Quantile Search: A Distance-Penalized Active Learning Algorithm for Spatial Sampling

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1 Quantile Search: A Distance-Penalized Active Learning Algorith for Spatial Sapling John Lipor 1, Laura Balzano 1, Branko Kerkez 2, and Don Scavia 3 1 Departent of Electrical and Coputer Engineering, 2 Departent of Civil and Environental Engineering, 3 School of Natural Resources and Environent University of Michigan, Ann Arbor {lipor,girasole,bkerkez,scavia}@uich.edu Abstract Adaptive sapling theory has shown that, with proper assuptions on the signal class, algoriths exist to reconstruct a signal in R d with an optial nuber of saples. We generalize this proble to when the cost of sapling is not only the nuber of saples but also the distance traveled between saples. This is otivated by our work studying regions of low oxygen concentration in the Great Lakes. We show that for one-diensional threshold classifiers, a tradeoff between nuber of saples and distance traveled can be achieved using a generalization of binary search, which we refer to as quantile search. We derive the expected total sapling tie for noiseless easureents and the expected nuber of saples for an extension to the noisy case. We illustrate our results in siulations relevant to our sapling application. I. INTRODUCTION In any signal reconstruction probles, the cost associated with taking a saple is in soe way dependent on the distance between saples. For exaple, consider sapling the Great Lakes with a sall autonoous watercraft in order to reconstruct a spatial signal. Lakes such as Lake Erie span an area on the order of thousands of kiloeters; hence the tie it takes for the boat to reach the next saple location is a ajor part of the sapling cost. In the traditional or passive sapling scenario, the saple locations (or saple ties are chosen a priori, often on a unifor grid. Contrasting this is the field of active sapling or active learning wherein saple locations ay be chosen as a function of all previous locations and their corresponding easureents. Active learning theory has shown that binary search guarantees optial sapling rates [1], but this theory ignores any sapling cost other than the nuber of saples taken. Our proble of interest is one in spatial sapling where the spatial region of interest is relatively large, and the total sapling tie is a function of both the nuber of saples required and the distance traveled throughout the sapling procedure. Our contribution is an algorith, tered quantile search, that provides a tradeoff between the nuber of saples required and the distance traveled in finding the change point of a one-diensional threshold classifier. At its two extrees, quantile search iniizes either the nuber of saples or the distance traveled, with a tradeoff achieved by varying a This work was supported by NSF ECCS and F Graduate Research Fellowship Progra. Fig. 1: Dissolved oxygen concentrations in Lake Erie. Points represent saple locations and solid black lines delineate the central basin. Source: search paraeter. We derive and analyze this algorith in the noise-free and noisy cases and study it in siulations with an application to environental onitoring, the study of oxygen levels in the Great Lakes. The reconstruction of this signal is of interest to water quality experts in order to ensure the balance and quality of water in this region [2]; we show an estiate interpolated fro historical easureents in Figure 1. Where as binary search takes saples at the bisection of a probability distribution, or the 2-quantile, quantile search instead saples at the first -quantile. For a preview of our results, suppose that we wish to estiate a threshold θ on a one diensional interval. For noiseless saples, our results show that the expected estiation error after n easureents is ( ( E[ ˆθ n θ ] = ( 2 n 1 +, (1 4 which generalizes to binary search E[ ˆθ n θ ] = 2 (n+2. The expected distance traveled is E[D] = 2 2, (2 which also generalizes to binary search where E[D] = 1. Therefore, larger values of allow for a tradeoff between nuber of saples and distance traveled. The reainder of this paper is organized as follows. In Section II, we otivate and forulate the proble of interest. In Section III we place this work into the context of the current literature on active learning. Section IV describes

