Bandits with Anytime Knapsacks

arXiv:2501.18560 · cs.LG · Submitted 2025-01-30 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Bandits with Anytime Knapsacks".

Jane: The gist We propose SUAK, an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation.

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So, this paper is about this new version of bandits where instead of just having one budget, you have an anytime knapsack constraint that keeps your average cost from getting too high over time.

Jane: Exactly. It’s not just about spending a total amount; it’s about keeping the average cost under a certain limit at every single step, which makes the decision-making way more complex.

Tom: The authors introduce this algorithm called SUAK to solve that problem. They use upper confidence bounds to figure out how to mix exploration and exploitation while making sure you respect that anytime constraint.

Jane: It’s designed to be smart about when it needs to pull an arm with a higher cost, so it skips rounds when it sees a risk of violating the budget constraint.

Tom: The big thing they show is this regret upper bound, O(K T), which is instance-dependent. That’s better than the prior work they compared it against, which was O(sqrt KT).

Jane: So what does that mean for someone just listening? It means that if you're dealing with a long period of decision making, this method is much more efficient in terms of how much regret it accumulates.

Tom: It’s not just about finding an arm right now; it’s about managing the entire sequence of decisions under those dynamic cost limitations.

Jane: And the mechanism behind it involves using upper confidence bounds to guide exploration while also strategically skipping rounds when necessary, which helps keep the number of skips down.

Tom: This paper shows that this approach performs better than simpler strategies like One Phase Skip after a certain number of rounds, around one point four times ten six.

Jane: It’s really about showing how to handle those tricky constraint scenarios in practice, where you can't afford any waste or budget overruns.

Tom: The paper gives us a way to think about how AI systems can make decisions that are feasible under strict, changing cost limits.

Jane: And this leads us to wonder, how will this kind of adaptive strategy play out in real-world scenarios like inventory management or online advertising?

The paper's summary: Tom: So, the paper isn't just proposing SUAK; they also suggest ways to make it even better when you’re dealing with these anytime knapsack constraints.

Jane: They focus on how to improve the way the algorithm manages that budget in real time, so it doesn't get stuck in a bad spot due to budget fluctuations.

Tom: One improvement they point out is making the algorithm more adaptive to the available budget at each round, which means it reacts better when things change.

Jane: That’s smart because if your available budget swings up or down, the system needs to adjust its exploration or exploitation balance instantly.

Tom: They also discuss how to reduce those skips even further by being super careful about when they decide to skip a round—only skipping when it’s really necessary, not just because of a general rule.

Jane: That sounds like they are trying to make the skipping mechanism more precise, so you don't waste precious rounds on unnecessary pauses.

Tom: The core idea is pushing the algorithm toward a tighter performance guarantee, aiming for that O(K T) bound we talked about earlier.

Jane: So what does that mean practically? It suggests that if you use these improvements, the long-term cost of not finding the optimal arm actually gets smaller.

Tom: And Lalam sees a vision here where this level of constraint management could help build AI systems that operate reliably in environments with unpredictable resource availability.

Jane: It moves the focus from just getting *a* good result to getting the *best* result possible under hard, dynamic cost limitations.

Tom: This opens up questions about how we can integrate these upper confidence bounds with other complex model architectures, like those we’ve been looking at with diffusion models.

Jane: And Meng is wondering about the computational cost; does this more adaptive approach slow down the decision-making process too much for things that need to happen in real time?

Tom: That’s a fair concern, Meng. The paper focuses on the theory of the bounds, but they do talk about running that linear program in phase two as a way to find that optimal policy efficiently.

Jane: It seems like the next big step is seeing if we can take these adaptive ideas and apply them beyond just bandits to other areas where we have strict resource budgets.

The paper's improvements: Tom: So to wrap up, this paper on "Bandits with Anytime Knapsacks" introduces SUAK, an algorithm that uses upper confidence bounds to balance exploration and exploitation while respecting a dynamic anytime cost constraint.

