Multiplier Bootstrap-based Exploration

The bandit problem has found wide applications in various areas such as clinical trials (Durand et al., 2018), finance (Shen et al., 2015), recommendation systems (Zhou et al., 2017), among others. Accurate uncertainty quantification is the key to address the exploration-exploitation trade-off. Most existing bandit algorithms critically rely on certain analytical property of the imposed model (e.g., linear bandits) to quantify the uncertainty and derive the exploration strategy. Thompson Sampling (TS, Thompson, 1933) and Upper Confidence Bound (UCB, Auer et al., 2002) are two prominent examples, which are typically based on explicit-form posterior distributions or confidence sets, respectively.

However, in many real problems, the reward model is fairly complex: e.g., a general graphical model (Chapelle & Zhang, 2009) or a pipeline with multiple prediction modules and manual rules. In these cases, it is typically impossible to quantify the uncertainty in an analytical way, and frameworks such as TS or UCB are either methodologically not applicable or computationally infeasible. Motivated by the real needs, we are concerned with the following question:

Can we design a practical bandit algorithm framework that is general, adaptive, and computationally tractable, with certain theoretical guarantee?

A straightforward idea is to apply the bootstrap method (Efron, 1992), a widely applicable data-driven approach for measuring uncertainty. However, as discussed in Section 2, most existing bootstrap-based bandit algorithms are either heuristic without a theoretical guarantee, computationally intensive, or only applicable in limited scenarios. To address these limitations, we propose a new exploration strategy based on multiplier bootstrap (Van Der Vaart & Wellner, 1996), an easy-to-adapt bootstrap framework that only requires randomly weighted data points. We further show that a naive application of multiplier bootstrap may result in linear regret, and we introduce a suitable way to add additional perturbations for sufficient exploration.

Contribution. Our contributions are three-fold. First, we propose a general-purpose bandit algorithm framework, Multiplier Bootstrap-based Exploration (MBE). The main advantage of MBE is that it is general: it is applicable to any reward model amenable to weighted loss minimization, without need of analytical-form uncertainty quantification or case-by-case algorithm design. As a data-driven exploration strategy, MBE is also adaptive to different environments.

Second, theoretically, we prove near-optimal regret bounds for MBE under sub-Gaussian multi-armed bandits (MAB), in both the instance-dependent and the instance-independent sense. Compared with all existing results for bootstrap-based bandit algorithms, our result is strictly more general (see Table 1), since existing results only apply to some special cases of sub-Gaussian distributions. To overcome the technical challenges, we proved a novel concentration inequality for some function of sub-exponential variables, and also developed the first finite-sample concentration and anti-concentration analysis for multiplier bootstrap, to the best of our knowledge. Given the broad applications of multiplier bootstrap in statistics and machine learning, we believe our theoretical analysis has separate interest.

Third, with extensive simulation and real data experiments, we demonstrate that MBE yields comparable performance with existing algorithms in different MAB settings and three real-world problems (online learning to rank, online combinatorial optimization, and dynamic slate optimization). This supports that MBE is easily generalizable, as it requires minimal modifications and derivations to match the performance of those near-optimal algorithms specifically designed for each problem. Moreover, we also show that MBE adapts to different environments and is relatively robust, due to its data-driven nature.

The most popular bandit algorithms, arguably, include $\epsilon$-greedy (Watkins, 1989), TS, and UCB. $\epsilon$-greedy is simple and thus widely used. However, its exploration strategy is not aware of the uncertainty in data and thus is known to be statistically sub-optimal. TS and UCB reply on posteriors and confidence sets, respectively. Yet, their closed forms only exist in limited cases, such as MAB or linear bandits. For a few other models (such as generalized linear model or neural nets), we know how to construct the approximate posteriors or confidence sets (Filippi et al., 2010; Li et al., 2017; Phan et al., 2019; Kveton et al., 2020b; Zhou et al., 2020), though the corresponding algorithms are usually costly or conservative. In more general cases, it is often not clear how to adapt UCB and TS in a valid and efficient way. Moreover, the dependency on the probabilistic model assumptions also pose challenges to being robust.

To enable wider applications of bandit algorithms, several bootstrap-based (and related perturbation-based) methods have been proposed in the literature. Most algorithms are TS-type, by replacing the posterior with a bootstrap distribution. We next review the related papers, and summarize those with near-optimal regret bounds in Table 1.

