Testing Juntas Nearly Optimally

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1 Testing Juntas Nearly Optimally Eric Blais Carnegie Mellon University School o Computer Science Pittsburgh, PA eblais@cscmuedu ABSTRACT A unction on n variables is called a k-junta i it depends on at most k o its variables In this article, we show that it is possible to test whether a unction is a k-junta or is ar rom being a k-junta with O(k/ + k log k) queries, where is the approximation parameter This result improves on the previous best upper bound o Õ(k3/ )/ queries and is asymptotically optimal, up to a logarithmic actor We obtain the improved upper bound by introducing a new algorithm with one-sided error or testing juntas Notably, the algorithm is a valid junta tester under very general conditions: it holds or unctions with arbitrary inite domains and ranges, and it holds under any product distribution over the domain A key component o the analysis o the new algorithm is a new structural result on juntas: roughly, we show that i a unction is ar rom being a k-junta, then is ar rom being determined by k parts in a random partition o the variables The structural lemma is proved using the Eron-Stein decomposition method Categories and Subject Descriptors G3 [Probability and Statistics]: Probabilistic algorithms; Fm [Analysis o Algorithms and Problem Complexity]: Miscellaneous General Terms Algorithms, Theory Keywords Property Testing, Juntas, Eron-Stein decomposition Research supported in part by a scholarship rom the Fonds québécois de recherche sur la nature et les technologies (FQRNT) c ACM, (009) This is the author s version o the work It is posted here by permission o ACM or your personal use Not or redistribution The deinitive version will be published in the proceedings o STOC 09 1 INTRODUCTION In many areas o science, data collection methods are rapidly becoming more sophisticated As a result, datasets obtained rom experiments contain an increasing number o eatures For example, until recently biologists were only able to measure the expression o a handul o genes at a time; now they can measure the expression o tens o thousands o genes simultaneously [1] Datasets with large numbers o eatures provide new opportunities, but they also introduce new challenges For the task o learning a target unction, a large number o eatures causes naïve learning algorithms to overit the data, and can lead to the ormulation o hypotheses that are hard to interpret The eature subset selection method is commonly used to avoid both those challenges [11, 34] The eature subset selection method is appropriate when the target unction depends on only k o the n eatures in a dataset, or some k n We call such target unctions k-juntas In many cases, the target unction may be quite ar rom being a k-junta When this is the case, the eature subset selection method is bound to ail It is thereore preerable to test a target unction to see i it is a k-junta beore attempting to learn it via the eature subset selection method In this article, we study the problem o testing k-juntas in the property testing ramework Inormally, we seek to determine the minimum number o queries to a unction required to distinguish k-juntas rom unctions that are ar rom being k-juntas, or some appropriate notion o distance (See Section or ormal deinitions) 11 Previous work The irst result explicitly related to testing juntas was obtained by Parnas, Ron, and Samorodnitsky [17], who generalized a result o Bellare, Goldreich, and Sudan [] on testing long codes to obtain an algorithm or testing 1-juntas (ie, dictators) with only O(1/) queries Soon aterwards, Fischer et al [9] introduced algorithms or testing k-juntas with Õ(k )/ queries The original analysis o Fischer et al only applied to unctions with a boolean range; Diakonikolas et al [7] extended the analysis to handle unctions with arbitrary inite ranges The junta-testing algorithms o Fischer et al remained the most query-eicient ways to test juntas until very recently, when the current author introduced an algorithm or testing boolean unctions or the property o being k-juntas with Õ(k3/ )/ queries [3]

