [5]), [4, 14] and [16] considered pricing policies with a given budget in the number of price changes .

Original: Motivated by clinical trials, [6] studied the best arm identiﬁcation problem when the reward of the arms are monotone and the goal is to identify the arm closest to a threshold.
Revised: Motivated by clinical trials, [6] studied the best arm identiﬁcation problem when the reward of the arms are monotone and the goal is to identify the arm closest to a threshold.

Original:  Model and Assumptions
Revised:  Model and Assumptions

Original: We begin by formally stating our model.
Revised: We begin by formally stating our model.

Original: In this work we assume an unlimited supply of a single product.
Revised: In this work we assume an unlimited supply of a single product.

Original: Given a discrete time horizon of T rounds, in each round t , the policy (representing the “seller”) selects a price p t (the particular interval [0 , 1] is without loss of generality, by scaling).
Revised: Given a discrete time horizon of T rounds, in each round t , the policy (representing the “seller”) selects a price p t (the particular interval [0 , 1] is without loss of generality, by scaling).

Original: The demand D t in this round is then independently drawn from a ﬁxed distribution with unknown mean
Revised: The demand D t in this round is then independently drawn from a ﬁxed distribution with unknown mean

Original: D ( p t ) , and the policy receives revenue (or reward , which we will use interchangeably) p t for each unit sold, and hence a total of p t · D t revenue in this round.
Revised: D ( p t ) , and the policy receives revenue (or reward , which we will use interchangeably) p t for each unit sold, and hence a total of p t · D t revenue in this round.

Original: The only constraint the policy must satisfy is the markdown constraint: p 1  · · ·  p T with probability 1 .
Revised: The only constraint the policy must satisfy is the markdown constraint: p 1  · · ·  p T with probability 1 .

Original: The function D ( p ) which maps each price p to the mean demand at this price is known as the demand function .
Revised: The function D ( p ) which maps each price p to the mean demand at this price is known as the demand function .

Original: For any policy 1 ⇡ and demand function D ( · ) , we use r ( ⇡ , D ) to denote the expected total reward of ⇡ under D .
Revised: For any policy 1 ⇡ and demand function D ( · ) , we use r ( ⇡ , D ) to denote the expected total reward of ⇡ under D .

Original: Rather than evaluating policies directly in terms of r ( ⇡ , D ) , it is more informative (and ubiquitous in the literature on MAB) to measure performance using the notion of regret with respect to a certain idealized benchmark.
Revised: Rather than evaluating policies directly in terms of r ( ⇡ , D ) , it is more informative (and ubiquitous in the literature on MAB) to measure performance using the notion of regret with respect to a certain idealized benchmark.

Original: Speciﬁcally, since we assumed unlimited supply, when the true reward function is known, the seller simply always selects a revenue-maximizing price p ⇤ D = arg max p 2 [0 , 1] p · D ( p ) at each round, and we denote this maximal reward rate to be r ⇤ D = max p 2 [0 , 1] p · D ( p ) .
Revised: Speciﬁcally, since we assumed unlimited supply, when the true reward function is known, the seller simply always selects a revenue-maximizing price p ⇤ D = arg max p 2 [0 , 1] p · D ( p ) at each round, and we denote this maximal reward rate to be r ⇤ D = max p 2 [0 , 1] p · D ( p ) .

Original: The regret of a policy is then deﬁned with respect to this quantity, and we seek to bound the worst-case value over a given family of demand functions.
Revised: The regret of a policy is then deﬁned with respect to this quantity, and we seek to bound the worst-case value over a given family of demand functions.

Original: Deﬁnition 1 (Regret) .
Revised: Deﬁnition 1 (Regret) .

Original: For any policy ⇡ and demand function D , deﬁne the regret of policy ⇡ under
Revised: For any policy ⇡ and demand function D , deﬁne the regret of policy ⇡ under

Original: D as Reg( ⇡ , D ) :
Revised: D as Reg( ⇡ , D ) :

Original: = r ⇤ D · T  r ( ⇡ , D ) .
Revised: = r ⇤ D · T  r ( ⇡ , D ) .

Original: For any given family F of demand functions
Revised: For any given family F of demand functions

Original: 2.1 Basic Assumptions
Revised: 2.1 Basic Assumptions

Original: Now we state the common assumptions that all of our results rely on.
Revised: Now we state the common assumptions that all of our results rely on.

Original: A demand function D ( · ) is associated with a revenue function (or, reward function ) R ( p ) =
Revised: A demand function D ( · ) is associated with a revenue function (or, reward function ) R ( p ) =

Original: p · D ( p ) .
Revised: p · D ( p ) .

Original: Sometimes it is more convenient to work directly with the revenue functions, in which cases we abuse the notations and denote by Reg( ⇡ , R ) the regret of under reward function R .
Revised: Sometimes it is more convenient to work directly with the revenue functions, in which cases we abuse the notations and denote by Reg( ⇡ , R ) the regret of under reward function R .

Original: Deﬁnition 2 (Optimal Price Mapping) .
Revised: Deﬁnition 2 (Optimal Price Mapping) .

Original: Let F be a set of functions deﬁned on S ✓ R .
Revised: Let F be a set of functions deﬁned on S ✓ R .

Original: For any function R : S !
Revised: For any function R : S !

Original: R , let M ( R ) be the subset of its global maxima on S .
Revised: R , let M ( R ) be the subset of its global maxima on S .

Original: The optimal price mapping p
Revised: The optimal price mapping p

Original: ⇤ : F ! S is deﬁned as p ⇤ ( R ) =
Revised: ⇤ : F ! S is deﬁned as p ⇤ ( R ) =

Original: inf M ( R ) .
Revised: inf M ( R ) .

Original: We ﬁrst introduce a standard assumption (see e.g. [2]) that the derivative of R vanishes at p
Revised: We ﬁrst introduce a standard assumption (see e.g. [2]) that the derivative of R vanishes at p

Original: ⇤ ( R ) .
Revised: ⇤ ( R ) .

Original: Assumption 1 (First Order Optimality) .
Revised: Assumption 1 (First Order Optimality) .

Original: We assume that every reward function R is differentiable on its domain, and moreover, R 0 ( p ⇤ ( R ))
Revised: We assume that every reward function R is differentiable on its domain, and moreover, R 0 ( p ⇤ ( R ))

Original: = 0 .
Revised: = 0 .

Original: If, in addition, R is twice continuously-differentiable, then by Taylor expansion around p ⇤ , | R ( p ⇤ ) 
Revised: If, in addition, R is twice continuously-differentiable, then by Taylor expansion around p ⇤ , | R ( p ⇤ )

Original: R ( p ⇤ + " ) | = 12 R 00 ( p ⇤ ) · " 2 + o ( " 2 ) .
Revised: R ( p ⇤ + " ) | = 12 R 00 ( p ⇤ ) · " 2 + o ( " 2 ) .

Original: Thus, if a policy overshoots the optimal price by " , then only an O ( " 2 ) loss is incurred in each round.
Revised: Thus, if a policy overshoots the optimal price by " , then only an O ( " 2 ) loss is incurred in each round.

Original: We next introduce a distributional assumption.
Revised: We next introduce a distributional assumption.

Original: Deﬁnition 3 (Subgaussian Random Variable) .
Revised: Deﬁnition 3 (Subgaussian Random Variable) .

