Logarithmic Regret in Feature-based Dynamic Pricing
Year: 2022

Abstract:
Feature-based dynamic pricing is an increasingly popular model of setting prices for highly differentiated products with applications in digital marketing, online sales, real estate and so on. The problem was formally studied as an online learning problem [Javanmard and Nazerzadeh, 2019] where a seller needs to propose prices on the fly for a sequence of T products based on their features x while having a small regret relative to the best —“omniscient”— pricing strategy she could have come up with in hindsight. We revisit this problem and provide two algorithms (EMLP and ONSP) for stochastic and adversarial feature settings, respectively, and prove the optimal O(d log T ) regret bounds for both. In comparison, the best existing results are O ( min { 1 λmin log T, √ T }) and O(T ) respectively, with λmin being the smallest eigenvalue of E[xx ] that could be arbitrarily close to 0. We also prove an Ω( √ T ) information-theoretic lower bound for a slightly more general setting, which demonstrates that “knowing-the-demand-curve” leads to an exponential improvement in feature-based dynamic pricing.

text: ( min { 1
λ2min log T,
√ T }) and O(T 2/3) respectively, with
λmin being the smallest eigenvalue of E[xxT ] that could be arbitrarily close to 0. We also prove an Ω( √ T ) information-theoretic lower bound for a slightly more general setting, which demonstrates that “knowing-the-demand-curve” leads to an exponential improvement in feature-based dynamic pricing.

heading: 1 Introduction
text: The problem of pricing — to find a high-and-acceptable price — has been studied since Cournot [1897]. In order to locate the optimal price that maximizes the revenue, a firm may adjust their prices of products frequently, which inspires the dynamic pricing problem. Existing works [Kleinberg and Leighton, 2003, Broder and Rusmevichientong, 2012, Chen and Farias, 2013, Besbes and Zeevi, 2015] primarily focus on pricing a single product, which usually will not work well in another setting when thousands of new products are being listed every day with no prior experience in selling them. Therefore, we seek methods that approach an acceptable-and-profitable price with only observations on this single product and some historical selling records of other products.
In this work, we consider a “feature-based dynamic pricing” problem, which was studied by Amin et al. [2014], Cohen et al. [2020], Javanmard and Nazerzadeh [2019]. In this problem setting, a sales session (product, customer and other environmental variables) is described by a feature vector, and the customer’s expected valuation is modeled as a linear function of this feature vector.
Feature-based dynamic pricing. For t = 1, 2, ..., T : 1. A feature vector xt ∈ Rd is revealed that describes a sales session (product, customer and context). 2. The customer valuates the product as wt = x⊤t θ∗ +Nt. 3. The seller proposes a price vt > 0 concurrently (according to xt and historical sales records). 4. The transaction is successful if vt ≤ wt, i.e., the seller gets a reward (payment) of rt = vt · 1(vt ≤ wt).
Here T is unknown to the seller (and thus can go to infinity), xt’s can be either stochastic (e.g., each sales session is drawn i.i.d.) or adversarial (e.g., the sessions arrive in a strategic sequence), θ∗ ∈ Rd is a fixed parameter for all time periods, Nt is a zero-mean noise, and 1t = 1(vt ≤ wt) is an indicator that equals 1 if vt ≤ wt and 0 otherwise. In this online-fashioned setting, we only see
35th Conference on Neural Information Processing Systems (NeurIPS 2021).
and sell one product at each time. Also, the feedback is Boolean Censored, which means we can only observe 1t instead of knowing wt directly. The best pricing policy for this problem is the one that maximizes the expected reward, and the regret of a pricing policy is accordingly defined as the difference of expected rewards between this selected policy and the best policy.
Summary of Results. Our contributions are threefold.
1. When xt’s are independently and identically distributed (i.i.d.) from an unknown distribution, we propose an “Epoch-based Max-Likelihood Pricing (EMLP)” algorithm that guarantees a regret bound at O(d log T ). The design of EMLP is similar to that of the RMLP algorithm in Javanmard and Nazerzadeh [2019], but our new analysis improves their regret bound at O( √ T ) when E[xx⊤] is near singular.
2. When xt’s are adversarial, we propose an “Online-Newton-Step Pricing (ONSP)” algorithm that achieves O(d log T ) regret on constant-level noises for the first time, which exponentially improves the best existing result of O(T 2/3) [Cohen et al., 2020].1
3. Our methods that achieve logarithmic regret require knowing the exact distribution of Nt in advance, as is also assumed in Javanmard and Nazerzadeh [2019]. We prove an Ω( √ T ) lower
bound on the regret if Nt ∼ N (0, σ2) where σ is unknown, even with θ∗ given and xt fixed for all t.
The O(log T ) regret of EMLP and ONSP meets the information-theoretical lower bound [Theorem 5, Javanmard and Nazerzadeh, 2019]. In fact, the bound is optimal even when wt is revealed to the learner [Mourtada, 2019]. From the perspective of characterizing the hardness of dynamic pricing problems, we generalize the classical results on “The Value of Knowing a Demand Curve” [Kleinberg and Leighton, 2003] by further dividing the random-valuation class with an exponential separation of: (1) O(log T ) regret for knowing the demand curve exactly (even with adversarial features), and (2) Ω( √ T ) regret for almost knowing the demand curves (up to a one-parameter parametric family).

