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Conservative Bandits

Yifan Wu, Roshan Shariff, Tor Lattimore, Csaba Szepesvári

arXiv:1602.04282v1stat.MLcs.LG

TL;DR

The paper studies how bandit algorithms can explore for better rewards while keeping cumulative return above a fixed default-based threshold throughout learning. It develops conservative strategies for stochastic and adversarial settings; in the stochastic case, a lower bound shows the additive regret penalty is unavoidable, while the adversarial algorithm’s penalty is higher.

  • Problem

    Existing approaches addressed return constraints only at a fixed future time, leaving the problem of maintaining a revenue floor uniformly during exploration.

    Source: §1 · pp. 1–2

  • Method

    The paper formulates conservative bandit problems and designs strategies for stochastic and adversarial settings that maintain the return constraint uniformly over time.

    Source: §1 · p. 2

  • Results

    Conservative UCB pays an additive rather than BudgetFirst’s multiplicative regret penalty, and a lower bound shows the stochastic additive penalty is unavoidable; the adversarial algorithm’s penalty is higher.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The proposed strategies demonstrate ways to explore while maintaining return constraints in both stochastic and adversarial bandit settings.

    Source: §1 · p. 2

    For the adversarial setting, it remains unclear whether the higher penalty reflects a suboptimal algorithm, a loose analysis, or intrinsic difficulty.

    Source: §4 · p. 7

Abstract

from arXiv · show

We study a novel multi-armed bandit problem that models the challenge faced by a company wishing to explore new strategies to maximize revenue whilst simultaneously maintaining their revenue above a fixed baseline, uniformly over time. While previous work addressed the problem under the weaker requirement of maintaining the revenue constraint only at a given fixed time in the future, the algorithms previously proposed are unsuitable due to their design under the more stringent constraints. We consider both the stochastic and the adversarial settings, where we propose, natural, yet novel strategies and analyze the price for maintaining the constraints. Amongst other things, we prove both high probability and expectation bounds on the regret, while we also consider both the problem of maintaining the constraints with high probability or expectation. For the adversarial setting the price of maintaining the constraint appears to be higher, at least for the algorithm considered. A lower bound is given showing that the algorithm for the stochastic setting is almost optimal. Empirical results obtained in synthetic environments complement our theoretical findings.

1. Introduction

The paper formalizes bandit exploration under a return constraint that must hold uniformly over time, addressing the tension between learning better strategies and protecting ongoing revenue. It proposes conservative algorithms and contrasts this requirement with prior approaches that do not guarantee the constraint throughout learning.

1. Introduction

  • The core challenge is exploring for higher revenue without losing more than 10% of current revenue at any point during day-to-day operations.The same tension arises in applications such as dialogue systems, educational games, and factory production, where experimentation with weaker strategies may be considered too risky.Source: §1 · p. 1
  • A build-budget-then-learn strategy can guarantee the constraint, but Lattimore’s studied approach cannot guarantee it uniformly in time, making it unsuitable for this requirement.The paper notes that modifying that approach to enforce the stronger requirement may substantially increase worst-case regret.Source: §1 · p. 2
  • Conservative UCB satisfies the uniform return constraint while paying an additive regret penalty, unlike BudgetFirst’s multiplicative penalty; a lower bound makes the additive penalty unavoidable in the stochastic setting.The paper also considers high-probability and expectation guarantees, unknown default-arm payoff, and an adversarial algorithm whose additive penalty is higher than in the stochastic setting.Source: §1 · p. 2

1.1. Previous Work

  • Earlier full-information work found that constant regret to a default action can be maintained with essentially no increase in regret to the best action, while bandit exploration makes the constraint more costly.The paper situates its question within work on regret to a default action and notes the more difficult bandit setting.Source: §1.1 · p. 2
  • Prior bandit algorithms can achieve strong regret guarantees but do not maintain the return constraint uniformly in time, and they rely heavily on knowing the default action’s payoff.The cited stochastic and adversarial alternatives also fail to maintain the constraint with high probability.Source: §1.1 · p. 3
  • The paper positions its work as a theoretical approach to safe exploration, which prior reinforcement-learning literature had not addressed by maintaining return constraints during learning.It describes its bandit setting as simpler and suggests contextual bandits and reinforcement learning as directions for future work.Source: §1 · p. 2, §1.1 · p. 3

2. Conservative Multi-Armed Bandits

This section defines the bandit setting and the conservative return requirement. It frames the objective as minimizing regret while keeping cumulative reward above a fraction of the default arm’s return at every time step.

