Nearly Dimension-Independent Sparse Linear Bandit over Small Action Spaces via Best Subset Selection

We briefly review existing works on linear stochastic bandit problems under both low-dimensional and high-dimensional settings. Under the classical low-dimensional setting, Auer (2002) pioneers the use of upper-confidence-bound (UCB) type algorithms for the linear stochastic bandit, which is one of the most powerful and fundamental algorithms for this class of problems, and is also considered in Chu et al. (2011). Dani et al. (2008); Rusmevichientong and Tsitsiklis (2010) study linear stochastic bandit problems with large or infinite action spaces, and derive corresponding lower bounds. Under the high-dimensional setting, where we assume that $\theta^{*}$ is sparse, when the action spaces $\mathcal{A}_{t}$’s are hyper-cube spaces $[-1,1]^{d}$, Lattimore et al. (2015) develop the SETC algorithm that attains nearly dimension-independent regret bounds. We point out that this algorithm exploits the unique structure of hyper-cubes and is unlikely to be applicable for general action spaces including the ones of our interest where the action spaces are finite. Abbasi-Yadkori et al. (2012) consider a UCB-type algorithm for general action sets and obtain a regret upper bound of $\widetilde{\mathcal{O}}(\sqrt{sdT})$, which depends polynomially on the ambient dimension $d$. Note that here the $\widetilde{\mathcal{O}}(\cdot)$ notation is the big-O notation that ignores all logarithmic factors. Carpentier and Munos (2012) consider a different reward model and obtain an $\widetilde{\mathcal{O}}(\|\theta^{*}\|_{2}^{2}s\sqrt{T})$ regret upper bound. Goldenshluger and Zeevi (2013); Bastani and Bayati (2015) study a variant of the linear stochastic bandit problem in which only one contextual covariate $X_{t}$ is observed at each time period $t$, while each action $i_{t}$ corresponds to a different unknown model $\theta_{i_{t}}^{*}$. We point out that this model is a special case of our model (1), as discussed in Foster et al. (2018).

Another closely related problem is the online sparse prediction problem (Gerchinovitz, 2013; Foster et al., 2016), in which sequential predictions $\widehat{Y}_{t}$’s of $Y_{t}=\langle X_{t},\theta^{*}\rangle+\varepsilon_{t}$ are of the interest, and the regret is measured in mean-square error $\sum_{t}|\widehat{Y}_{t}-Y_{t}|^{2}$. It can be further generalized to online empirical-risk minimization (Langford et al., 2009) or even the more general derivative-free/bandit convex optimization (Nemirovsky and Yudin, 1983; Flaxman et al., 2005; Agarwal et al., 2010; Shamir, 2013; Besbes et al., 2015; Bubeck et al., 2017; Wang et al., 2017). Most existing works along this direction have continuous (infinite) action spaces $\{\mathcal{A}_{t}\}$. They allow small-perturbation type methods like estimating gradient descent. We summarize the existing theoretical results on stochastic linear bandit problems in Table 1.

From the application perspective of finding the optimal treatment regime, existing literatures focus on achieving the optimality through batch settings. General approaches include model-based methods such as Q-learning (Watkins and Dayan, 1992; Murphy, 2003; Moodie et al., 2007; Chakraborty et al., 2010; Goldberg and Kosorok, 2012; Song et al., 2015) and A-learning (Robins et al., 2000; Murphy, 2005) and model-free policy search methods (Robins et al., 2008; Orellana et al., 2010a, b; Zhang et al., 2012b; Zhao et al., 2012, 2014). These methods are all developed based on batch settings where we use the whole dataset to estimate the optimal treatment regime. This approach is applicable after the clinical trial is completed, or when the observational dataset is fully available. However, the batch setting approach is not applicable when it is emerging to identify the optimal treatment regime. For a contemporary example, during the recent outbreak of coronavirus disease (COVID-19), it is extremely important to quickly identify the optimal or nearly optimal treatment regime to assign each patient to the best treatment among a few choices. However, since the disease is novel, there is no or very little historical data. Thus, the batch setting approaches mentioned above are not applicable. On the other hand, our model naturally provides a “learn while optimizing” alternative approach to sequentially improve the policy/treatment regime. This provides an important motivating application of the proposed method.