heading: 2 Related Works
text: In this section, we discuss our results relative to existing works on feature-based dynamic pricing, and highlight the connections and differences to the related settings of contextual bandits and contextual search (for a broader discussion, see Appendix A).
Feature-based Dynamic Pricing. There is a growing body of work on dynamic pricing with linear features [Amin et al., 2014, Qiang and Bayati, 2016, Cohen et al., 2020, Javanmard and Nazerzadeh, 2019]. Table 1 summarizes the differences in the settings and results2. Among these work, our paper directly builds upon [Cohen et al., 2020] and [Javanmard and Nazerzadeh, 2019], as we share the same setting of online feature vectors, linear and noisy valuations and Boolean-censored feedback. Relative to the results in [Javanmard and Nazerzadeh, 2019], we obtain O(d log T ) regret under weaker assumptions on the sequence of input features — in both distribution-free stochastic feature
1Previous works [Cohen et al., 2020, Krishnamurthy et al., 2021] did achieve polylog regrets, but only for negligible noise with σ = O( 1
T log T ).
2We only concern the dependence on T since there are various different assumptions on d.
setting and the adversarial feature setting. It is to be noted that [Javanmard and Nazerzadeh, 2019] also covers the sparse high-dimensional setting, and handles a slightly broader class of demand curves. Relative to [Cohen et al., 2020], in which the adversarial feature-based dynamic pricing was first studied, our algorithm ONSP enjoys the optimal O(d log T ) regret when the noise-level is a constant. In comparison, Cohen et al. [2020] reduces the problem to contextual bandits and applies the (computationally inefficient) “EXP-4” algorithm [Auer et al., 2002] to achieve a Õ(T 2/3) regret. The “bisection” style-algorithm in both Cohen et al. [2020] and Krishnamurthy et al. [2021] could achieve Õ(poly(d)poly log(T )) regrets but requires a small-variance subgaussian noise satisfying σ = O( 1T log T ).
Lower Bounds. Most existing works focus on the lower regret bounds of non-feature-based models. Kleinberg and Leighton [2003] divides the problem setting as fixed, random, and adversarial valuations, and then proves each a Θ(log log T ), Θ( √ T ), and Θ(T 2/3) regret, respectively. Broder and Rusmevichientong [2012] further proves a Θ( √ T ) regret in general parametric valuation models. In this work, we generalize the methods of Broder and Rusmevichientong [2012] to our feature-based setting and further narrow it down to a linear-feature Gaussian-noisy model. As a complement to Kleinberg and Leighton [2003], we further separate the exponential regret gap between: (1) O(log T ) of the hardest (adversarial feature) totally-parametric model, and (2) Ω( √ T ) of the simplest (fixed known expectation) unknown-σ Gaussian model.
Contextual Bandits. For readers familiar with the online learning literature, our problem can be reduced to a contextual bandits problem [Langford and Zhang, 2007, Agarwal et al., 2014] by discretizing the prices. But this reduction only results in O(T 2/3) regret, as it does not capture the special structure of the feedback: an accepted price indicates the acceptance of all lower prices, and vise versa. Moreover, when comparing to linear bandits [Chu et al., 2011], it is the valuation instead of the expected reward that we assume to be linear.
Contextual Search. Feature-based dynamic pricing is also related to the contextual search problem [Lobel et al., 2018, Leme and Schneider, 2018, Liu et al., 2021, Krishnamurthy et al., 2021], which often involves learning from Boolean feedbacks, sometimes with a “pricing loss” and “noisy” feedback. These shared jargons make this problem appearing very similar to our problem. However, except for the noiseless cases [Lobel et al., 2018, Leme and Schneider, 2018], contextual search algorithms, even with “pricing losses” and “Noisy Boolean feedback” [e.g., Liu et al., 2021], do not imply meaningful regret bounds in our problem setup due to several subtle but important differences in the problem settings. Specifically, the noisy-boolean feedback model of [Liu et al., 2021] is about randomly toggling the “purchase decision” determined by the noiseless valuation x⊤θ∗ with probability 0.5 − ϵ. This is incompatible to our problem setting where the purchasing decision is determined by a noisy valuation x⊤θ∗ + Noise. Ultimately, in the setting of [Liu et al., 2021], the optimal policy alway plays x⊤θ∗, but our problem is harder in that we need to exploit the noise and the optimal price could be very different from x⊤θ∗. 3 Krishnamurthy et al. [2021] also discussed this issue explicitly and considered the more natural noisy Boolean feedback model studied in this paper. Their result, similar to Cohen et al. [2020], only achieves a logarithmic regret when the noise on the valuation is vanishing in an Õ(1/T ) rate.