2 The irst non-trivial lower bound on the query complexity o the testing juntas problem was provided by Fischer et al [9], who showed that Ω(log k) queries are necessary to test k-juntas 1 That lower bound was subsequently improved to Ω(k) by Chockler and Gutreund [6] 1 Our results The research presented in this article was motivated by the desire to close the gap between the upper and lower bounds on the query complexity o the junta testing problem Our main result is a new algorithm or testing juntas that signiicantly improves the upper bound Theorem 11 The number o queries required to -test k-juntas is bounded above by O (k/ + k log k) Furthermore, this result holds or testing unctions that have arbitrary inite product domains and arbitrary inite ranges, and it also holds under any product distribution over the domain Combined with the lower bound o Chockler and Gutreund [6], this completely characterizes the asymptotic query complexity or -testing k-juntas (up to a logarithmic actor) or constant values o The new algorithm or testing juntas, presented in Section 31, is surprisingly simple It is, however, quite general As Theorem 11 indicates, it can test unctions with arbitrary inite product domains and arbitrary inite ranges or the property o being a k-junta, and it also is a valid tester under the general property testing ramework where distance is measured by any product distribution over the input Furthermore, the algorithm has one-sided error: it always accepts k-juntas The analysis o the algorithm constitutes the main technical contribution o this current research At the heart o the analysis lies a undamental structural lemma about juntas: roughly, the lemma states that i a unction is ar rom being a k-junta, then it will also be ar rom being determined by the coordinates in k parts in a (suiciently ine) random partition o the coordinates The lemma is presented in Section 3 and its proo is presented in Section 4 13 Our techniques The analysis o the junta testing algorithm and the proo o our main structural lemma rely on the analysis o the inluence o coordinates in a unction The main tool we use to do this is the Eron-Stein decomposition method The Eron-Stein decomposition o a unction is a coarser version o the Fourier decomposition o a unction While both decompositions share many similarities, it can be more convenient to work with the Eron-Stein decomposition when the range o the unction is not boolean In particular, the Eron-Stein decomposition provides a much simpler analysis o junta tests or unctions with non-boolean ranges than the approach o Diakonikolas et al [7] The Eron-Stein decomposition method has ound numerous applications in statistics [8, 13, 19], hardness o approximation [1, 14, 15], learning theory [4], and social choice theory [15] As we see below, the method is also particularly well suited or the analysis o juntas 1 In act, Fischer et al proved the stronger statement that Ω( k) queries are required to test k-juntas non-adaptively PRELIMINARIES Throughout this article, we consider unctions o the orm : Y, where = 1 n is a inite set and Y is an arbitrary inite set We deine Ω = Ω 1 Ω n to be a product probability space over, where Ω i = ( i, µ i) is deined by an arbitrary probability measure µ i on i For the elements x = (x 1,, x n), y = (y 1,, y n) and the set S [n], we let x S represent the ordered list (x i : i S) and use the notation x Sy S to represent the element z = (z 1,, z n) where z S = x S and z S = y S Throughout the rest o this article, I = {I 1,, I s} denotes a random partition o the coordinates in [n] obtained by uniormly and independently assigning each coordinate at random to one o the parts I 1,, I s 1 Juntas Deinition 1 (Inluence) The inluence o the set S [n] o coordinates in the unction : Y under the probability space Ω is In (S) de = Pr x,y Ω [(x) (ysx S )] When In (S) > 0, we say that the set S o coordinates is relevant to, or alternatively that depends on the coordinates in S Deinition (Juntas) The unction : Y is a k-junta i it has at most k relevant coordinates Conversely, is -ar rom being a k-junta under Ω i or every k-junta g : Y, Pr [(x) g(x)] The analysis o the junta testing algorithm relies on the ollowing characterization o non-juntas Proposition 3 I : Y is -ar rom being a k-junta, then or every set J [n] o size J k, In `[n] \ J In our analysis o the junta testing algorithm, we also show that there is a close connection between juntas and partition juntas Deinition 4 (Partition juntas) Let I be a partition o [n] The unction : Y is a k-part junta with