2 the quantile search algorith for both the noise-free and noisy cases. In Section V we verify the theory presented via siulation and apply the algoriths described to a proble of interest. Conclusions and directions of future research are given in Section VI. II. MOTIVATION AND PROBLEM FORMULATION The proble of interest for this work is the initial otivating proble described in the Introduction, where we wish to deterine hypoxic regions in the Great Lakes. The spatial extent of low oxygen concentration or hypoxia is a strong indicator of the health of the lakes [3]. Studies perfored by the National Oceanic and Atospheric Adinistration on Lake Erie utilize a nuber of easureent stations on the order of tens, as well as hydrodynaic odels, in order to provide a warning syste for severe hypoxia in the central basin. We wish to deterine the spatial extent of the hypoxic region in the central basin of Lake Erie, a phenoenon that occurs yearly in the late suer onths. A region is deeed hypoxic if the dissolved oxygen concentration is less than 0.2 pp [3]. Further, we assue the hypoxic zone is one connected region with a sooth boundary. The proble can then be considered a binary classification proble, with spatial points receiving a label 0 if they are hypoxic and 1 otherwise, where the desired spatial extent corresponds to the Bayes decision boundary. To perfor sapling, an autonoous watercraft with a speed ranging fro /s will be used. Further, since the axiu spatial extent ay occur at one of any depths, a depth profile ust be taken at each point, increasing the tie per saple significantly. Finally, the hypoxic zone can only be considered approxiately stationary for a period of tie less than one week. We therefore seek to estiate the decision boundary while siultaneously iniizing the total tie spent sapling. Following [4], we split our proble into several one diensional intervals, a process that is described further in Section V. In each interval we ust find a threshold beyond which the lake is hypoxic. Define the step function class F = {f : [0, 1] R f(x = 1 [0,θ (x} where θ [0, 1] is the change point and 1 S (x denotes the indicator function which is 1 on the set S and 0 elsewhere. In contrast to the active learning scenario, our goal is to estiate θ while iniizing the total tie required for sapling, a function of both the nuber of saples taken and the distance traveled during sapling. Denote the observations {Y i } n i=1 {0, 1}n as saples of an unknown function f θ F taken at saple locations on the unit interval {X i } n i=1. With probability p, 0 p < 1/2, we observe an erroneous easureent. Thus { f θ (X i with probability 1 p Y i = = f(x i U i, 1 f θ (X i with probability p where denotes suation odulo 2, and U i {0, 1} are Bernoulli rando variables with paraeter p. While other noise scenarios are coon, here we assue the U i are f(x θ = x Fig. 2: Threshold classifier in one diension with change point θ = 1 3. independent and identically distributed and independent of {X i }. This noise scenario is of interest as the otivating data (hypoxia is a thresholded value in {0, 1}, where Gaussian noise results in iproper thresholding of the easureents. The extension to nonunifor noise (e.g., a Tsybakov-like noise condition as studied in [17] reains as a topic for future work. See Figure 2 for an illustration of f θ (X. As stated previously, our goal is to estiate θ using as little tie as possible. To this end, the saple location at step n is chosen as a function of all previous saple locations and easureents as well as the noise. III. RELATED WORK A nuber of active learning algoriths exist to estiate θ while iniizing the nuber of saples taken. Binary search or binary bisection has been studied on this proble and on any other active learning probles. Consider binary bisection applied to the signal in Figure 2, where θ = 1/3. Initially, the hypothesis space is the unit interval. The binary bisection algorith takes its first easureent at X 1 = 1/2, easuring Y 1 = 0, indicating that θ lies to the left of 1/2, and thus reducing the hypothesis space to the interval [0, 1/2]. Continuing in this anner, the algorith reduces the hypothesis space by half after each easureent, achieving what has been shown to be an optial rate of convergence [1]. In fact, copared to its passive counterpart (in this case, saple locations distributed uniforly throughout the interval, binary bisection has been shown to achieve an exponential speedup [1], [5]. A large body of work has answered questions such as what type of iproveent is attainable in higher diensions [5], [6], for ore coplex decision boundaries and under easureent noise [4], [7] [9], and whether algoriths exist that achieve such iproveents [10] [13]. Extensions of the binary bisection algorith continue to be an iportant topic of research in active learning and related fields. In [14], the author presents a odified version of binary search that can handle noisy easureents, referred to as probabilistic binary search (PBS. A discretized version of this algorith was presented in [7] and analyzed, yielding

