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AJ16-Taming the Uncertainty Budget Limited Robust Crowdsensing Through Online Learning

AJ16-Taming the Uncertainty Budget Limited Robust Crowdsensing Through Online Learning
AJ16-Taming the Uncertainty Budget Limited Robust Crowdsensing Through Online Learning

Taming the Uncertainty:Budget Limited Robust Crowdsensing Through Online Learning

Kai Han,Chi Zhang,and Jun Luo

Abstract—Mobile crowdsensing has been intensively explored recently due to its?exible and pervasive sensing ability.Although many crowdsensing platforms have been built for various applica-tions,the general issue of how to manage such systems intelligently remains largely open.While recent investigations mostly focus on incentivizing crowdsensing,the robustness of crowdsensing toward uncontrollable sensing quality,another important issue, has been widely neglected.Due to the non-professional personnel and devices,the quality of crowdsensing data cannot be fully guaranteed,hence the revenue gained from mobile crowdsensing is generally uncertain.Moreover,the need for compensating the sensing costs under a limited budget has exacerbated the situation: one does not enjoy an in?nite horizon to learn the sensing ability of the crowd and hence to make decisions based on suf?cient statistics.In this paper,we present a novel framework,Budget LImited robuSt crowdSensing(BLISS),to handle this problem through an online learning approach.Our approach aims to minimize the difference on average sense(a.k.a.regret)between the achieved total sensing revenue and the(unknown)optimal one, and we show that our BLISS sensing policies achieve logarithmic regret bounds and Hannan-consistency.Finally,we use extensive simulations to demonstrate the effectiveness of BLISS.

Index Terms—Crowdsourcing,machine learning algorithms.

I.I NTRODUCTION

G IVEN the pervasive availability of hand-held mobile

devices(in particular the increasingly powerful smart phones),the concept of Mobile Crowdsensing[1]has started a new sensing paradigm,where human crowds(along with their mobile devices)are not only consumers of the sensed data but also their producers.Thanks to the huge number of pervasively available mobile sensors(those embedded in smart phones)and their virtually unlimited spatial-temporal cov-erage,the ef?ciency(in gathering a suf?cient amount of data) and the ubiquity(in capturing relevant events)are the major strengths of this new sensing paradigm.Consequently,there

Manuscript received May20,2014;revised October29,2014,January22, 2015;accepted March09,2015;approved by IEEE/ACM T RANSACTIONS ON N ETWORKING Editor L.Ying.Date of publication April23,2015;date of current version June14,2016.This work was supported in part by the National Natural Science Foundation of China(No.61472460)and AcRF Tier1Grant RGC5/13.

A preliminary version of this paper appeared in IEEE SECON'2014,Singapore, June30–July3,2014.

K.Han is with the School of Computer Science and Technology/Suzhou In-stitute for Advanced Study,University of Science and Technology of China, Suzhou,China(e-mail:hankai@https://www.sodocs.net/doc/40408556.html,).

C.Zhang and J.Luo are with School of Computer Engineering, Nanyang Technological University,Singapore(e-mail:czhang8@https://www.sodocs.net/doc/40408556.html,.sg; junluo@https://www.sodocs.net/doc/40408556.html,.sg).

Color versions of one or more of the?gures in this paper are available online at https://www.sodocs.net/doc/40408556.html,.

Digital Object Identi?er10.1109/TNET.2015.2418191have recently emerged many interesting mobile crowdsensing applications across a wide variety of research and application domains[2].

However,compared with the traditional remote sensing systems,the mobile crowdsensing paradigm has posed several unique challenges.While an owner of a crowdsensing task can save the expenditures of buying and deploying specialized sensors,substantial(preferably monetary)compensation is necessary to drive mobile crowdsensing[3]–[7].This is so because a participant to a mobile crowdsensing task needs to i) move to speci?c areas where sensing is required,2)consume his/her smart phone,mostly in terms of the embedded sensors and battery,and iii)probably pay for the3G access to upload sensing data[6].Furthermore,as the crowdsensing participants are usually unprofessional(hence resulting in high missing data rate and low sensing quality),the data readings acquired from

a single participant may be noisy and of poor data quality[8],

[9].This makes it necessary to require a minimum number of participants for improving sensing robustness.Actually,such a requirement is essential in a lot of crowdsensing applications [3],[10].

Based on the above observations,an astute sensing task owner has to seriously set up a budget,and to carefully choose participants so that it can harvest the most from informa-tion-gathering under that budget.Whereas this problem seems to fall in a conventional combinatorial optimization framework, the uncertainness of data quality in mobile crowdsensing makes it much more complicated.As neither the involved sensors nor their operators(the crowdsensing participants)are professional, the quality of sensing data cannot be perfectly guaranteed at a certain level.As one typical example,the amount of useful (or quali?ed)data gathered by a certain participant during a given time span may well be a random number instead of a deterministic function of the sampling rate[9],[11]–[13]. Consequently,the value of the sensing data to the owner can be random,and the owner would certainly need to seek robustness against such an uncertainty subject to the budget limit.To the best of our knowledge,this issue has never been tackled in the literature so far.

We study in this paper a novel robust sensing problem im-posed by mobile crowdsensing:an owner aims to repetitively conduct a sensing task under a limited budget,by choosing from a set of available participants whose individual sensing values are random with unknown probabilistic distributions. The problem is combinatorial in nature due to the budget limited selection process and is related to the NP-hard knap-sack problem.However,it is made far more challenging than the knapsack problem as the sensing values of individual

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participants are non-deterministic with unknown statistical information,whereas the value of each item is deterministic and known in advance in the knapsack problem.To this end, we propose Budget LImited robuSt crowdSensing(BLISS) as a general framework to tackle the problem.To keep the selection process robust to the uncertainty,we adopt an online learning approach to acquire the statistical information about the sensing values throughout the selection process.Due to the uncontrollable sensing quality,the objective of our robust crowdsensing is to minimize the difference on average sense (a.k.a.regret)between the achieved total sensing revenue and the optimal one computed by a genie.In summary,we make the following major contributions in this paper:

?We introduce the robust crowdsensing problem,a realistic yet open issue for many crowdsensing scenarios,and pro-pose Budget LImited robuSt crowdSensing(BLISS)as a general framework to tackle this problem.