Original: Deﬁne the subgaussian norm of a random variable X as k X k 2 :=
Revised: Deﬁne the subgaussian norm of a random variable X as k X k 2 :=

Original: inf { c > 0 : E [ e X 2 /c 2 ]  2 } .
Revised: inf { c > 0 : E [ e X 2 /c 2 ]  2 } .

Original: Say X is subgaussian if k X k 2 < 1 .
Revised: Say X is subgaussian if k X k 2 < 1 .

Original: Assumption 2 (Subgaussian noise) .
Revised: Assumption 2 (Subgaussian noise) .

Original: There exists a constant C sg > 0 such that under any true demand function and any price p , the random demand X at price p satisﬁes k X k 2  C sg .
Revised: There exists a constant C sg > 0 such that under any true demand function and any price p , the random demand X at price p satisﬁes k X k 2  C sg .

Original: 2.2 Warm-up: Markdown Pricing on Linear Demand
Revised: 2.2 Warm-up: Markdown Pricing on Linear Demand

Original: To introduce the key concept, markdown dimension, let’s take linear demand as a warm-up.
Revised: To introduce the key concept, markdown dimension, let’s take linear demand as a warm-up.

Original: Consider D ( x ; ✓ ) =
Revised: Consider D ( x ; ✓ ) =

Original: ✓ 1  ✓ 2 x for x⇥ 12 ,⇤ and the following natural markdown policy for
Revised: ✓ 1  ✓ 2 x for x⇥ 12 ,⇤ and the following natural markdown policy for

Original: F =
Revised: F =

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ⇤ : collect a number of samples at two prices p 1 , p 2 close to 1 , and denote by ¯ d 1 , ¯ d 2 the empirical mean demands.
Revised: ⇤ : collect a number of samples at two prices p 1 , p 2 close to 1 , and denote by ¯ d 1 , ¯ d 2 the empirical mean demands.

Original: By the Fundamental Theorem of Algebra, since a degree- d polynomial can be uniquely determined by its value at d distinct points, there is a ˆ ✓ 2 Rsatisfying D ( p i ; ˆ ✓ ) =
Revised: By the Fundamental Theorem of Algebra, since a degree- d polynomial can be uniquely determined by its value at d distinct points, there is a ˆ ✓ 2 Rsatisfying D ( p i ; ˆ ✓ ) =

Original: ¯ d i for i = 1 , 2 .
Revised: ¯ d i for i = 1 , 2 .

Original: Finally, select the optimal price of D ( · ; ˆ ✓ ) in remaining rounds.
Revised: Finally, select the optimal price of D ( · ; ˆ ✓ ) in remaining rounds.

Original: This policy is “robust” in the following sense.
Revised: This policy is “robust” in the following sense.

Original: Suppose ¯ d i deviates from the mean D ( p i ; ✓ ) by ⇠  , then the estimation error in ✓ is ⇠
Revised: Suppose ¯ d i deviates from the mean D ( p i ; ✓ ) by ⇠  , then the estimation error in ✓ is ⇠

Original: [Equation]
Revised: [Equation]

Original: i.e., proportional to  and inverse proportional to the gap between the two sample prices.
Revised: i.e., proportional to  and inverse proportional to the gap between the two sample prices.

Original: For more complex demand models, the dependence on the “divergence” of the sample prices changes.
Revised: For more complex demand models, the dependence on the “divergence” of the sample prices changes.

Original: To characterize such dependence, we introduce the notion of markdown dimension .
Revised: To characterize such dependence, we introduce the notion of markdown dimension .

Original: Informally, a parameterized family has markdown dimension d if for any set of d prices at distance h apart, an
Revised: Informally, a parameterized family has markdown dimension d if for any set of d prices at distance h apart, an

Original: O (  ) error on the estimated demands results in an O   h d  estimation error on the model parameter.
Revised: O (  ) error on the estimated demands results in an O   h d  estimation error on the model parameter.

Original: We make this idea formal in the subsequent sections.
Revised: We make this idea formal in the subsequent sections.

Original: 2.3 Identiﬁability and Robust Parametrization
Revised: 2.3 Identiﬁability and Robust Parametrization

Original: Intuitively, the exploration-exploitation trade-off for markdown pricing becomes harder to manage as the given family becomes more complex.
Revised: Intuitively, the exploration-exploitation trade-off for markdown pricing becomes harder to manage as the given family becomes more complex.

Original: Consider, for example, linear demand functions.
Revised: Consider, for example, linear demand functions.

Original: If each function takes the form D ( p ; c ) = 1 + cp where only c 2 (0 , 1) is unknown and p 2 [0 , 1] , then the seller simply needs to estimate the price elasticity c by sampling sufﬁciently many times at p = 1 .
Revised: If each function takes the form D ( p ; c ) = 1 + cp where only c 2 (0 , 1) is unknown and p 2 [0 , 1] , then the seller simply needs to estimate the price elasticity c by sampling sufﬁciently many times at p = 1 .

Original: In contrast, if the demand function is D ( p ; a, b ) = a  bp where both a, b are unknown, then sampling at one price would not sufﬁce.
Revised: In contrast, if the demand function is D ( p ; a, b ) = a  bp where both a, b are unknown, then sampling at one price would not sufﬁce.

Original: Rather, one needs to select (at least) two distinct prices, say p and p 0 , to ﬁt a linear function, thereby facing the following dilemma.
Revised: Rather, one needs to select (at least) two distinct prices, say p and p 0 , to ﬁt a linear function, thereby facing the following dilemma.

Original: Suppose p < p 0 are far apart, then, p may be far away from the optimal price p
Revised: Suppose p < p 0 are far apart, then, p may be far away from the optimal price p

Original: ⇤ if p ⇤ is close to p 0 , resulting in high regret.
Revised: ⇤ if p ⇤ is close to p 0 , resulting in high regret.

Original: Otherwise, when p, p 0 are close-by, the demand learning requires a high volume of samples, which also leads to a high regret.
Revised: Otherwise, when p, p 0 are close-by, the demand learning requires a high volume of samples, which also leads to a high regret.

Original: Thus we reach a natural question: can we introduce a notion of complexity to measure the difﬁculty of performing markdown pricing on a given family, and then provide tight regret bounds in terms for each level of complexity?
Revised: Thus we reach a natural question: can we introduce a notion of complexity to measure the difﬁculty of performing markdown pricing on a given family, and then provide tight regret bounds in terms for each level of complexity?

Original: In this work, we propose a novel concept of markdown dimension and provide nearly-optimal regret bounds in terms of this complexity metric.
Revised: In this work, we propose a novel concept of markdown dimension and provide nearly-optimal regret bounds in terms of this complexity metric.

Original: The formal deﬁnition relies on two other concepts which we introduce in the next two subsections: the identiﬁability of a family, and the robustness of a parametrization.
Revised: The formal deﬁnition relies on two other concepts which we introduce in the next two subsections: the identiﬁability of a family, and the robustness of a parametrization.

Original: Identiﬁability Our notion of identiﬁability generalizes a key property of single-variable polynomials, that every degree- d polynomial can be uniquely determined by its values at any ( d + 1) points.
Revised: Identiﬁability Our notion of identiﬁability generalizes a key property of single-variable polynomials, that every degree- d polynomial can be uniquely determined by its values at any ( d + 1) points.

