Paper: Dynamic Assortment Selection and Pricing with Learning (ICLR-2024)

Reviewer comment (Novelty):
One main concern is the novelty in the analysis of the proposed algorithm. While this problem is proposed by the authors and is new, the algorithm design is very similar to the previous work in [Oh & Iyenger 2021]. Specifically, in both works, the algorithm has two phases with the first phase doing uniform exploration on the decision variables to ensure good conditioning on the covariance matrix and the second phase making the decisions based on UCB. While in this work, the decisions involve both the assortment and pricing, given the proposition proposed in [Wang 2013], this UCB step is also not hard to obtain. From the analysis perspective, the proof idea is also similar to [Oh & Iyenger 2021] by showing the consistency and normality of MLE and then the regret bound follows from the standard term decompositions, which in my opinion does not introduce much challenge in the analysis.

Author response:
Our work considers a scenario with pricing optimization as well as assortment selections. Therefore, the utlitity terms are functions of the corresponding prices rather than being constants as usually considered in the online assortment selection literature. As a result, our algorithm and analysis differ from the previous literature since it must construct functional upper bounds. The main challenge is in this construction since the upper bound should hold true for all price levels.

Reviewer comment (Theory):
Another concern is about the tightness of the upper bound and the dependency on problem-dependent constants. Specifically, the lower bound has $\Omega(\frac{1}{L_0})$ dependence on the minimum price sensitivity but the dependency on $L_0$ is not explicitly written in the upper bound. With some check, I think $P=\Theta(\frac{1}{L_0})$ (according to Lemma 1), $\kappa = \Theta(\exp(-\frac{1}{L_0}))$ (according to Lemma 3), and $T_0=\Theta(\frac{1}{L_0^2} \exp(\frac{2}{L_0}))$ (according to equation (16)), leading to regret dependent on $\Theta(PT_0)=\Theta(\frac{1}{L_0^3}\exp^{\frac{2}{L_0}})$. This constant can be much larger than the one shown in the lower bound.

Author response:
We have concentrated on tightness of upper and lower bounds in terms of $T$ and $d$ in our work. We agree that there might be room for improvement in terms of regret dependency to other problem parameters $K$ and $L_0$.

Reviewer comment (Novelty):
Can the author highlight the main technical challenge in handling both the price decision and the assortment decision? Currently, it seems to me that as long as the uniform exploration phase is done, the remaining analysis follows smoothly from [Oh & Iyenger 2021] and [Wang 2013].

Author response:
Our work considers a scenario with pricing optimization as well as assortment selections. Therefore, the utlitity terms are functions of the corresponding prices rather than being constants as usually considered in the online assortment selection literature. As a result, our algorithm and analysis differ from the previous literature since it must construct functional upper bounds. The main challenge is in this construction since the upper bound should hold true for all price levels.

Reviewer comment (Theory):
One question for the analysis: while I understand the general idea of the algorithm and the analysis, for Lemma 6, I wonder how is $|\langle \theta_t,\tilde{x}_t\rangle - \langle\theta^\star, \tilde{x}_t\rangle|\leq (1+p)g_t$ obtained? I suppose that this is due to $||\hat{\theta}_t-\theta^\star|| _{V_t} \leq \alpha_t$ but $||(x_t,-px_t)|| _{M}\leq (1+p)||(x_t,x_t)|| _{M}$ does not hold in general for PSD matrix $M$. Can the authors explain more on the proof of this lemma?

Author response:
Regarding your comment on the proof of Lemma 6, we agree that we need to modify the definition of $g_{ti}$ as $ g_{ti} = \alpha_t \max_{1 \leq p \leq P} \||(x_{ti}, -p x_{ti})\||\_{V_t^{-1}}$ . Then, we would obtain $h_{ti}(p) - u_{ti}(p) \leq g_{ti}$ and the given results would still follow.

Reviewer comment (Theory):
**Dependence on $kappa$**: Dependence on $kappa$ should be stated in the regret bound. The paper does not even state the assumption on $kappa$ which commonly exists in almost all the existing literature on contextual MNL bandits (Chen et al., 2020; Oh & Iyengar, 2021; Perivier & Goyal, 2022). I encourage the authors to state the required assumptions clearly in the main text. Rather, the paper shows such dependence on $kappa$ only in the appendix (in Lemma 3). This dependence should be discussed. Particularly, given that the improvement in its dependence in the recent literature such as Perivier & Goyal, 2022 has been already shown.

Author response:
In our work, we do not directly make an assumption of the $\kappa$ value. However, our assumption regarding the $L_0$ parameter directly translates into Lemma 3, which states that there is a sufficiently large probability (i.e., at least $\kappa$ probability) of selection for any item $i \in S_t$ under any parameter $\boldsymbol{\theta}$ sufficiently close to the true parameter $\boldsymbol{\theta}^*$.

Reviewer comment (Theory):
**No dependence on $K$ at all?** The previous results (Chen et al., 2020; Oh & Iyengar, 2021; Perivier & Goyal, 2022) have shown logarithmic dependence on $K$. Yet, this paper does not show any dependence on $K$ even logarithmically, which is questionable. How was this independence achieved?

