The Symmetry between Arms and Knapsacks: A Primal-Dual Approach for Bandits with Knapsacks

We begin our discussion with deriving a new upper bound for a generic BwK algorithm. First, we define the knapsack process as the remaining resource capacity at each time $t$. Specifically, we define $\bm{B}^{(0)}\coloneqq\bm{B}$ and

for $t\in[T].$ Recall that $\bm{C}_{t}$ is the (random) resource consumption at time $t$. The process $\bm{B}^{(t)}$ is pertaining to the BwK algorithm. In addition, we follow the convention of the bandits literature and define the count process $n_{i}(t)$ as the number of times the $i$-th arm is played up to the end of time period $t$.

Here $\Delta_{i}$ is known as reduced cost/profit in LP literature and it quantifies the cost-efficiency of each arm (each basic variable in LP). The upper bound in Proposition 1 is new to the existing literature and it can be generally applicable to all BwK algorithms. It consists of two parts: (i) the number of times for which a sub-optimal arm is played multiplied by the corresponding reduced cost; (ii) the remaining resource at time $\tau$, either when any of the resource is depleted or at the end of time horizon $T$. The first part is consistent with the classic bandits literature in that we always want to upper bound the number of sub-optimal arms being played throughout the horizon. At each time a sub-optimal arm $i\in\mathcal{I}^{\prime}$ is played, a cost of $\Delta_{i}$ will be induced. Meanwhile, the second part is particular to the bandits with knapsacks setting and can easily be overlooked. Recall that the definition of $\tau$ refers to the first time that any resource $j\in[d]$ is exhausted (or the end of the horizon). It tells that the left-over of resources when the process terminates at time $\tau$ may also induce regret. For example, for two binding resources $j,j^{\prime}\in\mathcal{J}^{*}$, it would be less desirable for one of them $j$ to have a lot of remaining while the other one $j^{\prime}$ is exhausted. Since the binding resources are critical in determining optimal objective value for LP (1), intuitively, it is not profitable to waste any of them at the end of the procedure.

From Proposition 1, we see the importance of dual problem (2) in bounding an algorithm’s regret. Now, we pursue further along the path and propose a new notion of sub-optimality for the BwK problem. Our sub-optimality measure is built upon both the primal and dual LPs, and it reveals the combinatorial structure of the problem. In the following, we define two classes of LPs, one for the arm $i\in[m]$ and the other for the constraints $j\in[d].$

First, for each arm $i\in[m],$ define

By definition, $\text{OPT}_{i}$ denotes the optimal objective value of an LP that takes the same form as the primal LP (1) except with an extra constraint $x_{i}=0$. It represents the optimal objective value if the $i$-th arm is not allowed to use. For a sub-optimal arm $i\in\mathcal{I}^{\prime},$ $\text{OPT}_{i}=\text{OPT}_{\text{LP}}$, while for an optimal arm $i\in\mathcal{I}^{*}$, $\text{OPT}_{i}<\text{OPT}_{\text{LP}}$. In this way, $\text{OPT}_{i}$ characterizes the importance of arm $i$.

Next, for each constraint $j\in[d]$, define

where $\bm{C}_{j,\cdot}$ denotes the $j$-th row of the constraint matrix $\bm{C}.$ Though it may not be as obvious as the previous case of OPT_i, the definition of $\text{OPT}_{j}$ aims to characterize the bindingness/non-bindingness of a constraint $j$. The point can be illustrated by looking at the primal problem for (5). From LP’s strong duality, we know

Compared to the original primal LP (1), there is an extra term in the objective function in LP (6). The extra term is a penalization for the left-over of the $j$-th constraint, and thus it encourages the usage of the $j$-th constraint. For a binding constraint $j\in\mathcal{J}^{*},$ it will be exhausted under the optimal solution $\bm{x}^{*}$ to LP (1) so the penalization term does not have any effect, i.e., OPT${}_{j}=$OPT${}_{\text{LP}}$. In contrast, for a non-binding constraint $j\in\mathcal{J}^{\prime},$ the extra term will result in a reduction in the objective value, i.e., $\text{OPT}_{j}<\text{OPT}_{\text{LP}}$. We note that one can introduce any positive weight to the penalization term so as to trade off between the reward and the left-over of the $j$-th constraint in (6), but its current version suffices our discussion.