3 an upper bound on the expected rate of convergence of the algorith. This discretized algorith, referred to as the B- Z algorith after its authors, has been analyzed fro an inforation-theoretic perspective [15], [16], as well as under Tsybakov s noise condition [17]. Further, in [17], the authors show that for the boundary fragent class in diension d 2, the B-Z algorith can be used to perfor a series of one-diensional searches, achieving an optial rate of convergence up to a logarithic factor. The algorith has also been used ore recently in [18] to deonstrate the relationship between active learning and stochastic convex optiization. As we noted before, however, binary search travels on average the entire length of the interval under consideration in order to estiate the threshold. In the case of Lake Erie, one interval corresponds to approxiately 58 k, aking the use of binary search undesirable. A nuber of active learning algoriths exist aside fro binary search, any of which have been shown to be optial under various boundary and noise conditions [4], [6], [8], [10] [13]. The work of [6], [10] [13] presents active linear classifiers that are increasingly adaptive to noise conditions and coputationally feasible. In [13], the authors present an algorith that akes use of epirical risk iniization of convex losses. Another approach relies on the use of recursive dyadic partitions (RDPs [19], in which the d- diensional unit hypercube is recursively divided into 2 d equal sized hypercubes, foring a tree that is then pruned to give rise to an optial estiator. An active learning algorith based on this idea is shown to be nearly optial in [8], and this algorith is used in a lake sapling context in [9]. While these algoriths provide optial saple rates, they both begin by sapling uniforly at rando throughout the entire feature space. This would require traversing the entire lake before proceeding to locations where the boundary is likely to lie, rendering these algoriths infeasible. A sall aount of literature exists in which the cost of sapling is generalized beyond the nuber of saples [20] [22]. The probles considered in [20], [21] are siilar to the setting here, in that the distance traveled to obtain saples is taken into account. In [20], the authors eploy a traveling salesan with profits odel to iniize the distance traveled while obtaining labels for points of axiu uncertainty. The authors of [21] iniize an objective function that takes into account both label uncertainty and label cost. In both cases, the saples are taken batch-wise before being labeled by a huan expert. As every individual saple is inforative, we wish to take advantage of all available data by choosing our saple locations on a saple-by-saple basis, rather than updating after each batch. Further, neither algorith is accopanied by theoretical analysis. In [22], the authors investigate perforance bounds for active learning in the case where label costs are nonunifor. The quantile search algoriths presented here extend this idea to the case where label costs are both nonunifor and dynaic. As a final note, one ay wonder about the relation to what is known as -ary search [24]. In contrast to quantile search, -ary search is tree-based and thus requires integer values of. In contrast, quantile search does not require to be an integer and therefore gives ore flexibility in the resulting tradeoff. Further, quantile search as described is the natural generalization of PBS and lends itself to the analysis of [4], [7] in the case where the easureents are noisy. A coparison to noisy -ary search is a topic for future work. IV. QUANTILE SEARCH In this section, we extend the ideas of [4], [7] to account for a distance penalization in addition to saple coplexity, resulting in what we refer to as quantile search. The basic idea behind this algorith is as follows. We wish to find a tradeoff between the nuber of saples required and the total distance traveled to achieve a given estiation error for the change point of a step function on the unit interval. As we know, binary bisection iniizes the nuber of required saples. On the other hand, continuous spatial sapling iniizes the required distance to estiate the threshold. Binary search