Original: To present the formal deﬁnition, we ﬁrst introduce a mapping which, for a ﬁxed subset of prices, assigns each demand function a proﬁle based on its values at those prices.
Revised: To present the formal deﬁnition, we ﬁrst introduce a mapping which, for a ﬁxed subset of prices, assigns each demand function a proﬁle based on its values at those prices.

Original: Deﬁnition 4 (Proﬁle Mapping) .
Revised: Deﬁnition 4 (Proﬁle Mapping) .

Original: Consider a set F of real-valued functions deﬁned on A ✓ R .
Revised: Consider a set F of real-valued functions deﬁned on A ✓ R .

Original: For any p = ( p 0 , p 1 , ..., p d ) 2 A d +1 , the proﬁle-mapping at prices p is deﬁned as
Revised: For any p = ( p 0 , p 1 , ..., p d ) 2 A d +1 , the proﬁle-mapping at prices p is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: We may subsequently call  p ( D ) the proﬁle of function D at p .
Revised: We may subsequently call  p ( D ) the proﬁle of function D at p .

Original: Informally, a family of functions is said to be d -identiﬁable, if the graphs of any two functions intersect for at most d times.
Revised: Informally, a family of functions is said to be d -identiﬁable, if the graphs of any two functions intersect for at most d times.

Original: For example, the family of all degree d polynomials is d -identiﬁable.
Revised: For example, the family of all degree d polynomials is d -identiﬁable.

Original: Deﬁnition 5 (Identiﬁability) .
Revised: Deﬁnition 5 (Identiﬁability) .

Original: A family F of functions deﬁned on S ✓ R is d -identiﬁable , if for any distinct p 0 , p 1 ..., p d 2 S , the proﬁle mapping  p 0 ,...,p d is injective,
Revised: A family F of functions deﬁned on S ✓ R is d -identiﬁable , if for any distinct p 0 , p 1 ..., p d 2 S , the proﬁle mapping  p 0 ,...,p d is injective,

Original: i.e. F ( f ) 6 = F ( ˜ f ) for any distinct f, ˜ f 2 F .
Revised: i.e. F ( f ) 6 = F ( ˜ f ) for any distinct f, ˜ f 2 F .

Original: We will soon use the following fact: if a family is d -identiﬁable, then for any distinct p 0 , p 1 ..., p d the inverse proﬁle-mapping   1 p : R p !
Revised: We will soon use the following fact: if a family is d -identiﬁable, then for any distinct p 0 , p 1 ..., p d the inverse proﬁle-mapping   1 p : R p !

Original: F exists, where R p is the range of the mapping  p .
Revised: F exists, where R p is the range of the mapping  p .

Original: Robust Parametrization We ﬁrst formally deﬁne a parametrization .
Revised: Robust Parametrization We ﬁrst formally deﬁne a parametrization .

Original: Deﬁnition 6 (Parametrization) .
Revised: Deﬁnition 6 (Parametrization) .

Original: An order- m parametrization for a family F of functions is a one-toone mapping from some set ⇥ ✓ R m to F .
Revised: An order- m parametrization for a family F of functions is a one-toone mapping from some set ⇥ ✓ R m to F .

Original: Moreover, each ✓ 2 ⇥ is called a parameter .
Revised: Moreover, each ✓ 2 ⇥ is called a parameter .

Original: We use D ( p ; ✓ ) to denote the function in F that parameter ✓ corresponds to.
Revised: We use D ( p ; ✓ ) to denote the function in F that parameter ✓ corresponds to.

Original: For example, for
Revised: For example, for

Original: D ( p ; ✓ 1 , ✓ 2 ) =
Revised: D ( p ; ✓ 1 , ✓ 2 ) =

Original: ✓ 1  ✓ 2 · p is a parametrization of the family of linear functions.
Revised: ✓ 1  ✓ 2 · p is a parametrization of the family of linear functions.

Original: As a standard assumption (see e.g. [2]), we assume ⇥ to be compact, which leads to many favorable properties.
Revised: As a standard assumption (see e.g. [2]), we assume ⇥ to be compact, which leads to many favorable properties.

Original: Assumption 3 (Compact Domain) .
Revised: Assumption 3 (Compact Domain) .

Original: The domain ⇥ of the parametrization is compact.
Revised: The domain ⇥ of the parametrization is compact.

Original: Under this assumption, the demand functions in F are bounded, and thus we may without loss of generality scale the range (i.e. target space) of those functions to be [0 , 1] .
Revised: Under this assumption, the demand functions in F are bounded, and thus we may without loss of generality scale the range (i.e. target space) of those functions to be [0 , 1] .

Original: Assumption 4 (Smoothness) .
Revised: Assumption 4 (Smoothness) .

Original: The mapping D : [0 , 1] ⇥ ⇥ !
Revised: The mapping D : [0 , 1] ⇥ ⇥ !

Original: R is twice continuously differentiable.
Revised: R is twice continuously differentiable.

Original: In particular, since [0 , 1] ⇥ ⇥ is compact, under the above assumption, there exist constants C ( j ) >such that | D ( j ) ( p, ✓ ) |  C ( j ) for any ( p, ✓ ) 2 [0 , 1] ⇥ ⇥ and j = 0 , 1 , 2 .
Revised: In particular, since [0 , 1] ⇥ ⇥ is compact, under the above assumption, there exist constants C ( j ) >such that | D ( j ) ( p, ✓ ) |  C ( j ) for any ( p, ✓ ) 2 [0 , 1] ⇥ ⇥ and j = 0 , 1 , 2 .

Original: By abuse of notation, we view the optimal price mapping p
Revised: By abuse of notation, we view the optimal price mapping p

Original: ⇤
Revised: ⇤

Original: as being deﬁned on ⇥ , rather than on F as before.
Revised: as being deﬁned on ⇥ , rather than on F as before.

Original: Formally, let M ( ✓ ) be the set of global maxima of R ( x ; ✓ ) , then p ⇤ ( ✓ ) =
Revised: Formally, let M ( ✓ ) be the set of global maxima of R ( x ; ✓ ) , then p ⇤ ( ✓ ) =

Original: inf R ( x ; ✓ ) .
Revised: inf R ( x ; ✓ ) .

Original: For example, for D ( p ; ✓ 1 , ✓ 2 ) = ✓ 1  ✓ 2 · p one can verify that p ⇤ =
Revised: For example, for D ( p ; ✓ 1 , ✓ 2 ) = ✓ 1  ✓ 2 · p one can verify that p ⇤ =

Original: ✓ 1 2 ✓ 2 .
Revised: ✓ 1 2 ✓ 2 .

Original: The next assumption allows us to “propagate” the estimation error in ✓ to that of p ⇤ ( ✓ ) .
Revised: The next assumption allows us to “propagate” the estimation error in ✓ to that of p ⇤ ( ✓ ) .

Original: Assumption 5 (Lipschitz Optimal Price Mapping) .
Revised: Assumption 5 (Lipschitz Optimal Price Mapping) .

Original: The optimal price mapping p
Revised: The optimal price mapping p

Original: ⇤ : ⇥ !
Revised: ⇤ : ⇥ !

Original: [0 , 1] is
Revised: [0 , 1] is

Original: C ⇤ -Lipschitz for some constant C ⇤ > 0 .
Revised: C ⇤ -Lipschitz for some constant C ⇤ > 0 .

