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Knapsack based Optimal Policies for Budget-Limited Multi-Armed Bandits
Long Tran-Thanh, Archie Chapman, Alex Rogers, Nicholas R. Jennings
TL;DR
Budget-limited MABs require learning under costly pulls and a fixed budget, where the best policy may be a sequence of arms rather than repeated use of one arm. The paper develops KUBE and fractional KUBE, which jointly explore and exploit through knapsack-based choices. Both achieve asymptotically optimal logarithmic regret bounds, while KUBE typically performs up to 40% better but costs more computationally.
Problem
Budget-limited MABs require policies that maximize reward from costly arm pulls without exceeding a fixed budget, and existing ε-first methods depend on a difficult-to-select exploration fraction.
Method
KUBE and fractional KUBE adaptively choose arms using current reward estimates and knapsack-based best-arm sets rather than explicitly separating exploration from exploitation.
Results
Both algorithms achieve O(ln B) regret upper bounds that are asymptotically optimal, while KUBE typically outperforms fractional KUBE by up to 40%.
Takeaways & Limitations
KUBE offers better experimental performance, whereas fractional KUBE is computationally less expensive.
Takeaways & Limitations
Both algorithms assume that expected reward values are static, so extending the model to dynamically changing reward distributions remains future work.
Abstract
from arXiv · showhide
In budget-limited multi-armed bandit (MAB) problems, the learner's actions are costly and constrained by a fixed budget. Consequently, an optimal exploitation policy may not be to pull the optimal arm repeatedly, as is the case in other variants of MAB, but rather to pull the sequence of different arms that maximises the agent's total reward within the budget. This difference from existing MABs means that new approaches to maximising the total reward are required. Given this, we develop two pulling policies, namely: (i) KUBE; and (ii) fractional KUBE. Whereas the former provides better performance up to 40% in our experimental settings, the latter is computationally less expensive. We also prove logarithmic upper bounds for the regret of both policies, and show that these bounds are asymptotically optimal (i.e. they only differ from the best possible regret by a constant factor).
1 Introduction
Budget-limited MABs differ from standard bandits because costly arm pulls constrain both exploration and exploitation, requiring policies that optimize a sequence of arms within the budget. The paper introduces KUBE and fractional KUBE, which jointly explore and exploit while achieving logarithmic regret bounds.
- Motivation: Budget-limited MABs make arm pulling costly and restrict exploration and exploitation to a fixed budget.Unlike the standard model, the agent may need to choose a sequence of arms rather than repeatedly pull one arm.
- Motivation: ε-first methods split the budget between exploration and exploitation, but their performance depends on choosing ε for the problem instance.Different budget limits or unknown arm costs can require different ε values, and poor choices create inefficient exploration or exploitation.
- Proposed methods: KUBE and fractional KUBE adaptively select arms using current reward estimates instead of explicitly separating exploration from exploitation.KUBE uses an unbounded-knapsack model to find the best estimated arm set that fits the residual budget.
- Results: KUBE outperforms fractional KUBE by up to 40% in experiments, but incurs higher computational cost.The paper reports this comparison alongside numerical evaluation against the budget-limited ε-first approach.
- Results: O(ln B) regret upper bounds are proved for KUBE and fractional KUBE, while budget-limited ε-first approaches fail to achieve logarithmic regret bounds.Here B denotes the budget limit, which determines the number of pulls.
2 Model Description
The budget-limited MAB model has costly arms, bounded rewards, unknown mean rewards, and a fixed budget that the total pulling cost cannot exceed. Its objective is to learn a pulling sequence that maximizes expected total reward or minimizes regret relative to a theoretical optimum.
- Model: The model contains K arms, each with pulling cost c_i and a non-negative reward drawn from its arm-specific distribution.The total cost of all pulls must remain within budget B.
- Model: Reward distributions have bounded supports, and μ_i denotes the mean reward for arm i.The bounded-support assumption reflects the paper's modeling of typically bounded real-world reward values.
- Objective: The agent does not initially know the arms' μ_i values and must learn them to construct a policy maximizing expected total reward under budget B.The objective is defined over the expected sum of rewards without exceeding the cost budget.
- Formalization: An arm-pulling algorithm A produces a finite pull sequence whose arm counts satisfy the budget constraint.N_A^i(B) represents the number of pulls of arm i with respect to budget B.
- Objective: The theoretical optimum A* maximizes expected total reward but is generally unachievable because it requires prior knowledge of μ_i.Regret compares an algorithm's expected cumulative reward with that theoretical optimum.
3 The Algorithms
The paper introduces KUBE and fractional KUBE, which adaptively select arms by approximating an unbounded knapsack problem under the residual budget. KUBE uses density-ordered greedy selection, while fractional KUBE uses a cheaper fractional relaxation.