heading: 3 Problem Setup
text: Symbols and Notations. Now we introduce the mathematical symbols and notations involved in the following pages. The game consists of T rounds. xt ∈ Rd, vt ∈ R+ and Nt ∈ R denote the feature vector, the proposed price and the noise respectively at round t = 1, 2, ..., T .4 We denote the product ut := x ⊤ t θ
∗ as an expected valuation. At each round, we receive a payoff (reward) rt = vt · 1t, where the binary variable 1t indicates whether the price is accepted or not, i.e., 1t = 1(vt ≤ wt). As we may estimate θ∗ in our algorithms, we denote θ̂t as an estimator of θ∗, which we will formally define in the algorithms. Furthermore, we denote some functions that are related to noise distribution: F (ω) and f(ω) denote the cumulative distribution function (CDF) and probability density function (PDF) sequentially. We know that F ′(ω) = f(ω) if we assume differentiability. To concisely denote all data observed up to round τ (i.e., feature, price and payoff of all past rounds), we define
3As an explicit example, suppose the valuation x⊤θ∗ = 0, then the optimal price must be > 0 in order to avoid zero return.
4In an epoch-design situation, a subscript (k, t) indicates round t of epoch k.
hist(τ) = {(xt, vt,1t) for t = 1, 2, ..., τ}. hist(τ) represents the transcript of all observed random variables before round (τ + 1).
We define lt(θ) := −1t · log ( 1− F (vt − x⊤t θ) ) − (1− 1t) log ( F (vt − x⊤t θ) ) (1)
as a negative log-likelihood function at round t. Also, we define an expected log-likelihood function Lt(θ): Lt(θ) := ENt [lt(θ)|xt] (2) Notice that we will later define an L̂k(θ) which is, however, not an expectation.
Definitions of Key Quantities. We firstly define an expected reward function g(v, u). g(v, u) := E[rt|vt = v, x⊤t θ∗ = u] = v · P [v ≤ x⊤t θ∗ +Nt] = v · (1− F (v − u)). (3)
This indicates that if the expected valuation is u and the proposed price is v, then the (conditionally) expected reward is g(v, u). Now we formally define the regret of a policy (algorithm) A as is promised in Section 1.
Definition 1 (Regret). Let A : Rd × ( Rd,R, {0, 1} )t−1 → R be a policy of pricing, i.e. A(xt, hist(t− 1)) = vt. The regret of A is defined as follows.
RegA = T∑ t=1 max v g(v, x⊤t θ ∗)− g(A(xt, hist(t− 1)), x⊤t θ∗). (4)
Here hist(t− 1) is the historical records until (t− 1)th round. Summary of Assumptions. We specify the problem settings by proposing three assumptions.
Assumption 1 (Known, bounded, strictly log-concave distribution). The noise Nt is independently and identically sampled from a distribution whose CDF is F . Assume that F ∈ C2 is strictly increasing and that F and (1− F ) are strictly log-concave. Also assume that f and f ′ are bounded, and denote Bf := supω∈R f(ω), Bf ′ := supω∈R |f ′(ω)| as two constants. Assumption 2 (Bounded convex parameter space). The true parameter θ∗ ∈ H, where H ⊆ {θ : ||θ||2 ≤ B1} is a bounded convex set and B1 is a constant. Assume H is known to us (but θ∗ is not). Assumption 3 (Bounded feature space). Assume xt ∈ D ⊆ {x : ||x||2 ≤ B2}, ∀t = 1, 2, . . . , T . Also, 0 ≤ x⊤θ ≤ B, ∀x ∈ D,∀θ ∈ H, where B = B1 ·B2 is a constant. Assumption 2 and 3 are mild as we can choose B1 and B2 large enough. In Section 4.1, we may add further complement to Assumption 3 to form a stochastic setting. Assumption 1 is stronger since we might not know the exact CDF in practice, but it is still acceptable from an informationtheoretic perspective. There are at least three reasons that lead to this assumption: Primarily, this is necessary if we hope to achieve an O(log(T )) regret. We will prove in Section 5.3 that an Ω( √ T ) is unavoidable if we cannot know one parameter exactly. Moreover, the pioneering work of Javanmard and Nazerzadeh [2019] also assumes a known noise distribution with log-concave CDF, and many common distributions are actually strictly log-concave, such as Gaussian and logistic.5 Besides, although we did not present a method to precisely estimate σ in this work, it is a reasonable algorithm to replace with a plug-in estimator estimated using historical offline data. As we have shown, not knowing σ requires O( √ T ) regret in general, but the lower bound does not rule out the plug-in approach achieving a smaller regret for interesting subclasses of problems in practice.
Finally, we state a lemma and define an argmax function helpful for our algorithm design.
Lemma 2 (Uniqueness). For any u ≥ 0, there exists a unique v∗ ≥ 0 such that g(v∗, u) = maxv∈R g(v, u). Thus, we can define a greedily pricing function that maximizes the expected reward:
J(u) := argmax v g(v, u) (5)
Please see the proof of Lemma 2 in Appendix B.1.