bisects the feasible interval (hypothesis space at each step. In contrast, one can think of continuous sapling as dividing the feasible interval into infinitesial subintervals at each step. With this in ind, a tradeoff becoes clear: one can divide the feasible interval into subintervals of size 1/, where ranges between 2 and. Intuition would tell us that increasing would increase the nuber of saples required but decrease the distance traveled in sapling. In what follows, we show that this intuition is correct in both the noise-free and noisy cases, resulting in two novel search algoriths. A. Deterinistic Quantile Search We first describe and analyze quantile search in the noisefree case (p = 0, here referred to as deterinistic quantile search (DQS and given as Algorith 1. In the following subsections, we analyze the expected saple coplexity and distance traveled for the algorith and show how these results can be used to iniize the total sapling tie. Further, the results show that the required nuber of saples increases onotonically with and the distance traveled decreases onotonically with, indicating that the desired tradeoff is achieved. 1 Rate of Convergence: Recalling fro Section II, our goal is to estiate the threshold θ of a one-diensional threshold classifier. We analyze the expected error of our estiate ˆθ n after a fixed nuber of saples for the DQS algorith. The ain result and a sketch of the proof are provided here. An expanded proof can be found in [23]. Theore 1: Consider a deterinistic quantile search with paraeter and let ρ = 1. Begin with a unifor prior on θ. The expected estiation error after n easureents is then E[ ˆθ n θ ] = 1 [ ρ 2 + (1 ρ 2] n. (3 4 Proof: (Sketch The proof proceeds fro the law of total expectation. Let Z n = ˆθ n θ. The first easureent is

4 Algorith 1 Deterinistic Quantile Search (DQS 1: input search paraeter, saple budget N 2: initialize X 0 0, Y 0 1, n 1, a 0, b 1 3: while n N do 4: if Y n 1 = 1 then 5: X n X n (b a 6: else 7: X n X n 1 1 (b a 8: end if 9: Y n f(x n 10: a = ax {X i : Y i = 1, i n} 11: b = in {X i : Y i = 0, i n} 12: ˆθn a+b 2 13: end while taken at 1/, and hence the expected error can be calculated when θ 1/ and θ > 1/. [ E[Z 1 ] = E Z 1 θ 1 ] ( Pr θ 1 + [ E Z 1 θ > 1 ] ( Pr θ > 1 = 1 [ (1 ρ 2 + ρ 2]. 4 Siilarly, after the second easureent is taken, there are four intervals, two that partition the interval [0, 1/], and two that partition (1/, 1]. These intervals contribute four onoials of degree 4 to the error, one of which is (1 ρ 4, one which is ρ 4, and two which are (1 ρ 2 ρ 2. The basic idea is that each parent interval integrates to (1 ρ 2i ρ 2j and in the next step gives birth to two child intervals, one evaluating to (1 ρ 2i+2 ρ 2j and the other (1 ρ 2i ρ 2j+2. The proof of the theore then follows by induction. Consider the result of Theore 1 when = 2. In this case, the error becoes E[ ˆθ n θ ] = 2 (n+2. Coparing to the worst case, we see that the average case saple coplexity is exactly one saple better than the worst case, atching the well-known theory of binary search. In Section V we confir this result through siulation. 2 Distance Traveled: Next, we analyze the expected distance traveled by the DQS algorith in order to converge to the true θ. The proof is siilar to that of the previous section in that it follows by the law of total expectation. After each saple, we analyze the expected distance given that the true θ lies in a given interval. The result and a proof sketch are given below, with the full proof included in [23]. Theore 2: Let D denote a rando variable representing the distance traveled during a deterinistic quantile search with paraeter. Begin with a unifor prior on θ. Then E[D] = 2 2. (4 Proof: (Sketch The DQS algorith for our class of functions chooses saples oving further into the interval until it akes a zero easureent. It then turns back and takes saples in the opposite direction until a one is easured. This behavior allows us to analyze the total distance by splitting it up into stages first the expected distance traveled before the algorith reaches a point x 1 > θ, and then x 2 < θ, etc. Let D n be a rando variable denoting the distance required to ove to the right of θ for the n 2 th tie when n is odd, and to the left of θ for the n 2 th tie when n is even. In this case, we have that E[D] = E[D n ]. n=1 First, [ we would like to find E[D 1 ]. Let A i denote the interval 1 1 i 1 ( p p=0, 1 i ( 1 p p=0, where A 0 = [ 0, 1, so that the Ai s for a partition of the unit interval whose endpoints are possible values of the saple locations X j. Now note that E[D 1 ] = E[D 1 θ A i ] Pr(θ A i. i=0 Then since we assue θ is distributed uniforly over the unit interval, Pr(θ A i = 1 i ( p 1 1 i 1 ( p 1 p=0 p=0 = 1 ( i 1. Next, note that Thus we have E[D 1 θ A i ] = 1 E[D 1 ] = i ( 1 p=0 p E[D 1 θ A i ] Pr(θ A i i=0 = 2 1. The proof proceeds by rewriting the above in ters of ρ = ( 1/ and then calculating E[D n ]. This is done by dividing each A i into subintervals that for partitions of A i. The result then follows by induction on E[D n ]. 