2. Conservative Multi-Armed Bandits

  • A multi-armed bandit is a sequential task in which an agent chooses an arm and observes its reward, but not the rewards of unchosen arms.The setup uses K + 1 arms, with arm 0 designated as the default action.Source: §2 · p. 3
  • Regret measures the reward lost relative to always choosing the best arm in hindsight; learning is sublinear when regret grows slower than linearly.The section gives O(√n) and O(log n) as examples of regret growth for good agents.Source: §2 · p. 3

2.1. Conservative Exploration

  • The learner must earn at least a 1 − α fraction of the reward from the default arm, with smaller α imposing a more conservative constraint.For the introductory example, α = 0.1 corresponds to allowing at most a 10% revenue loss relative to the default website.Source: §2.1 · p. 3
  • The budget Z_n tracks proximity to constraint violation, and the constraint holds only if Z_t ≥ 0 at every time t.The requirement is uniform over time, not just at the end of the horizon.Source: §2.1 · p. 3
  • The objective is to minimize regret while satisfying the constraint; the paper treats stochastic and adversarial settings and finds the stochastic regret guarantee cannot improve even for an end-only constraint.The lower-bound result concerns Z_n ≥ 0, whereas the algorithms are designed for the anytime constraint Z_t ≥ 0.Source: §2.1 · p. 3

3. The Stochastic Setting

In the stochastic setting, Conservative UCB maintains a uniform return constraint by exploring only when a confidence-based budget check says it is safe. Its constraint cost is an additive regret penalty, and the same order is achievable in expectation and when the default arm’s mean is unknown.

3.1. The Budget Constraint

  • The stochastic constraint requires cumulative expected reward to remain at least (1 − α)µ0t at every time, with default-arm plays increasing the budget and some other arms potentially reducing it.The guarantees use high probability 1 − δ; the reward model assumes independent 1-subgaussian noise.Source: §3 · p. 3, §3.1 · p. 4

3.2. BudgetFirst — A Naive Algorithm

  • BudgetFirst builds a budget by playing the default arm for an initial period before switching to high-probability UCB, but its constraint cost is significantly worse than the additive penalty of Conservative UCB.The initial default-only period scales as ˜O(Rworst/αµ0).Source: §3.2 · p. 4

3.3. Conservative UCB

  • The algorithm follows UCB when a lower confidence bound on the post-play budget is nonnegative, and otherwise plays the default arm to replenish the budget.The safety check uses confidence intervals for arm means and the proposed UCB arm.Source: §3.3 · p. 5
  • In a stochastic environment with arm means μ_i ∈ [0,1] and 1-subgaussian noise, Conservative UCB, with probability at least 1−δ, ensures μ_Is ≥ (1−α)μ_0t for all t=1,…,n and pays additive regret of order O(KL/(αμ_0)).Source: §3.3 · pp. 5–6

3.4. Considering the Expected Regret and Budget

  • For α ≥ c/n with c > 1, running Algorithm 1 with δ = 1/n and α′ = (α−δ)/(1−δ) gives the expected constraint and regret bound of the same order as Theorem 2.The expected constraint is E[∑ Is] ≥ (1 − α)µ0t for all t.Source: §3.4 · p. 6

3.5. Learning an Unknown µ0

  • When µ0 is unknown, a modified Algorithm 1 still satisfies the constraint with probability 1 − δ for all horizons and achieves the same order of regret.The modification estimates the default arm using confidence intervals and adjusts the lower bound on the budget.Source: §3.5 · pp. 6–7

4. The Adversarial Setting

The adversarial setting places no assumptions on how non-default rewards are generated, while the default arm has a fixed reward. A safe-playing adaptation preserves the anytime return constraint with a regret guarantee, but its penalty is higher than in the stochastic setting and the reason remains unresolved.