heading: 4 Algorithms
text: In this section, we propose two dynamic pricing algorithms: EMLP and ONSP, for stochastic and adversarial features respectively.
5In fact, F and (1− F ) are both log-concave if its PDF is log-concave, according to Prekopa’s Inequality.
Algorithm 1 Epoch-based max-likelihood pricing (EMLP)
Input: Convex and bounded set H Observe x1, randomly choose v1 and get r1. Solve θ̂1 = argminθ∈H l1(θ); for k = 1 to ⌊log2 T ⌋+ 1 do
Set τk = 2k−1; for t = 1 to τk do
Observe xk,t; Set price vk,t = J(x⊤k,tθ̂k); Receive rk,t = vkt · 1t;
end for Solve: θ̂k+1 = argminθ∈H L̂k(θ), where L̂k(θ) =
1 τk ∑τk t=1 lk,t(θ).
end for
Algorithm 2 Online Newton Step Pricing (ONSP)
Input: Convex and bounded set H, θ1, parameter γ, ϵ > 0 Set A0 = ϵ · Id; for t = 1 to T do
Observe xt; Set price vt = J(x⊤t θt); Receive rt = vt · 1t; Set surrogate loss function lt(θ); Calculate ∇t = ∇lt(θ); Rank-1 update: At = At−1 +∇t∇⊤t ; Newton step: θ̂t+1 = θt − 1γA −1 t ∇t;
Projection: θt+1 = ∏At
H (θ̂t+1). end for

heading: 4.1 Pricing with Distribution-Free Stochastic Features
text: Assumption 4 (Stochastic features). Assume xt ∼ D ⊆ D are independently identically distributed (i.i.d.) from an unknown distribution, for any t = 1, 2, . . . , T .
The first algorithm, Epoch-based Max-Likelihood Pricing (EMLP) algorithm, is suitable for a stochastic setting defined by Assumption 4. EMLP proceeds in epochs with each stage doubling the length of the previous epoch. At the end of each epoch, we consolidate the observed data and solve a maximum likelihood estimation problem to learn θ. A max likelihood estimator (MLE) obtained by minimizing L̂k(θ) := 1τk ∑τk t=1 lk,t(θ), which is then used in the next epoch as if it is the true parameter vector. In the equation, k, τk denotes the index and length of epoch k. The estimator is computed using data in hist(k), which denotes the transcript for epoch 1 ∼ k. The pseudo-code of EMLP is summarized in Algorithm 1. In the remainder of this section, we discuss the computational efficiency and prove the upper regret bound of O(d log T ).
Computational Efficiency. The calculations in EMLP are straightforward except for argmin L̂k(θ) and J(u). As g(v, u) is proved unimodal in Lemma 2, we may efficiently calculate J(u) by binary search. We will prove that lk,t is exp-concave (and thus also convex). Therefore, we may apply any off-the-shelf tools for solving convex optimization.
MLE and Probit Regression. A closer inspection reveals that this log-likelihood function corresponds to a probit [Aldrich et al., 1984] or a logit model [Wright, 1995] for Gaussian or logistic noises. See Appendix C.2.1.
Affine Invariance. Both optimization problems involved depend only on x⊤θ, so if we add any affine transformation to x into x̃ = Ax, the agent can instead learn a new parameter of θ̃∗ = (A⊤)−1θ∗ and achieve the same ut = x⊤t θ ∗. Also, the regret bound is not affected as the upper bound B over x⊤θ does not change 6. Therefore, it is only natural that the regret bound does not depend on the distribution x, nor the condition numbers of E[xx⊤] (i.e., the ratio of λmax/λmin).

