Title: True Impact of Cascade Length in Contextual Cascading Bandits
Venue: NeurIPS 2025
Accepted: yes
Keywords: contextual bandit

cascading bandit

upper confidence bound

online leaning
Abstract: We revisit the contextual cascading bandit, where a learning agent recommends an ordered list (\emph{cascade}) of items, and a user scans the list sequentially, stopping at the first attractive item. Although cascading bandits underpin various applications including recommender systems and search engines, the role of the cascade length $K$ in shaping regret has remained unclear. Contrary to prior results that regret grows with $K$, we prove that regret actually \emph{decreases} once $K$ is large enough.
Leveraging this insight, we design a new upper-confidence-bound algorithm built on online mirror descent that attains the sharpest known regret upper bound, $\tilde{\mathcal{O}}\bigl(\min \lbrace K\bar{p}^{K-1}, 1 \rbrace d \sqrt{T}\bigr)$ for contextual cascading bandits. To complement this new regret upper bound, we provide a nearly matching lower bound of $\Omega \bigl(\min \lbrace K\underline{p}^{K-1}, 1 \rbrace d \sqrt{T}\bigr)$, where $0 \leq \underline{p} \leq \bar{p} < 1$. Together, these results fully characterize how regret truly scales with $K$, thereby closing the theoretical gap for contextual cascading bandits.
Finally, comprehensive experiments validate our theoretical results and show the effectiveness of our proposed method.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe cascading bandit problem is widely used in recommendation systems, search engines, and online advertising. In this model, an agent selects an ordered list (cascade) of length $ K $ from $ N $ options (arms), and users sequentially browse the list, stopping at the first appealing item. Traditional studies suggest that regret—defined as the difference in expected rewards between the optimal and selected cascades—increases or remains constant with larger $ K $. However, this contradicts the intuition that longer cascades offer more feedback opportunities, potentially reducing regret. The paper challenges this view, proving that in contextual cascading bandits, regret decreases when $ K $ is sufficiently large.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper challenges existing research on the impact of cascade length $ K $ on regret in contextual cascading bandits, demonstrating that regret decreases when $ K $ is sufficiently large. Specifically, the authors propose a new regret upper bound of $\\mathcal{O}(K \\bar{p}^{K-1} d \\sqrt{T})$, where $\\bar{p}$ is the upper bound of the feedback probability, $d$ is the dimension of the context vector, and $T$ is the total number of rounds. This bound shows that regret diminishes with increasing $K$ due to the exponential decay of $\\bar{p}^{K-1}$ ($\\bar{p} < 1$).\n\n2. Additionally, the authors derive a matching lower bound of $\\Omega(K \\underline{p}^{K-1} d \\sqrt{T})$, where $\\underline{p}$ is the lower bound of the feedback probability. Together, these bounds fully characterize the true trend of regret with respect to $K$, addressing a theoretical gap in contextual cascading bandits.\n\n3. The paper introduces UCB-CLB (Upper Confidence Bound for Cascading Logistic Bandits), an algorithm based on Upper Confidence Bound (UCB) that integrates Online Mirror Descent (OMD) and a cascade restructuring technique (DO-SWAP). Operating under a logistic model, UCB-CLB achieves efficient parameter estimation and cascade selection. By avoiding the high computational cost of traditional maximum likelihood estimation (MLE), UCB-CLB significantly improves efficiency while maintaining theoretically optimal regret bounds. The DO-SWAP technique optimizes the exploration-exploitation trade-off by prioritizing items with higher uncertainty in the cascade.\n\n\n**Weaknesses**\n\n1. The theoretical results rely on specific assumptions, such as Assumption 5.1 (bounded norms of feature vectors and parameters) and Assumption 5.3 (a positive lower bound $\\kappa$ in the logistic model). These assumptions may not always hold in practice. For instance, a small $\\kappa$ (indicating significant deviation from a linear model) could impact algorithm performance, despite claims that UCB-CLB remains effective in such cases. However, the paper does not thoroughly explore how violations of these assumptions affect performance, potentially limiting the generalizability of its theoretical findings.\n\n2. The analysis depends on $\\bar{p} \\leq 1/e \\approx 0.37$, where the regret upper bound decreases monotonically with increasing $K$. When $\\bar{p} > 1/e$, regret may initially increase before decreasing (as $K \\bar{p}^{K-1}$ peaks at $K = \\frac{1}{\\log(1/\\bar{p})}$). The paper does not adequately address the implications of $\\bar{p} > 1/e$ in practical high-feedback-probability scenarios, which may restrict the applicability of its theoretical results.\n\n3. Experiments only test cascade lengths $K = 5$ and $K = 10$, without exploring a broader range of $K$ values.\n\n4. The UCB-CLB algorithm relies on the DO-SWAP technique to restructure cascades, prioritizing items with higher uncertainty. While effective, this increases algorithmic complexity, potentially introducing additional computational overhead or implementation challenges in practice. The paper does not provide a detailed analysis of DO-SWAP’s computational costs or potential failure cases across different scenarios.\n\nQUESTIONS:\n1. The UCB-CLB algorithm leverages Online Mirror Descent (OMD) and DO-SWAP techniques to enhance computational efficiency and optimize regret. Can this approach effectively extend to dynamic environments, such as those with time-varying contextual features?\n\n2. The paper introduces the combination of OMD and DO-SWAP as a novel proof technique to address the sequential feedback characteristics of cascading bandits. What new challenges might this integration introduce?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the role of the cascade length $K$ in contextual cascading bandits—a learning framework relevant to search and recommendation systems. Contrary to previous works suggesting that regret increases with $K$, the authors theoretically and empirically demonstrate that regret can decrease as $K$ grows large. They propose a novel UCB-based algorithm, UCB-CLB, which uses Online Mirror Descent (OMD) for efficient parameter estimation, achieving a regret upper bound of $\\tilde{O}(K^{\\bar{p}} d \\sqrt{T})$ that diminishes with increasing $K$. They also provide the first matching lower bound in the contextual setting, resolving a long-standing open problem about the impact of cascade length.\n\nSTRENGTHS AND WEAKNESSES:\n#### **Strengths**\n\n* The paper challenges and overturns a widely held belief in cascading bandits literature—that regret necessarily increases with cascade length. This is an important and surprising contribution.\n* The UCB-CLB algorithm leverages OMD for parameter estimation, significantly improving computational efficiency compared to MLE-based approaches.\n* The paper provides both upper and lower bounds that match up to logarithmic factors, a rare accomplishment in bandit theory.\n* UCB-CLB is shown to be scalable and effective on real-world datasets like MovieLens, and avoids non-convex optimization bottlenecks.\n*  The paper is generally well-organized, with a thorough theoretical exposition, detailed algorithms, and compelling empirical results.\n\n#### **Weaknesses**\n\n* Some results rely on regularity conditions such as bounded features and logistic noise parameters, which may not hold in all real-world deployments.\n* While the MovieLens experiments are convincing, broader validation across datasets or task domains would strengthen the claims.\n* The lower bound uses a slightly different constant $p$ than the upper bound's $\\bar{p}$, though the authors do acknowledge and justify this.\n\nQUESTIONS:\n1. **Clarification on $p$ vs $\\bar{p}$**: Can the authors comment on how large the gap between $p$ and $\\bar{p}$ might be in practice, and whether it's observable in experiments?\n2. **OMD vs MLE Trade-offs**: Could the authors elaborate on potential limitations of OMD versus MLE, especially in terms of statistical efficiency in low-noise regimes?\n3. **Impact of Arm Pool Size $N$**: Even though the bounds are independent of $N$, does increasing $N$ affect empirical performance (e.g., convergence speed)?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper addresses a long-standing and misunderstood problem in the field of recommendation systems: how the length of a recommended list (the cascade length.