The following proposition summarizes the properties of OPT_i and OPT${}_{j}.$

In this way, the definition of $\text{OPT}_{i}$ distinguishes optimal arms $\mathcal{I}^{*}$ from sub-optimal arms $\mathcal{I}^{\prime}$, while the definition of $\text{OPT}_{j}$ distinguishes the binding constraints $\mathcal{J}^{*}$ from non-binding constraints $\mathcal{J}^{\prime}$. The importance of such a distinguishment arises from the upper bound in Proposition 1: on one hand, we should avoid playing sub-optimal arms, and on the other hand, we should exhaust the binding resources. A second motivation for defining both OPT_i and OPT_j can be seen after we present our algorithm. Furthermore, we remark that a measurement of the sub-optimality of the arms has to be defined through the lens of LP due to the combinatorial nature of the problem. The effect of the $i$-th arm’s removal on the objective value can only be gauged by solving an alternative LP of OPT${}_{i}.$ Similarly, a measurement of the bindingness of the constraints should also take into account the combinatorial relation between constraints.

Next, define

where the factor $\frac{1}{T}$ is to normalize the optimality gap by the number of time periods. Under Assumption 1, all the objective values in above should scale linearly with $T.$

To summarize, $\delta$ characterizes the hardness of distinguishing optimal arms from non-optimal arms (and binding constraints from non-binding constraints). It can be viewed as a generalization of the sub-optimality measure $\delta_{\text{MAB}}=\min_{i\neq i^{*}}\mu_{i^{*}}-\mu_{i}$ for the MAB problem (Lai and Robbins, 1985). $\delta_{\text{MAB}}$ characterizes the hardness of an MAB problem instance, i.e., the hardness of distinguishing the optimal arm from sub-optimal arms. In the context of BwK, $\delta$ is a more comprehensive characterization in that it takes into account both the arms and the constraints. Imaginably, it will be critical in both algorithm design and analysis for the BwK problem.

Now, we define two LP-related quantities that will appear in our regret bound:

• •

  Linear Dependency between Arms: Define $\sigma$ be the minimum singular value of the matrix $\bm{C}_{\mathcal{I}^{*},\mathcal{J}^{*}}.$ Specifically,

  $$\sigma\coloneqq\sigma_{\min}\left(\bm{C}_{\mathcal{J}^{*},\mathcal{I}^{*}}\right)=\sigma_{\min}\left((c_{ji})_{j\in\mathcal{J}^{*},i\in\mathcal{I}^{*}}\right).$$

  In this light, $\sigma$ represents the linear dependency between optimal arms across the binding constraints. For a smaller value of $\sigma$, the optimal arms are more linearly dependent, and then it will be harder to identify the optimal numbers of plays. Under Assumption 2, the uniqueness of optimal solution implies $\sigma>0.$

• •

  Threshold on the optimal solution:

  $\displaystyle\chi$

  $\displaystyle\coloneqq\frac{1}{T}\cdot\min\{x_{i}^{*}\neq 0,i\in[m]\}$

  If the total resource $\bm{B}=T\cdot\bm{b},$ both the optimal solution $(x_{1}^{*},...,x_{m}^{*})$ should scale linearly with $T$. The factor $\frac{1}{T}$ normalizes the optimal solution into a probability vector. $\chi$ denotes the smallest non-zero entry for the optimal solution. Intuitively, a small value of $\chi$ implies that the optimal proportion of playing an arm $i\in\mathcal{I}^{*}$ is small and thus it is more prone to “overplay” the arm.