3 Sapling Tie: Given the results in Theores 1 and 2, we can now choose the optial value of to iniize the expected sapling tie. First, let N be a rando variable denoting the nuber of saples required to achieve an error ε and denote its expectation E[N] = log(4ε log ( ( n, which is found by solving (3 for n. Now let T be a rando variable denoting the total tie required to achieve an error ε. Then the expected value is E[T ] = γe[n] + ηe[d] ( = γn + η (2 2(2 1 n γn + η 2 2, (5

5 Algorith 2 Probabilistic Quantile Search (PQS 1: input search paraeter, saple budget N, probability of error p 2: initialize π 0 (x = 1 for all x [0, 1], n 0 3: while n N do 4: choose X n+1 such that X n+1 0 π n (xdx = 1 5: observe Y n+1 f(x n+1 U n+1, where U n+1 Bern(p 6: if Y n+1 = 0 then 7: ( (1 p π n+1 (x = ( p 8: else 9: 1+( 2p ( p π n+1 (x = ( (1 p 1+( 2p 1+( 2p π n (x, x X n+1 π n (x, x > X n+1 π n (x, x X n+1 π n (x, x > X n+1 1+( 2p 10: end if 11: end while 12: estiate ˆθ n such that ˆθ n 0 π n (x = 1/2 where γ denotes the tie required to take one saple, and η denotes the tie required to travel one unit of distance. Finding the optial is difficult. However, (5 can be used to solve for nuerically using standard techniques. We show exaples of such values in Section V. B. Probabilistic Quantile Search We now extend the idea behind Section IV-A to the case where easureents ay be noisy (i.e., p 0. In [7], the authors present the probabilistic binary search (PBS algorith. The basic idea behind this algorith is to perfor Bayesian updating in order to aintain a posterior distribution on θ given the easureents and locations. Rather than bisecting the interval, at each step the algorith bisects the posterior distribution. This process is then iterated until convergence and has been shown to achieve optial saple coplexity [16], [17]. We now extend this idea to achieve a tradeoff between saple coplexity and distance traveled. We refer to the noise-tolerant algorith as probabilistic quantile search (PQS, which proceeds as follows. Starting with a unifor prior, the first saple is taken at X 1 = 1/. The posterior density π n (x is then updated as described below, and ˆθ n is chosen as the edian of this distribution. The algorith proceeds by taking saples X n such that Xn+1 0 π n (xdx = 1, i.e., X n is the first -quantile of π n 1. For = 2, the above denotes the edian of the posterior distribution and reduces to PBS. A foral description is given in Algorith 2. The Bayesian update on π n can be derived as follows. Begin with the first saple. We have π 0 (x = 1 for all x and wish to find π 1 (x. Let f 1 (x X 1, Y 1 be the conditional density of θ given X 1, Y 1. Applying Bayes rule, the posterior becoes: f 1 (x X 1, Y 1 = Pr(X 1, Y 1 θ = xπ 0 (x Pr(X 1, Y 1 For illustration, consider the case where θ = 0. We now take the first easureent at X 1 = 1/. Then and Pr (X 1 = 1/, Y 1 = 0 θ = 0 = 1 p Pr (X 1 = 1/, Y 1 = 1 θ = 0 = p. In fact, this holds for any θ < 1/. Now exaine the denoinator: 1 + ( 2p Pr(X 1 = 1/, Y 1 = 0 =. Notice that for = 2 this siplifies to 1/2, which is the case in [7]. We then update the posterior distribution to be ( (1 p x 1/ π 1 (x = ( p 1+( 2p 1+( 2p x > 1/. The equivalent posterior density can be found for when Y 1 = 1. The process of aking an observation and updating the prior is then repeated, yielding general forula for the posterior update. When Y n+1 = 0, we have ( (1 p π n+1 (x = ( p 1+( 2p 1+( 2p π n (x π n (x x 1/ x > 1/. Siilarly, for Y n+1 = 1, we have ( p 1+( 2p π n (x x 1/ π n+1 (x = ( (1 p π n (x x > 1/. 1+( 2p 1 Rate of Convergence: Analysis of the above algorith has proven difficult since its inception in 1974, with a first proof of a geoetric rate of convergence appearing only recently in [25]. Instead, the authors of [4], [7], [16], [17] use a discretized version involving inor odifications. In this case, the unit interval is divided into bins of size, such that 1 N. The posterior distribution is paraeterized, and a paraeter α is used instead of p in the Bayesian update, where 0 < p < α. The analysis of rate of convergence then centers around the increasing probability that at least half of the ass of π n (x lies in the correct bin. A foral description of the algorith can be found in [23]. Given a version of PQS discretized in the sae way, we arrive at the following result. Theore 3: Under the assuptions given in Section II, the discretized PQS algorith satisfies sup E[ ˆθ n θ ] 2 θ [0,1] ( n/2 p(1 p. (6