Original: This assumption has appeared in the previous literature on parametric demand learning, see e.g.
Revised: This assumption has appeared in the previous literature on parametric demand learning, see e.g.

Original: Assumption 1(c) of [2].
Revised: Assumption 1(c) of [2].

Original: Moreover, it is satisﬁed by many basic demand functions such as linear, exponential and logit demand.
Revised: Moreover, it is satisﬁed by many basic demand functions such as linear, exponential and logit demand.

Original: For instance, let D ( p ; c ) =
Revised: For instance, let D ( p ; c ) =

Original:  cp where p, c 2 [0 , 1] , then p ⇤ ( c ) =
Revised:  cp where p, c 2 [0 , 1] , then p ⇤ ( c ) =

Original: min { 12 c , 1 } , which is 1 -Lipschitz.
Revised: min { 12 c , 1 } , which is 1 -Lipschitz.

Original: Nonetheless, this assumption is somewhat unnatural as it is not made on the parametrization directly, we leave it open whether there is an assumption imposed directly on F that implies this assumption.
Revised: Nonetheless, this assumption is somewhat unnatural as it is not made on the parametrization directly, we leave it open whether there is an assumption imposed directly on F that implies this assumption.

Original: The ﬁnal ingredient for robust parametrization is motivated by a nice property of the natural parametrization D ( p ; ✓ ) =
Revised: The ﬁnal ingredient for robust parametrization is motivated by a nice property of the natural parametrization D ( p ; ✓ ) =

Original: P dj =

Revised: P dj =

Original: ✓ j p j for polynomials.
Revised: ✓ j p j for polynomials.

Original: Consider any distinct p 0 , p 1 , ..., p d , and distinct real numbers y 0 , y 1 , ..., y d representing, for example, the mean reward at each p i .
Revised: Consider any distinct p 0 , p 1 , ..., p d , and distinct real numbers y 0 , y 1 , ..., y d representing, for example, the mean reward at each p i .

Original: We can then uniquely determine a degree- d polynomial by solving the linear equation V p ✓ = y where
Revised: We can then uniquely determine a degree- d polynomial by solving the linear equation V p ✓ = y where

Original: ✓ = ( ✓ 0 , ✓ 1 ..., ✓ d ) T , y =
Revised: ✓ = ( ✓ 0 , ✓ 1 ..., ✓ d ) T , y =

Original: ( y 0 , y 1 , ..., y d ) T and V p :
Revised: ( y 0 , y 1 , ..., y d ) T and V p :

Original: = V ( p 0 , ..., p d ) is the Vandermonde matrix.
Revised: = V ( p 0 , ..., p d ) is the Vandermonde matrix.

Original: One can easily verify that V p is invertible if and only if p i ’s are distinct, in which case we have ✓ = V  1 p y .
Revised: One can easily verify that V p is invertible if and only if p i ’s are distinct, in which case we have ✓ = V  1 p y .

Original: Next we consider the effect of a perturbation on y , in terms of the following separability parameter.
Revised: Next we consider the effect of a perturbation on y , in terms of the following separability parameter.

Original: Deﬁnition 7 (Separability) .
Revised: Deﬁnition 7 (Separability) .

Original: For any p = ( p 0 , ..., p d ) 2 R d +1 , deﬁne h ( p ) :
Revised: For any p = ( p 0 , ..., p d ) 2 R d +1 , deﬁne h ( p ) :

Original: = min i 6 =
Revised: = min i 6 =

Original: j | p i  p
Revised: j | p i  p

Original: j | .
Revised: j | .

Original: To motivate the general deﬁnition of robustness, ﬁrst consider a result speciﬁc to polynomials.
Revised: To motivate the general deﬁnition of robustness, ﬁrst consider a result speciﬁc to polynomials.

Original: Recall that R p is range of the proﬁle mapping  p .
Revised: Recall that R p is range of the proﬁle mapping  p .

Original: Proposition 1 (The Natural Parametrization for Polynomials is Robust) .
Revised: Proposition 1 (The Natural Parametrization for Polynomials is Robust) .

Original: There exist constants
Revised: There exist constants

Original: C 1 , C 2 > 0 such that for any p 2 [0 , 1] d +1 with 0 < h ( p )  C 1 , and y , ˆy 2 R p with k y  ˆy k 1 
Revised: C 1 , C 2 > 0 such that for any p 2 [0 , 1] d +1 with 0 < h ( p )  C 1 , and y , ˆy 2 R p with k y  ˆy k 1 

Original: C 1 , it holds that k V  1 p
Revised: C 1 , it holds that k V  1 p

Original: y  V  1 p ˆy k 1  C 2 · k y  ˆy k 1 · h ( p )  d .
Revised: y  V  1 p ˆy k 1  C 2 · k y  ˆy k 1 · h ( p )  d .

Original: More concretely, let D ( p ; ✓ ) be the underlying polynomial demand function, then y
Revised: More concretely, let D ( p ; ✓ ) be the underlying polynomial demand function, then y

Original: = V p
Revised: = V p

Original: ✓ are the mean demands at the prices in p .
Revised: ✓ are the mean demands at the prices in p .

Original: Suppose we observe empirical mean demands ˆy =
Revised: Suppose we observe empirical mean demands ˆy =

Original: (ˆ y 0 , ˆ y 1 , ..., ˆ y d ) at prices p , then we have a reasonable estimation ˆ ✓ = V  1 p ˆy .
Revised: (ˆ y 0 , ˆ y 1 , ..., ˆ y d ) at prices p , then we have a reasonable estimation ˆ ✓ = V  1 p ˆy .

Original: Our Proposition 1 can then be viewed as an upper bound on the estimation error k ✓  ˆ ✓ k 1 in terms of h ( p ) and y  ˆy .
Revised: Our Proposition 1 can then be viewed as an upper bound on the estimation error k ✓  ˆ ✓ k 1 in terms of h ( p ) and y  ˆy .

Original: In order to achieve sublinear regret, we need to ensure h ( p ) =
Revised: In order to achieve sublinear regret, we need to ensure h ( p ) =

Original: o (1) as T ! 1 .
Revised: o (1) as T ! 1 .

Original: Moreover, the rate of this convergence crucially affects our regret bounds.
Revised: Moreover, the rate of this convergence crucially affects our regret bounds.

Original: Proposition 1 establishes a nice property for polynomials, that the estimation error scales as h  d , as h ! 

Revised: Proposition 1 establishes a nice property for polynomials, that the estimation error scales as h  d , as h !

Original: + .
Revised: + .

Original: We introduce robust parametrization by generalizing this property beyond polynomials.
Revised: We introduce robust parametrization by generalizing this property beyond polynomials.

Original: Loosely, an order- d parametrization is robust , if the bound in Proposition 1 holds.
Revised: Loosely, an order- d parametrization is robust , if the bound in Proposition 1 holds.

Original: Deﬁnition 8 (Robust Parametrization) .
Revised: Deﬁnition 8 (Robust Parametrization) .

Original: An order- d parametrization ✓ : ⇥ !
Revised: An order- d parametrization ✓ : ⇥ !