In this paper, we propose new algorithms, which iteratively learn the parameter $\theta^{*}$ while optimizing the regret. Our algorithms use the “doubling trick” and modern optimization techniques, which carefully balance the randomization for exploration to fully learn the parameter and maximizing the reward to achieve the near-optimal regret. In particular, our algorithms fall under the general UCB-type algorithms (Lai and Robbins, 1985; Auer et al., 2002). Briefly speaking, we start with some pure exploration periods that we take random actions uniformly, and we estimate the parameter. Then, for the next certain periods, which we call an epoch, we take the action at each time period by optimizing some upper confidence bands using the previous estimator. At the end of each epoch, we renew the estimator using new information. We then enter the next epoch using the new estimator and renew the estimator at the end of the next epoch. We repeat this until the $T$-th time period.

The high-dimensional regime (i.e., $d\gg T$) poses significant challenges in our setting, which cannot be solved by exiting works. First, unlike in the low-dimensional regime where ordinary least squares (OLS) always admits closed-form solutions and error bounds, in the high-dimensional regime, most existing methods like the Lasso (Tibshirani, 1996) or the Dantzig selector (Candes and Tao, 2007) require the sample covariance matrix to satisfy certain restricted eigenvalue”conditions (Bickel et al., 2009), which do not hold under our setting for sequentially selected covariates. Additionally, our action spaces $\{\mathcal{A}_{t}\}$ are finite. This rules out several existing algorithms, including the SETC method (Lattimore et al., 2015) that exploits the specific structure of hyper-cube actions sets and finite-difference type algorithms in stochastic sparse convex optimization (Wang et al., 2017; Balasubramanian and Ghadimi, 2018). We adopt the best subset selector estimator (Miller, 2002) to derive valid confidence bands only using ill-conditioned sample covariance matrices. Note that while the optimization for best subset selection is NP-hard in theory (Natarajan, 1995), by the tremendous progress of modern optimization, solving such problems is practically efficient as discussed in Pilanci et al. (2015); Bertsimas et al. (2016). In addition, the renewed estimator may correlate with the previous one. This decreases the efficiency. We let the epoch sizes grow exponentially, which is known as the “doubling trick” (Auer et al., 1995). This “removes” the correlation between recovered support sets by best subset regression. Our theoretical analysis is also motivated from some known analytical frameworks such as the elliptical potential lemma (Abbasi-Yadkori et al., 2011) and the SupLinUCB framework (Auer, 2002) in order to obtain sharp regret bounds.

We summarize our main theoretical contribution in the following corollary, which is essentially a simplified version of Theorem 2 in Section 3.3.

Note that this corollary holds even if $T\ll d$. Theorem 2 provides more general results than this corollary, and can be applied for a broader family of distributions for contextual covariates $\{X_{t,i}\}$. A simpler and more implementable algorithm, with a weaker regret guarantee of $\widetilde{\mathcal{O}}(s\sqrt{T})$, is given in Algorithm 1 and Theorem 1, respectively. The form of regret guarantee in this Corollary 1 looks particularly similar to its low-dimensional counterparts in Table 1, with the ambient dimension $d$ replaced by the intrinsic dimension $s$.

As we mentioned above, our algorithm is inspired by the standard LinUCB algorithm, which balances the tradeoff between exploration and exploitation following the principle of “optimism in the face of uncertainty”. In particular, the LinUCB algorithm repeatedly constructs upper confidence bands for the potential rewards of the actions. The upper confidence bands are optimistic estimators. We then pick the action associated with the largest upper confidence band. This leads to the optimal regret under the low-dimensional setting. However, under the high-dimensional setting, directly applying the LinUCB algorithm incurs some suboptimal regret since we only get lose confidence bands under the high-dimensional regime. Thus, it is desirable to construct tight confidence bands under the high-dimensional and sparse setting to achieve the optimal regret.