4. The Adversarial Setting

  • The safe-playing strategy uses the adversarial algorithm when the budget is safe and otherwise plays the default arm.The default arm has fixed reward µ0, and the return constraint must hold at every round.Source: §4 · p. 7
  • The guarantee requires an admissible any-time high-probability adversarial algorithm; Neu’s any-time algorithm is given as an achievable example.The paper states that this performance requirement is stronger than the typical high-probability bound.Source: §4 · p. 7
  • With probability at least 1−δ, the safe-playing adaptation satisfies constraint (2) and bounds regret by t0 + Rhat_n^δ.Here t0 is the largest t for which αµ0t ≤ Rhat_t^δ.Source: §4 · p. 7
  • The adversarial strategy pays a higher regret penalty than in the stochastic setting, and whether this reflects the algorithm, analysis, or intrinsic difficulty remains open.The learner faces worst-case reward sequences but may randomize its arm choice.Source: §4 · p. 7

5. Lower Bound on the Regret

The stochastic-setting lower bound establishes that Conservative UCB is near-optimal in the worst case up to a logarithmic factor. It also shows that the regret guarantee cannot be improved merely by requiring the return constraint at the last round.

5. Lower Bound on the Regret

  • The lower-bound analysis assumes normally distributed unit-variance rewards; other subgaussian distributions work when their parameter is fixed independently of the means.The means of the non-default arms are free to vary, with α, µ0, and n fixed.Source: §5 · p. 7
  • Conservative UCB is near-optimal in worst-case stochastic regret, up to a logarithmic factor L.The lower-bound result uses an expected total-return constraint at the last round.Source: §5 · p. 8
  • The regret guarantee cannot be improved by requiring the return constraint only at the last round, although anytime enforcement can prevent early exploration.Unbalanced MOSS performs well in expected regret but does not satisfy the constraint in early rounds.Source: §5 · p. 8
  • The lower bound is stronger than Lattimore (2015a)’s up to constant factors because it applies when the default-arm mean may be known.The cited result in Lattimore (2015a) assumes µ0 is unknown.Source: §5 · p. 8

6. Experiments

Simulations compare Conservative UCB with UCB, BudgetFirst, and Unbalanced MOSS across two regimes. They show how regret changes with constraint severity and horizon, while also comparing Conservative UCB variants that know or must learn the default-arm mean.

6. Experiments

  • As α approaches zero, Conservative UCB’s average regret degrades relative to UCB; with α=0.1, it eventually becomes nearly as good as UCB as n grows.These are the two reported regimes: fixed horizon with varying α, and varying n with α=0.1.Source: §6 · p. 8, Figure 2 · p. 8, Figure 3 · p. 8
  • The Conservative UCB variant that does not know µ0 performs relatively close to the version that knows it, though the known-µ0 version also performs better when α=1.The reported comparison also notes that µ0 knowledge is useful in the unconstrained setting.Source: §6 · p. 8
  • BudgetFirst is so conservative that it must first build a significant budget even when α is nearly zero; Unbalanced MOSS can explore early but lacks uniform-in-time constraint guarantees.Unbalanced MOSS also requires µ0 and other inputs in advance.Source: §6 · p. 8

7. Conclusion

The conclusion presents conservative exploration as a new family of multi-armed bandit frameworks and identifies broader settings as possible extensions.

