Batched Multi-armed Bandits Problem

The multi-armed bandits problem is an important class of sequential optimization problems which has been extensively studied in various fields such as statistics, operations research, engineering, computer science and economics over the recent years BCB (12). In the fully adaptive scenario, the regret analysis for stochastic bandits can be found in Vog (60); LR (85); BK (97); ACBF (02); AB (09); AMS (09); AB (10); AO (10); GC (11); BPR (13); PR (13).

There is less attention on the batched setting with limited rounds of interaction. The batched setting is studied in CBDS (13) under the name of switching costs, where it is shown that $O(\log\log T)$ batches are sufficient to achieve the optimal minimax regret. For small number of batches $M$, the batched two-armed bandit problem is studied in PRCS (16), where the results of Theorems 1 and 2 are obtained for $K=2$. However, the generalization to the multi-armed case is not straightforward, and more importantly, the practical scenario where the grid is adaptively chosen based on the historic data is excluded in PRCS (16). For the multi-armed case, a different problem of finding the best $k$ arms in the batched setting has been studied in JJNZ (16); AAAK (17), where the goal is pure exploration, and the error dependence on the time horizon decays super-polynomially. We also refer to DRY (18) for a similar setting with convex bandits and best arm identification. The regret analysis for batched stochastic multi-armed bandits still remains underexplored.

We also review some literature on general computation with limited rounds of adaptivity, and in particular, on the analysis of lower bounds. In theoretical computer science, this problem has been studied under the name of parallel algorithms for certain tasks (e.g., sorting and selection) given either deterministic Val (75); BT (83); AA (88) or noisy outcomes FRPU (94); DKMR (14); BMW (16). In (stochastic) convex optimization, the information-theoretic limits are typically derived under the oracle model where the oracle can be queried adaptively NY (83); AWBR (09); Sha (13); DRY (18). However, in the previous works, one usually optimizes the sampling distribution over a fixed sample size at each step, while it is more challenging to prove lower bounds for policies which can also determine the sample size. There is one exception AAAK (17), whose proof relies on a complicated decomposition of near-uniform distributions. Hence, our technique of proving Theorem 3 is also expected to be an addition to these literatures.

The rest of this paper is organized as follows. In Section 2, we introduce the BaSE policy for general batched multi-armed bandit problems, and show that it attains the upper bounds in Theorem 1 under two specific grids. Section 3 presents the proofs of lower bounds for both the minimax and problem-dependent regrets, where Section 3.1 deals with the static grids and Section 3.2 tackles the adaptive grids. Experimental results are presented in Section 4. The auxiliary lemmas and the proof of main lemmas are deferred to supplementary materials.

For a positive integer $n$, let $[n]\triangleq\{1,\cdots,n\}$. For any finite set $A$, let $|A|$ be its cardinality. We adopt the standard asymptotic notations: for two non-negative sequences $\{a_{n}\}$ and $\{b_{n}\}$, let $a_{n}=O(b_{n})$ iff $\limsup_{n\to\infty}a_{n}/b_{n}<\infty$, $a_{n}=\Omega(b_{n})$ iff $b_{n}=O(a_{n})$, and $a_{n}=\Theta(b_{n})$ iff $a_{n}=O(b_{n})$ and $b_{n}=O(a_{n})$. For probability measures $P$ and $Q$, let $P\otimes Q$ be the product measure with marginals $P$ and $Q$. If measures $P$ and $Q$ are defined on the same probability space, we denote by $\mathsf{TV}(P,Q)=\frac{1}{2}\int|dP-dQ|$ and $D_{\text{KL}}(P\|Q)=\int dP\log\frac{dP}{dQ}$ the total variation distance and Kullback–Leibler (KL) divergences between $P$ and $Q$, respectively.

The policy that achieves the optimal regrets is essentially adapted from Successive Elimination (SE). The original version of SE was introduced in EDMM (06), and PR (13) shows that in the $M=T$ case SE achieves both the optimal minimax and problem-dependent rates. Here we introduce a batched version of SE called Batched Successive Elimination (BaSE) to handle the general case $M\leq T$.