Inspired by the remarkable success of the best subset selection (BSS) in high-dimensional regression problems, we propose to incorporate this powerful tool into the LinUCB algorithm. Meanwhile, since the BSS procedure is computationally expensive, it is impractical to execute the BSS method during every time period. In addition, as we will discuss later, it is not necessary.

Before we present our algorithms, we briefly discuss the support of parameters. Given a $d$-dimensional vector $\theta$, we denote by $\mathrm{supp}(\theta)$ the support set of $\theta$, which is the collection of dimensions of $\theta$ with nonzero coordinates that

This definition agrees with that of most literatures. However, for the BSS procedure, it is desirable to generalize this definition. We propose the concept of “generalized support” as follows.

The generalized support $\mathrm{supp}^{+}(\theta)$ is a relaxation of the normal support, since any support is a generalized support (but not vice versa). Moreover, the generalized support is not unique. Any subset including the support is a valid generalized support.

We distinguish the difference between support and generalized support in order to define the best subset selection without causing confusion. For example, we consider a linear model $\theta^{*}\in\mathbb{R}^{d},X_{t}\in\mathbb{R}^{d}$, and $Y_{t}=\langle X_{t},\theta^{*}\rangle+\varepsilon_{t}$. Calculating the ordinary least square estimator restricted on the generalized support $S\in[d]$, which is denoted by

means that we consider a low-dimensional model only using the information in $S$ and set the coordinates of estimator except in $S$ as zeros. Formally, let ${[\widehat{\theta}]}_{S}\in\mathbb{R}^{|S|}$ be

Then we have

Since we do not guarantee $[\widehat{\theta}\big{]}_{j}\neq 0,\ \forall j\in S$, we call $S$ the generalized support instead of support.

We are ready to present the details of SLUCB algorithm now. Our algorithm works as follows. We first apply the “doubling trick”, which partitions the whole $T$ decision periods into several consecutive epochs such that the lengths of the epochs increase doubly. We only implement the BSS procedure at the end of each epoch to recover the support of the parameter $\theta^{*}$. Within each epoch, we start with a minimal epoch length of $n_{0}$ for pure exploration, which is called the pure-exploration-stage. At each time period $t$ of this stage, we pick an action within $\mathcal{A}_{t}$ uniformly at random. Intuitively speaking, in pure-exploration-stage, we explore the unknown parameters at all possible directions through uniform randomization. From a technical perspective, the pure-exploration-stage is desirable since it maintains the smallest sparse eigenvalue of the empirical design matrix lower bounded away from zero and hence, help us obtain an efficient estimator for $\theta^{*}$. After the pure-exploration-stage, we keep the estimated support, say of $s$ nonzero components, from the previous epoch unchanged, and treat the problem as an $s$-dimensional regression. Specifically, at each time period, we use the ridge estimator with penalty weight $\lambda$ to estimate $\theta^{*}$ and construct corresponding confidence bands to help us make decisions in the next time period.

In summary, we partition the time horizon $[T]$ into consecutive epochs $\{E_{\tau}\}_{\tau=1}^{\widetilde{\tau}}$ such that

The length of the last epoch $E_{\widetilde{\tau}}$ may be less than $2^{\widetilde{\tau}}$ or $n_{0}$. By definition, the number of epochs $\widetilde{\tau}\asymp\log(T)$. Hence, in the SLUCB algorithm, we run the BSS procedure $\widetilde{\mathcal{O}}(\log(T))$ times. In our later simulations studies, we find that this is practical for moderately large dimensions.