Remarks. By the definition, it seems that the above parameters $\delta$ and $\chi$ both involve a factor of $T$. But if we replace $\bm{B}$ (the right-hand-side of LP) with $\bm{b}$ from Assumption 1, then the factor $T$ disappears, and the parameters $\chi$ and $\delta$ are essentially dependent on $\bm{\mu}$, $\bm{C}$, and $\bm{b}$ which are inherent to the problem instance but bear no dependency on the horizon $T$. In other words, Assumption 1 frees the dependency on $T$ by introducing the quantity $b$. Practically, the assumption states the resource capacity should be sufficiently large and it is natural in many application contexts (for example, the small bids assumption in AdWords problem Mehta et al. (2005)). Theoretically, in two previous works Flajolet and Jaillet (2015); Sankararaman and Slivkins (2020), either a factor of $1/T$ appears in the parameter definition Sankararaman and Slivkins (2020) or the assumption is explicitly imposed Flajolet and Jaillet (2015). Such an assumption might be inevitable for a logarithmic regret to be derived.

Throughout this paper, we denote the reward and resource consumption of the $s$-th play of the $i$-th arm as $r_{i,s}$ and $\bm{C}_{i,s}=(C_{1i,s},...,C_{di,s})^{\top}$ respectively, for $i\in[m]$ and $s\in[T].$ Let $n_{i}(t)$ be the number of times the $i$-th arm is played in the first $t$ time periods. Accordingly, we denote the estimators at time $t$ for the $i$-th arm as

for $i\in[m]$ and $j\in[d].$ In a similar manner, we define the estimator for the $i$-th arm’s resource consumption vector as $\hat{\bm{C}}_{i}(t)\coloneqq(\hat{C}_{1i}(t),...,\hat{C}_{di}(t))^{\top}$, and the resource consumption matrix as $\hat{\bm{C}}(t)\coloneqq(\hat{\bm{C}}_{1}(t),...,\hat{\bm{C}}_{m}(t))$. Specifically, without changing the nature of the analysis, we ignore the case that when $n_{i}(t)=0$. Then, we define the lower confidence bound (LCB) and upper confidence bound (UCB) for the parameters as

where $proj_{[0,1]}(\cdot)$ projects a real number to interval $[0,1].$ The following lemma is standard in bandits literature and it characterizes the relation between the true values and the LCB/UCB estimators. It states that all the true values will fall into the intervals defined by the corresponding estimators with high probability.

With the UCB/LCB estimators for the parameters, we can construct UCB/LCB estimators for the objective of the primal LP. Specifically,

The following lemma states the relation between $\text{OPT}_{\text{LP}}^{U}$, $\text{OPT}_{\text{LP}}^{L}$, and $\text{OPT}_{\text{LP}}.$ Intuitively, if we substitute the original constraint matrix $\bm{C}$ with its LCB (or UCB) and the objective coefficient $\bm{\mu}$ with its UCB (or LCB), the resultant optimal objective value will be a UCB (or LCB) for $\text{OPT}_{\text{LP}}$.

A similar approach is used in (Agrawal and Devanur, 2014) to develop an UCB-based algorithm for the BwK problem. For our algorithm presented in the following section, we will construct estimates not only for the primal LP (1), but also for the LPs (4) and (5). By comparing the estimates of OPT${}_{\text{LP}}$, OPT_i, and OPT_j, we will be able to identify the optimal arms $\mathcal{I}^{*}$ and the non-binding constraints $\mathcal{J}^{\prime}.$

Proposition 3 provides an upper bound on the number of time periods within which Phase I will terminate. It also states that the identification of $\mathcal{I}^{*}$ and $\mathcal{J}^{\prime}$ will be precise conditional on the high probability event in Lemma 2.