heading: 4.2 Pricing with Adversarial Features
text: In this part, we propose an “Online Newton Step Pricing (ONSP)” algorithm that deals with adversarial {xt} series and guarantees O(d log T ) regret. The pseudo-code of ONSP is shown as Algorithm 2. In each round, it uses the likelihood function as a surrogate loss and applies “Online Newton Step”(ONS) method to update θ̂. In the next round, it adopts the updated θ̂ and sets a price greedily. In the remainder of this section, we discuss some properties of ONSP and prove the regret bound.
The calculations of ONSP are straightforward. The time complexity of calculating the matrix inverse A−1t is O(d
3), which is fair as d is small. In high-dimensional cases, we may use Woodbury matrix identity7 to reduce it to O(d2) as we could get A−1 directly from the latest round.
6Here A is assumed invertible, otherwise the mapping from x̃t to ut does not necessarily exist. 7(A+ xx⊤)−1 = A−1 − 1
1+x⊤A−1x A−1x(A−1x)⊤.

heading: 5 Regret Analysis
text: In this section, we mainly prove the logarithmic regret bounds of EMLP and ONSP corresponding to stochastic and adversarial settings, respectively. Besides, we also prove an Ω( √ T ) regret bound on fully parametric F with one parameter unknown.
5.1 O(d log T ) Regret of EMLP In this part, we present the regret analysis of Algorithm 1. First of all, we propose the following theorem as our main result on EMLP.
Theorem 3 (Overall regret). With Assumptions 1, 2, 3 and 4, the expected regret of EMLP can be bounded by:
E[RegEMLP] ≤ 2Csdlog T , (6)
where Cs is a constant that depends only on F (ω) and is independent to D.
The proof of Theorem 3 is sophisticated. For the sake of clarity, we next present an inequality system as a roadmap toward the proof. After this, we formally illustrate each line of it with lemmas.
Since EMLP proposes J(x⊤k,tθ̂k) in every round of epoch k, we may denote the per-round regret as Regt(θ̂k), where:
Regt(θ) := g(J(x ⊤ t θ ∗), x⊤t θ ∗)− g(J(x⊤t θ), x⊤t θ∗). (7)
Therefore, it is sufficient to prove the following Theorem:
Theorem 4 (Expected per-round regret). For the per-round regret defined in Equation (7), we have:
E[Regk,t(θ̂k)] ≤ Cs · d
τk .
The proof roadmap of Theorem 4 can be written as the following inequality system.
E[Regk,t(θ̂k)] ≤ C ·E[(θ̂k−θ∗)⊤xk,tx⊤k,t(θ̂k−θ∗)] ≤ 2C
Cdown E[L̂k(θ̂k)− L̂k(θ∗)] ≤ 2C · Cexp C2down d τk .
(8)
We explain Equation (8) in details. The first inequality comes from the following Lemma 5.
Lemma 5 (Quadratic regret bound). We have:
Regt(θ) ≤ C · (θ − θ∗)⊤xtx⊤t (θ − θ∗),∀θ ∈ H,∀xt ∈ D. (9)
Here C = 2Bf + (B + J(0)) ·Bf ′ .
The intuition is that function g(J(u), u) is 2nd-order-smooth at (J(u∗), u∗). A detailed proof of Lemma 5 is in Appendix B.2.1. Note that C is highly dependent on the distribution F . After this, we propose Lemma 6 that contributes to the second inequality of Equation (8).
Lemma 6 (Quadratic likelihood bound). For the expected likelihood function Lt(θ) defined in Equation (2), we have:
Lt(θ)− Lt(θ∗) ≥ 1
2 Cdown(θ − θ∗)⊤xtx⊤t (θ − θ∗),∀θ ∈ H,∀x ∈ D, (10)
where Cdown := inf ω∈[−B,B+J(0)] min
{ d2 log(1− F (ω))
dω2 , d2 log(F (ω)) dω2
} > 0. (11)
Proof. Since the true parameter always maximizes the expected likelihood function [Murphy, 2012], by Taylor Expansion we have ∇L(θ∗) = 0, and hence Lt(θ)−Lt(θ∗) = 12 (θ−θ
∗)⊤∇2Lt(θ̃)(θ−θ∗) for some θ̃ = αθ∗ + (1− α)θ. Therefore, we only need to prove the following lemma:
Lemma 7 (Strong convexity and Exponential Concavity). Suppose lt(θ) is the negative log-likelihood function in epoch k at time t. For any θ ∈ H, xt ∼ D, we have:
∇2lt(θ) ⪰ Cdownxtx⊤t ⪰ Cdown Cexp ∇lt(θ)∇lt(θ)⊤ ⪰ 0, (12)
where Cexp := sup ω∈[−B,B+J(0)] max
{ f(ω)2
F (ω)2 ,
f(ω)2
(1− F (ω))2
} < +∞. (13)
Proof of Lemma 7 is in Appendix B.2.2. With this lemma, we see that Lemma 6 holds.
With Lemma 5 and Lemma 6, we can immediately get the following Lemma 8.
Lemma 8 (Surrogate Regret). The relationship between Reg(θ) and likelihood function can be shown as follows:
Regt(θ) ≤ 2 · C Cdown (Lt(θ)− Lt(θ∗)) , (14)
∀θ ∈ H,∀x ∈ D, where C and Cdown are defined in Lemma 5 and 6 respectively. Lemma 8 enables us to choose the negative log-likelihood function as a surrogate loss. This is not only an important insight of EMLP regret analysis, but also the foundation of ONSP design.
The last inequality of Equation (8) comes from this lemma:
Lemma 9 (Per-epoch surrogate regret bound). Denoting θ̂k as the estimator coming from epoch (k − 1) and being used in epoch k, we have:
Eh[L̂k(θ̂k)− L̂k(θ∗)] ≤ Cexp Cdown · d τk + 1 . (15)
Here Cexp is defined in Equation 13, and Eh[·] = E[·|hist(k − 1)]. Proof of Lemma 9 is partly derived from the work Koren and Levy [2015], and here we give a proof sketch without specific derivations. A detailed proof lies in Appendix B.2.3.
Proof sketch of Lemma 9. We list the four main points that contribute to the proof:
• Notice that lk,t(θ) is strongly convex w.r.t. a seminorm xk,tx⊤k,t, we know L̂k(θ) is also strongly convex w.r.t. ∑τk t=1 xk,tx ⊤ k,t.
• For two strongly convex functions g1 and g2, we can upper bound the distance between their arg-minimals (scaled by some norm || · ||) with the dual norm of ∇(g1 − g2).
• Since a seminorm has no dual norm, we apply two methods to convert it into a norm: (1) separation of parameters and likelihood functions with a “leave-one-out” method (to separately take expectations), and (2) separation of the spinning space and the null space.
• As the dual data-dependent norm offsets the sum of xx⊤ to a constant, Lemma 9 holds.
We have so far proved Inequality (8) after proving Lemma 5, 6, 9. Therefore, Theorem 4 holds.
5.2 O(d log T ) Regret of ONSP Here we present the regret analysis of Algorithm 2 (ONSP). Firstly, we state the main theorem.
Theorem 10. With Assumptions 1, 2, 3, the regret of Algorithm 2 (ONSP) satisfies: RegONSP ≤ Ca · d log T, (16)
where Ca is a function only dependent on F .
Proof. Proof of Theorem 10 here is more concise than Section 5.1, because the important Lemma 7 and 8 have been proved there. From Lemma 8, we have:
g(J(u∗t ), u ∗ t )− g(J(ut), u∗t ) ≤ 2 · C Cdown · ENt [lt(θt)− lt(θ∗)]. (17)
With Equation 17, we may reduce the regret of likelihood functions as a surrogate regret of pricing. From Lemma 7 we see that the log-likelihood function is CdownCexp -exponentially concave 8. This enables
8A function f(µ) is α-exponentially concave iff ∇2f(µ) ⪰ α∇f(µ)∇f(µ)⊤.
an application of Online Newton Step method to achieve a logarithmic regret. Therefore, by citing from the Online Convex Optimization [Hazan, 2016], we have the following Lemma.
Lemma 11 (Online Newton Step). With parameters γ = 12 min{ 1 4GD , α} and ϵ = 1 γ2D2 , and T > 4 guarantees:
sup {xt}
{ T∑
t=1
lt(θt)−min θ∈H T∑ t=1 lt(θ)
} ≤ 5 ( 1
α +GD
) d log T.
Here α = CdownCexp , D = 2 ·B1 and G = √ Cexp ·B2.
With Equation 17 and Lemma 11, we have:
Reg = T∑ t=1 ( g(J(u∗t ), u ∗ t )− EN1,N2,...,Nt−1 [g(J(ut), u∗t )] ) ≤ 2 · C Cdown ·5 ( 1 α +GD ) d log T. (18)
Therefore, we have proved Lemma 10.