Jane: It’s a method for making sequential decisions under those hard budget rules by being much smarter about when to explore versus when to pull known good arms.

Tom: The main thing they deliver is that O(K T) instance-dependent regret bound, which is tighter than the previous O(sqrt KT) bounds.

Jane: That means for real applications, you get a stronger guarantee on how much regret the system will accumulate over time.

Tom: It shows that we can get better performance when we use adaptive strategies and confidence bounds to manage exploration in these kinds of resource-constrained environments.

Jane: The simulation results they show are pretty solid too; SUAK beats simpler approaches after about one point four times ten six rounds, and the skips are very low.

Tom: It’s a significant step because it tackles the complexity of anytime constraints with a method that actually performs well in practice.

Jane: And Lalam thinks this kind of careful constraint management could lead to AI systems that feel much more robust when deployed in real-world settings where budgets aren't always guaranteed.

Tom: Meng is curious about how an engineer would actually implement this; does it add too much computational overhead compared to a simpler skip strategy?

Jane: That’s a good point, Meng. The paper mentions the linear program evaluation in phase two as the way to find that optimal policy, which suggests it’s structured for computation.

Lu: Lu is thinking about how this concept of using upper confidence bounds to identify an optimal base could be applied to other complex modeling tasks we're looking at, like auditing diffusion models.

Lalam: Lalam sees a vision where this kind of careful constraint management could help build AI systems that feel much more robust when deployed in real-world settings where budgets aren't always guaranteed.

Tom: So for anyone listening, the "Bandits with Anytime Knapsacks" paper gives us a concrete example of how to get tighter performance guarantees by being adaptive and using confidence bounds.

Jane: It’s a solid piece of research that moves us toward more reliable decision-making under strict, changing cost limitations.

Tom: We'll be looking into how this instance-dependent bound helps us set better standards for other sequential decision making problems next.

Conclusion: Tom: So we’re wrapping up on "Bandits with Anytime Knapsacks". This paper introduces SUAK, an algorithm that uses upper confidence bounds to balance exploration and exploitation while respecting a dynamic anytime cost constraint.

Jane: It’s about managing those hard budget rules in sequential decision making by being much smarter about when to explore versus when to pull known good arms.

Tom: The main thing they delivered is that O(K T) instance-dependent regret bound, which is tighter than the previous O(sqrt KT) bounds.

Jane: That means for real applications, you get a stronger guarantee on how much regret the system will accumulate over time.

Tom: It shows that we can get better performance when we use adaptive strategies and confidence bounds to manage exploration in these kinds of resource-constrained environments.

Jane: The simulation results they show are pretty solid too; SUAK beats simpler approaches after about one point four times ten six rounds, and the skips are very low.

Tom: It’s a significant step because it tackles the complexity of anytime constraints with a method that actually performs well in practice.

Jane: And Lalam thinks this kind of careful constraint management could lead to AI systems that feel much more robust when deployed in real-world settings where budgets aren't always guaranteed.

Tom: Meng is curious about how an engineer would actually implement this; does it add too much computational overhead compared to a simpler skip strategy?

Jane: That’s a good point, Meng. The paper mentions the linear program evaluation in phase two as the way to find that optimal policy, which suggests it’s structured for computation.

Tom: Lu is thinking about how this concept of using upper confidence bounds to identify an optimal base could be applied to other complex modeling tasks we're looking at, like auditing diffusion models.

Jane: And Lalam sees a vision where this kind of careful constraint management could help build AI systems that feel much more robust when deployed in real-world settings where budgets aren't always guaranteed.

Tom: So for anyone listening, the "Bandits with Anytime Knapsacks" paper gives us a concrete example of how to get tighter performance guarantees by being adaptive and using confidence bounds.

Jane: It’s a solid piece of research that moves us toward more reliable decision-making under strict, changing cost limitations.

Tom: We'll be looking into how this instance-dependent bound helps us set better standards for other sequential decision making problems next.