respect to I i the relevant coordinates in are all contained in at most k parts o I Conversely, is -ar rom being a k-part junta with respect to I under Ω i or every set J ormed by taking the union o k parts in I, In ([n] \ J) The proo o Proposition 3 is included in Appendix A We examine the problem o testing juntas in the property testing model introduced by Goldreich, Goldwasser, and Ron [10], which is a generalized version o the property testing model o Rubineld and Sudan [18] Deinition 5 (Junta testers) A randomized algorithm A that queries a given unction on a small number o inputs is an -tester or k-juntas under the distribution Ω i it accepts k-juntas with probability at least /3, and rejects unctions that are -ar rom being k-juntas under Ω with probability at least /3 The query complexity o the algorithm A is the number o queries it makes to the unction beore accepting or rejecting it

3 I a junta tester accepts k-juntas with probability 1, then we say that it has one-sided error Eron-Stein decomposition Let R Y be the vector space generated by the set o all ormal linear combinations o elements in Y Given two vectors v = P y Y vyy and w = P y Y wyy in RY, we deine their inner product to be v, w R Y = P y Y vywy The set o all unctions o the orm : R Y orms the inner product space L (Ω, R Y ) under the inner product, g = E ˆ (x), g(x) R Y The norm o a unction L (Ω, R Y ) is deined by = p q, = ˆ (x), (x) R Y E By identiying elements in Y with elements in R Y in the natural way (ie, by identiying y Y with the ormal linear combination 1 y R Y ), we observe that the set o unctions o the orm : Y orms a subset o L (Ω, R Y ) We call such unctions pure-valued unctions The norm o a purevalued unction is = 1 Theorem 6 (Eron-Stein [8]) Every unction in L (Ω, R Y ) has a unique decomposition o the orm (x) = S (x) where or every S [n], S S, and y : 1 S depends only on the coordinates in S, and E ˆ S (x) xs = y S = 0 The Eron-Stein decomposition is an orthogonal decomposition o unctions As a result, Parseval s identity holds in this context Theorem 7 (Parseval s identity) For every unction L (Ω, R Y ), S = In particular, when is a pure-valued unction, S = 1 Remark 8 When : { 1, 1} n { 1, 1} is a boolean unction, the Eron-Stein decomposition o is the same as its Fourier decomposition (ie, or every set S [n], S = ˆ(S)χ S) This can be easily veriied by noting that the unctions ˆ(S)χ S satisy the two conditions o Theorem 6 3 Inluence There is a natural connection between the inluence o coordinates in a unction and the Eron-Stein decomposition o that unction That is, the sets o projections H S = { S : L (Ω, R Y )} orm orthogonal subspaces o L (Ω, R Y ) Proposition 9 For every pure-valued unction in L (Ω, R Y ) and every set S [n], In (S) = T T : S T For completeness, we include the proo o Proposition 9 in Appendix A1 The monotonicity and subadditivity properties o inluence ollow directly rom the proposition Corollary 10 (Monotonicity & Subadditivity) For any pure-valued unction L (Ω, R Y ) and any sets S, T [n], In (S) In (S T ) In (S) + In (T ) Another key contribution o Proposition 9 to the proo o our main lemma is that it suggests two natural extensions to the deinition o inluence: low-order and high-order inluence Deinition 11 (Low- & High-order inluence) The inluence o order at most k o a set S [n] o coordinates in the pure-valued unction L (Ω, R Y ) is In k (S) = T T k : S T and the inluence o order greater than k o the set S on is In >k (S) = T T >k : S T Proposition 9, along with Parseval s identity, makes it easy to show that the sum o the low-order inluence o each coordinate in a pure-valued unction cannot be too large, and that only a ew coordinates can have signiicant loworder inluence in a pure-valued unction Proposition 1 For every pure-valued unction in L (Ω, R Y ) and any k n, the sum o the low-order inluence o each coordinate in is bounded above by In k (i) k i [n] Corollary 13 For every pure-valued unction in L (Ω, R Y ), any k n, and any θ > 0, n i [n] : In k o (i) θ k θ 3 MAIN RESULT 31 The algorithm The JuntaTest algorithm is based on a simple but useul observation o Blum, Hellerstein, and Littlestone [5]: i we have two inputs x, y such that (x) (y), then the set S o coordinates in which x and y disagree contains a coordinate that is relevant in Furthermore, by perorming a binary search over the hybrid inputs ormed rom x and y, we can identiy the relevant coordinate with O(log S ) queries We build on this observation by noting that i we have a partition o the coordinates into s parts and only care to

