X-Armed Bandits: Optimizing Quantiles and Other Risks

04/17/2019 ∙ by Léonard Torossian, et al. ∙ Inra 0

We propose and analyze StoROO, an algorithm for risk optimization on stochastic black-box functions derived from StoOO. Motivated by risk-averse decision making fields like agriculture, medicine, biology or finance, we do not focus on the mean payoff but on generic functionals of the return distribution, like for example quantiles. We provide a generic regret analysis of StoROO. Inspired by the bandit literature and black-box mean optimizers, StoROO relies on the possibility to construct confidence intervals for the targeted functional based on random-size samples. We explain in detail how to construct them for quantiles, providing tight bounds based on Kullback-Leibler divergence. The interest of these tight bounds is highlighted by numerical experiments that show a dramatic improvement over standard approaches.

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1 Introduction

We consider an unknown function , where and

denotes the probability space representing some uncontrollable variables. For any fixed

is a random variable of law

and we consider with , a real-valued functional defined on probability measures. We assume that there exists at least one such that . Using a set of sequential observations , our goal is to minimize the simple regret with the value returned after using a budget .

Different families of algorithms have been developed to treat this problem. Some are for example of Bayesian flavor (see [Shahriari et al., 2016] for instance), some are inspired by the bandit literature. Here we focus our interest on the bandit framework.

In the classical -armed bandit problem, a forecaster selects repeatedly a point in the input space and receives a reward distributed according to an unknown distribution . Historically, the main goal was to minimize the cumulative regret, i.e. the sum of the difference between his collected rewards and the ones that would have been brought by optimal actions. In the last decade, other works focused on the simple regret. These can be divided in two: algorithms that optimize an unknown function with the knownledge of the smoothness, for example StoOO [Munos et al., 2014], HOO [Bubeck et al., 2011] Zooming [Kleinberg et al., 2008] or HCT [Azar et al., 2014], and others focusing on the optimization of unknown functions without the knowledge of the smoothness, such as POO [Grill et al., 2015], StroquOOL [Bartlett et al., 2018], GPO [Shang et al., 2019] StoSOO [Valko et al., 2013] or [Locatelli and Carpentier, 2018].

Those algorithms focus on the optimization of the conditional expectation of

. This choice is questionable in some situations. For example if the shape and variance of the reward distribution depend on the input, a forecaster may be interested in different aspects of the unknown distribution in order to modulate its risk exposure. In the literature, some measures of risk have been proposed to replace the expectation: for instance quantiles (also referred to as Value-at-Risk, see

[Artzner et al., 1999, McNeil and Frey, 2000] for instance), the Conditional Value-at-Risk [Rockafellar et al., 2000], the entropy Value-at-Risk [Ahmadi-Javid, 2012], or expectiles [Bellini and Di Bernardino, 2017]. The purpose of this paper is to present a risk optimization framework of an unknown stochastic function with the knowledge of the smoothness using only pointwise sequential observations and a finite budget .

-armed bandit algorithms rely on optimistic strategies that associate with each point of the space an upper confidence bound (UCB), that is, an “optimistic” prediction of the outcome. Adapting the classical setting to the optimization of risk measures implies being able to create high-probability confidence bounds for that particular measure. This problem has been tackled in the multi-armed bandit setting ( when the input space is discrete and finite). For instance, [Audibert et al., 2009, Sani et al., 2012] focused on the empirical variance, [Galichet et al., 2013, Kolla et al., 2019, Hepworth, 2017] on the CVaR while in [David and Shimkin, 2016, Szorenyi et al., 2015] the authors based their policies on the quantile. However, the literature is scarce in the continuous input space case.

In this paper we provide a new version of the Stochastic Optimistic Optimization (StoOO) algorithm [Munos et al., 2014], named StoROO (Stochastic Risk Optimistic Optimization), which is designed to optimize any function . In a first part, we provide an analysis of the simple regret from a generic point of view (that is, for any ). Then, we apply StoROO to optimize the conditional quantile. Using only the assumption that the output distribution support is connected and bounded in and admits a continuous density, we first propose an upper bound on the simple regret using Hoeffding’s inequality. Next we derive confidence intervals that take into account the order of the quantile respectively based on Bernstein’s and Chernoff’s inequalities. Finally, we present numerical experiments that illustrate the ability of our method to optimize conditional quantiles of a black-box function and the relevance to use confidence bounds derived from Chernoff’s inequality. Due to space limitation, technical proofs are deferred to Supplementary Material.