Arguably, the non-parametric bootstrap is the most well-known bootstrap method, which works by re-sampling data with replacement. Vaswani et al. (2018) proposes a version of non-parametric bootstrap with forced exploration to achieve a $\mathcal{O}(T^{2/3})$ regret bound in Bernoulli MAB. GIRO proposed in Kveton et al. (2019c) successfully achieves a rate-optimal regret bound in Bernoulli MAB, by adding Bernoulli perturbations to non-parametric bootstrap. However, due to the re-sampling nature of non-parametric bootstrap, it is hard to be updated efficiently other than in Bernoulli MAB (see Section 4.3). Specifically, the computational cost of re-sampling scales quadratically in $T$.

Another line of research is the residual bootstrap-based approach (ReBoot) (Hao et al., 2019; Wang et al., 2020; Tang et al., 2021; Wu et al., 2022). For each arm, ReBoot randomly perturbs the residuals of the corresponding observed rewards with respect to the estimated model to quantify the uncertainty for its mean reward. We note that, although these methods also use random weights, they are applied to residuals and hence are in a way fundamentally different from ours. The limitation is that, by design, this approach is only applicable to problems with a fixed and finite set of arms, since the residuals are attached closely to each arm (see Appendix LABEL:sec:ReBoot for more details).

The perturbed history exploration (PHE) algorithm (Kveton et al., 2019a, b, 2020a) is also related. PHE works by adding additive noise to the observed rewards. Osband et al. (2019) applies similar ideas to reinforcement learning. However, PHE has two main limitations. First, for models where adding additive noise is not feasible (e.g., decision trees), PHE is not applicable. Second, as demonstrated in both (Wang et al., 2020) and our experiments, the fact that PHE relies on only the extrinsically injected noise for exploration makes it less robust. For a complex structured problem, it may not be clear how to add the noise in a sound way (Wang et al., 2020). In contrast, it is typically more natural (and hence easier to be accepted) to leverage the intrinsic randomness in the observed data.

Finally, we note that multiplier bootstrap has been considered in the bandit literature, mostly as a computationally efficient approximation to non-parametric bootstrap studied in those papers. Eckles & Kaptein (2014) studies the direct adaption of multiplier bootstrap (see Section 4.1) in simulation, and its empirical performance in contextual bandits is studied later (Tang et al., 2015; Elmachtoub et al., 2017; Riquelme et al., 2018; Bietti et al., 2021). However, no theoretical guarantee is provided in these works. In fact, as demonstrated in Section 4.1, such a naive adaptation may have a linear regret. Osband & Van Roy (2015) shows that, in Bernoulli MAB, a variant of multiplier bootstrap is mathematically equivalent to TS. No further theoretical or numerical results are established except for this special case. Our work is the first systematic study of multiplier bootstrap in bandits. Our unique contributions include: we identify the potential failure of naively applying multiplier bootstrap, highlight the importance of additional perturbations, design a general algorithm framework to make this heuristic idea concrete, provide the first theoretical guarantee in general MAB settings, and conduct extensive numerical experiments to study its generality and adaptivity.

Setup. We consider a general stochastic bandit problem. For any positive integer $M$, let $[M]=\{1,\dots,M\}$. At each round $t\in[T]$, the agent observes a context vector ${\bm{x}}_{t}$ (it is empty in non-contextual problems) and an action set $\mathcal{A}_{t}$, then chooses an action $A_{t}\in\mathcal{A}_{t}$, and finally receives the corresponding reward $R_{t}=f({\bm{x}}_{t},A_{t})+\epsilon_{t}$, Here, $f$ is an unknown function and $\epsilon_{t}$ is the noise term. Without loss of generality, we assume $f({\bm{x}}_{t},A_{t})\in[0,1]$. The goal is to minimize the cumulative regret

At time $t$, with an existing dataset $\mathcal{D}_{t}=\{({\bm{x}}_{l},A_{l},R_{l})\}_{l\in[t]}$, to decide the action $A_{t+1}$, most algorithms typically first estimate $f$ in some function class $\mathcal{F}$ by solving a weighted loss minimization problem (also called weighted empirical risk minimization or cost-sensitive training)