The surprising point of Proposition 3 lies in that there are $O(2^{m+d})$ possible configurations of $(\mathcal{I}^{*},\mathcal{J}^{\prime})$ and, without any prior knowledge, the true configuration can be identified within $O(\log T)$ number of plays for each arm. In contrast, (Flajolet and Jaillet, 2015) does not utilize the primal-dual structure of the problem and conducts a brute-force search in all possible configurations which results in an $O(2^{m+d}\log T)$ regret. In addition, the search therein requires the knowledge of a non-degeneracy parameter a priori. The result also explains why (Sankararaman and Slivkins, 2020) imposes a single-best-arm condition for the BwK problem, which assumes $\mathcal{I}^{*}=1$. This additional greatly simplifies the combinatorial structure and reduces the BwK problem more closely to the standard MAB problem. In this light, Proposition 3 can be viewed as a best-arm-identification result (Audibert and Bubeck, 2010) for the BwK problem in full generality and without any prior knowledge.

Proposition 4 provides an upper bound on the remaining resource for binding constraints when the procedure terminate, which corresponds to the second part of Proposition 1. Notably, the upper bound has no dependency on $T$. In a comparison with the $\Omega(\sqrt{T})$ fluctuation under the static design, it demonstrates the effectiveness of our adaptive design.

The idea of proof is to introduce an auxiliary process $b_{j}^{(t)}=\frac{B_{j}^{(t)}}{T-t}$ for $t\in[T-1]$ and $j\in\mathcal{J}^{*}$. Recall that $B_{j}^{(t)}$ is the $j$-th component of the knapsack process $\bm{B}^{(t)}$, we know its initial value $b_{j}^{(0)}=b$. Then define

for a fixed $\epsilon>0.$ With the definition, $b_{j}^{(t)}$ can be interpreted as average remaining resource (per time period) and $\tau_{j}$ can be interpreted as the first time that $b_{j}^{(t)}$ deviates from its initial value $b$ by a small amount. It is easy to see that $\tau_{j}\leq\tau$ with a proper choice of $\epsilon.$ Next, we aim to upper bound $\mathbb{E}[T-\tau_{j}]$ by analyzing the process $\{b_{j}^{(t)}\}_{t=0}^{T}.$ From the dynamic of the knapsack process $\bm{B}_{t},$ we know that

where $C_{j,t}$ as defined earlier is the resource consumption of $j$-th constraint at time $t$. The above formula (10) provides a technical explanation for the motivation of the adaptive design in (9). Intuitively, when the right-hand-side of (9) is $\bm{B}^{(t-1)},$ it will lead to a solution that (approximately and on expectation) consumes $b_{j}^{(t-1)}$ of the $j$-th resource for each of the following time periods. Ideally, this will make the second term in (10) have a zero expectation. However, due to estimation error for the LP’s parameters, this may not be the case. The idea is to first provide an upper bound for the “bias” term

where $\mathcal{H}_{t-1}=\{(r_{s},\bm{C}_{s},i_{s})\}_{s=1}^{t-1}$ encapsulates all the information up to time $t-1.$ Unsurprisingly, the upper bound is on the order of $O\left(\frac{1}{\sqrt{t}}\right)$. Next, with this bias upper bound, we can construct a super-martingale (sub-martingale) based on the dynamic (10) and employ Azuma–Hoeffding inequality to provide a concentration result for the value of the martingale. Through the analysis of the process $b_{j}^{(t)}$, we can derive an upper bound for $\mathbb{E}[T-\tau_{j}]$, and consequently, it leads to an upper bound on $\mathbb{E}[B^{(\tau)}_{j}]$.