2 Problem setup

2.1 Hierarchical partitioning

The upper confidence bounds on which optimistic algorithms are based are surrogate functions larger than the objective (in a sense detailed below) with high probability. At each round , the point having the highest UCB is sampled and a reward is collected.

In the classical multi-armed bandit problem, computing and sorting the UCB can be done without major issues. But dealing with continuous input spaces ( infinitely many arms) implies maximizing a UCB function over a continuous space, which can be both computational intensive and algorithmically challenging. For example, Piyavskii’s algorithm (see [Bouttier, 2017] and references therein) defines

using a global Lipschitz assumption on the targeted function. Because of the Lipschitz hypothesis, the UCB maximizer is at an intersection of hyperplanes, i.e. where the UCB is non-differentiable. Thus a gradient-based algorithm cannot be used, implying that finding the point with the highest UCB is a very hard problem to solve.

To overcome the computational difficulties, a popular alternative is to rely on hierarchical partitions [Bubeck et al., 2011, Munos et al., 2014]. Let us consider an infinite hierarchical space structure of such that

with the number of sub-regions obtained after expanding a cell and the -th cell at depth . In the following we assume that:

Assumption : There exists a decreasing sequence , such that for any and for any cell , , with the center of .

Assumption : There exists such that all cells of depth contain a ball of radius .

Starting with and following an optimistic strategy, at time the algorithm has expanded some cells and the result is a tree that is a subset of and a partition of . In this setting is taken as a piecewise constant function. Indeed for any we define such that for all , .

In the literature of -armed bandits there are two ways to select a cell of at each round. In [Bubeck et al., 2011], the algorithm follows an optimistic path from the root to the leaves. In [Munos et al., 2014], StoOO selects the cell having the highest UCB among all the cells of that have not been expanded, the set of leaves of . We consider here this second alternative. Hence, to find the maximizer of at time , we only need to evaluate and sort a finite number of values .

2.2 Upper and lower confidence bounds, bias

To create confidence bounds for , the idea of StoOO is to get a sample of every node cell center . Thanks to the fact that all observed values are independent, we can use a deviation inequality to create a UCB for , that we denote . Finally to create the UCB over the cell , a bias term is added that takes into account how can potentially increase from the center of the cell to its edges.

To ensure the convergence of StoOO (and StoROO), the function only needs to be a UCB of for the cell containing , as is detailed in the proof of Proposition 1 (see also [Munos et al., 2014]). Bounding by how much can potentially increase from the center to the edge of the optimal cell requires a regularity assumption on . Following [Munos et al., 2014, Azar et al., 2014], we assume the following smoothness property:

(1)

Note that this condition is less restrictive than a global Lipschitz condition. It does not exclude functions that are very irregular (possibly discontinuous), except close to global maxima. Based on (1) we define

The algorithm also needs a quantity that bounds from below in order to provide guaranties on the value of over each cell. We thus construct a lower confidence bound, termed , for , and use it as an LCB for the maximum of on . In particular, on the cell containing the optimum , it holds that

with high probability. To summarize, the estimation of

is altered by two sources of error: the local estimation error made at the center of the cell, and the bias term . Balancing those two terms naturally provides a trade-off between exploration and exploitation.

3 Stochastic Risk Optimistic Optimization

3.1 The StoROO algorithm

StoROO starts by sampling one time each sub-region of the root node. Then, at each time the algorithm selects having the highest UCB. To reduce the estimation error, StoROO can either get more samples from (to reduce the variance), or split the cell in order to reduce its diameter (to reduce the bias). The good balance between these two options is found by dividing a cell as soon as the local estimation error is smaller than the bias, that is when

(2)

If Condition (2) is satisfied, StoROO expands and requires a new sample at the center of each sub-region. If Condition (2) is not satisfied, then StoROO requires a new sample at the center which is used to update and .