STRENGTHS AND WEAKNESSES:
Strengths:
They propose a new Upper Confidence Bound (UCB) algorithm for cascading logistic bandits. It uses Online Mirror Descent (OMD) for parameter estimation, which makes it computationally efficient and scalable, with per-round costs that are independent of the total time horizon. Specifically, they novelly achieve the decreasing for large K regret bound compared to others.

Weaknesses:
1. They assume a lower bound for the vairances. This assumptions are not so standard. Could the authors discuss about this assumption, why do they need it and could their analysis degrade to deterministic settings where variances are zero?

QUESTIONS:
1. In the experiment, the K values are only 5, 10. Could the authors try a larger K, such as 50, 100? Also, it would be interesting and convincing if they could plot the curve between the regret and a range of K to validate their theories.

2. For \bar p in the bound, \bar p is small when all the arms in the space has a relatively high logit. It seems a little strict since there may exist x such that x^T\theta is very small. In this case, the bound only decreases when K is sufficienlty large. Could the authors disscuss more about the limitations of their result?

LIMITATIONS:
no
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

comments: SUMMARY:
This paper considers contextual cascading bandits where an agent select ordered lists of $K$ arms from $N$ total arms, receiving sequential feedback until a criterion is met. 

The authors propose UCB with online mirror descent for cascading logistic bandits, which requires only $O(1)$ per-round computation, independent of $T$. 

This work is the first to prove that longer cascades can improve performance

STRENGTHS AND WEAKNESSES:
Strengths:
- With the help of OMD subroutine, the proposed algorithm is computationally efficient.
- The first to prove that longer cascades can improve performance
- Provides first lower bound derivation for contextual cascading bandits

Weaknesses:
- There is still s gap between the upper and lower bound. It would be better if the authors discuss more on how to close the gap in the future.

- The regret bound increases when $K$ is relatively small, especially when $\bar p$ is large.

QUESTIONS:
Q1. In the description of Algorithm 2 on page 5, there is no explanation on how to choose $i_t^{(1)}$ and $i_t^{(2)}$.

Q2. When $K$ is not large enough, how can you argue that your regret bound is the tightest compared to previous results (on Page 2)?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03