heading: 5.3 Lower Bound for Unknown Distribution
text: In this part, we evaluate Assumption 1 and prove that an Ω( √ T ) lower regret bound is unavoidable with even a slight relaxation: a Gaussian noise with unknown σ. Our proof is inspired by Broder and Rusmevichientong [2012] Theorem 3.1, while our lower bound relies on more specific assumptions (and thus applies to more general cases).
We firstly state Assumption 5 covering this part, and then state Theorem 12 as a lower bound:
Assumption 5. The noise Nt ∼ N (0, σ2) independently, where 0 < σ ≤ 1 is fixed and unknown. Theorem 12 (Lower bound with unknown σ). Under Assumption 2, 3, 4 and 5, for any policy (algorithm) Ψ : Rd × ( Rd,R, {0, 1}
)t−1 → R+ and any T > 2, there exists a Gaussian parameter σ ∈ R+, a distribution D of features and a fixed parameter θ∗, such that: RegΨ ≥ 124000 · √ T .
Remark: Here we assume xt to be i.i.d., which also implies the applicability on adversarial features. However, the minimax regret of the stochastic feature setting is Θ( √ T ) [Javanmard and Nazerzadeh, 2019], while existing results have not yet closed the gap in adversarial feature settings.
Proof sketch of Theorem 12. Here we assume a fixed valuation, i.e. u∗ = x⊤t θ ∗,∀t = 1, 2, . . .. Equivalently, we assume a fixed feature. The main idea of proof is similar to that in Broder and Rusmevichientong [2012]: we assume σ1 = 1, σ2 = 1− T− 1 4 , and we prove that: (1) it is costly for an algorithm to perform well in both cases if the σ’s are different by a lot, and (2) it is costly for an algorithm to distinguish the two cases if σ’s are close enough to each other. We put the detailed proof in Appendix B.3.