6 Proof: (Sketch In the discretized algorith, a i (j denotes the posterior ass at the ith bin after j easureents and a(j denotes the vectorized collection of all bins after j easureents. Let k(θ be the index of the bin containing θ. Note that a i (0 = for all i. We define and N θ (j + 1 = M θ(j + 1 M θ (j M θ (j = 1 a k(θ(j, a k(θ (j = a k(θ(j(1 a k(θ (j + 1 a k(θ (j + 1(1 a k(θ (j. As a brief bit of intuition, note that M θ (j is a easure of how uch ass is in the bin actually containing θ, while N θ (j + 1 is a easure of iproveent fro one iteration to the next and is strictly less than one when an iproveent is ade [4]. Fro [4] and following a straightforward application of Markov s inequality and the law of total expectation, we have that ( Pr θ n θ > E[M θ (n], (7 and { } n E[M θ (n] M θ (0 ax ax E[N θ(j + 1 a(j]. j 0,...,n 1 a(j The reainder of the proof consists of bounding E[N θ (j + 1 a(j] by considering the three cases: (i k(j = k(θ; (ii k(j > k(θ; and (iii k(j < k(θ. In all cases, we show that E[N θ (j+1 a(j] q 2β + p 2α + ( q 2β p 2α (β α where q = 1 p and β = 1 α. Using (7, we have ( Pr ˆθ n θ > 1 [ q 2β + p ( q 2α + 2β p 2α ( ] n 2 (β α. Next, we bound the expected error ( E[ ˆθ n θ ] = Pr ˆθ n θ > t dt 0 [ (1 2 q + 2β + p 2α ( q + 2β p ( ] n 2 (β α, 2α and the result follows by iniizing with respect to and α. The full proof can be found in [23]. In the case where = 2, the above result atches that of [4], [7] as desired. One iportant fact to note is that in contrast to the deterinistic case, the result here is an upper bound on the nuber of saples required for convergence as opposed to an expected value. As this sees to be the case for all analyses of siilar algoriths [4], [7], [25], we instead rely on Monte Carlo siulations to choose the optial value of. Finally, the bound here is loose. For clarity, consider the case where p = 0 and = 2. Then the above becoes ( n/2 1 sup E[ ˆθ n θ ] 2. θ [0,1] 2 As noted in [17] and entioned in Section IV-A, ( n+1 1 sup E[ ˆθ n θ ], θ [0,1] 2 indicating that the bound is quite loose, even for the previously considered PBS algorith. However, in [17], the authors use this result when = 2 to show rate optiality of the PBS algorith. This fact suggests that despite the discrepancy, the result of Theore 3 ay still be useful in proving optiality for the PQS algorith. While the rate of convergence for PQS can be derived using standard techniques, the expected distance or a useful bound on the distance is ore difficult. The technique used in Section IV-A becoes intractable as the values of X i are no longer deterinistic given θ. We have considered the approach of exaining the posterior distribution after each step and calculating the possible locations, but at the nth easureent, there are 2 n 1 possible distributions. Deterining the expected distance traveled by PQS reains a subject of future work. V. SIMULATIONS A. Verification of Algoriths ( 2 In this section, we verify through siulation the theoretical, rate of convergence and distance traveled derived in Section IV-A. Further, we present siulated results for the PQS algorith and copare with the bound derived in Section IV- B. We first siulate the the DQS algorith over a range of fro 2 to 20, where θ is swept over a 1000-point grid on the unit interval. The resulting average error after 20 saples is shown in Fig. 3a, while the average distance before convergence to an error of ε = is shown in Fig. 3b. The figures show the theoretical values for expected error and distance atch the siulated values. Further, our intuition is confired; by inverting the error, one can see that the nuber of saples required to achieve a given error is onotonically increasing in, while the distance traveled is onotonically decreasing. This indicates that DQS achieves a tradeoff in the noise-free case. Fig. 3c shows a plot of the expected sapling tie as a function of, with the optial value of indicated in the case where γ = 60 s and η = 4 /s. Next, we siulate the PQS algorith over a range of fro 2 to 20, where θ ranges over a 100 point grid on the unit interval with 500 Monte Carlo trials run for each value of θ. Fig. 4a shows the average nuber of saples required to converge to an error of As in the deterinistic case, the required nuber of saples increases onotonically with. Fig. 4b shows the average distance traveled before converging to the sae error value. Again,