When the budget is exhausted, several choices are possible for the return value: they have the same theoretical guarantees. Following [Munos et al., 2014], one can return the deepest node among those that have been expanded. Here we propose a different, more conservative choice. Denoting by the set of nodes having the highest LCB among those that have been expanded after a budget , StoROO returns the node with the highest value (an estimator of ) among the deepest nodes of .

Input: error probability ; number of children ; time horizon ; ; ;
Define: UCB and LCBInitialization ; ;;
Expand into sub-regions the root node and sample one time each child;
while   do
       foreach   do
             compute ;
            
      Select ;
       Compute the LCB ;
       if  then
             expand the node, remove from , add to the sub-cells of and sample each new node once, , ;
            
      else
            Sample the state and collect the observation , ,
      
Return the node according to the returning rule.;
Algorithm 1 StoROO

The pseudo-code of the full algorithm is given in Algorithm 1. It requires the parameters and of Condition (1), but of course the inequality do not have to be tight.

3.2 Analysis of the algorithm

In this section we provide a theoretical analysis of StoROO. It is inspired by [Munos et al., 2014], but differs most notably by the fact that the analysis is suited for any and not only for the conditional expectation.

The analysis relies on the possibility to construct, for any , upper- and lower-confidences bounds and such that the event

has probability at least . We defer to Section 4 their specific expression for the case of the quantile.

Contrary to the framework of [Munos et al., 2014], in our setting the magnitude of the confidence bound () associated to each node is not explicit. We thus need to introduce the following definition to quantify how many times a node needs to be sampled before satisfying the expansion condition (Eq. 2).

Definition 1

Let

The vector of safe constants is composed of the constants and such that the event

has probability at least .

Note that in the case of the conditional expectation, [Munos et al., 2014] take , and

We first prove (Proposition 1) that any point at the center of an expanded cell of depth belongs to

(3)

Next, we show that using a budget , the tree reaches at least a depth given below (Proposition 2). This implies that the point returned by the algorithm belongs to (Proposition 3). Finally, using an assumption on the size of that can be formalized by the so-call near-optimality dimension [Bubeck et al., 2011, Munos et al., 2014], we provide an upper bound on the regret (Theorem 1).

Proposition 1

Conditionally on , StoROO only expands cells such that .

Given the value and the total budget , the deeper the algorithm builds the tree, the better are the guarantees on the final point returned. So the goal of the following proposition is to provide a lower bound on the depth of .

Proposition 2

Define the largest such that

with the cardinal of . The deepest node expanded by StoROO is such that

Intuitively, is the budget needed to expand all the nodes in for all . It may be that some of this nodes will not be visited, but in the worst case they are and they need to be considered in order to obtain a valid bound. Putting Propositions 1 and 2 together, yields a first upper bound on the simple regret:

Proposition 3

Running StoROO with budget , with probability the regret is bounded as

A more explicit bound for the regret can be obtained by quantifying the volume of

for small values of . Introducing the Holderian semi-metric

that is associated with its regularity constants and , the near-optimality dimension of the function is defined as follows, see [Munos et al., 2014, Bubeck et al., 2011].

Definition 2

The -near optimality dimension is the smallest such that for all , there exists such that the maximal number of disjoint -balls of radius with center in is less than .

To evaluate we need to bound for all . The following proposition makes the link between the near optimality dimension and .

Proposition 4

Let be the -near-optimality dimension, and the corresponding constant. Then

Finally, combining Propositions 3 and 4 with an hypothesis on the decreasing sequence , it is possible to provide the speed of convergence of .

Theorem 1

Assume that for some and , and assume that . Thus with probability , the regret of StoOO is bounded as

where is the near optimality dimension and the corresponding near optimality constant.

Remark: In the particular case where each cell is a hypercube and the sub-regions are created by the division of the parent-cell into sub-regions of equal size, then , is equal to and is equal to .

4 Optimizing Quantiles

In this section, we focus on the optimization of quantiles, which are well-established tools in (risk-averse) decision theory (see [Rostek, 2010]

for instance). In particular, they benefit from interesting robustness properties, with respect to outliers or heavy tails. Let

now denote the -quantile of , where

is the cumulative distribution function (CDF) of 

.