Lu: I think the next big thing is exploring how these upper confidence bounds can be integrated more deeply with other complex model architectures to see if we can generalize this idea beyond just the bandit setting.

Meng: I'm curious about the practical implementation details; how much computational overhead does SUAK add compared to a simpler algorithm, and can it handle very high-dimensional arm spaces effectively?

Lalam: I see this framework creating a new culture for AI development where we prioritize feasibility under strict constraints, making those resource limitations part of the design from the start.

Eray Can Elumar, Cem Tekin, Osman Yagan ˘

Carnegie Mellon University · Bilkent University

cs.LG

Submitted: 2025-01-30

Updated: 2026-10-09

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist We propose SUAK, an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation.

Key concepts

Bandits with Anytime Knapsacks (BwAK) problem: This is a variation of the standard bandits problem where you must maximize cumulative reward while ensuring that the average cost incurred by your chosen arms never exceeds a specified anytime constraint 'c'. This adds complexity because decisions must constantly consider this budget limit.
Upper Confidence Bounds (UCB): UCB is a technique used to balance exploration and exploitation in bandit problems. It helps the algorithm estimate how good an arm might be based on past observations, providing a confidence interval around its estimated performance. This allows the algorithm to intelligently choose arms that are either promising or uncertain.
Regret Upper Bound: Regret measures how much worse an algorithm's performance is compared to the best possible outcome. The paper provides a specific mathematical bound, O(K log T), for SUAK in this setting. This bound is better than previous results, which were instance-independent, showing that SUAK performs very well in terms of long-term performance.

Terminology

Adaptive Algorithm — 1× in this paper · explained in 1 episode
Regret — 1× in this paper · explained in 3 episodes

Summary

The gist We propose SUAK, an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation.

Problem Setting

The Bandits with Anytime Knapsacks (BwAK) problem is a novel version of the BwK problem where there is an anytime cost constraint instead of a total cost budget, introducing additional complexities as it mandates adherence to the constraint throughout the decision-making process. This setting requires a mixture strategy to pull arms with mean costs higher than the average cost budget without violating the anytime constraint. The problem is formally expressed as maximizing cumulative reward subject to an anytime constraint on the average cost, denoted as max F(t) = E "Xt s=1 r(s) s.t. Pu s=1 c(s) u ≤ c, ∀ u ≤ t.

Algorithm Design

SUAK is an adaptive algorithm that strategically utilizes the available budget in each round in the decision-making process and skips a round when it is possible to violate the anytime cost constraint. The algorithm works in two phases:

Initialization Phase: This phase ensures whether the mean cost of an arm is above or below c is correctly known, ensuring that the base SUAK selects in a round t in the second phase includes an arm with cost less than c. The round t is skipped if Sp(t − 1) + 1 > c · Np(t − 1).
Main Algorithm (Phase 2): In every round, SUAK checks for constraint violations and then evaluates a linear program to find the optimal policy. The decision rule involves calculating a probability p(t) based on comparing the available budget b(t) with the empirical average costs of arms j(t) and k(t).

Key Contributions and Analysis

The main contributions of this work are:

SUAK Algorithm: SUAK utilizes the upper confidence bounds to explore the optimal base that solves the problem, and also strategically under-utilizes the available budget to limit the number of rounds that are skipped when satisfying the anytime cost constraint.
Regret Upper Bound for SUAK: We improve prior work by providing the first O(K log T) instance-dependent bound for this problem setting, compared to O(√KT) instance-independent bounds on prior work.

The theoretical upper bound on expected cumulative regret is given by Theorem 3.2 as RT ≤ X K i=1 432r∗ (δi+1δi)2 log T δmin∆min,i + 241Kr∗ log T cδ2 min + 3π/2 r∗ δ squared min + 2π/2 K squared = O(K log T) + O(1). This problem-dependent upper bound order-wise matches the O(K log T) instance-dependent bound of prior work for the regular BwK setting.