Original: F is robust , if
Revised: F is robust , if

Original: (1) it satisﬁes Assumptions 3, 4 and 5, and (2) there exist constants C 1 , C 2 > 0 such that for any p 2 R d +1 with 0 < h ( p )  C 1 and any y, y 0 2 R p with k y  y 

Revised: (1) it satisﬁes Assumptions 3, 4 and 5, and (2) there exist constants C 1 , C 2 > 0 such that for any p 2 R d +1 with 0 < h ( p )  C 1 and any y, y 0 2 R p with k y  y

Original: k 1  C 1 , it holds that k   1 p ( y )    1 p ( y 0 )
Revised: k 1  C 1 , it holds that k   1 p ( y )    1 p ( y 0 )

Original: k 

Revised: k

Original:  C 2 · k y  y 0 k 1 · h ( p )  d .
Revised:  C 2 · k y  y 0 k 1 · h ( p )  d .

Original: In
Revised: In

Original: particular, when d = 0 , this inequality simply says  p  1 is C 2 -Lipschitz.
Revised: particular, when d = 0 , this inequality simply says  p  1 is C 2 -Lipschitz.

Original: 2.4 Markdown Dimension
Revised: 2.4 Markdown Dimension

Original: Now we are ready to deﬁne the markdown dimension.
Revised: Now we are ready to deﬁne the markdown dimension.

Original: Deﬁnition 9 (Markdown Dimension) .
Revised: Deﬁnition 9 (Markdown Dimension) .

Original: The markdown dimension (or simply dimension ) for a family
Revised: The markdown dimension (or simply dimension ) for a family

Original: F of functions, denoted d ( F ) , is the minimum integer d  0 such that F is (i) d -identiﬁable, and (ii) admits a robust order-
Revised: F of functions, denoted d ( F ) , is the minimum integer d  0 such that F is (i) d -identiﬁable, and (ii) admits a robust order-

Original: d parametrization.
Revised: d parametrization.

Original: If no ﬁnite d satisﬁes the above conditions, then d ( F ) = 1 .
Revised: If no ﬁnite d satisﬁes the above conditions, then d ( F ) = 1 .

Original: We further illustrate our deﬁnition by considering the dimensions of some basic families.
Revised: We further illustrate our deﬁnition by considering the dimensions of some basic families.

Original: As the simplest family, one may verify that our deﬁnition of 0 -dimensional family is equivalent to the separable family as deﬁned in Section 4 of [2].
Revised: As the simplest family, one may verify that our deﬁnition of 0 -dimensional family is equivalent to the separable family as deﬁned in Section 4 of [2].

Original: We provide more concrete examples below.
Revised: We provide more concrete examples below.

Original: Details can found in Appendix A.
Revised: Details can found in Appendix A.

Original: Proposition 2.
Revised: Proposition 2.

Original: Let D ( x ; ✓ ) =
Revised: Let D ( x ; ✓ ) =

Original: P dj =

Revised: P dj =

Original: ✓ j x j and suppose F = { D ( x ; ✓ ) : ✓ 2 ⇥ } satisﬁes Assumptions 3, 4 and 5.
Revised: ✓ j x j and suppose F = { D ( x ; ✓ ) : ✓ 2 ⇥ } satisﬁes Assumptions 3, 4 and 5.

Original: Then, d ( F ) =
Revised: Then, d ( F ) =

Original: d .
Revised: d .

Original: Proposition 3 ( 0 -Dimensional Families) .
Revised: Proposition 3 ( 0 -Dimensional Families) .

Original: The following families F 1 , F 2 and F 3 are 0 -dimensional.
Revised: The following families F 1 , F 2 and F 3 are 0 -dimensional.

Original: (1) [Single-Parameter Linear Demand] Let D a ( x ) =
Revised: (1) [Single-Parameter Linear Demand] Let D a ( x ) =

Original:  ax for x 2 [ 12 , 1] and F 1 = { D a ( x ) : a⇥ 12 ,⇤ } .
Revised:  ax for x 2 [ 12 , 1] and F 1 = { D a ( x ) : a⇥ 12 ,⇤ } .

Original: (2) [Exponential Demand] Let D a ( x ) =
Revised: (2) [Exponential Demand] Let D a ( x ) =

Original: e 1  ax for x 2 [ 12 , 1] and F 2 =  D a ( x ) : a⇥ 12 ,⇤ .
Revised: e 1  ax for x 2 [ 12 , 1] and F 2 =  D a ( x ) : a⇥ 12 ,⇤ .

Original: (3) [Logit Demand] Let D a ( x ) =
Revised: (3) [Logit Demand] Let D a ( x ) =

Original: e 1  ax 1+ e 1  ax for x⇥ 12 ,⇤ and F 3 =  D a ( x ) : a⇥ 12 ,⇤ .
Revised: e 1  ax 1+ e 1  ax for x⇥ 12 ,⇤ and F 3 =  D a ( x ) : a⇥ 12 ,⇤ .

Original: Finally, we observe that by our deﬁnition, if a family of functions is not d -identiﬁable for any d , then d ( F ) = 1 .
Revised: Finally, we observe that by our deﬁnition, if a family of functions is not d -identiﬁable for any d , then d ( F ) = 1 .

Original: For example, for any d distinct prices, there exist multiple (more precisely, inﬁnitely many) Lipschitz functions having the same values on these d prices.
Revised: For example, for any d distinct prices, there exist multiple (more precisely, inﬁnitely many) Lipschitz functions having the same values on these d prices.

Original: Proposition 4.
Revised: Proposition 4.

Original: Let F be the set of 1 -Lipschitz functions from [0 , 1] to [0 , 1] , then d ( F ) = 1 .
Revised: Let F be the set of 1 -Lipschitz functions from [0 , 1] to [0 , 1] , then d ( F ) = 1 .

Original: 2.5 Sensitivity
Revised: 2.5 Sensitivity

Original: Consider the Taylor expansion of a reward function R ( x ) around an optimal price p
Revised: Consider the Taylor expansion of a reward function R ( x ) around an optimal price p

Original: ⇤ : R ( p ) =
Revised: ⇤ : R ( p ) =

Original: R ( p ⇤ )
Revised: R ( p ⇤ )

Original: + 

Revised: +

Original: + 12!
Revised: + 12!

Original: R 

Revised: R

Original: ( p ⇤ )
Revised: ( p ⇤ )

Original: ( p  p ⇤ ) 2 + 13!
Revised: ( p  p ⇤ ) 2 + 13!

Original: R (3) ( p ⇤ )
Revised: R (3) ( p ⇤ )

Original: ( p  p ⇤ ) 3 + ...
Revised: ( p  p ⇤ ) 3 + ...

Original: Suppose the ﬁrst nonzero derivative is R ( k ) ( p ⇤ ) .
Revised: Suppose the ﬁrst nonzero derivative is R ( k ) ( p ⇤ ) .

Original: Then, the higher k , the less sensitive the revenue is to the estimation error in p ⇤ .
Revised: Then, the higher k , the less sensitive the revenue is to the estimation error in p ⇤ .

Original: We capture this aspect in the following notion of sensitivity .
Revised: We capture this aspect in the following notion of sensitivity .

Original: Deﬁnition 10 (Sensitivity) .
Revised: Deﬁnition 10 (Sensitivity) .