?We propose two online learning algorithms for BLISS based on the Multi-Armed Bandit(MAB)paradigm

[14];they handle the static-sensing-cost case and the dy-

namic-sensing-cost case,respectively.We show that both our learning algorithms achieve logarithmic regret bounds and Hannan-consistency,with one of them also achieving asymptotical optimality.

?We perform extensive simulations to demonstrate the ef-fectiveness of our BLISS algorithms.

From an algorithmic point of view,our algorithms are,to the best of our knowledge,the?rst to consider an MAB model that multiple arms with(static or dynamic)costs must be played under the constraints of a limited budget and a combinatorial arm-selection space.

The remaining of our paper is organized as follows.We introduce the background and models in Section II,where we also formulate the BLISS framework.Then we present our?rst BLISS online learning algorithm for the static-sensing-cost case in Section III,and analyze its theoretical performance in Section IV.In Section V,we provide another online learning algorithm for the dynamic-sensing-cost case and give the regret analysis for it.We report the results of our extensive simulations in Section VI.We?nally discuss the related work in Section VII,before concluding our paper in Section IX.In order to maintain?uency,we postpone all the(sketched)proofs to the Appendix A.

II.M ODELING AND P ROBLEM F ORMULATION

We shall?rst give a brief discussion on the application sce-narios,before diving into the mathematical formulations.

A.Background and Scenarios

One of the major revolutions brought by mobile crowd-sensing is urban scale information gathering[15]–[17]. Traditionally,these information gathering procedures always rely on professional operators and specialized(high-end)sen-sors(e.g.,traf?c cameras)that have limited coverage.Mobile crowdsensing,on the contrary,makes use of the pervasive availability of human participants,so it can be made more scal-able both spatially and temporally.Nevertheless,the sensing data quality(in terms of data timeliness,relevancy,coverage, etc.[11])in crowdsensing cannot be perfectly guaranteed due to the unprofessional sensors and the casual behaviors of the participants.This problem gets even more prominent given that many crowdsensing systems are designed to involve the least user intervention(a.k.a.opportunistic sensing)[1],[12]. The sensing in these systems is autonomously activated if prede?ned conditions have been satis?ed,which can result in uncertain data quality and high data-missing rate because suf?cient exposure time for sensors may not be guaranteed [9],[12].Therefore,we cannot fully bene?t from mobile crowdsensing in revolutionizing our urban living and working without handling this uncertainty on sensing data quality.

Let us take the dust level sensing task as an example,for which a certain number of participants are chosen in a city to gather information on the dust levels,and?nally get remuner-ated for the data gathered by them.One typical constraint is that a minimum number of participants have to be chosen.This re-quirement is meaningful to many crowdsensing tasks,for ex-ample,the dust level sensing should involve a suf?cient amount of participants to improve the overall value of the collected data and/or to cover a sensing area.While the participants may al-ways deliver a sequence of readings either online or of?ine to claim their remunerations,it is highly possible that not all the readings are quali?ed due to,for example,a wrong placement of the sensing device.Although other sensors can be used to identify and remove unquali?ed data(e.g.,accelerometers can be used to detect if a smartphone is put into pocket[18]where the dust level readings are not quali?ed),the amount of quali-?ed data from each participant(hence the sensing value of this participant to the owner)becomes uncontrollable and random. At the meantime,the task owner has at its disposal a budget and a set of participants to recruit[19].The owner bears the wish that the total sensing revenue got from all participants is max-imized,but it cannot achieve this by a one-time participant se-lection due to the randomness in data quality.Given the amount of budget to support its mobile crowdsensing task for a cer-tain period of time(e.g.,tens of days),a reasonable strategy taken by the owner is to learn the sensing values gradually, smartly reshuf?e the selected participants every day,and aim to minimize the gap on average sense between the achieved total sensing revenue and the one obtained by a genie.To summarize, the robust sensing problem raised by the realistic crowdsensing applications share the following features:

F1:At least a minimum number of participants have to be involved for a crowdsensing task.

F2:The sensing values of individual participants are random.

F3:The owner of a crowdsensing task has a limited budget to recruit participants.

B.Models and Assumptions

Suppose that the owner of a crowdsensing task has a budget to conduct its task that often lasts for a certain amount of time slots(e.g.,tens of days).The owner also has a set of participants

indexed by1at its disposal to actually per-form the sensing.

For each participant,we de?ne his/her sensing value as the amount of quali?ed data that he/she collected within one time slot(e.g.,a day).According to the discussions in Section II.A, this quantity for participant during the-th time slot is ob-viously a random variable,which we denote by

.The value of can only be revealed after participant is selected and has?nished his/her sensing for the-th time slot.Without loss of generality,we assume that the sequence

are non-negative random variables satisfying the condition that for any time slot,but is unknown.Note that no assumptions are made here for different users,allowing the sensing values of different users to be arbitrarily correlated or follow different dis-tributions.We also assume that has normalized supports in,but our results can be easily extended to the case of arbitrary supports of.To evaluate the total sensing revenue,the owner has,for each participant,a weight.This weight may represent several factors related to the prior infor-mation on this participant's ability of performing the sensing task,such as the types and sampling resolution of the partici-pant's sensors.As a result,the sensing revenue obtained by the owner from participant in the-th time slot can be expressed as.

During each time slot,the owner selects a certain number of participants to conduct the crowdsensing task.Without loss of generality,we assume that there exists a basic cost for the owner at each time slot,which is independent of the selection of the users and is the necessary expenditure to keep the system running,such as the cost for renting a server or hiring engineers to maintain the system.We also assume that selecting partici-pant for one time slot costs the owner,which is the cost of rewarding.For the moment,we assume that is static,and we shall handle the case of dynamic sensing costs in Section V. As the sensing costs are assumed static here,we can employ some pre-processing procedures to know them in advance.For example,a Vickery auction can be used to handle the truthful-ness problem and reveal the users'true sensing costs beforehand if the costs are reported by the users.2

To accomplish a meaningful sensing task,the number of par-ticipants selected by the owner during each time slot must be no less than a prede?ned positive integer(see F1of Section II.A).For this requirement,we formally introduce the concept of Feasible Sensing Engagement in De?nition1:

De?nition1(FSE):A Sensing Engagement(SE)is a vector

,where indicates whether par-ticipant is selected for sensing.If,then we term a Feasible Sensing Engagement The set of all FSEs is denoted by.