Given a pre-specified grid ${\mathcal{T}}=\{t_{1},\cdots,t_{M}\}$, the idea of the BaSE policy is simply to explore in the first $M-1$ batches and then commit to the best arm in the last batch. At the end of each exploration batch, we remove arms which are probably bad based on past observations. Specfically, let ${\mathcal{A}}\subseteq{\mathcal{I}}$ denote the active arms that are candidates for the optimal arm, where we initialize ${\mathcal{A}}={\mathcal{I}}$ and sequentially drop the arms which are “significantly” worse than the “best” one. For the first $M-1$ batches, we pull all active arms for a same number of times (neglecting rounding issues¹¹1There might be some rounding issues here, and some arms may be pulled once more than others. In this case, the additional pull will not be counted towards the computation of the average reward $\bar{Y}^{i}(t)$, which ensures that all active arms are evaluated using the same number of pulls at the end of any batch. Note that in this way, the number of pulls for each arm is underestimated by at most half, therefore the regret analysis in Theorem 4 will give the same rate in the presence of rounding issues.) and eliminate some arms from ${\mathcal{A}}$ at the end of each batch. For the last batch, we commit to the arm in ${\mathcal{A}}$ with maximum average reward.

Before stating the exact algorithm, we introduce some notations. Let

denote the average rewards of the arm $i$ up to time $t$, and $\gamma>0$ is a tuning parameter associated with the UCB bound. The algorithm is described in detail in Algorithm 1.

Note that the BaSE algorithm is not fully specified unless the grid ${\mathcal{T}}$ is determined. Here we provide two choices of static grids which are similar to PRCS (16) as follows: let

where parameters $a,b$ are chosen appropriately such that $t_{M}=t_{M}^{\prime}=T$, i.e.,

For minimizing the minimax regret, we use the “minimax” grid defined by ${\mathcal{T}}_{\mathrm{minimax}}=\{t_{1},\cdots,t_{M}\}$; as for the problem-dependent regret, we use the “geometric” grid which is defined by ${\mathcal{T}}_{\mathrm{geometric}}=\{t_{1}^{\prime},\cdots,t_{M}^{\prime}\}$. We will denote by $\pi_{\text{BaSE}}^{1}$ and $\pi_{\text{BaSE}}^{2}$ the respective policies under these grids.

The performance of the BaSE policy is summarized in the following theorem.

Note that Theorem 4 implies Theorem 1. In the sequel we sketch the proof of Theorem 4, where the main technical difficulity is to appropriately control the number of pulls for each arm under batch constraints, where there is a random number of active arms in ${\mathcal{A}}$ starting from the second batch. We also refer to a recent work EKMM (19) for a tighter bound on the problem-dependent regret with an adaptive grid.

The proof of Theorem 2 relies on the following lemma.

We first show that Lemma 2 implies Theorem 2 by choosing the smallest gap $\Delta>0$ appropriately. By definitions of the minimax regret $R_{\text{min-max}}^{\star}$ and the problem-dependent regret $R_{\text{pro-dep}}^{\star}$, choosing $\Delta=\Delta_{j}=\sqrt{(K-1)/(t_{j-1}+1)}\in[0,\sqrt{K}]$ in Lemma 2 yields that

for some numerical constant $c_{0}>0$. Since $t_{0}=0,t_{M}=T$, the lower bounds in Theorem 2 follow.

Next we employ the general idea of the multiple hypothesis testing to prove Lemma 2. Consider the following $K$ candidate reward distributions:

We remark that this construction is not entirely symmetric, where the reward distribution of the first arm is always ${\mathcal{N}}(\Delta,1)$. The key properties of this construction are as follows:

1. 1.

   For any $i\in[K]$, arm $i$ is the optimal arm under reward distribution $P_{i}$;

2. 2.

   For any $i\in[K]$, pulling a wrong arm incurs a regret at least $\Delta$ under reward distribution $P_{i}$.

As a result, since the average regret serves as a lower bound of the worst-case regret, we have

where $P_{i}^{t}$ denotes the distribution of observations available at time $t$ under $P_{i}$, and $R^{t}(\pi)$ denotes the instantaneous regret incurred by the policy $\pi_{t}$ at time $t$. Hence, it remains to lower bound the quantity $\frac{1}{K}\sum_{i=1}^{K}P_{i}^{t}(\pi_{t}\neq i)$ for any $t\in[T]$, which is the subject of the following lemma.

The proof of Lemma 3 is deferred to the supplementary materials, and we make some remarks below.