7 theoretical siulated theoretical siulated ˆθ20 θ average distance (a (b (a expected sapling tie (s = (c Fig. 3: Siulated and theoretical vales for (a expected error after 20 saples and (b distance traveled for DQS algorith. (c Optial value of for γ = 60 s and η = 4 /s. (b (c average saples (a average distance Fig. 4: Siulated average (a saples required for convergence and (b distance traveled for PQS algorith with probability of error p = 0.1. the distance decreases onotonically with, indicating that the algorith achieves the desired tradeoff in the noisy case. B. Application to Spatial Sapling In this section, we apply the DQS and PQS algoriths to the proble of sapling hypoxic regions in Lake Erie. Fig. 5a shows the lake with an exaple hypoxic zone pictured in gray. In [17], the authors show that for the set of distributions such that the Bayes decision set is a boundary fragent, a variation on PBS can be used to estiate the boundary while achieving optial rates up to a logarithic factor. The authors of [17] inforally describe the boundary fragent class on [0, 1] d as the collection of sets in which the Bayes decision boundary is a Hölder sooth function of the first d 1 coordinates. In [0, 1] 2, this iplies that the boundary is crossed at ost one tie when traveling on a path along the second coordinate. For a foral definition, see [17], Defn. 1 and the surrounding text. (b (d Fig. 5: Exaple splitting of hypoxic zone in Lake Erie to achieve two boundary fragent sets. (a Lake erie with hypoxic region shown in gray. (b Exaple splitting of hypoxic zone to achieve two boundary fragent sets. (c Division of boundary fragent into strips. (d Piecewise linear estiation of boundary. Given one location x = (a, b inside the hypoxic zone, the proble can be transfored into two probles, each with a function belonging to the boundary fragent class. Using odels and easureents fro previous years, this is a reasonable assuption. Splitting the lake along the line y = b yields the two sets shown in Fig. 5b. Now we can further divide the proble into strips along the first diension, as shown in Fig. 5c. Along each of these strips, the proble reduces to change point estiation of a onediensional threshold classifier as we have studied thus far. After estiating the change point at each strip, the boundary is estiated as a piecewise linear function of the estiates, as shown in Fig. 5d. We apply this procedure to the hypoxic region shown in Fig. 5a using 16 strips for a variety of values for tie per saple and speed of watercraft. These values are used with Theores 1 and 2 in order to select the optial value of. The total tie required for both DQS and binary search can be seen in Table I. The results are as expected. A higher sapling tie biases the algorith toward lower values of in order to iniize the nuber of saples required, while a lower speed biases the algorith toward higher values of in order to iniize the distance traveled. We see that in the case of a low sapling tie and low speed, the total tie saved copared to binary search is on the order of 60

8 Sapling Tie (s Speed (/s Total Tie (hrs TABLE I: Coparison of deterinistic quantile search with binary search for a variety of sapling ties and speeds. hours. This brings the sapling tie fro roughly 5 days down to 2-3, decreasing the possibility that the spatial extent of the hypoxic region will change significantly before the easureent is coplete. Siilar results hold for PQS, as shown in Table II, where we show the sapling ties with a probability of easureent error p = 0.1 averaged over 100 Monte Carlo siulations. Note that in this case, the chosen value of ay not be optial. VI. CONCLUSIONS & FUTURE WORK The algoriths described here deonstrate the desired tradeoff between saples required and distance traveled for one-diensional threshold classifier estiation. We have shown how this idea can be used to estiate a twodiensional region of hypoxia under certain soothness assuptions on the boundary, and epirical results indicate the benefits of quantile search over traditional binary search. Several open questions reain. Deriving or bounding the expected distance for the PQS algorith is an iportant next step. Further, while the choice of shown here is optial for the DQS algorith, the question reains whether this algorith