Here, $\mathcal{L}$ is a loss function (e.g., $\ell_{2}$ loss or negative log-likelihood), $\omega_{l}$ is the weight of the $l$th data point, and $J$ is an optional penalty function. We consider the weighted problem as it is general and related to our proposal below. One can just set $\omega_{l}\equiv 1$ to get the unweighted problem. As the simplest example, consider the $K$-armed bandit problem where ${\bm{x}}_{l}$ is empty and $\mathcal{A}_{l}=[K]$. Let $\mathcal{L}$ be the $\ell_{2}$ loss, $J\equiv 0$, and $f({\bm{x}}_{l},A_{l})\equiv r_{A_{l}}$ where $r_{k}$ is the mean reward of the $k$-th arm. Then, (1) reduces to $\operatorname*{arg\,min}_{\{r_{1},\dots,r_{K}\}}\sum_{l=1}^{t}\omega_{l}(R_{l}-r_{A_{l}})^{2}$, which gives the estimator $\widehat{r}_{k}=(\sum_{l:A_{l}=k}\omega_{l})^{-1}\sum_{l:A_{l}=k}\omega_{l}R_{l}$, i.e., the arm-wise weighted average. Similarly, in linear bandits, (1) reduces to the weighted least-square problem (see Appendix LABEL:sec:MBTS_LB for details).

Challenges. The estimation of $f$, together with the related uncertainty quantification, forms the foundation of most bandit algorithms. In the literature, $\mathcal{F}$ is typically a class of models that permit closed-form uncertainty quantification (e.g., linear models, Gaussian processes, etc.). However, in many real applications, the reward model can yield a fairly complicated structure, e.g., a hierarchical pipeline with both classification and regression modules. Manually specified rules are also commonly part of the model. It is challenging to quantify the uncertainty of these complicated models in analytical forms. Even when feasible, the dependency on the probabilistic model assumptions also pose challenges to being robust.

Therefore, in this paper, we focus on the bootstrap-based approach due to its generality and data-driven nature. Bootstrapping, as a general approach to quantify the model uncertainty, has many variants. The most popular one, arguably, is non-parametric bootstrap (used in GIRO), which constructs bootstrap samples by re-sampling the dataset with replacement. However, due to the re-sampling nature, it is computationally intense (see Section 4.3 for more discussions). In contrast, multiplier bootstrap (Van Der Vaart & Wellner, 1996), as an efficient and easy-to-implement alternative, is popular in statistics and machine learning.

Multiplier bootstrap. The main idea of multiplier bootstrap is to learn the model using randomly weighted data points. Specifically, given a multiplier weight distribution $\rho(\omega)$, for every bootstrap sample, we first randomly sample $\{\omega_{t}^{MB}\}_{t=1}^{t^{\prime}}\sim\rho(\omega)$, and then solve (1) with $\omega_{t}=\omega_{t}^{MB}$ to obtain $\widehat{f}^{MB}$. Repeat the procedure and the distribution of $\widehat{f}^{MB}$ forms the bootstrap distribution that quantifies our uncertainty over $f$. The popular choices of $\rho(\omega)$ include $\mathcal{N}(1,\sigma_{\omega}^{2})$, $\text{Exp}(1)$, $\text{Poisson}(1)$, and the double-or-nothing distribution $2\times\text{Bernoulli}(0.5)$.

In this section, we provide the regret bound for Algorithm 1 under MAB with sub-Gaussian rewards. We regard this as the first step towards the theoretical understanding of MBE, and leave the analysis of more general settings to future research. We call a random variable $X$ as $\sigma$-sub-Gaussian if $\mathbb{E}\exp\{t(X-\mathbb{E}X)\}\leq\exp\{{t^{2}\sigma^{2}}/{2}\}$ for any $t\in\mathbb{R}$.

The two regret bounds are known as near-optimal (up to a logarithm term) in both the problem-dependent and problem-independent sense (Lattimore & Szepesvári, 2020). Notably, recall that the Gaussian distribution and all bounded distributions belong to the sub-Gaussian class. Therefore, as reviewed in Table 1, our theory is strictly more general than all existing results for bootstrap-based MAB algorithms.