If a FSE is selected in the-th time slot,the owner would get a sensing revenue,but at a total cost of,where is the cost for remu-nerating the users.As the owner is not sure about the random 1We use to denote the set for any.

2Similar ideas/procedures also appear in other work such as[10],where a separate bargaining process is used to know the sensing costs in advance.sensing values of individual participants,it would adaptively select different FSEs at different points in time.However,due to the budget limit,the total time span for the owner to play this“trial-and-error”procedure is limited.We formally de?ne this procedure as a Robust Sensing Policy:

De?nition2(RSP):A Robust Sensing Policy of the owner with the budget is a sequence for some,where

is the FSE selected for the-th time slot.Also,the policy satis?es.The total sensing revenue of is.The expected total revenue of is:

(1) For convenience,we sometimes omit in our notations if it is clear from the context,e.g.,writing instead of.

Let denote the vector.For,we denote the set of selected participants in by

and use the function to represent the expected revenue of selecting for in one time slot. The minimum and maximum total costs of sensing using an FSE are denoted by and,respectively,where is the sum of and the smallest's and.We summarize some globally used notations in

C.Problem Formulation

Under our Budget LimIted robuSt crowdSensing(BLISS) framework described in Section II.B,a desired objective is to?nd an RSP such that is maximized.In terms of combinatorial optimization,is an optimal solution to the following integer linear programming(ILP)problem [BLISS-ILP].In[BLISS-ILP],constraints(3)and(4)are due to the above de?nitions of FSE and RSP.The variable

denotes whether the sensing task is performed for the-th time slot.Obviously,the number of non-zero is bounded by.Constraint(6)is arti?cially introduced to force these non-zero elements appearing only at the beginning of the time sequence:it con?nes the problem dimension without sacri?cing generality.Constraint(5)states that participants are chosen only when the task is performed.

(2)

(3)

(4)

(5)

(6) Assuming that the expected sensing values are known, BLISS-ILP can be proven as NP-hard:

Theorem1:BLISS-ILP is NP-hard when is known. However,as the expected sensing values are not known in the BLISS problem,it is impossible to solve BLISS-ILP either optimally or approximately;it only serves as a benchmark in our performance evaluation.Consequently,we shall instead

TABLE I

S OME F REQUENTLY U SED N

OTATIONS

adopt an online learning approach to tackle the robust sensing

problem,i.e.,the owner repetitively learns the participants'sensing values and chooses the next FSE accordingly until running out of budget.The goal is then to minimize the differ-ence with respect to the optimal solution computed by a genie,essentially a standard optimization objective in the ?eld of online learning [20].Formally speaking,we aim at an RSP such that the regret is minimized.

III.B LISS O NLINE L EARNING A LGORITHM

A.Motivations

According to Section II.C,we are confronting an “explo-ration vs.exploitation”dilemma under the BLISS framework,i.e.,balancing revenue maximization based on the already acquired empirical knowledge of the sensing values with at-tempting new FSEs to acquire further knowledge.A popular model for solving such a kind of dilemma is the Multi-Armed Bandit (MAB)problem in the area of reinforcement learning [21].In [21],Auer et al.study the problem of regret minimiza-tion for pulling a row of slot machines (or one-armed bandits )with unknown i.i.d.rewards over time,and the rule is to pull exactly one arm each time.The UCB algorithm proposed in

[21]for multi-armed bandits achieves the storage and regret bounds that both grow linearly in the number of arms.

Directly applying UCB to our problem faces two major ob-stacles.Firstly,we have to model each FSE as an arm,hence re-sulting in

arms in total.Consequently,the

regret bound and required storage can grow exponentially in the number of participants.Secondly,pulling arms is assumed to be free in UCB (hence the arms can be pulled for ever),whereas our BLISS framework has a budget limit in selecting FSEs.Ac-tually,in the BLISS problem,we aim to achieve a small re-gret bound while simultaneously respecting the budget limit and the combinatorial nature of FSE selection,which makes the problem extremely challenging.B.Algorithm Details

To conquer the above dif?culties,we propose a BLISS on-line learning algorithm shown by Algorithm 1.In this algo-rithm,we maintain two vectors and

as the empirical knowledge learned from the

history.More speci?cally,is the sample mean of partici-pant 's sensing value at the end of the -th time slot and is the number of time slots that is selected (sampled)by then.At the initialization stage (lines 1–3),the algorithm selects all participants to acquire the initial information and .Then Algorithm 2is invoked for each of the later time slots to se-lect FSEs based on current and (line 7).Instead of di-rectly using the sample means,we introduce a new vector by amending each with an additive factor (line 6),and is actually used for selecting .In fact,serves as an upper con?dence bound on ,i.e.,is no more than with high probability when gets large.This property comes from the Chernoff-Hoeffding bound and will be used in the re-gret analysis for Algorithm 1.

Algorithm 2adopts a greedy strategy that selects an FSE with the maximum Revenue-Cost Ratio (RCR henceforth)pa-rameterized by .In other words,Algorithm 2should return

so that

is maximized.This

is essentially a fractional programming [22]problem and we solve it optimally in polynomial time by Algorithm 2based on a parametric sorting method.More detailed explanations and analysis of Algorithm 2are given in Section IV.A.The intu-ition behind the greedy strategy adopted by Algorithm 2can be explained as follows.We observe that a trivial upper bound for

is ,where is the maximum

RCR with respect to .As is unknown,we have to select at each time slot to maximize the RCR using the estimated (i.e.,using ),so that is most likely to be close to .This may potentially get us a total revenue close to hence a small regret.

After identifying ,Algorithm 1checks if its cost is bigger than the current leftover budget (lines 8–11).If so,the algo-rithm stops and returns the FSEs selected so far as well as the number of time slots during which the crowdsensing task has been performed.Otherwise,it employs the participants indi-cated by to perform sensing for the -th time slot and subtracts from the current budget.The sensing values

learned during this time slot are then used to update the empir-ical knowledge.Finally,Algorithm1outputs and,where

is the number of time slots during which crowdsensing has been performed by Algorithm1.