is optial in soe general sense. Beyond this, several interesting generalizations exist. The boundary fragent class entioned here is restrictive [4], and the extension to ore general cases would be of interest. The recent work of [26] describes a graph-based algorith that eploys PBS to higher-diensional nonparaetric estiation. Extending this idea to penalize distance traveled is a proising avenue for practical applications of quantile search. Finally, the PQS algorith requires knowledge of the noise paraeter p in order to update the posterior. The algoriths presented in [13], [18] enjoy the property that they are adaptive to unknown noise levels. The developent of a noise-adaptive probabilistic search would certainly be of great interest, with potential applications in areas such as stochastic optiization [18] beyond direct applicability to this proble. REFERENCES [1] S. Dasgupta, Analysis of a greedy active learning strategy, in Proc. Advances in Neural Inforation Processing Systes, [2] D. Beletsky, D. Schwab, and M. McCorick, Modeling the suer circulation and theral structure in Lake Michigan, Journal of Geophys. Res., vol. 111, [3] G. L. E. R. Laboratory, Lake Erie hypoxia warning syste, [4] R. Castro and R. Nowak, Active learning and sapling, in Foundations and Applications of Sensor Manageent, 1st ed. New York, NY: Springer, 2008, ch. 8. Sapling Tie (s Speed (/s Total Tie (hrs TABLE II: Coparison of probabilistic quantile search with binary search for a variety of sapling ties and speeds with probability of error p = 0.1. [5] S. Dasgupta, Coarse saple coplexity bounds for active learning, in Proc. Advances in Neural Inforation Processing Systes, [6] S. Hanneke, A bound on the label coplexity of agnostic active learning, in Proc. ICML, [7] M. V. Burnashev and K. S. Zigangirov, An interval estiation proble for controlled observations, Probles in Inforation Transission, vol. 10: , 1974, translated fro Probley Peredachi Inforatsii, 10(3:51-61, July-Septeber, [8] R. Castro, R. Willett, and R. Nowak, Faster rates in regression via active learning, in Proc. Advances in Neural Inforation Processing Systes, [9] A. Singh, R. Nowak, and P. Raanathan, Active learning for adaptive obile sensing networks, in Proc. Inforation Processing in Sensor Networks, [10] M.-F. Balcan, A. Beygelzier, and J. Langford, Agnostic active learning, in Proc. International Conference on Machine Learning, [11] S. Dasgupta, D. Hsu, and C. Monteleoni, A general agnostic active learning algorith, in Proc. Advances in Neural Inforation Processing Systes, [12] M.-F. Balcan, A. Broder, and T. Zhang, Margin based active learning, Learning Theory, p. 350, [13] Y. Wang and A. Singh, Noise-adaptive argin-based active learning for ulti-diensional data, arxiv Preprint, vol v1, [14] M. Horstein, Sequential decoding using noiseless feedback, IEEE Trans. Inf. Theory, vol. 9, [15] R. M. Karp and R. Kleinberg, Noisy binary search and its applications, in Proc. ACM-SIAM Syposiu on Discrete Algoriths, [16] M. B. Or and A. Hassidi, The bayesian learner is optial for noisy binary search (and pretty good for quantu as well, in Proc. IEEE Syposiu of Foundations of Coputer Science, [17] R. Castro and R. Nowak, Miniax bounds for active learning, IEEE Trans. Inf. Theory, vol. 54, pp , May [18] Y. R. Wang and A. Singh, Algorithic connections between active learning and stochastic convex optiization, in Proc. Algorithic Learning Theory, [19] C. Scott and R. Nowak, Miniax-optial classification with dyadic decision trees, IEEE Trans. Inf. Theory, vol. 52, pp , Apr [20] A. Liu, G. Jun, and J. Ghosh, Spatially cost-sensitive active learning, in Proc. SIAM Conf. on Data Mining, [21] B. Deir, L. Minello, and L. Bruzzone, A cost-sensitive active learning technique for the definition of effective training sets for supervised classifiers, in Proc. SIAM Conf. on Data Mining, [22] A. Guillory and J. Blies, Average-case active learning with costs, in Proc. Algorithic Learning Theory, [23] J. Lipor and L. Balzano, Quantile search: A distance-penalized active learning algorith for spatial sapling, arxiv Preprint, vol , [24] B. Schlegel, R. Geulla, and W. Lehner, k-ary search on odern processors, in Proc. Fifth Int. Workshop on Data Manageent on New Hardware, [25] R. Waeber, P. I. Frazier, and S. G. Henderson, Bisection search with noisy responses, SIAM Journal on Control and Optiization, vol. 51, pp , [26] G. Dasarathy and R. Nowak, S 2 : An efficient graph-based active learning algorith with application to nonparaetric learning, in Proc. Conf. on Learning Theory, 2015.