To lower bound (9), we apply Lemma 3 with the star tree $T=([n],\{(1,i):2\leq i\leq n\})$. For $i\in[K]$, denote by $T_{i}(t)$ the number of pulls of arm $i$ anterior to the current batch of $t$. Hence, $\sum_{i=1}^{K}T_{i}(t)=t_{j-1}$ if $t\in(t_{j-1},t_{j}]$. Moreover, since $D_{\text{KL}}(P_{1}^{t}\|P_{i}^{t})=2\Delta^{2}\mathbb{E}_{P_{1}^{t}}T_{i}(t)$, we have

Now combining (9) and (10) completes the proof of Lemma 2.

Now we investigate the case where the grid may be randomized, and be generated sequentially in an adaptive manner. Recall that in the previous section, we construct multiple fixed hypotheses and show that no policy under a static grid can achieve a uniformly small regret under all hypotheses. However, this argument breaks down even if the grid is only randomized but not adaptive, due to the non-convex (in $(t_{1},\cdots,t_{M})$) nature of the lower bound in Lemma 2. In other words, we might not hope for a single fixed multiple hypothesis testing problem to work for all policies. To overcome this difficulty, a subroutine in the proof of Theorem 3 is to construct appropriate hypotheses after the policy is given (cf. the proof of Lemma 4). We sketch the proof below.

We shall only prove the lower bound for the minimax regret, where the analysis of the problem-dependent regret is entirely analogous. Consider the following time $T_{1},\cdots,T_{M}\in[1,T]$ and gaps $\Delta_{1},\cdots,\Delta_{M}\in(0,\sqrt{K}]$ with

Let ${\mathcal{T}}=\{t_{1},\cdots,t_{M}\}$ be any adaptive grid, and $\pi$ be any policy under the grid ${\mathcal{T}}$. For each $j\in[M]$, we define the event $A_{j}=\{t_{j-1}<T_{j-1},t_{j}\geq T_{j}\}$ under policy $\pi$ with the convention that $t_{0}=0,t_{M}=T$. Note that the events $A_{1},\cdots,A_{M}$ form a partition of the entire probability space. We also define the following family of reward distributions: for $j\in[M-1],k\in[K-1]$ let

where the $k$-th component of $P_{j,k}$ has a non-zero mean. For $j=M$, we define

Note that this construction ensures that $P_{j,k}$ and $P_{M}$ only differs in the $k$-th component, which is crucial for the indistinguishability results in Lemma 5.

We will be interested in the following quantities:

where $P_{j,k}(A)$ denotes the probability of the event $A$ given the true reward distribution $P_{j,k}$ and the policy $\pi$. The importance of these quantities lies in the following lemmas.

The detailed proofs of Lemma 4 and Lemma 5 are deferred to the supplementary materials, and we only sketch the ideas here. Lemma 4 states that, if any of the events $A_{j}$ occurs with a non-small probability in the respective $j$-th world (i.e., under the mixture of $(P_{j,k}:k\in[K-1])$ or $P_{M}$), then the policy $\pi$ has a large regret in the worst case. The intuition behind Lemma 4 is that, if the event $t_{j-1}\leq T_{j-1}$ occurs under the reward distribution $P_{j,k}$, then the observations in the first $(j-1)$ batches are not sufficient to distinguish $P_{j,k}$ from its (carefully designed) perturbed version with size of perturbation $\Delta_{j}$. Furthermore, if in addition $t_{j}\geq T_{j}$ holds, then the total regret is at least $\Omega(T_{j}\Delta_{j})$ due to the indistinguishability of the $\Delta_{j}$ perturbations in the first $j$ batches. Hence, if $A_{j}$ occurs with a fairly large probability, the resulting total regret will be large as well.

Lemma 5 complements Lemma 4 by stating that at least one $p_{j}$ should be large. Note that if all $p_{j}$ were defined in the same world, the partition structure of $A_{1},\cdots,A_{M}$ would imply $\sum_{j\in[M]}p_{j}\geq 1$. Since the occurrence of $A_{j}$ cannot really help to distinguish the $j$-th world with later ones, Lemma 5 shows that we may still operate in the same world and arrive at a slightly smaller constant than $1$.

Finally we show how Lemma 4 and Lemma 5 imply Theorem 3. In fact, by Lemma 5, there exists some $j\in[M]$ such that $p_{j}\geq(2M)^{-1}$. Then by Lemma 4 and the arbitrariness of $\pi$, we arrive at the desired lower bound in Theorem 3.