IV.P ERFORMANCE A NALYSIS

In this section,we provide theoretical performance anal-ysis for the BLISS online learning algorithm proposed in Section III.We will?rst prove the optimality of Algorithm2 in Section IV.A,and then prove the regret bound of Algorithm 1in Section IV.B.

A.Optimality of the FSE Selection

As we mentioned before,the objective of Algorithm2is to ?nd an FSE that maximizes RCR based on the estimated sensing value vector.A critical building block of Algorithm 2is the function,which is a parametric comparison func-tion called by the sorting process in Algorithm2.To prove the optimality of,we?rst reveal two important features of the function,as shown by Lemma1:

Lemma1:Suppose that the function

returns.If.Then we must have

i).

ii)

Clearly,hence is guaran-teed to be in after line1of Algorithm2is executed.As the parametric(bubble)sorting process in lines3–6of Algo-rithm2repetitively calls the function to compare partic-ipants while at the same time to shrink,we know that

is never excluded from due to i)of Lemma1.Moreover, after the sorting is completed,the sequence is a permutation of that satis?es the following property due to ii)of Lemma1:

(7) where is any pair of integers that satis?es. Based on these,lines7–10of Algorithm2then?nd the optimal ,and the correctness of Algorithm2is proved by Theorem 2:

Theorem2:.

We also show that Algorithm2is a polynomial-time algo-rithm.

Theorem3:The average time complexity of Algorithm2is .

B.Regret Bound for BLISS

Now we are ready to prove the regret bound of Algorithm 1.The overall idea of the proofs is the following:we show that the expected participant sensing values learned by Algorithm 1do not deviate much from the real values,so we do not suffer a big loss by using them for selecting FSEs,compared with using the real sensing value expectations for selection.Note that the number of time slots during which Algorithm1performs crowdsensing(i.e.,)is a random variable,hence most of our proofs are based on conditional probabilities with respect to. Let for https://www.sodocs.net/doc/40408556.html,ing the Chernoff-Hoeffding bound[21],[23],we can show that if a participant is selected for a suf?cient number of times in the history(i.e.,

is suf?ciently large),then the sample mean of's sensing value will be close to the real expected sensing value with high probability.In other words,

(8) Let,i.e.,is the set of sub-op-timal FSEs with RCRs less than,computed by using the real expected sensing values of the participants.If,then let and be the smallest and largest discrepancy between and the RCR of any FSE in,i.e.,

and.Let

.Let the event

.Based on(8)and Theorem2,the following lemma reveals that,if all the selected participants for the-th time slots have been selected in the past for a suf?cient number of times(greater than),then with high proba-bility the FSE chosen for the-th time slot would maximize the RCR with respect to the real expected sensing values. Lemma2:For any,we have

.

Lemma2will be used by Lemma3to bound the expected number of sub-optimal FSEs selected by the algorithm when the involved users have been selected in the history for suf?-cient number of times(i.e.,bigger than the threshold). However,it is also possible that some sub-optimal FSEs can be played when the involved users have not been suf?ciently se-lected in the history.Fortunately,the proof of Lemma3shows that the expected number of such kind of FSEs is no more than ,hence the expected number of all sub-optimal FSEs selected by BLISS can be bounded by: Lemma3:Let.We have

.

Bounding the number of sub-optimal FSEs chosen by Algo-rithm1through Lemma3allows us to further bound the total regret of choosing these FSEs.Moreover,repetitively choosing optimal FSEs is an approximation policy to BLISS-ILP,whose regret can also be derived.Based on these ideas,we prove the regret bound of Algorithm1in Theorem4.Note that the factor in the regret bound actually accounts for the regret caused by playing sub-optimal FSEs when the involved users are insuf-?ciently sampled.

Theorem4:The regret of Algorithm1is no more than

.

As a special case,note that if, then we have and,hence the regret bound shown in Theorem4would be a constant.

Note that we only consider as a variant in the regret bound as it determines how many time slots the BLISS learning algo-rithm can run.The asymptotical optimality of this regret bound can be further proven as follows.

Theorem5:Any algorithm for BLISS has a regret of at least .

Note that when,the average regret of Algorithm1 per time slot goes to0.This implies that Algorithm1is Hannan consistent[20].

V.D YNAMIC S ENSING C OSTS

So far,we have assumed that the sensing costs of the users are static in the BLISS online learning algorithm.This is suit-able under the case that the users'willingness for participating crowdsensing is stable,i.e.,once a user is satis?ed with a re-muneration amount for him/her to participate in crowdsensing, he/she will not change his/her mind all the time.However,in some cases the users may decide different acceptable remunera-tion amounts in different time slots,either due to some practical reasons or simply due to the users'temperamental moods.For example,the users may demand higher remunerations in their busy days and lower remunerations in their holidays(or vice versa).In such cases,the sensing costs for rewarding the users become dynamic,and are not known to the owner in advance. In this section,we tackle the robust crowdsensing problem formulated in Section II.C under the scenario where both the sensing revenue and the sensing costs of the users are uncertain. We will propose a new online learning problem called D-BLISS for this case,where the users'sensing costs(i.e.,willingness)in different sensing periods are assumed to be i.i.d random vari-ables with a known support.Furthermore,as it is natural for the users to monetarize their willingness on partici-pating in crowdsensing,the possible values of the users'sensing costs can be further represented by a?nite set, where.For example,if we know that all users are willing to perform crowdsensing in one time slot for1dollar but no one is likely to do the job for1cent, then we can set,and obviously.

A.The D-BLISS Algorithm

Our D-BLISS algorithm is based on a posted-price scheme described as follows.At the beginning of each time slot,the sensing task owner selects no less than users(according to F1)as a sensing group and offers them a take-it-or-leave-it price .The selected users then report their individual decisions about if they accept the price or not,and any user would accept the posted price if and only if his/her willingness is no more than that price.If all the selected users accept the posted price, then the owner has made a good choice and it uses the selected users to perform crowdsensing in that time slot as well as re-wards each of them according to the posted price.Otherwise, the owner has failed on selecting the posted price for recruiting the users,hence the crowdsensing will not be performed in that time slot.