Next, we introduce the details of constructing upper confidence bands in the SLUCB algorithm. We assume that at period $t\in E_{\tau}$, we pick action $i_{t}\in\mathcal{A}_{t}$ and observe the associated covariate $X_{t,i_{t}}$ and reward $Y_{t}$. We also abbreviate $X_{t,i_{t}}$ as $X_{t}$, if there is no confusion. We denote by $\widehat{\theta}_{\tau-1,\lambda}$ the BSS estimator for the true parameter $\theta^{*}$ at the end of previous epoch $E_{\tau-1}$, and let

be its generalized support, i.e., the generalized support recovered by epoch $E_{\tau-1}$. For periods in the pure-exploration-stage of $E_{\tau}$, we pick arms uniformly at random and do not update the estimator of $\theta^{*}$. Beyond the pure-exploration-stage, we estimate $\theta^{*}$ by a ridge estimator. Let $\widehat{\theta}^{t-1}_{\tau,\lambda}$ be the most updated ridge estimator of $\theta^{*}$ by $t\in E_{\tau}$, which is estimated by restricting its generalized support on $S_{\tau-1}$ using data $\{X_{t^{\prime}},Y_{t^{\prime}}\}_{t^{\prime}\in E^{t-1}_{\tau}}$. In particular, all components of $\widehat{\theta}^{t-1}_{\tau,\lambda}$ outside $S_{\tau-1}$ are set as zeros and

Given $\widehat{\theta}^{t-1}_{\tau,\lambda}$, we calculate the upper confidence band of potential reward $\langle X_{t,i},\theta^{*}\rangle$ for each possible action $i\in\mathcal{A}_{t}$. In particular, let the tuning parameters $\alpha$ and $\beta$ be

which correspond to the confidence level and upper estimate of potential reward, respectively. Then we calculate the upper confidence band associated with action $i$ as

where

After that, we pick the arm $i_{t}$ corresponding to the largest upper confidence band to play and observe the corresponding reward $Y_{t}=\langle X_{t,i_{t}},\theta^{*}\rangle+\varepsilon_{t}$. We repeat this process until the end of epoch $E_{\tau}$.

Then we run the BSS procedure using all data collected in $E_{\tau}$ to recover the support of $\theta^{*}$. We also enlarge the size of generalized support by $s$. To be specific, let $S_{\tau}$ be the generalized support recovered in this step. We require that $S_{\tau}$ satisfies constraints

and obtain the BSS estimator $\widehat{\theta}_{\tau,\lambda}$ as

where $\text{Proj}_{r}\{\cdot\}$ denotes the projection on a centered $\ell_{2}$-ball with radius $r$, the $\ell_{2}$-norm of $\theta^{*}$. Note that in comparison with the standard BSS estimator, we project the minimizer for regularization. The boundedness also simplifies our later theoretical analysis. We also add the inclusion restriction $S_{\tau}\supseteq S_{\tau-1}$ for some technical reasons, which does not lead to any fundamental difference. As a result, we need to consider the sparsity $\tau s$ instead of $s$. It boosts the probability of recovering the true support.

A pseudo-code description of the SLUCB algorithm is presented in Algorithm 1. We remark that although computing the BSS estimator is computationally expensive, many methods exist that can solve this problem very efficiently in practice. See, for example, (Pilanci et al., 2015; Bertsimas et al., 2016) for more details. We also conduct extensive numerical experiments to demonstrate that our proposed algorithm is practical in Section 4.