4 JuntaTest(, k, ) Additional parameters: s = 10 0 k 9 / 5, r = 1(k + 1)/ 1 Randomly partition the coordinates in [n] into s sets I 1,, I s Initialize S [n], l 0 3 For each o r rounds, 31 Generate a pair (x, y) Ω Ω 3 I (x) (y Sx S ), then 31 Use binary search to ind a set I j that contains a relevant variable 3 Update S S \ I j 33 Set l l I l > k, then reject the unction 4 Accept the unction Figure 1: The algorithm or -testing k-juntas identiy a part that contains a relevant coordinate (rather than the coordinate itsel), then we can optimize the binary search to only take O(log s) queries The JuntaTest algorithm applies the above observation in the obvious way It maintains a set S o coordinates that may or may not be relevant to the unction, generates pairs o inputs x, y at random, and checks i (x) (x S y S) When such a pair is ound, the algorithm identiies a part that contains a relevant coordinate, and removes all the coordinates in that part rom S I the algorithm identiies k + 1 dierent parts with relevant coordinates, it rejects the unction; otherwise, it accepts the unction The details o the algorithm are presented in Figure 1 3 Main lemma To establish the correctness o the JuntaTest algorithm, we want to show that the algorithm does not lose too much accuracy by identiying parts that contain relevant coordinates instead o identiying the relevant coordinates individually In other words, we want to show that under a random partition I, (1) a unction that is a k-junta is also a k-part junta with respect to I, and () a unction that is -ar rom being a k-junta is Θ()-ar rom being a k-part junta with respect to I The ormer statement is clearly always true; the ollowing lemma shows that the latter statement holds with high probability Lemma 31 Let I be a random partition o [n] with s = 10 0 k 9 / 5 parts obtained by uniormly and independently assigning each coordinate to a part With probability at least 5/6, a unction : Y that is -ar rom being a k-junta is also -ar rom being a k-part junta with respect to I The proo o Lemma 31 is presented in Section 4 Beore proceeding with that proo, we irst show how the rest o the proo o Theorem 11 is constructed 33 Proo o Theorem 11 We prove Theorem 11 by showing that the JuntaTest algorithm -tests k-juntas with only O(k/+k log k) queries Theorem 11 (Restated) The number o queries required to -test k-juntas is bounded above by O (k/ + k log k) Furthermore, this result holds or testing unctions that have arbitrary inite product domains and arbitrary inite ranges, and it also holds under any product distribution over the domain Proo We begin by determining the query complexity o the JuntaTest algorithm At most r = 4(k+1)/ queries are made in the execution o line 3 o the algorithm, and at most (k + 1) log s = O(k log(k/)) queries are made in line 31 o the algorithm So the algorithm makes a total o O(k/ + k log k) queries to the input unction The completeness o the JuntaTest algorithm is easy to establish: when the input unction is a k-junta, it contains at most k parts with relevant coordinates, so the algorithm must accept the unction Thereore, the JuntaTest algorithm has one-sided error Finally, we analyze the soundness o the JuntaTest algorithm By Lemma 31, with probability at least 5/6 a unction that is -ar rom being a k-junta is also /-ar rom being a k-part junta with respect to the random partition o the coordinates When this is the case, the inluence o S is at least / until k+1 parts with relevant coordinates are identiied So the expected number o rounds required to identiy k + 1 parts with relevant variables is (k + 1)/ By Markov s Inequality, the probability that the algorithm does not identiy k + 1 relevant parts in 1(k + 1)/ rounds is at most 1/6, and the overall probability that the JuntaTest algorithm ails to reject is at most 1/3 4 PROOF OF THE MAIN LEMMA 41 Overview o the proo To establish Lemma 31, we want to show that with high probability every set J ormed by taking the union o k parts in a random partition o the coordinates satisies S 1 (1) S J We show this with a combination o three arguments First, In Section 4, we examine the Eron-Stein coeicients S or sets o size S > k Under a random partition o the coordinates, most o these sets have elements distributed over more than k parts Thereore, with high probability the contribution o those sets to the sum in (1) is small We then examine the coordinates with large low-order inluence In Section 43, we show that or a suiciently ine