heading: 6 Numerical Result
text: In this section, we conduct numerical experiments to validate EMLP and ONSP. In comparison with the existing work, we implement a discretized EXP-4 [Auer et al., 2002] algorithm for pricing, as is introduced in Cohen et al. [2020] (in a slightly different setting). We will test these three algorithms in both stochastic and adversarial settings.
Basically, we assume d = 2, B1 = B2 = B = 1 and Nt ∼ N (0, σ2) with σ = 0.25. In both settings, we conduct EMLP and ONSP for T = 216 rounds. For ONSP, we empirically select γ and ϵ that accelerates the convergence, instead of using the values specified in Lemma 11. Since EXP-4 consumes exponential time and requires the knowledge of T in advance to discretize the policy and valuation spaces, we execute EXP-4 for a series of T = 2k, k = 1, 2, . . . , 12. We repeat every experiment 5 times for each setting and then take an average.
Stochastic Setting. We implement and test EMLP, ONSP and EXP-4 with stochastic {xt}’s. The numerical results are shown in Figure 1a on a log-log diagram, with the regrets divided by log(t). It shows log(t)-convergences on EMLP and ONSP, while EXP-4 is in a tα rate with α ≈ 0.699. Adversarial Setting. We implement the three algorithms and test them with an adversarial {xt}’s: for the k-th epoch, i.e. t = 2k−1, 2k−1 + 1, . . . , 2k − 1, we let xt = [1, 0]⊤ if k ≡ 1( mod 2) and
xt = [0, 1] ⊤ if k ≡ 0( mod 2). The numerical results are shown in Figure 1b on a log-log diagram, with the regrets divided by log(t). The log-log plots of ONSP and EXP-4 are almost the same as those in Figure 1a. However, EMLP shows an almost linear (tα rate with α ≈ 0.912) regret in this adversarial setting. This is because the adversarial series only trains one dimension of θ in each epoch, while the other is arbitrarily initialized and does not necessarily converge. However, in the next epoch, the incorrect dimension is exploited. Therefore, a linear regret originates.

heading: 7 Discussion
text: Here we discuss the coefficients on our regret bounds as a potential extension of future works. In Appendix C we will discuss more on algorithmic design, problem modeling, and ethic issues.
Coefficients on Regret Bounds. The exact regret bounds of both EMLP and ONSP contain a constant Cexp Cdown that highly depends on the noise CDF F and could be large. A detailed analysis in Appendix C.1 shows that CexpCdown is exponentially large w.r.t. B σ (see Equation 39 and Lemma 21) for Gaussian noise N (0, σ2), which implies that a smaller noise variance would lead to a (much) larger regret bound. This is very counter-intuitive as a larger noise usually leads to a more sophisticated situation, but similar phenomenons also occur in existing algorithms that are suitable for constant-variance noise, such as RMLP in Javanmard and Nazerzadeh [2019] and OORMLP in Wang et al. [2020]. In fact, it is because a (constantly) large noise would help explore the unknown parameter θ∗ and smoothen the expected regret. In this work, this can be addressed by increasing T since we mainly concern the asymptotic regrets as T → ∞ with fixed noise distributions. However, we admit that it is indeed a nontrivial issue for finite T and small σ situations. There exists a “ShallowPricing” method in Cohen et al. [2020] that can deal with a very-small-variance noise setting (when σ = Õ( 1T )) and achieve a logarithmic regret. Specifically, its regret bound would decrease as the noise variance σ decreases (but would still not reach O(log log T ) as the noise vanishes). We might also apply this method as a preprocess to cut the parameter domain and decrease Bσ within logarithmic trials (see Cohen et al. [2020] Thm. 3), but it is still open whether a log(T ) regret is achievable when σ = Θ(T−α) for α ∈ (0, 1).