Although the SLUCB algorithm is intuitive and easy to implement, we are unable to prove the optimal upper bound for its regret due to some technical reasons. Specifically, as discussed in the next section, we can only establish an $\widetilde{\mathcal{O}}(s\sqrt{T})$ upper bound for the regret of the SLUCB algorithm, while the optimal regret should be $\widetilde{\mathcal{O}}(\sqrt{sT})$. Here we omit all the constants and logarithmic factors and only consider the dependency on horizon length and dimension parameters. The obstacle leading to sub-optimality is the dependency of covariates on random noises. Recall that at each period $t\in E_{\tau}$, the SLUCB algorithm constructs the ridge estimator $\widehat{\theta}^{t-1}_{\tau,\lambda}$ using all historical data, where designs $\{X_{t^{\prime}}\}_{t^{\prime}\in E^{t-1}_{\tau}}$ are correlated with noises $\{\varepsilon_{t^{\prime}}\}_{t^{\prime}\in E^{t-1}_{\tau}}$ due to the UCB-type policy. Such a complicated correlation impedes us from establishing tight confidence bands for predicted rewards, which results in suboptimal regret.

To close the aforementioned gap and achieve the optimality, we modify the seminal SupLinUCB algorithm (Auer, 2002; Chu et al., 2011), which is originally proposed to attain the optimal regret for classic stochastic linear bandit problem, as a subroutine in our framework. Then we propose the Sparse-SupLinUCB (SSUCB) algorithm. Specifically, we replace the ridge estimator and UCB-type policy with a modified SupLinUCB algorithm. The basic idea of the SupLinUCB algorithm is to separate the dependent designs into several groups such that within each group, the designs and noises are independent of each other. Then the ridge estimators of the true parameters are calculated based on group individually. Thanks to the desired independency, now we can derive tighter confidence bands by applying sharper concentration inequality, which gives rise to the optimal regret in the final.

In the next, we present the details of SupLinUCB algorithm and show how to embed it in our framework. For each period $t\in E_{\tau}$ that belongs to the UCB-stage, the SupLinUCB algorithm partitions the historical periods $E^{t-1}_{\tau}$ into $\widetilde{\zeta}$ disjoint groups

where $\widetilde{\zeta}=\lceil\log(\beta T)\rceil$. These groups are initialized by evenly partitioning periods from the pure-exploration-stage and are updated sequentially following the rule introduced as follows. For each period $t$, we screen the groups $\{\Psi^{t-1,\zeta}_{\tau}\}$ one by one (in an ascending order of index $\zeta$) to determine the action to take or eliminate some obvious sub-optimal actions.

Suppose that we are at the $\zeta$-th group now. Let ${\mathcal{N}}^{t-1,\zeta}_{\tau}$ be the set of candidate actions that are still kept by the $\zeta$-th step, which is initialized as the whole action space $\mathcal{A}_{t}$ when $\zeta=1$. We first calculate the ridge estimator $\widehat{\theta}^{t-1,\zeta}_{\tau,\lambda}$ restricted on the generalized support $S_{\tau-1}$, using data from group $\Psi^{t-1,\zeta}_{\tau}$. Then for each action $i\in{\mathcal{N}}^{t-1,\zeta}_{\tau}$, we calculate $\omega_{\tau,\lambda}^{t-1,\zeta}(i)$, the width of confidence band of the potential reward. Specifically, we have

where the tuning parameter

corresponds to the confidence level. Our next step depends on the values of $\omega_{\tau,\lambda}^{t-1,\zeta}(i)$. If

which means that the widths of confidence bands are uniformly small, we pick the action associated with the largest upper confidence band

where $\beta=r\sigma\cdot\sqrt{s\log(kTd/\delta)}$ is an upper estimate of potential rewards. In this case, we discard the data point $(X_{t},Y_{t})$ and do not update any group, i.e., setting $\Psi^{t,\zeta}_{\tau}=\Psi^{t-1,\zeta}_{\tau}$, for all $\zeta\in[\widetilde{\zeta}].$

Otherwise, if there exists some $i\in{\mathcal{N}}^{t-1,\zeta}_{\tau}$ such that $\omega_{\tau,\lambda}^{t-1,\zeta}(i)\geq 2^{-\zeta}\beta,$ which means that the width of confidence band is not sufficiently small, then we pick such an action $i$ to play for exploration. In this case, we add the period $t$ into the $\zeta$-th group while keeping all other groups unchanged, i.e.,