5 random partition, with high probability those coordinates are completely separated by the partition and thereore provide a limited contribution to the sum in (1) Lastly, we examine the coordinates with small low-order inluence In Section 44, we use a Hoeding bound argument to show that their contribution to the sum in (1) is also negligible We combine the above arguments to complete the proo o Lemma 31 in Section 45 4 High-order coeicients Deinition 41 (Covered sets) Let I = {I 1,, I s} be a partition o [n] For any subset S [n], we say that S is k-covered by the partition I, denoted by S k I, i there exist k indices i 1,, i k such that S I i1 I ik Proposition 4 Let L (Ω, R Y ) be a pure-valued unction, let s 7ek/, and let I = {I 1,, I s} be a random partition o [n] Then with probability at least 17/18, the high-level inluence o contained in sets that are k- covered by I is at most S /4 S k I : S >k Proo Let S [n] be a subset o size S > k The probability that all the elements in S are sent to k or ewer parts in I is Pr[S k I]! «k+1 s k k s es «k k+1 «k+1 k = e k k k s s 7 The expected weight o all large sets that are covered by I is 3 E 4 S 5 = S Pr[S k I] 7, S k I : S >k S >k where the last inequality uses the above upper bound on Pr[S k I] and Parseval s identity The proposition then ollows rom Markov s Inequality 43 Coords with large low-order inluence For any pure-valued unction in L (Ω, R Y ), let us deine H de = {i [n] : In k (i) θ} to be the set o coordinates with large low-order inluence in With high probability, a random partition o the coordinates completely separates the set H Proposition 43 Let L (Ω, R Y ) be a pure-valued unction, let θ > 0, and let s 7k /θ Then with probability at least 17/18, a random partition I = {I 1,, I s} satisies the condition that or all i [s], H I i 1 Proo By Corollary 13, H k/θ So the probability that there exists a part I i that contains at least elements rom H is!! «H s (1/s) k 1 1 θ s Coords with small low-order inluence Since every coordinate in [n]\h has small low-order inluence, we can expect these coordinates to have little impact on the total low-order inluence o each part Indeed, this is what the next proposition shows Proposition 44 Let L (Ω, R Y ) be a pure-valued unction, let s 16k /, let θ /64k 3 log(18s), and let I = {I 1,, I s} be a random partition o [n] Then with probability at least 17/18, In k I i \ H 4k or every i [s] Proo Fix i [s] For every j [n], deine j to be a random variable that takes the value ( In k (j), i j I i \ H, j = 0, otherwise By the subadditivity o inluence, I i \ H In k In k j I i \H (j) = j By Proposition 1, P In k (j) k Furthermore, Pr[j I i] = 1/s or every j [n] So 3 E 4 j 5 = (j) Pr[j I i] k s \H In k By our choice o s, we have that k/s /8k We can apply Hoeding s inequality to obtain 3 Pr 4! j 8k + t 5 t exp P\H In k (j) Applying the elementary inequality P i x i max i x i Pi xi to the summation on the right-hand side o the equation and recalling that max \H In k (j) < θ, we get that So h Pr In k I i \ H In k \H > i 4k (j) kθ 3 Pr 4 j 8k + 5 8k e log(18s) = 1 18s Applying the union bound over all i [s] completes the proo o the proposition

6 45 Proo o Lemma 31 Lemma 31 (Restated) Let I be a random partition o [n] with s = 10 0 k 9 / 5 parts obtained by uniormly and independently assigning each coordinate to a part With probability at least 5/6, a unction : Y that is -ar rom being a k-junta is also -ar rom being a k-part junta with respect to I Proo Let J be the union o any k parts in I By Proposition 9, In ([n] \ J) = S S S J Let θ = log(k/)/10 9 k 4 Note that the values o s and θ satisy the conditions o Propositions 4 44 Deine the ollowing amilies: H = {S J : S k, S J H }, L = {S J : S k, S J H }, and B = {S J : S > k} The sets H, L, B orm a partition o the subsets o J, so In ([n] \ J) = S S S H S S S L S B In `[n] \ (J H ) S S S L S B As Proposition 43 shows, with probability at least 17/18, H is completely separated by the partition I, so every set J ormed by