The importance of the adaptive design has been widely recognized in other constrained online learning problems, such as online matching problem Manshadi et al. (2012), online assortment problem Golrezaei et al. (2014), online linear programming problem Li and Ye (2019), network revenue management problem Jasin and Kumar (2012), etc. The common pattern of these problem is to allocate limited resources in a sequential manner, and the idea of adaptive design is to adjust the allocation rule dynamically according to the remaining resource/inventory. This is in parallel with LP (9) where the solution at each time $t$ is contingent on the remaining resources $\bm{B}^{(t)}.$ The significance of our algorithm design and analysis lies in that (i) to the literature of BwK, our paper is the first application of the idea of adaptive design; (ii) to the existing literature of adaptive design in constrained online learning problems, our work provides its first application and analysis in a partial-information environment. For the second aspect, all the existing analysis on the adaptive design fall in the paradigm of “first-observe-then-decide” while the BwK problem is “first-decide-then-observe”. Specifically, in matching/resource allocation/revenue management problems, at each time period, a new agent arrives, and upon the observation, we decide the matching for the agent; or a new customer arrives, and upon the observation of her preference, we decide the assortment decision for the customer. So, the existing analyses are analogous to a BwK “setting” where the reward and resource consumption of playing an arm are first observed (magically), and then we decide whether we want to play the arm or not.

Combining the two parts, we have the following result on the regret of Algorithm 1.

The result reduces the exponential dependence on $m$ and $d$ in (Flajolet and Jaillet, 2015) to polynomial, and also it does not rely on any prior knowledge. Specifically, the authors consider several settings for BwK problem, many of which assume special structures such as one or two resource constraints. The most general settings therein, which is comparable to ours, allows arbitrary number of constraints and number of optimal arms. In terms of the key parameters, the way we define $\delta$ is the same as their definition of $\Delta_{x}$. However, the regret bound (Theorem 8 therein) involves a summation of exponentially many $\frac{1}{\Delta_{x}}$’s (the same as the total number of the bases of the LP). Our parameter $\sigma$ is related to their $\epsilon$ (Assumption 8 therein) while the latter is more restrictive. Because $\sigma$ in our paper represents the minimal singular value of the matrix corresponding to only the optimal basis of the primal LP, whereas the parameter $\epsilon$ therein represents a lower bound of the determinant of the matrices corresponding to all the possible (exponentially many) bases of the primal LP. In this light, if they adopt our parameter $\sigma$, their bound would be improved by a factor of $d!$ ($C!$ therein). Moreover, $\epsilon$ is a lower bound for our parameter $\chi$ and (Flajolet and Jaillet, 2015) explicitly requires the knowledge of $\epsilon$ a priori.

The proposition also generalizes the one-constraint bound in (Sankararaman and Slivkins, 2020) and relaxes the deterministic resource consumption assumption therein. Specifically, the authors assume there is one single optimal arm and one single resource (other than time), i.e., the optimal solution to the primal LP has only one non-zero entry ($|\mathcal{I}^{*}|=|\mathcal{J}^{*}|=1$ and $d=2$). They also assume the underlying LP is non-degenerate. Our results generalize their work in allowing arbitrary $d$, $|\mathcal{I}^{*}|$ and $|\mathcal{J}^{*}|.$ In terms of the key parameters, under their assumption, our parameter $\sigma=1$ because it is defined by a 1-by-1 matrix. Our parameter $\chi$ is deemed as a constant in their paper. For $\delta$, under their assumptions, our definition of OPT_i can be adjusted accordingly. Specifically, we can replace the constraint $x_{i}=0$ in defining OPT_i with $x_{i^{\prime}}=0,i^{\prime}\neq i$. Then our definition of $\delta$ would reduce to their definition of $G_{LAG}(a)$, both of which characterize the sub-optimality gap of an arm.

The above comparison against the existing literature highlights that the parameters $\sigma,$ $\delta$, and $\chi$ or other LP-based parameters might be inevitable in characterizing logarithmic regret bound. Our paper makes some preliminary efforts along this line, but we do not believe our bound is tight: parameters such as $\chi$ and $\sigma$ may be replaced with some tighter parameter through a better algorithm and sharper analysis. As remarked in Section 3.3, the parameters are not dependent on $T$ under Assumption 1. But to characterize the dependency of these parameters (such as $\delta$ and $\chi$) on the problem size $m$ and $d$ remains an open question. Moreover, our algorithm and analysis highly rely on the structural properties of the underlying LP, which might not be the unique method to handle the BwK problem.