7. Conclusion

  • The paper introduces a family of multi-armed bandit frameworks for exploring while maintaining revenue constraints.The conclusion presents this as the beginning of a broader research direction.Source: §7 · p. 8
  • The authors expect similar strategies may generalize to contextual bandits and reinforcement learning.This is framed as a possibility, not a demonstrated result.Source: §7 · p. 8
  • The conclusion calls for improving current results, closing open problems, and broadening the model’s applications.Source: §7 · p. 8

A. Proof of Theorem 2

Theorem 2 establishes that Conservative UCB meets the return constraint at every time horizon with high probability in the stated stochastic setting. Its proof uses valid confidence intervals to verify safety and then bounds regret by analyzing pulls of non-default and default arms.

A. Proof of Theorem 2

  • With probability at least 1 −δ, the return constraint µIs ≥ (1 −α)µ0t holds for every t from 1 to n.The theorem assumes expected rewards in [0, 1] and 1-subgaussian noise; its confidence intervals are valid across rounds and arms on an event of probability at least 1 −δ.Source: A. Proof of Theorem 2 · p. 10
  • The proof verifies safety directly from the confidence intervals and Algorithm 1’s construction.It conditions on the event that the intervals are valid, which has probability at least 1 −δ.Source: A. Proof of Theorem 2 · p. 10
  • The proof bounds suboptimal-arm selection using valid confidence intervals, then combines the resulting regret bounds for non-default and default arms.For the default arm, it uses max{∆i, ∆0 −∆i} ≥ ∆0/2 to bound its contribution.Source: A. Proof of Theorem 2 · pp. 10–11

B. Proof of Theorem 5

Theorem 5 extends the stated constraint and regret guarantee to a modification of Algorithm 1 that does not know µ0. Its proof uses confidence intervals that include the default arm and bounds how often that arm is played.

B. Proof of Theorem 5

  • The modified Algorithm 1 satisfies the return constraint and regret bound with probability at least 1 −δ for every time horizon, without knowing µ0.The theorem otherwise uses the same conditions as Theorem 2.Source: B. Proof of Theorem 5 · p. 11
  • The proof assumes confidence intervals are valid across rounds and arms, including the default arm, on an event of probability at least 1 −δ.It then checks that the modified algorithm satisfies the return constraint.Source: B. Proof of Theorem 5 · p. 11
  • The regret analysis bounds pulls of suboptimal arms and separately controls default-arm pulls using the last round in which the default arm is played.The proof handles the case where the default arm is selected because the modified budget condition calls for it.Source: B. Proof of Theorem 5 · pp. 11–12

C. Proof of Theorem 7

Theorem 7 adapts an admissible bandit algorithm with a safe-playing strategy. The strategy always satisfies the return constraint, while the regret guarantee adds a bound on the duration of safe play to the base algorithm’s regret.

C. Proof of Theorem 7

  • With probability at least 1−δ, the adapted algorithm satisfies the constraint and has regret Rn ≤ t0 + ˆRδn.Here t0 = max{t|αµ0t ≤ ˆRδ}; the bound applies to a ˆRδt-admissible algorithm.Source: C. Proof of Theorem 7 · p. 12
  • The safe-playing strategy always satisfies the return constraint by playing safe when a lower bound on the budget is negative.The proof bounds the last safe-play round by t0, yielding the stated additive regret term.Source: C. Proof of Theorem 7 · p. 12

D. Proof of Theorem 9

Theorem 9 proves a lower bound under an expected return constraint by considering multiple reward environments. It concludes that some environment must incur expected regret at least B, so no algorithm can have worst-case regret below B.

D. Proof of Theorem 9

  • Under the expected return constraint, some environment has expected regret at least B, ruling out any algorithm with worst-case regret smaller than B.The proof contrasts an environment where all non-default arms have reward µ0 −∆ with alternatives where one arm has reward µ0 +∆.Source: D. Proof of Theorem 9 · pp. 12–13
  • The lower-bound argument uses a change-of-measure comparison between reward distributions that differ on one arm.The proof defines the corresponding KL divergence and applies a change-of-measure argument.Source: D. Proof of Theorem 9 · p. 13
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