Author response:
As stated above Theorem 3, our regret upper bound does not capture the dependency with respect to the assortment size parameter K. Since the maximum assortment size is typically small (i.e., K = O(1)) in many real-world applications, we concentrate on the dependency to $T$ and $d$ as an initial step in obtaining matching upper and lower regret upper bounds.

Reviewer comment (Novelty):
Comparison with the most recent and related work, Perivier & Goyal (2022), is very limited. Perivier & Goyal (2022) also considered both assortment pricing and selection under the MNL choice model. Although Perivier & Goyal (2022) do not jointly address pricing and assortment selection as this paper did, there is no sufficient comparison with Perivier & Goyal (2022) both in theory and even in the simulation.

Author response:
Thank you for pointing out the existing work by Miao and Chao that addresses assortment pricing and selection jointly in a non-contextual setting. We will cite their work and provide a comparison. Similarly, we will elaborate more on the comparison with Perivier & Goyal (2022).

Reviewer comment (Novelty):
The authors assert that "[they] are the first to address the problem of dynamic contextual assortment selection and pricing." However, there appears to be an existing work by Miao and Chao (link below) that addresses assortment pricing and selection jointly. Miao and Chao's work is in a non-contextual setting. Yet, the authors at least need to cite their work and provide a comparison.

Author response:
Thank you for pointing out the existing work by Miao and Chao that addresses assortment pricing and selection jointly in a non-contextual setting. We will cite their work and provide a comparison. Similarly, we will elaborate more on the comparison with Perivier & Goyal (2022).

Reviewer comment (Theory):
the assortment optimization in (5) and Line 5 in Algorithm 1 both require finding an assortment that maximizes revenue under given prices. This doesn't seem trivial to me, and I don't know whether this is even an NP-hard problem. More discussions on this is necessary.

Author response:
Each step of algorithm 1 requires computing $v_{ti}(B)$ value for each item $i$ and finding the top-$k$ among these numbers. Therefore, each step has a computational complexity of $O(n)$ as described in Section 3. On the other hand, since Algorithm 1 is running a binary search to find the fixed point $B$, it only needs $\log(1/\epsilon)$ iterations to reach $\epsilon$ accuracy. For some constant $\epsilon$ of our choice, the number of iterations also become a small consant.

Reviewer comment (Theory):
The main algorithm (Algorithm 2) requires initialization rounds involving uniformly random selection of assortments and prices, which could be a practical challenge. Such fluctuations in prices and assortments can be undesirable in practice. As noted in the proof sketch of Theorem 3, these rounds just aim to ensure the invertibility of the design matrix. Then, would an L2-regularized/ridge-type MLE objective solve such an invertibility issue?

Author response:
We agree on that initialization rounds in Algorithm 2 lead to fluctuations in the prices. As you noted out, one solution approach might be to consider a regularized MLE problem. However, it will lead to major changes in the analysis of the algorithm and therefore it is out of scope of our current work.

Reviewer comment (Novelty):
The contribution of this paper is mainly on the modeling part, while the technical contribution is limited. The proof skills are standard. The author should highlight the challenge in the proof and the technical contribution.

Author response:
Our work considers a scenario with pricing optimization as well as assortment selections. Therefore, the utlitity terms are functions of the corresponding prices rather than being constants as usually considered in the online assortment selection literature. As a result, our algorithm and analysis differs from the previous literature since it must construct functional upper bounds.

Reviewer comment (Evaluation):
It seems Algorithm 1 is very computationally expensive. Could the authors have more discussion of the running time of Algorithm 1?

Author response:
Each step of algorithm 1 requires computing $v_{ti}(B)$ value for each item $i$ and finding the top-$k$ among these numbers. Therefore, each step has a computational complexity of $O(n)$ as described in Section 3. On the other hand, since Algorithm 1 is running a binary search to find the fixed point $B$, it only needs $\log(1/\epsilon)$ iterations to reach $\epsilon$ accuracy. For some constant $\epsilon$ of our choice, the number of iterations also become a small consant.

Reviewer comment (Theory):
The assumptions look strong. For example, $L_0$ is not observable in practice. I wonder whether the algorithm or a modified algorithm still works if $L_0$ is unknown.

Author response:
The assumption regarding $L_0$ directly translates into Lemma 3, which states that there is a sufficiently large probability (i.e., at least $\kappa$ probability) of selection for any item $i \in S_t$ under any parameter $\boldsymbol{\theta}$ sufficiently close to the true parameter $\boldsymbol{\theta}^*$. The statement of this lemma is usually assumed to be true as an assumption in the literature of MNL bandits (Oh and Iyengar 2021) and the algorithms depend on the value $\kappa$. Following a similar approach here, we make use of our assumption on the $L_0$ parameter. Even though it might be posssible to estimate (or estimate a lower bound for) the parameter $L_0$ using choice observations, it is out of scope of our work here.