Original: A reward function R is s -sensitive if it is ( s + 1)
Revised: A reward function R is s -sensitive if it is ( s + 1)

Original: -times continuously differentiable with R (1) ( p ⇤ ( R ))
Revised: -times continuously differentiable with R (1) ( p ⇤ ( R ))

Original: = · · · = R ( s  1) ( p ⇤ ( R ))
Revised: = · · · = R ( s  1) ( p ⇤ ( R ))

Original: = 0 and R ( s ) ( p ⇤ ( R ))
Revised: = 0 and R ( s ) ( p ⇤ ( R ))

Original: < 0 .
Revised: < 0 .

Original: A family F of reward functions is called s -sensitive if
Revised: A family F of reward functions is called s -sensitive if

Original: (a) every R 2 F is s -sensitive, (b) it admits a parametrization R ( x ; ✓ ) satisfying Assumptions 3 to 5, (c) there is a constant C 6 > 

Revised: (a) every R 2 F is s -sensitive, (b) it admits a parametrization R ( x ; ✓ ) satisfying Assumptions 3 to 5, (c) there is a constant C 6 >

Original: such that R ( s ) ( p ⇤ ( R ))
Revised: such that R ( s ) ( p ⇤ ( R ))

Original:   C 6 < 0 for any R 2 F , and (d) for each 0  j  s , there exists a constant C ( j ) s.t.
Revised:   C 6 < 0 for any R 2 F , and (d) for each 0  j  s , there exists a constant C ( j ) s.t.

Original: | R ( j ) ( x ; ✓ ) |  C ( j ) , 8 x 2 [0 , 1] , ✓ 2 ⇥ .
Revised: | R ( j ) ( x ; ✓ ) |  C ( j ) , 8 x 2 [0 , 1] , ✓ 2 ⇥ .

Original: For example, let s  3 and R ( x ; ✓ ) =
Revised: For example, let s  3 and R ( x ; ✓ ) =

Original: ✓  | 12  x | s for x 2 [0 , 1] and any ✓ 2 R , then { R ( x ; ✓ ) : ✓ 2 [ 12 , 1] } is an s -sensitive family.
Revised: ✓  | 12  x | s for x 2 [0 , 1] and any ✓ 2 R , then { R ( x ; ✓ ) : ✓ 2 [ 12 , 1] } is an s -sensitive family.

Original: Note that by Taylor’s Theorem, | R ( p ⇤ + " )  R
Revised: Note that by Taylor’s Theorem, | R ( p ⇤ + " )  R

Original: ( p ⇤ ) |  O ( " s ) .
Revised: ( p ⇤ ) |  O ( " s ) .

Original: Consequently, if a policy errs by " in the optimal price, the regret per round is O ( " s ) , lower than the per-round regret O ( " 2 ) in the basic case.
Revised: Consequently, if a policy errs by " in the optimal price, the regret per round is O ( " s ) , lower than the per-round regret O ( " 2 ) in the basic case.

Original: In each of the next three sections, we will ﬁrst present the regret bounds for the basic case s = 2 , and then characterize how this bound improves as s increases.
Revised: In each of the next three sections, we will ﬁrst present the regret bounds for the basic case s = 2 , and then characterize how this bound improves as s increases.

Original:  Zero-Dimensional Family
Revised:  Zero-Dimensional Family

Original: We start with the simplest case, 0 -dimensional demand functions.
Revised: We start with the simplest case, 0 -dimensional demand functions.

Original: As opposed to the optimism in the face of uncertainty in UCB policies, our Cautious Myopic policy (Algorithm 1) adopts conservatism in the face of uncertainty.
Revised: As opposed to the optimism in the face of uncertainty in UCB policies, our Cautious Myopic policy (Algorithm 1) adopts conservatism in the face of uncertainty.

Original: More precisely, we partition the time horizon into phases where phase j consists of t j = d 9 j log T e rounds, and thus in total there are K
Revised: More precisely, we partition the time horizon into phases where phase j consists of t j = d 9 j log T e rounds, and thus in total there are K

Original: = O (log T  log log T ) phases.
Revised: = O (log T  log log T ) phases.

Original: In each phase, the policy builds a conﬁdence interval I j for the true parameter using the samples collected in this phase, and sets the price for phase j + 1 to be the largest optimal price of any “surviving” parameter ✓ 2 I j .
Revised: In each phase, the policy builds a conﬁdence interval I j for the true parameter using the samples collected in this phase, and sets the price for phase j + 1 to be the largest optimal price of any “surviving” parameter ✓ 2 I j .

Original: We write t ( j ) :=
Revised: We write t ( j ) :=

Original: P jk =

Revised: P jk =

Original: t k and for convenience t (0)
Revised: t k and for convenience t (0)

Original: = 0 .
Revised: = 0 .

Original: % Empirical mean demand in phase j
Revised: % Empirical mean demand in phase j

Original: % Estimate parameter 10: q
Revised: % Estimate parameter 10: q

Original: % Width of conﬁdence o % Conservative estimation of the optimal price
Revised: % Width of conﬁdence o % Conservative estimation of the optimal price

Original: Ensure monotonicity
Revised: Ensure monotonicity

Original: Theorem 1 (Zero-Dimensional Upper Bound) .
Revised: Theorem 1 (Zero-Dimensional Upper Bound) .

Original: Let F be any 0 -dimensional, s -sensitive family of demand functions.
Revised: Let F be any 0 -dimensional, s -sensitive family of demand functions.

Original: Then the Cautious Myopic (CM) Policy has regret
Revised: Then the Cautious Myopic (CM) Policy has regret

Original: Reg(CM , F ) = ⇢
Revised: Reg(CM , F ) = ⇢

Original: O (log 2 T ) , if s = 2 ,
Revised: O (log 2 T ) , if s = 2 ,

Original: O (log T ) , if s > 2 .
Revised: O (log T ) , if s > 2 .

Original: Remark 1.
Revised: Remark 1.

Original: It is worth noting that this bound is asymptotically higher than the O (log T ) upper bound for unconstrained pricing ([2]).
Revised: It is worth noting that this bound is asymptotically higher than the O (log T ) upper bound for unconstrained pricing ([2]).

Original: Intuitively, this is because the CM policy purposely makes conservative choices of prices to avoid overshooting.
Revised: Intuitively, this is because the CM policy purposely makes conservative choices of prices to avoid overshooting.

Original: But can we achieve o (log 2 T ) regret by behaving less conservatively?
Revised: But can we achieve o (log 2 T ) regret by behaving less conservatively?

Original: We answer this question negatively, and provide a separation between markdown and unconstrained pricing for dimension 0 .
Revised: We answer this question negatively, and provide a separation between markdown and unconstrained pricing for dimension 0 .

Original: Theorem 2 (Zero-Dimensional Lower Bound) .
Revised: Theorem 2 (Zero-Dimensional Lower Bound) .

Original: There exists a 0 -dimensional 2 -sensitive family F such that for any policy ⇡ , we have Reg( ⇡ , F ) =
Revised: There exists a 0 -dimensional 2 -sensitive family F such that for any policy ⇡ , we have Reg( ⇡ , F ) =

Original: ⌦ (log 2 T ) .
Revised: ⌦ (log 2 T ) .

Original: For dimension d , the learner needs d + 1 distinct sample prices , as opposed to just one price when d = 0 .
Revised: For dimension d , the learner needs d + 1 distinct sample prices , as opposed to just one price when d = 0 .

Original: This, however, incurs extra regret, since the optimal price may lie between these sample prices.
Revised: This, however, incurs extra regret, since the optimal price may lie between these sample prices.