It is worth noting that such a posted-price scheme also en-forces truthfulness,i.e.,no user has the incentive to act“strate-gically”in order to maximize his/her own utility.This can be further explained as follows.If any user's sensing cost is bigger than the posted price,then he/she is better off acting truthfully (i.e.,denying the task to get a0utility)because otherwise he/she will get a negative utility for accepting the sensing task.By similar reasoning,if the user's sensing cost is no more than the posted price,then he/she is better off accepting the task,as denying the task brings him/her no more utility than accepting the task.

However,the owner is confronted with the dif?culty that the users'willingness is dynamically changing according to

unknown distributions;and it has to beat the tradeoff between price and revenue:if the posted price is set too high,a waste on the budget will be caused because a lower price could have been suf?cient for recruiting the users;on the other hand,if the posted price is set too low,then it might happen that the users will deny it and hence the owner gets 0sensing revenue for the time slot but still suffers a basic cost for keeping the system running.

To tackle the challenge described above,D-BLISS learns the sensing revenue and sensing costs of the users simultaneously ,and uses the knowledge learned to select the users and prices.To implement this idea,some new notations are introduced,com-pared with Algorithm 1presented in Section III.Firstly,we in-troduce the variable to denote the sample-mean esti-mation of by the th time slot,where is the probability that a user's willingness is no more than ;and the variable is introduced to denote the total number of users that has been offered to at the end of the th time slot.Secondly,we replace the set in BLISS by the set

;each is still called a FSE,but the

th component of denotes the index of a selected price

in .For notational simplicity,we abuse the function a little by setting

.Finally,according to our posted-price scheme,we replace the function in BLISS by a new function which is de?ned

as

;this function calculates the expected revenue of a selected FSE given the vec-tors and ,where is actually the probability that all users in accept the price

.Similarly,the function in BLISS is also substituted by the function for the expected cost of enrolling the users,which

is de?ned as

.In the initializing stage of D-BLISS (lines 1–5),the algorithm tries every user and every price to acquire the initial knowledge of and .Then the algorithm uses a while loop (lines 6–12)to select an FSE for each sensing period,and updates the variables accordingly using a procedure (lines 15–30)similar to that in Algorithm 1.

The most salient difference between D-BLISS and BLISS is the FSE-selection method.Instead of adjusting each by an “exploration factor”to ?nd the FSE for the -th time slot as that in lines 6–7of Algorithm 1,the D-BLISS algorithm directly ?nds the FSE as the solution to an optimization problem (i.e.,maximizing the function Ξ,see line 8).Given and

,the function Ξis de?ned as follows:

Ξ

Actually,the design of Ξis closely associated with the regret analysis of the D-BLISS algorithm,and more insights about it can be revealed by checking the proofs of the theorems proposed in the next section.

Note that maximizing Ξis not a fractional programming problem,hence the method presented in Algorithm 2cannot

be applied.A naive solution for maximizing Ξis to enumerate all the FSEs in ,but with exponential time.Fortunately,we ?nd that we can maximize Ξin polynomial time using Algorithm 4,whose idea is roughly explained as follows:given a posted price and the minimum value of the 's of all the selected users ,the optimal FSE for maximizing Ξmust be the one that has the largest values on the 's;hence we can enumerate on all the posted prices to solve the problem optimally.To implement this idea ef?ciently,we generate some ordered sequences in Algorithm 4(lines 2–6)before the enumeration (lines 7–14)begins.For example,for each ,we select different numbers from

to form a sequence which satis?es:(i);(ii)For any is at least the biggest element in the set .Using these ordered sequences,Algorithm 4achieves a time complexity of .B.Regret Analysis for D-BLISS

We denote the robust sensing policy output by Algorithm 3by ,where is the FSE selected for the -th slot.The expected revenue of

can be written as .Suppose that the optimal policy for the owner with budget under the dynamic-cost case is ,we will give an upper bound for the regret in this section.

The regret analysis of D-BLISS is more complex than that of BLISS,as both the sensing values and the sensing costs are un-certain.Actually,it is non-trivial even to derive an upper bound of ,which is tackled by Theorem 6.

Theorem6:For any,let. Let.We have

.

As Theorem6has bounded the the optimal revenue using,we then try to quantify the number of FSEs selected by Algorithm3which are inferior to the optimal FSE.Intuitively,the more times the users and prices are sam-pled,the more accurate knowledge the owner will get,hence the probability that the owner selects an“inferior”FSE can also be reduced accordingly.In the following,we convert this in-sight into a concrete mathematical formulation by Lemma4and Lemma5.

Lemma4:For any and any,we have and

.

Lemma5:Let be the set

Let the event where

.Let.For any,we have. Using Lemma5,we can further bound the expected number of inferior FSEs selected by Algorithm3,as shown by Lemma 6.Based on Lemma6,the regret of our algorithm is proved by Theorem7.

Lemma6:Let.We have

.

Theorem7:Let and

.The regret of Algorithm1is no more than.

As the number of running time of3is ,it can be seen from Theorem7that the average regret of D-BLISS per time slot converges to0when. which means that Algorithm3is also a Hannan-consistent(or “no-regret”)learning algorithm.

VI.S IMULATIONS

In this section we evaluate the performance of our online learning algorithm through extensive simulations.The simula-tions focus on the effect of various crowdsensing conditions on the performance of sensing policies generated by Algorithm1, Algorithm3(denoted by BLISS and D-BLISS in the simula-tions,respectively)and other related algorithms.

A.Performance Evaluation for BLISS

To the best of our knowledge,the closest algorithm that can be adapted to the setting of BLISS is the LLR algorithm pro-posed by[24]:it solves network optimization problems(e.g., maximum weighted matching)under a stochastic MAB model. However,as the costs for pulling arms are neglected in LLR,its policy behaves in a“myopic”way in our scenario by selecting all the participants in every time slot to maximize the short-term sensing revenue.We also implement a straightforward policy, RANDOM,that randomly selects an FSE in each time slot.

In all our simulations for BLISS,the weight of any partici-pant(i.e.,)is randomly generated from the uniform distri-bution,and is generated by the same method.The sensing value of participant in any time slot is randomly sampled from two candidate distributions:the?rst one is the truncated Gaussian distribution with mean,stan-dard deviation,and support,and the second one is the uniform with support.