In this section we detail how to construct the UCB and LCB for quantiles. First, we provide bounds based on Hoeffding’s inequality and we use them to adapt the regret bounds of Theorem 3. Then we provide two more refined bounds that take into account the order of the quantile based respectively on the Bernstein’s inequality and on the Kullback-Leibler divergence.

Let us first introduce some notation. For all , , and we denote

the empirical CDF of the reward inside the cell , where is the (random) number of times the cell was sampled up to time (see Definition 1). The generalized inverse of the piecewise constant function is defined as

that is the order statistic of the sample that has been collected from the node until time .

To define confidence bounds on the conditional quantile we proceed in two steps. First we propose confidence bounds on

. To do so, we simply use deviation bounds for Bernoulli distributions, since for all

, for all , the random variables are independent and identically distributed with a Bernoulli law of parameter , if denotes the time when the node has been sampled for the -th time. Then we use the properties

(4)
(5)

to create confidence bounds on using bounds on . The first equivalence in illustrated on Figure 1.

Figure 1: Illustration of the equivalence (4).

4.1 Hoeffding’s bound and regret analysis

Let , and let

(6)
(7)

The next proposition motivates the choice of the above quantities as a UCB and a LCB for the quantile of order at the points .

Proposition 5

For any , for all , for all and for all , if and are defined according to (7) and (6), respectively, then the event has probability at least .

Now, analyzing the regret requires a high probability bound on the number of time a node is sampled before being expanded:

Proposition 6

Assuming that for all , has a density supported in , , such that . If and are defined according to (7) and (6), respectively then a vector of safe constants is given as

and for any

According to the previous proposition, if we have sampled a node at depth more than

(8)

times, then with probability Condition (2) is satisfied and thus the node is expanded.

Equation (8) reflects that the smaller the minimum (taken over the whole support) of the density, the larger the upper bound on the number of samples needed before being expanded. Actually the bound is crude. It is rather clear, in fact, that the local minimum of around is the crucial quantity. Here we chose to write the results in terms of the global minimum to simplify the proof of Proposition (6). A more precise way to understand the behaviour of StoROO is that the number of time a node needs to be sampled before expansion depends on the pdf value in a neighborhood (of decreasing size with ) of the targeted quantile.

To obtain an upper bound on the simple regret, we now just need to combine Theorem 1 with Proposition 6 that provides the following theorem.

Theorem 2

Assume that for some and , then with probability , the regret of StoROO for minimizing the quantile is bounded as

with the near-optimality dimension and the near-optimality corresponding constant.

Note that the speed of convergence is the same as the one obtained in the conditional expectation optimization setting; only the constant varies.

4.2 Tight bounds

Using Hoeffding’s inequality is convenient because it leads to explicit lower and upper confidence bounds, which simplifies the deriviation of bounds on the regret. However, it implicitly upper-bounds the variance of all -valued random variables by , which is overly pessimistic when the inequality is applied to variables whose expectations are far from . This is in particular the case for quantile estimation, when the quantile is of order close to or . To take into account the order of the quantile, following [David and Shimkin, 2016], a first possibility is to derive confidence intervals from Bernstein’s inequality as presented in the following theorem.

Proposition 7

For any , for all , and , define

and

with

Then the event has probability at least .

Although Bernstein’s inequality takes into account the order of the quantile, it is possible to do something better. In order to create tighter confidence bound, we thus go back to Chernoff’s inequality and derive less explicit, but more accurate upper- and lower- confidence bounds on the -quantiles. We follow here  [Garivier and Cappé, 2011], but a close inspection at the proofs shows however a difference in the order of the marginals of the KL functions. Recall that the binary relative entropy is defined for as:

with by convention, , and

Proposition 8

For any , for all , and , define

if

Define

if

Then the event has probability at least .

Contrary to Bernstein’s inequality, Chernoff’s bound is always tighter than Hoeffding’s inequality , which follows from Pinsker’s inequality (see e.g.[Garivier et al., 2018]):

For example, given and an i.i.d. sample of size , one can see that

with (resp. ) the UCB associated to Chernoff’s inequality (resp. Hoeffding’s inequality). Berstein’s inequality is tighter than Hoeffding’s when is different from and sufficiently large, but always looser than Chernoff. It follows in particular that the regret of StoROO using confidence bounds derived from Chernoff’s inequality has, at least, the guarantees presented in Theorem 3.