heading: 8 Conclusion
text: In this work, we studied the problem of online feature-based dynamic pricing with a noisy linear valuation in both stochastic and adversarial settings. We proposed a max-likelihood-estimate-based algorithm (EMLP) for stochastic features and an online-Newton-step-based algorithm (ONSP) for
adversarial features. Both of them enjoy a regret guarantee of O(d log T ), which also attains the information-theoretic limit up to a constant factor. Compared with existing works, EMLP gets rid of strong assumptions on the distribution of the feature vectors in the stochastic setting, and ONSP improves the regret bound exponentially from O(T 2/3) to O(log T ) in the adversarial setting. We also showed that knowing the noise distribution (or the demand curve) is required to obtain logarithmic regret, where we prove a lower bound of Ω( √ T ) on the regret for the case when the noise is knowingly Gaussian but with an unknown σ. In addition, we conducted numerical experiments to empirically validate the scaling of our algorithms. Finally, we discussed the regret dependence on the noise variance, and proposed a subtle open problem for further study.

heading: Acknowledgments
text: The work is partially supported by the Adobe Data Science Award and a start-up grant from the UCSB Department of Computer Science. We appreciate the input from anonymous reviewers and AC as well as a discussion with Akshay Krishnamurthy for clarifying some details of Krishnamurthy et al. [2021].

Review summaries:
This paper considers the feature-based dynamic pricing setting. This setting is a sequential setting in which in each round a learner is to predict the price of a product based on a feature vector. Then, if the customer's private value exceeds the price predicted by the learner, the customer buys the product. The goal is to control the difference between the difference between the cumulative price of the sold products and the best offline pricing policy.   On of the main challenge in this setting lies in the feedback that the learner receives. Since the learner only sees whether or not a product is sold, he receives limited feedback about the customers private value for a product. To remediate this issue, the authors assume that the private value of the customer is the inner product between a unknown parameter vector and the feature vector plus some noise. Another important assumption is that the distribution of the the noise is known exactly and that both the cumulative distribution $F$ and $1-F$ are strictly log-concave.   With this set of assumptions (plus some additional standard assumptions) the authors provide two new algorithms, one for stochastic feature vectors and one for adversarial feature vectors. Both algorithms rely on the fact that as soon as both $F$ and $1-F$ are strictly log-concave and $F$ is known, one can derive a surrogate loss which is almost quadratic. Because of the nice properties of this surrogate loss functions to show that the regret is $O(d \ln(T))$.   The paper also contains a lower bound, roughly stating that as soon as the distribution of the noise is not exactly known, the regret has a $O(\sqrt{T})$ lower bound.   Finally, the paper contains some experiments which compare the performance of the two new algorithms to EXP-4. 

(Disclaimer: I have previously reviewed this paper for ICML 2021).  This paper develops algorithms for the problem of contextual dynamic pricing. This is an online learning problem that takes place over T rounds. There is a customer with a hidden value vector theta in R^d. Every round, the learner is given as context a feature vector x in R^d (both x and v are suitably bounded), representing the features of the current item. The customer is willing to pay up to v = <x, theta> + a Gaussian noise term for the item. The learner must set a price p for the item this round; if p > v the customer does not buy the item (and the learner receives 0 utility), otherwise the customer buys the item at price p and the learner receives p utility. The goal of the learner is to maximize their total learner (minimizing their regret compared to the optimal dynamic policy).  This paper presents two algorithms for two different settings. Their first algorithm (EMLP) applies when the feature vectors x_t are iid drawn from an unknown distribution. Their second algorithm (ONSP) applies when the feature vectors x_t are chosen adversarially. In both cases, they obtain O(d*log T) regret. Both algorithms require that the noise is Gaussian with a known variance sigma^2 (The authors show an O(sqrt(T)) lower bound if sigma is unknown). (In fact, this known variance sigma influences the constant in the O(d*log T) regret; see comments below).   Roughly, both algorithms work by maintaining an estimate of theta, and submitting the expected-utility-maximizing price p (assuming the theta estimate is accurate) per round. The two algorithms differ in how they maintain their estimate of theta. EMLP divides the rounds into log T geometrically-sized epochs, and uses a MLE to recompute the value of theta at the end of each epoch. ONSP instead runs an online Newton step method each round to obtain an estimate of theta.  The authors additionally present some empirical simulations of their algorithms (on synthetic data), showing that they outperform EXP4.

This paper studies the problem of feature-based dynamic pricing and offers strong new regret bounds. The headline result is the construction of Online Newton Step Pricing and the proof that it obtains regret $O(d \log T)$ even with adversarial features. This had been proven before only by assuming either (1) small noise or (2) iid features plus a minimum-eigenvalue assumption.