1)On Regret:We?rst compare the regrets of different al-gorithms.Unlike the other MAB algorithms such as[21],a major dif?culty for us to evaluate the regret of BLISS is the NP-hardness of computing the optimal solution(see Theorem 1).Therefore,we only compare the regrets of BLISS,LLR and RANDOM under a small case where,and the budget scales from10to300with an increment of10.For any participant,the expected sensing value is generated ran-domly such that the support of belongs to. In such a small case,BLISS-ILP can be solved optimally using an ILP solver(e.g.,CPLEX[25])in reasonable time,and we can compute the optimal solution and hence the regrets of all

Fig.1.Performance evaluations for BLISS and D-BLISS.(a)Testing the regret of BLISS.(b)Testing the regret of D-BLISS.(c)Testing the revenue of

D-BLISS.

Fig.2.Performance comparisons when the participants'random sensing values are drawn from homogeneous distributions.(a)Scaling .(b)Scaling .(c)Scaling

.

Fig.3.Performance comparisons when the participants'random sensing values are drawn from heterogeneous distributions.(a)Scaling .(b)Scaling .

(c)Scaling .

algorithms.The results are shown in Fig.1(a),where each al-gorithm's regret is normalized with respect to the logarithm of -the number of time slots during which the algorithm performs crowdsensing.

Obviously,the regret of BLISS is much lower than those of LLR and RANDOM (note the logarithmic scale of the -axis).Actually,the normalized regrets of both LLR and RANDOM grow linearly with respect to ,whereas that of BLISS levels off to a constant.Since (as ),the results in Fig.1(a)also strongly corroborate the theoretical regret bound proved in Section IV.B.

2)On Sensing Revenue:We then study the performance of different algorithms in terms of the total sensing revenue,which is the actual bene?t the owner gains in practice.The results are shown in Figs.2and 3,where is randomly generated from the uniform distribution for any participant .In Fig.2,all participants'sensing values are sampled from Gaussian distributions,whereas they are sampled from both Gaussian and uniform distributions with equal chance in Fig.3.All the ?gures show the statistical summaries (i.e.,means and standard deviations)of 100simulation results.

We study the impact of budget on the sensing revenue in Figs.2(a)and 3(a),where we set and scale the budget from 1000to 10000with an increment of 1000.

The sensing revenue of all the algorithms increasing with the budget can be easily understood:more participants can be em-ployed for sensing under a larger budget.In Figs.2(b)and 3(b),we set and scale from 100to 1000with an increment of 100.In this case,the revenue of BLISS exhibits an uptrend with the in-creasing of ,whereas those of LLR and RANDOM do not change much.This can be explained by the reason that BLISS intelligently selects participants based on their sensing values and costs,so a larger group of participants brings a larger space for selection and hence a higher revenue.On the contrary,LLR and RANDOM either myopically or blindly select participants,which makes their revenue insensitive to the enlargement of par-ticipant groups.

In Figs.2(c)and 3(c),we study the relation between the sensing revenue and by setting and scale from 10to 100with a step of 10.The results show that LLR and RANDOM give similar revenue under all the values of ,while the revenue obtained by BLISS drops with the in-crement of .The reasons for this phenomenon is that a larger results in a smaller selection space for BLISS.Actually,the revenue got by all algorithms would be similar when is very

close to,because all of them have to select all the participants at each time slot under the extreme case where.

We can further make the following observations by com-paring the three algorithms and contrasting Figs.2and3.?The sensing revenue obtained by BLISS is signi?cantly larger than those obtained by LLR and RANDOM under all the cases,demonstrating the superiority of BLISS under various crowdsensing conditions.

?BLISS is insensitive to the distributions of the sensing values:it outperforms other algorithms without affected by the speci?c sensing value distributions that the participants follows.

B.Performance Evaluation for D-BLISS

In this section we evaluate the performance of D-BLISS pre-sented in Section V.The methods for generating random param-eters for D-BLISS are mostly the same as that in Section VI.A, except that the sensing cost of each user is re-generated at each time slot by sampling from the uniform distribution. Without loss of generality,the basic cost is set to0.5.

To the best of our knowledge,D-BLISS is the?rst MAB algorithm that allows for the dynamic arm-playing costs and the combinatorial arm-selection strategy with a budget.As it is hard to adapt LLR to the dynamic-sensing-cost case,we only adapt RANDOM for comparisons in this section,and the adapted RANDOM algorithm is called D-RANDOM.At each time slot,D-RANDOM randomly selects a price in and no less than users in as an FSE.

1)On Regret:Due to the prohibitive time complexity of computing the optimal solution,}we compare the regret of D-BLISS and D-RANDOM under a small case where

, and the budget scales from10to300with an increment of 10.It can be seen from Fig.1(b)that the regret of D-BLISS signi?cantly outperforms D-RANDOM and grows slowly when increases,which is in accordance with the theoretical regret bound proved by Theorem7.

2)On Sensing Revenue:We study the performance of dif-ferent algorithms in terms of the total sensing revenue,where is set to and is randomly generated from the uniform distribution for any participant .The results are shown in Fig.1(a).Again,we can see that D-BLISS shows distinct superiority on the sensing revenue when the budget increases.

C.Performance Evaluation for Mixing Sensing Values

In this section,we study the performance of our algorithms when the observations on a user are temporally dependent,but the dependence weakens over time.This is a natural assump-tion as people's behavior can be more independent over a long time period than over a short time period.In fact,such a depen-dence-weakening assumption is widely adopted in the machine learning literature to model the non-i.i.d.cases

in practice,and it is usually known by the concept of“mixing”[26],[27].

In Fig.4(a),we compare the regrets of BLISS,LLR and RANDOM under a case of mixing sensing values,where the sequence of any user's sensing values is a correlated Gaussian Fig.4.Regret of BLISS and D-BLISS for mixing sensing values.(a)BLISS.