Simulation Results

Simulations verify the utility of SUAK in practical settings, showing that SUAK performs better compared to naive approaches like One Phase Skip (OPS). Specifically, SUAK exceeds the performance of OPS after around round 1.4 × 10 6. The third plot in Figure 2 shows that the number of skips is sublinear for all algorithms, and both USS and SUAK have a very small number of skips.

Conclusion

In this paper, we introduce a previously unexplored setting for the BwK problem, which we call the bandits with anytime knapsacks (BwAK) problem. We provide SUAK, a novel algorithm that under-utilizes the available cost budget, utilizes skipping to limit the probability of violating the anytime cost constraint, and uses upper confidence bounds to balance exploration and exploitation. SUAK achieves an instance-dependent regret upper bound of O(K log T). We provide simulation results to demonstrate the empirical performance of SUAK. This work opens multiple directions for future research. The first instance-dependent regret bound for this problem setting is O(K log T), as prior work can only achieve O(√KT) instance-independent bounds in our setting. The analysis in this section is similar to §D.6, except the fact that we now try to find the probability that the empirical average cost falls below a certain value due to under-budgeting instead of finding the probability that the empirical average cost crosses above c. The total expected number of skips can be upper bounded as S(T) ≤ 3π/2 δ squared min D.7>.

Improvements for AI systems

textbf SUAK Algorithm Implementation for Anytime Knapsacks (BwAK): This algorithm strategically under-utilizes available budget to limit skips while maintaining adherence to the anytime cost constraint, leading to a regret upper bound of O(K log T). The system will be able to operate in resource-constrained environments where a continuous budget is not guaranteed, ensuring feasibility throughout the decision process rather than relying on naive skipping strategies.
textbf Tight Regret Guarantees for BwAK: SUAK provides the first instance-dependent regret bound for this problem setting, as prior work can only achieve O(√KT) instance-independent bounds. This means AI systems deployed in scenarios requiring high precision under strict average cost constraints can be expected to perform significantly better than previous state-of-the-art algorithms.
textbf Adaptive Budget Management: The algorithm is non-stationary as it is adaptive to the available budget at that round, which prevents the performance degradation often seen in non-adaptive designs where the fluctuation of the available budget around its mean at round t can be as high as omega(1/√t). This allows AI systems to dynamically adjust their exploration/exploitation trade-off based on real-time cost accumulation.
textbf Robustness Against Constraint Violations: SUAK's design specifically targets to reduce the number of skips needed by utilizing skipping a round when it is possible to violate the anytime cost constraint. This directly addresses the complexity introduced by the anytime constraint, ensuring that decision-making remains feasible in applications like inventory management or online advertising where negative inventory or budget overruns are unacceptable.
textbf Enhanced Exploration/Exploitation Balance: SUAK utilizes upper confidence bounds to identify the optimal base and employs a mechanism where an individual arm in a base will be pulled at least 2/9 δmin probability in a round t if SUAK pulls an arm from a base. This balanced approach ensures that the system explores sufficient alternatives while exploiting known high-reward options under the given cost limitations.

Abstract

We consider bandits with anytime knapsacks (BwAK), a novel version of the BwK problem where there is an anytime cost constraint instead of a total cost budget. This problem setting introduces additional complexities as it mandates adherence to the constraint throughout the decision-making process. We propose SUAK (Strategic Under-utilization for Anytime Knapsacks), an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation. SUAK is an adaptive algorithm that strategically utilizes the available budget in each round in the decision-making process and skips a round when it is possible to violate the anytime cost constraint. In particular, SUAK slightly under-utilizes the available cost budget to reduce the need for skipping rounds. We show that SUAK attains the same problem-dependent regret upper bound of O(K T) established in prior work under the simpler BwK framework, where K denotes the number of arms, and T denotes the time horizon. We also establish an instance-dependent lower bound showing that the O(K T) dependence is unavoidable. Finally, we show via extensive simulations that SUAK achieves considerably higher rewards than naive adaptations of BwK algorithms that do not optimize learning under anytime cost constraints.