Original: A policy faces the following trade off.
Revised: A policy faces the following trade off.

Original: If the gap is large, the policy may learn the parameter efﬁciently, but there is potentially a higher regret due to overshooting, if the true optimal price lies between the sample prices.
Revised: If the gap is large, the policy may learn the parameter efﬁciently, but there is potentially a higher regret due to overshooting, if the true optimal price lies between the sample prices.

Original: On the other extreme, if the gap is small, there is less risk of overshooting but a slower rate of learning.
Revised: On the other extreme, if the gap is small, there is less risk of overshooting but a slower rate of learning.

Original: We introduce our Iterative Cautious Myopic (ICM) Policy (Algorithm 2) that hits the sweet spot.
Revised: We introduce our Iterative Cautious Myopic (ICM) Policy (Algorithm 2) that hits the sweet spot.

Original: The policy consists of m phases.
Revised: The policy consists of m phases.

Original: In phase j 2 [ m ] , it selects p j , p j  h, ..., p j  dh each for T j times.
Revised: In phase j 2 [ m ] , it selects p j , p j  h, ..., p j  dh each for T j times.

Original: Based on the observed demands at these prices, the policy computes an estimated optimal price ˆ p j and a conﬁdence interval [ L j , U j ] .
Revised: Based on the observed demands at these prices, the policy computes an estimated optimal price ˆ p j and a conﬁdence interval [ L j , U j ] .

Original: We determine the initial price p j +1 of the next phase by considering the relationship between the conﬁdence interval [ L j , U j ] and p j  dh .
Revised: To determine the initial price p j +1 of the next phase, we consider the relationship between the conﬁdence interval [ L j , U j ] and p j  dh .

Original: There are three cases:
Revised: There are three cases:

Original: Good Event.
Revised: Good Event.

Original: If p j  dh > U j , then we get closer to the optimal[-price by setting p j +
-]
Revised: If p j  dh > U j , then we get closer to the optimal[-price by setting p j +
-]

Original: {+price by setting p j +
+} =
Revised: {+price by setting p j +
+} =

Original: U j . 2.
Revised: U j . 2.

Original: Dangerous Event.
Revised: Dangerous Event.

Original: If L j  p j  dh  U j , i.e. the current price is already within the conﬁdence interval, the policy needs to behave conservatively by setting p j +1 = p
Revised: If L j  p j  dh  U j , i.e. the current price is already within the conﬁdence interval, the policy needs to behave conservatively by setting p j +1 = p

Original: j  dh to prevent from overshooting further.
Revised: j  dh to prevent from overshooting further.

Original: Overshooting Event.
Revised: Overshooting Event.

Original: If L j > p j  dh ,
Revised: If L j > p j  dh ,

Original: i
Revised: i

Original: e., the current price has already “overshot”, the policy exits the exploration phase and selects the current price in theremaining rounds
Revised: e., the current price has already “overshot”, the policy exits the exploration phase and selects the current price in the

Revised: remaining

Revised: rounds.

Original: Select price p j  kh for T j times and observe D 1 , ..., D T j 6: ¯ D k 1 T j
Revised: Select price p j  kh for T j times and observe D 1 , ..., D T j 6: ¯ D k 1 T j

Original: P
Revised: P

Original: T j i =
Revised: T j i =

Original: 1 D
Revised: 1 D

Original: i % Mean demand at p j  kh 7: end for 8: ˆ ✓   1 p j ,...,p j  dh  ¯ D 0 , ..., ¯ D d  % Estimate the parameter 9: w j 2 h  d · C 2 · C sg q d log T T j % Width of conﬁdence interval 10:
Revised: i % Mean demand at p j  kh 7: end for 8: ˆ ✓   1 p j ,...,p j  dh  ¯ D 0 , ..., ¯ D d  % Estimate the parameter 9: w j 2 h  d · C 2 · C sg q d log T T j % Width of conﬁdence interval 10:

Original: L j min n p ⇤ ( ✓ ) :  ✓  p ⇤ ⇣
Revised: L j min n p ⇤ ( ✓ ) :  ✓  p ⇤ ⇣

Original: ˆ ✓ ⌘ 2  w j o % Lower conﬁdence bound 11:
Revised: ˆ ✓ ⌘ 2  w j o % Lower conﬁdence bound 11:

Original: U j max n p ⇤ ( ✓ ) :  ✓  p ⇤ ⇣
Revised: U j max n p ⇤ ( ✓ ) :  ✓  p ⇤ ⇣

Original: ˆ ✓ ⌘ 2  w j o % Upper conﬁdence bound 12: if U j > p j  dh then 13: p j +
Revised: ˆ ✓ ⌘ 2  w j o % Upper conﬁdence bound 12: if U j > p j  dh then 13: p j +

Original: U
Revised: U

Original: j % Good event 14: end if 15: if
Revised: j % Good event 14: end if 15: if

Original: U j  p
Revised: U j  p

Original: j  dh  L j then 16: p j + p j  dh % Dangerous event 17: end if 18: if p j  dh <
Revised: j  dh  L j then 16: p j + p j  dh % Dangerous event 17: end if 18: if p j  dh <

Original: L j then 19:
Revised: L j then 19:

Original: Break % Overshooting event 20: end if 21: end for 22: Select the current price in every future round % Exploitation
Revised: Break % Overshooting event 20: end if 21: end for 22: Select the current price in every future round % Exploitation

Original: Theorem 3 (Upper Bound for Finite d  1 ) .
Revised: Theorem 3 (Upper Bound for Finite d  1 ) .

Original: Suppose s = 2 , then for h
Revised: Suppose s = 2 , then for h

Original: =
Revised: =

Original: T m m ( d +1)+1 , T j =
Revised: T m m ( d +1)+1 , T j =

Original: T md + j m ( d +1)+1 and m =
Revised: T md + j m ( d +1)+1 and m =

Original: log T , we have Reg (ICM , F ) =
Revised: log T , we have Reg (ICM , F ) =

Original: ˜
Revised: ˜

Original: O ⇣
Revised: O ⇣

Original: T dd + ⌘ .
Revised: T dd + ⌘ .

Original: More generally, for any s and m =
Revised: More generally, for any s and m =

Original: ˜ O (1) , there exist T 1 < ... <
Revised: ˜ O (1) , there exist T 1 < ... <

Original: T m such that Reg(ICM , F ) =
Revised: T m such that Reg(ICM , F ) =

Original: ˜ O 
Revised: ˜ O

Original: T ⇢ ( m,s,d )  where ⇢ ( m, s, d ) =
Revised: T ⇢ ( m,s,d )  where ⇢ ( m, s, d ) =

Original: 1 +  1 + s 2 + ...
Revised: 1 +  1 + s 2 + ...

Original: [Equation]
Revised: [Equation]

Original: We complement the upper bound with a nearly tight lower bound for s = 2 , as stated below.
Revised: We complement the upper bound with a nearly tight lower bound for s = 2 , as stated below.

Original: Theorem 4 (Lower Bound for Finite d  1 ) .
Revised: Theorem 4 (Lower Bound for Finite d  1 ) .