Finally, if neither one of the above scenarios happens, which implies that for all $i\in{\mathcal{N}}^{t-1,\zeta}_{\tau}$ $\omega_{\tau,\lambda}^{t-1,\zeta}(i)\leq 2^{-\zeta}\beta$, then we do not take any action for now. Instead, we eliminate some obvious sub-optimal actions and move to the next group $\Psi^{t-1,\zeta+1}_{\tau}$. Particularly, we update the set of candidate arms as

We repeat the above procedure until an arm is selected. Since the number of groups is $\widetilde{\zeta}=\lceil\log(\beta T)\rceil$ and $2^{-\widetilde{\zeta}}\beta=1/T\leq 1/\sqrt{T}$, the SupLinUCB algorithm stops eventually. Finally, by replacing the direct ridge regression and UCB-type policy with the SupLinUCB algorithm above, we obtain the SSUCB algorithm. The pseudo-code is presented in Algorithm 2.

We present the assumptions in our theoretical analysis and discuss their relevance and implications. We consider finite action spaces $\{\mathcal{A}_{t}\}_{t=1}^{T}$ and assume that there exists some constant $k$ such that

We also assume that for each period $t$, the covariates $\{X_{t,i}\}_{i\in\mathcal{A}_{t}}$ are sampled independently from an unknown distribution $P_{0}$. We further impose the following assumptions on distribution $P_{0}$.

We briefly discuss Assumption 1. First of all, (A1) is a standard assumption in literature, with sub-Gaussianity covering a broad family of distributions like Gaussian, Rademacher, and bounded distributions. Assumption (A2) is a non-degeneracy assumption which, together with (A3), implies that the smallest eigenvalue of the population covariance matrix $\mathbb{E}[XX^{\top}]$ is lower bounded by some constant $\rho>0$. Similar assumptions are also adopted in high-dimensional statistics literature in order to prove the “restricted eigenvalue” conditions of sample covariance matrices (Raskutti et al., 2010), which are essential in the analysis of penalized least square methods (Wainwright, 2009; Bickel et al., 2009). However, we emphasize that in this paper the covariates indexed by the selected actions $\{i_{t}\}$ do not satisfy the restricted eigenvalue condition in general, and therefore, we need novel and non-standard analyses of high-dimensional M-estimators. For Assumption (A3), at a higher level, independence among coordinates enables relatively independent explorations in different dimensions, which is similar to the key idea of the SETC method (Lattimore et al., 2015). Technically, (A3) is used to establish the key independence of sample covariance matrices restricted within and outside the recovered support. Due to such an independence, the rewards in the unexplored directions at each period are independent as well, which can be estimated efficiently.

We also impose the following assumptions on the unknown $d$-dimensional true parameter $\theta^{*}$.

Note that in Assumption 2, (B1) is the key sparsity assumption, which assumes that only $s\ll d$ components of the true parameter $\theta^{*}$ are non-zero. Assumption (B2) is a boundedness condition on the $\ell_{2}$-norm of $\theta^{*}$. This assumption is often imposed, either explicitly or implicitly, in contextual bandit problems for deriving an upper bound for rewards (Dani et al., 2008; Chu et al., 2011).

Finally, we impose the sub-Gaussian assumption on noises sequence $\{\varepsilon_{t}\}_{t=1}^{T}$, which is a standard assumption adopted in most statistics and bandit literatures.

In this section, we analyze the performance of the SLUCB algorithm. As discussed earlier, we measure the performance via the regret defined in (4). We show that with a tailored choice of pure-exploration-stage length $n_{0}$, tuning parameters $\alpha$, and $\beta$, the accumulated regret of the SLUCB algorithm is upper bounded by $\widetilde{\mathcal{O}}(s\sqrt{T})$ (up to logarithmic factors) with high probability. Formally, we have the following theorem.