Technical challenges. It is particularly challenging to analyze MBE due to two reasons. First, the probabilistic analysis of multiplier bootstrap itself is technically challenging, since the same random weights appear in both the denominator and the numerator (recall MBE uses the weighted averages $\big{\{}\overline{Y}_{k}=\sum_{\ell:A_{\ell}=k}(\omega_{\ell}R_{k,\ell}+\lambda\omega_{\ell}^{\prime})/\sum_{\ell:A_{\ell}=k}(\omega_{\ell}+\lambda\omega_{\ell}^{\prime}+\lambda\omega_{\ell}^{\prime\prime})\big{\}}_{k=1}^{K}$ to select actions in MAB). It is notoriously complicated to analyze the ratio of random variables, especially when they are correlated. Besides, existing bootstrap-based papers rely on the properties of specific parametric reward classes (e.g., Bernoulli in Kveton et al. (2019c) and Gaussian  in  Wang et al. (2020)), while we lose these nice structures when considering sub-Gaussian rewards.

To overcome these challenges, we start with carefully defining two good events on which the weighted average $\overline{Y}_{k}$, the non-weighted average (with pseudo-rewards)  $\overline{R}_{k}^{*}=\big{(}{\sum_{\ell:A_{\ell}=k}(R_{k,\ell}+1\times\lambda+0\times\lambda)}\big{)}/\big{(}{\sum_{\ell:A_{\ell}=k}(1+\lambda+\lambda)}\big{)}$, and the shifted asymptotic mean $(\mu_{k}+\lambda)/({1+2\lambda})$ are close to each other (see Appendix LABEL:proof_thm). To bound the probability of the bad event and to control the regret on the bad event, we face two major technical challenges. First, when transforming the ratio into an analyzable form, a summation of correlated sub-Gaussian and sub-exponential variables appears and is hard to analyze. We carefully design and analyze a novel event to remove the correlation and the sub-Gaussian terms (see proof of Lemma LABEL:lem_bounding_a_k_s_2). Second, the proof needs a new concentration inequality for functions of sub-exponential variables that do not exist in the literature. We obtain such a new concentration inequality (Lemma LABEL:lem_subE_ineq) via careful analysis of sub-exponential distributions.

We believe our new concentration inequality is of separate interest for the analysis of sub-exponential distributions. Moreover, to the best of our knowledge, our proof provides the first finite-sample concentration and anti-concentration analysis for multiplier bootstrap, which has broad applications in statistics and machine learning.

Tuning parameters. In Theorem 5.1, MBE has two tuning parameters $\lambda$ and $\sigma_{\omega}$. Intuitively, $\lambda$ controls the amount of external perturbation and $\sigma_{\omega}$ controls the magnitude of exploration from bootstrap. In general, higher values of these two parameters facilitate exploration but also lead to a slower convergence.  The condition $\lambda\geq\left(1+4/\sigma_{\omega}\right)+\sqrt{4\left(1+4/\sigma_{\omega}\right)/\sigma_{\omega}}$ requires that (i) $\lambda$ is not too small and (ii) the joint effect of $\lambda$ and $\sigma$ is not too small. Both are intuitive and reasonable. In practice, the theoretical condition could be loose: e.g., it requires $\lambda\geq 5+2\sqrt{5}$ when $\sigma_{\omega}=1$. As we observe in Section 6, MBE with a smaller $\lambda$ (e.g., $0.5$) still empirically performs well.

In this section, we study the empirical performance of MBE with both simulation (Section 6.1) and real datasets (Section 6.2).

In this paper, we propose a new bandit exploration strategy, Multiplier Bootstrap-based Exploration (MBE). The main advantage of MBE is its generality: for any reward model that can be estimated via weighted loss minimization, the idea of MBE is applicable and requires minimal efforts on derivation or implementation of the exploration mechanism. As a data-driven method, MBE also shows nice adaptivity. We prove near-optimal regret bounds for MBE in the sub-Gaussian MAB setup, which is more general compared with other bootstrap-based bandit papers. Numerical experiments demonstrate that MBE is general, efficient, and adaptive.

There are a few meaningful future extensions. First, the regret analysis for MBE (and more generally, other bootstrap-based bandit methods) in more complicated setups would be valuable. Second, adding pseudo-rewards at every round is needed for the analysis. We hypothesize that there exists a more adaptive way of adding them. Last, the practical implementation of MBE relies on an ensemble of models to approximate the bootstrap distribution and the online regression oracle to update the model estimation. Our numerical experiments show that such an approach works well empirically, but it would be still meaningful to have more theoretical understanding.