The online setting we consider in this article induces that, after steps, the set of nodes and the number of observations in each node are random. To cope with this, we thus need deviation bounds for random size samples. The most simple way to obtain such inequalities is to use a union bound on the possible number of observations in each node, as presented above. Tighter results can be obtained from a more thorough analysis (sometimes called peeling trick): this is what is presented below.

Proposition 9

For any let

and define

if

Define

if

Then the event has probability at least .

5 Experiments

We empirically highlight the capacity of StoROO to optimize the conditional quantile of a black-box function. Four versions of StoROO are compared, ( StoROO using confidence bounds derived from Hoeffding’s inequality), ( StoROO using confidence bounds derived from Bernstein’s inequality), ( StoROO using confidence bounds derived from Chernoff’s inequality) and ( StoROO using confidence bounds derived from Chernoff’s inequality and the peeling trick).

As a test-case, we use the function

with

following a log-normal distribution of parameter

and truncated at its

-quantile with the truncated mass following a uniform distribution between

and . Figure 2 (left) shows the shape of the and quantiles of , while Figure 2 (right) shows samples of .

The performance of each version of StoROO is evaluated for different values of and quantified according to the simple regret. In our experiments we fix the values and such that the condition (1) is satisfied. Note that these values do not correspond to the actual regularity conditions at optimum. In addition we fix and we choose to expand the nodes into three sub-region of equal sizes.

Figure 2: To the left: conditional quantiles of , to the right: one run of for the optimization of the -quantile with , and .
Figure 3: Evolution of the expectation of the simple regret for the optimization of the conditional quantile of : to the left , to the right .

Figure 2 reports the average of the simple regret over runs for and . For both values of all the variants of StoROO have a regret that decreases with the budget. However from our experiments a ranking can be created.

The less efficient method is . For its simple regret decreases slower than the three others methods and for does not reach the performance of the others variants. Sometimes to reach a fixed accuracy, needs a much larger budget than others variants. For example taking , needs a budget of to reach a simple regret of order , while and need a budget equals to .

Then there is . Using the maximal budget, on both experiments this variant reaches the same accuracy as and but its simple regret decreases slower. For some levels of performance needs a much larger budget than . For example, taking , to reach the value needs the budget while is enough for .

Finally, the most efficient methods are and . Both methods are always better or equal to and . The variants and are often equivalent but sometimes the regret of decreases slightly faster than the version without the peeling trick. This behaviour provides a small gain for .

6 Conclusion

In this work, we extended StoSOO to a generic algorithm applicable to any functional of the reward distribution. We proposed a tailored application to the problem of quantile optimization, with four variants: one based on the classical Hoeffding’s inequality, one based on Bernstein’s inequality, and two others based on Chernoff’s inequality. We showed that using Chernoff’s inequality to build confidence intervals resulted in a dramatic improvement, both in theory and practice.

For simplicity, we assumed in this paper that the local regularity (or at least, an upper bound) of the target function at the optimum was known to the user. However, we believe that it is possible to combine our results to the procedure defined in [Grill et al., 2015, Xuedong et al., 2019] so that creating an algorithm able to optimize without the knowledge of the smoothness near an optimal point: this is left for future work. A second possible extension is to leverage the results proposed here to design an algorithm for the cumulative regret, in the spirit of HOO [Bubeck et al., 2011] for example.

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Appendix A Proofs related to the generic analysis of StoROO

Proof of Proposition 1

Let us define the partition containing . Assume that the partition has been selected, thus

By definition , thus . Conditionally on , that implies

Note that the last inequality is obtained because the partition is expanded, which implies that

Finally:

thus belongs to .

Proof of Proposition 2

Thus so There is at least an expanded node of depth after a budget was used.

Proof of Proposition 4 According to the assumption , each cell contains ball of radius centered in that is a ball of radius centered in . If the is the near optimality dimension then there is at most disjoint balls of radius inside . Thus if this implies there is more than disjoint balls of radius with center in , that is a contradiction.

Proof of Therorem 1