(b)D-BLISS.

sequence with the covariance between any

and de?ned by the function

,and all the other parameters are set as the same with those in Fig.1(a).Similarly,in Fig.4(b)we compare the regrets of D-BLISS and D-RANDOM under a case where both the sensing values and sensing costs of any user are correlated Gaussian sequences with the covariances de?ned by the func-tion COV,and all the other parameter settings are the same with those in Fig.1(b).Clearly,both BLISS and D-BLISS outper-form the other compared algorithms.This proves that our algo-rithms are robust against temporally-dependent sensing values with the dependence weakening over time.

VII.R ELATED W ORK

Recently,there has been a substantial growth on designing crowdsensing systems for various applications,such as(ve-hicle)traf?c monitoring/prediction[15],localization[12], parking space allocation/searching[16],and ambient(e.g.,dust level)surveillance[17].At the same time,theoretical investiga-tions on managing crowdsensing has also been conducted,but most of them concentrate on the incentive problems of crowd-sensing[3]–[6].Although certain data-quality related problems for crowdsensing have been raised in[8],[9],[11],[13],none of them has considered the problem of handling data-quality uncertainty by intelligently recruiting the participants,as we have done in this paper.More detailed surveys on the literature of crowdsensing can be found in[1],[2].

The study on stochastic MAB problems is pioneered by Lai et al.[28]and Auer et al.[21],who provide algorithms with re-gret bounds growing logarithmically with respect to the number of arm-pullings.Following them,extensive proposals on mis-cellaneous MAB problems(e.g.,the restless bandits[29])have been proposed,and an excellent survey can be found in[14]. However,all these proposals assume that playing arms is free. The problem of considering arm-playing costs for MAB only starts to attract attention very recently in[30],[31];both of them are based on extensions of the renowned UCB1algorithm[21], under the assumption that the arm-playing costs are static and known.

We note that all the proposals in[28],[21],[30]and[31]adopt the classical MAB model,i.e.,exactly one single arm can be played at each step.This playing rule is revised by a recent work [24],where multiple arms with certain combinatorial structures can be played at the same time.The authors of[24]indicate that traditional MAB algorithms that play one arm at each step

(e.g.,UCB1in[21])perform poorly in such a scenario and pro-pose new algorithms with provable regret bounds.Another re-cent work[32]also allows for playing multiple arms simultane-ously,with a more stringent constraint that exact out of arms should be played at each step.

Nevertheless,as[24]and[32]still assume that playing arms is free(the same as[28],[21]),the arms can be played perpetu-ally to acquire suf?cient knowledge about them.Consequently, none of the existing work in[21],[24],[28],[30]–[32]?ts the BLISS problem studied in this paper.Actually,to the best of our knowledge,we are the?rst to consider a stochastic MAB model where multiple arms(participants)with(static or dy-namic)costs can be played simultaneously under a combina-torial structure and the regret should be minimized subject to a budget limit,and we believe that our framework can be applied in other scenarios in the networking area as well as motivate new online learning algorithms in the machine-learning area.

VIII.D ISCUSSION

We would like to indicate that our learning algorithms and re-gret analysis can also be easily extended to some cases where the observations on users are mixing random variables with weak-ening dependence.The key idea is to simply replace the Cher-noff-Hoeffding bound used in our paper by some existing con-centration inequalities for mixing random variables(e.g.,those proposed in[27],[33],[34]),whereas our algorithms,theorems and proofs remain almost the same.Due to the page limit,we leave a thorough study towards this direction as a topic for fu-ture work.Another interesting problem is that there may be mul-tiple requesters simultaneously asking for the sensing service. Studying the competition and equilibrium problems in this sce-nario will also be the topic of our future research.

IX.C ONCLUSION

We have raised and considered a novel but practical robust crowdsensing problem,where the quality of sensing data ac-quired by the participants are uncertain and a crowdsensing task owner aims to maximize its expected total sensing rev-enue under a limited budget for compensating the sensing costs. To ef?ciently tackle such a problem,we have formulated the Budget LImited robuSt crowdSensing(BLISS)as a high-level general framework to characterize it,and we have further pro-posed two online learning algorithms for the static-sensing-cost case and for the dynamic-sensing-cost case,respectively.We have shown that our learning algorithms have logarithmic re-gret bounds and achieve Hannan-consistency,and also achieve asymptotical optimality under the static-sensing-cost case.Fi-nally,we have conducted extensive simulations and the simula-tion results have strongly demonstrated the effectiveness of our approach.

A PPENDIX

Proof of Theorem1:The NP-hardness of BLISS-ILP can be proved by reduction from the NP-complete Knapsack problem or the Partition problem[35].We omit the detailed proof due to the lack of space.

Proof of Lemma1:We?rst prove i).Note that only lines 19–20of Algorithm2can make.If line19is executed,then there must exist such that

If in this case,then we must have

which yields;a contradiction.Hence

.Similarly,we can prove that the execution of line20also guarantees.This completes the proof for i). Now we prove ii).For simplicity,we assume and the case of can be proved by symmetry.Due line 13of Algorithm2,we have,and.When

,we get to line 15.in this case for any we have Hence ii)holds.Similarly,we can also prove that ii)is true for

(line16).If neither line15nor line16is executed,we must have according to line 17.In this case,if line19is executed and, we must have

hence ii)still holds.Similarly,the execution of line20allows ii)to hold for.

Proof of Theorem2:According to Lemma1,after com-pleting the parametric sorting process,we have. Moreover,for and any,we have

(9) https://www.sodocs.net/doc/40408556.html,ing(9)we know that,for any:

(10) Clearly,(10)equals0when.On the other hand,if(10) equals0and,then we have

and hence,a contradiction.Similarly we can prove that does not hold if(10)equals0.In other words, (10)is equal to0iff.Now the theorem follows from lines 7–10.

Proof of Theorem3:The dominant running time of the func-tion is spent on line18,done by a quick sort in

time.Therefore,the loop in lines3–6runs in time.

The time spent on lines7–10is.So the overall time com-plexity of Algorithm2is.However,if we replace the bubble sorting framework(used to simplify our presenta-tion)in lines3–6by a quick sorting framework,then Algorithm 2can be implemented in time on average. Proof of Lemma2:Suppose that holds.Let

.Using Theorem2we get

.Besides,

So we have

(11) Note that when,we must have

(12) Combining(11)and(12)yields

(13) Case1:;

In this case,we must have according to(13).Since

,we have

and hence.As

,we get

.