- 3 The Algorithms: KUBE and fractional KUBE determine the estimated best arm set at each time step using unbounded-knapsack techniques.The methods adapt their choices to current reward estimates rather than explicitly separating exploration and exploitation.
- 3.1 The Unbounded Knapsack Problem: O(K log K): KUBE uses density-ordered greedy approximation to select a near-optimal feasible arm set.The method sorts arms by estimated reward density and repeatedly adds feasible high-density arms.
- 3.2 KUBE: KUBE initially pulls each arm once, then samples the next arm according to frequencies in the approximate knapsack solution.It stops when no arm can be pulled within the residual budget and updates confidence bounds and budget after each pull.
- 3.3 Fractional KUBE: Fractional KUBE replaces KUBE’s knapsack approximation with fractional relaxation and pulls the arm with the highest estimated confidence-bound-to-cost ratio.Because the fractional solution selects one highest-density arm, fractional KUBE does not need random arm selection.
- 3.3 Fractional KUBE: O(K) per time step: fractional KUBE reduces computational cost relative to KUBE while retaining the same underlying adaptive knapsack approach.The paper states that both methods achieve asymptotically optimal regret bounds.
4 Performance Analysis
The analysis derives regret upper bounds for KUBE and fractional KUBE and establishes their asymptotic optimality. It also bounds suboptimal-arm pulls logarithmically and relates the algorithms’ performance to the fractional budget-limited optimum.
- 4 Performance Analysis: The analysis derives regret upper bounds for KUBE and fractional KUBE and shows that the bounds are asymptotically optimal.The bounds differ from the best possible regret only by a constant factor.
- 4 Performance Analysis: O(ln(B)): KUBE’s average performance regret is upper-bounded logarithmically in the budget.The analysis first bounds suboptimal-arm pulls and then relates the resulting pull count to the optimal policy.
- 4 Performance Analysis: O(ln(T)): KUBE pulls each arm j ≠ I* at most logarithmically many times.Here, I* is the arm with the highest true mean value density, µi/ci.
- 4 Performance Analysis: The number of KUBE pulls differs from the fractional budget-limited optimum by only a logarithmic function of the number of pulls.This pull-count comparison is used to derive KUBE’s regret bound.
- 4 Performance Analysis: For any arm-pulling algorithm, some budget-limited MAB instance has regret at least C ln B.This lower bound establishes the asymptotic optimality scale for the proposed logarithmic upper bounds.
- 4 Performance Analysis: In standard MABs, both algorithms achieve O(ln(T)) regret, while fractional KUBE has the smaller theoretical constant and KUBE performs up to 40% better empirically.The paper notes that the tighter fractional-KUBE bound does not imply better practical performance.
5 Performance Evaluation
Simulations compare KUBE and fractional KUBE across pulling-cost diversity, alongside budget-limited ε-first approaches. Both proposed algorithms achieve logarithmic regret behavior, while KUBE generally performs better in practice at higher computational cost.
- The evaluation uses homogeneous, moderately diverse, and extremely diverse pulling-cost cases, with costs sampled from [5, 10], [1, 10], and [1, 20], respectively.The experiments use 100 arms and truncated Gaussian reward distributions.
- Both algorithms’ performance regret is O(ln B) across the tested cases, differing from the best possible solution by small constant factors.The reported convergence limits typically range from 4 to 7 in the test cases.
- KUBE typically achieves lower regret than fractional KUBE, especially when pulling costs are diverse.Their performance is similar with homogeneous costs, whereas KUBE performs better in moderately and extremely diverse cases.
- 40%: KUBE outperforms fractional KUBE by up to 40% in the moderately diverse-cost case.
- KUBE’s advantage is attributed to its density-ordered greedy approximation, which is better than fractional relaxation for the underlying unbounded knapsack problem.
- Both KUBE and fractional KUBE outperform budget-limited ε-first approaches in the simulations.
6 Conclusions
The paper introduces KUBE and fractional KUBE for budget-limited MABs, using confidence-based best-arm sets without explicitly separating exploration and exploitation. Both achieve asymptotically optimal logarithmic regret bounds; KUBE is typically more effective in simulations but computationally more expensive.
- KUBE and fractional KUBE select arms using current reward estimates plus confidence intervals, without explicitly separating exploration from exploitation.
- Both algorithms have O(ln B) theoretical regret upper bounds that differ from the best possible regret by only a constant factor.
- KUBE typically outperforms fractional KUBE by up to 40% in simulations, but has higher computational cost.
- O(K ln K): KUBE’s average computational complexity per time step, compared with O(K) for fractional KUBE.
- Both algorithms assume static expected reward values, so extending the model to dynamically changing reward distributions is nontrivial.