Original: For any d  2 , there exists a d -dimensional family F of demand functions such that for any markdown policy ⇡ , we have Reg( ⇡ , F ) =
Revised: For any d  2 , there exists a d -dimensional family F of demand functions such that for any markdown policy ⇡ , we have Reg( ⇡ , F ) =

Original: ⌦ ( T dd +1 ) .
Revised: ⌦ ( T dd +1 ) .

Original: In our proof, for each d  1 we construct a family of ( d +1) -degree, decreasing polynomial functions – which is also d -dimensional – on which any policy suffers regret ⌦ ( T dd +1 ) .
Revised: In our proof, for each d  1 we construct a family of ( d +1) -degree, decreasing polynomial functions – which is also d -dimensional – on which any policy suffers regret ⌦ ( T dd +1 ) .

Original: Remark 2.
Revised: Remark 2.

Original: At ﬁrst glance, when d  4 this result seems to contradict the T 3 / 4 upper bound for unimodal, Lipschitz reward family ([3, 9]).
Revised: At ﬁrst glance, when d  4 this result seems to contradict the T 3 / 4 upper bound for unimodal, Lipschitz reward family ([3, 9]).

Original: This is not a contradiction because the unimodality assumption no longer holds.
Revised: This is not a contradiction because the unimodality assumption no longer holds.

Original: In fact, a degree- d polynomial with d  4 is in general not unimodal.
Revised: In fact, a degree- d polynomial with d  4 is in general not unimodal.

Original: Remark 3.
Revised: Remark 3.

Original: An interesting case is d = 1 .
Revised: An interesting case is d = 1 .

Original: In this case, the revenue function is unimodal, so the Tupper bound from [3, 9] applies, but Theorem 3 gives a stronger T 12 upper bound.
Revised: In this case, the revenue function is unimodal, so the Tupper bound from [3, 9] applies, but Theorem 3 gives a stronger T 12 upper bound.

Original: We conclude the section by pointing out a limitation of our results.
Revised: We conclude the section by pointing out a limitation of our results.

Original: For each d  1 , our regret bounds is tight in the exponent of T , however, the constants in the big-O have different dependence on d .
Revised: For each d  1 , our regret bounds is tight in the exponent of T , however, the constants in the big-O have different dependence on d .

Original: In fact, the upper and lower bound become exponentially higher and lower in d respectively as d grows.
Revised: In fact, the upper and lower bound become exponentially higher and lower in d respectively as d grows.

Original:  Inﬁnite Dimensional Family
Revised:  Inﬁnite Dimensional Family

Original: When d = 1 , it is more convenient to work with the reward function R ( x ): = x · D ( x ) instead of the demand function.
Revised: When d = 1 , it is more convenient to work with the reward function R ( x ) = x · D ( x ) instead of the demand function.

Original: In contrast to unconstrained pricing, there is no markdown policy that achieves o ( T ) regret on the family of Lipschitz reward functions ([3, 9]), since the reward function may have multiple local maxima.
Revised: In contrast to unconstrained pricing, there is no markdown policy that achieves o ( T ) regret on the family of Lipschitz reward functions ([3, 9]), since the reward function may have multiple local maxima.

Original: Nonetheless, [3] and [9] showed that if in addition we assume the reward function to be unimodal, then the optimal regret is T 3 / 4 .
Revised: Nonetheless, [3] and [9] showed that if in addition we assume the reward function to be unimodal, then the optimal regret is T 3 / 4 .

Original: In this setting, the problem boils down to ﬁnding the unique local optimum p ⇤ of the unknown revenue function.

Original: Speciﬁcally, the lower bound is established by considering reward functions that may change abruptly at p ⇤ .
Revised: Speciﬁcally, the lower bound is established by considering reward functions that may change abruptly at p ⇤ .

Original: We next show that the regret bound can be improved if we instead assume the reward function changes smoothly.
Revised: We next show that the regret bound can be improved if we instead assume the reward function changes smoothly.

Original: 1: Input
Revised: In spirit, our policy is similar to that of [3] and [9]: reduce the price at uniform speed and terminate when there is sufﬁcient evidence for “overshooting”, but we will achieve improved guarantee by choosing a different step size  that decreases in the smoothness s . Algorithm 3 Uniform Elimination Policy ( UE m,  ).

Original: 2: Initialize: q do 4: 
Revised: 2: Initialize q do 4:  5:

Original: Select price x j for the next m rounds and observe rewards Z j 1 , ..., Z jm 6: ¯ µ j  m
Revised: Select price x j for the next m rounds and observe rewards Z j 1 , ..., Z jm 6: ¯ µ j 1 m

Original: P mi =1 Z ji % Empirical mean reward 7:
Revised: P mi =1 Z ji % Empirical mean reward 7:

Original: [ L j , U j ] [¯ µ j  w, ¯ µ + w ] % Compute conﬁdence interval 8: if L j > L max then 9:
Revised: [ L j , U j ] [¯ µ j  w, ¯ µ + w ] % Compute conﬁdence interval 8: if L j > L max then 9:

Original: L max
Revised: L max

Original: L j % Keep track of the highest L j 10: end if 11: if U j < L max then 12: h j % Exploration ends, deﬁne the halting price x h 13:
Revised: L j % Keep track of the highest L j 10: end if 11: if U j < L max then 12: h j % Exploration ends, deﬁne the halting price x h 13:

Original: Break 14: end if 15: end for 16:
Revised: Break 14: end if 15: end for 16:

Original: Select price x h in all remaining rounds.
Revised: Select price x h in all remaining rounds.

Original: % Exploitation phase
Revised: % Exploitation phase

Original: Theorem 5 (Upper Bound for Inﬁnite-Dimensional Family) .
Revised: Theorem 5 (Upper Bound for Inﬁnite-Dimensional Family) .

Original: Let F Us be the family of unimodal, s -sensitive reward functions.
Revised: Let F Us be the family of unimodal, s -sensitive reward functions.

Original: For any s  2 , there exist m,  with Reg(UE m,  , F Us ) =
Revised: For any s  2 , there exist m,  with Reg(UE m,  , F Us ) =

Original: O ( T 2 s +1 3 s +1 ) .
Revised: O ( T 2 s +1 3 s +1 ) .

Original: We complement the above upper bound with a matching lower bound for every s  2 .
Revised: We complement the above upper bound with a matching lower bound for every s  2 .

Original: Theorem 6 (Lower Bound for s -Sensitive Family) .
Revised: Theorem 6 (Lower Bound for s -Sensitive Family) .

Original: For any s  2 , there is a family F of s -sensitive unimodal revenue functions such that any markdown policy ⇡ satisﬁes Reg( ⇡ , F ) = ⌦ ( T 2 s +1 3 s +1 ) .
Revised: For any s  2 , there is a family F of s -sensitive unimodal revenue functions such that any markdown policy ⇡ satisﬁes Reg( ⇡ , F ) = ⌦ ( T 2 s +1 3 s +1 ) .

Original: Remark 4.
Revised: Remark 4.

Original: As a ﬁnal remark, we point out that all our results assume d is given.
Revised: As a ﬁnal remark, we point out that all our results assume d is given.

Original: If instead the learner only has knows an upper bound d max on d , then she can simply choose the policy for d max .
Revised: If instead the learner only has knows an upper bound d max on d , then she can simply choose the policy for d max .

Original: We leave it open whether a best-of-all-worlds algorithm exists.
Revised: We leave it open whether a best-of-all-worlds algorithm exists.