Case2:;

In this case,we have. Synthesizing Case1and Case2,have

(14)

(15) where(14)holds because of the union bound and(15)is due to (8).So the lemma follows.

Proof of Lemma3:We de?ne a set of random variables

as follows.For any,let .For any,if,then we?nd

(breaking ties arbitrarily)and set

.Any not involved in

rule the same with.Let .We have

(16)Notice that if,we must have

,which implies

.Since and,we have Hence

(17) where(17)holds due to the Riemann zeta function

[36].

Proof of Theorem4:Let. Since,we have

(18) Note that for any,we have,hence

(19) On the other hand,

(20) Combining(18),(19)and(20)we get

(21) Note that and

(22) Therefore,using(21),(22)and Lemma3we get

Hence the theorem follows because.

Proof of Theorem5:Given an instance of the traditional MAB problem studied in[21]with arms,we can construct

an instance of BLISS as follows.Let the number of participants be and let.For any,let and .Suppose by contradiction that there exists an algorithm for BLISS that has a regret of.Then we can apply to the BLISS instance constructed above and get a policy that runs for at least time slots.More-over,we can convert into a valid solution to the traditional MAB problem by the following method:for each,we pull arm once in.It can be seen that the regret of is no more than that of,and has at least

rounds(i.e.,the number of arm-pullings).This contradicts the fact shown in[28]that the lower bound of the-round cumu-lative regret for the traditional MAB problem with arms is .

Proof of Theorem6:Let be the event that the price

is accepted by all the users in.Let

be the random variable that denotes the total sensing revenue gained by employing the policy.Clearly we have

(23) Now we prove the theorem by induction on the value of. We?rst consider the case that.In this case,we

that any policy runs for only time slot.Moreover,when

,using we get

and hence

When we have.Overall, we get

Suppose that the theorem holds for.For the case of,we have

(24)

(25) Combining(24),(25)with the inductive assumption gives us

(26) Combining(23)and(26),we get

Hence the theorem follows.

Proof of Lemma4:The lemma can be proved by an utiliza-tion of the Chernoff-Hoeffding bound[23].

Proof of Lemma5:For any,let

.Recall that

.Suppose that holds. According to line8of Algorithm3,we know that

, which implies that at least one of the following inequalities holds:

i)

ii)

iii)

Suppose that iii)holds when the event happens.Then we have

As,we get,a contradic-tion.

Now suppose that i)holds,then we must have:

or

.Otherwise let and ,we have

which is a contradiction.By similar reasoning,we can prove that ii)being true implies

or

Synthesizing the above results,we get

So the lemma follows.

Proof of Lemma6and Theorem7:The proofs are similar to those of Lemma3and Theorem4,respectively,hence are omitted.

A CKNOWLEDGMENT

This work is done when K.Han was a visiting fellow at Nanyang Technological University.

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Kai Han received the B.S.and Ph.D.degrees in computer science from the University of Science and Technology of China,Hefei,China,in1997and2004, respectively.

He is currently a Professor in the School of Computer Science and Tech-nology,University of Science and Technology of China.His research inter-ests include computer networks,combinatorial optimization,algorithmic game theory,and machine learning.

Chi Zhang received the B.S.degree from Zhejiang University,China,in2011. He is currently pursuing the Ph.D.degree in the School of Computer Engi-neering,Nanyang Technological University,Singapore.His research interests are wireless sensor networks and social networks.

Jun Luo received the Ph.D.degree in computer science from the EPFL(Swiss Federal Institute of Technology in Lausanne),Lausanne,Switzerland,in2006. He is currently an Associate Professor at the School of Computer Engi-neering,Nanyang Technological University,Singapore.His research interests include wireless networking,mobile and pervasive computing,as well as applied operations research.

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电动工具十大品牌榜中榜/名牌电工(2010) 1 博世BOSCH (电动工具十大品牌,1886年德国,世界品牌,博世电动工具(中国)有限公司) 2 牧田MAKITA (创于1915年日本东京,专业大型电动工具的制造商,牧田(中国)有限公司) 3 日立Hitachi (1910年日本,世界知名品牌,十电动工具十大品牌,日立(中国)有限公司) 4 百得Black&Decker (1910年美国,全球顶高品质工具类制造商,百得(苏州)电动工具有限公司) 5 得伟DEWALT (始于1910年美国,专业著名电动工具制造商,得伟(中国)有限公司) 6 博大BODA (中国名牌,浙江名牌,浙江省著名商标,浙江博大电器有限公司) 7 东成(国内综合规模较大的电动工具专业制造企业,江苏东成电动工具有限公司) 8 麦太保metabo (于1924年德国,著名专业电动工具制造商,麦太保电动工具(中国)有限公司) 9 国强(于1986年,拥有自营进出口经营权,江苏国强电动工具有限公司) 10 铁锚(十大电动工具品牌,行业知名品牌,产品远销国外,江苏铁锚电动工具有限公司) 2010电工十大品牌排名 1 松下Panasonic (于1918年日本,中国驰名商标,世界品牌,日本松下电器(中国)有限公司) 2 TCL-罗格朗 (中国驰名商标,电工十大品牌,TCL-罗格朗国际电工(惠州)有限公司) 3 霍尼韦尔-朗能Lonon (,广东名牌,广东省著名商标,霍尼韦尔朗能电器系统技术(广东)有限公司) 4 西蒙Simon (1916年西班牙,全球著名的通信布线品牌,西蒙电气中国有限公司) 5 西门子Siemens (电工十大品牌,始于1847年德国,全球最大的电气和电子公司) 6 梅兰日兰Merlin Gerin (施耐德电气旗下全球配电领域首选品牌之一,天津梅兰日兰电气集团) 7 松本SOBEN (电工十大品牌,广东省名牌产品,广东著名商标,松本电工实业有限公司) 8 德力西DELIXI 中国民营企业500强,国家大型工业企业,浙江德力西国际电工有限公司) 9 IDV (行业知名品牌,专业生产等高档电工产品,惠州爱帝威电工科技有限公司) 10 鸿雁 (中国驰名商标,浙江名牌,浙江省著名商标,杭州鸿雁电器有限公司)

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