taking the union o at most k parts satisies J H k Then, since is -ar rom being a k-junta, S L In `[n] \ (J H ) () By Proposition 44, with probability at least 17/18, every set I j in the partition satisies P i I j \H In k (i) /4k, so every set J ormed by taking the union o k parts satisies S = In k (J \ H ) k 4k = 4 (3) Finally, the amily B contains exclusively sets S that are k-covered by I, so B {S [n] : S > k, S k I} and we can apply Proposition 4 to obtain that with probability at least 17/18, or every set J S B S S k I : S >k S 4 (4) Combining equations (1) (4), we obtain that with probability at least 1 3(1/18) = 5/6, or every set J obtained by taking the union o at most k parts o I In `[n] \ J 4 4 = 5 ACKNOWLEDGMENTS The author wishes to thank Ryan O Donnell or much invaluable advice throughout the course o this research The author also thanks Kevin Matule, Krzyszto Onak, and the anonymous reerees or many helpul comments on an earlier drat o this article 6 REFERENCES [1] Per Austrin and Elchanan Mossel Approximation resistant predicates rom pairwise independence In Proc 3rd Con on Computational Complexity, 008 [] Mihir Bellare, Oded Goldreich, and Madhu Sudan Free bits, PCPs and non-approximability towards tight results SIAM J Comput, 7(3): , 1998 [3] Eric Blais Improved bounds or testing juntas In Proc 1th Workshop RANDOM, pages , 008 [4] Eric Blais, Ryan O Donnell, and Karl Wimmer Polynomial regression under arbitrary product distributions In Proc 1st Con on Learning Theory, pages , 008 [5] Avrim Blum, Lisa Hellerstein, and Nick Littlestone Learning in the presence o initely or ininitely many irrelevant attributes J o Comp Syst Sci, 50(1):3 40, 1995 [6] Hana Chockler and Dan Gutreund A lower bound or testing juntas Inormation Processing Letters, 90(6): , 004 [7] Ilias Diakonikolas, Homin K Lee, Kevin Matule, Krzyszto Onak, Ronitt Rubineld, Rocco A Servedio, and Andrew Wan Testing or concise representations In Proc 48th Symposium on Foundations o Computer Science, pages , 007 [8] Brad Eron and Charles Stein The jackknie estimate o variance Ann o Stat, 9(3): , 1981 [9] Eldar Fischer, Guy Kindler, Dana Ron, Shmuel Sara, and Alex Samorodnitsky Testing juntas J Comput Syst Sci, 68(4): , 004 [10] Oded Goldreich, Shari Goldwasser, and Dana Ron Property testing and its connection to learning and approximation J o the ACM, 45(4): , 1998 [11] Trevor Hastie, Robert Tibshirani, and Jerome Friedman The Elements o Statistical Learning: Data Mining, Inerence, and Prediction Springer, 001 [1] Timothy R Hugues et al Expression proiling using microarrays abricated by an ink-jet oligonucleotide synthesizer Nature Biotechnology, 19(4):34 347, 001 [13] Samuel Karlin and Yose Rinott Applications o ANOVA type decompositions or comparisons o conditional variance statistics including jack-knie estimates Ann o Statistics, 10(): , 198 [14] Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O Donnell Optimal inapproximability results or MA-CUT and other two-variable CSPs? SIAM J Comput, 37(1): , 007 [15] Elchanan Mossel Gaussian bounds or noise correlation o unctions and tight analysis o long codes In Proc 49th Symp on Foundations o Computer Science, 008 [16] Elchanan Mossel, Ryan O Donnell, and Krzyszto Oleszkiewicz Noise stability o unctions with low

7 inluences: invariance and optimality In Proc 46th Symp Foundations o Comp Sci, pages 1 30, 005 [17] Michal Parnas, Dana Ron, and Alex Samorodnitsky Testing basic boolean ormulae SIAM J Discret Math, 16(1):0 46, 003 [18] Ronitt Rubineld and Madhu Sudan Robust characterizations o polynomials with applications to program testing SIAM J Comput, 5():5 71, 1996 [19] J Michael Steele An Eron-Stein inequality or non-symmetric statistics Ann o Statistics, 14(): , 1986 APPENDI A ADDITIONAL PROOFS A1 Proo o Proposition 9 It is well-known that the identity In (S) = T T :S T holds or unctions in the space L (Ω, R) (see or example [15, 16, 19, 9]) Below, we show that the same identity holds or pure-valued unctions in L (Ω, R Y ) The proo itsel is a simple generalization o known proos o the above identity; we include it or the convenience o the reader Proposition 9 (Restated) For any pure-valued unction L (Ω, R Y ) and set S [n], In (S) = T T : S T Proo Since is pure-valued, the indicator unction 1[(x) = (y)] is equal to the inner product (x), (y) R Y, and so In (S) = Pr [(x) (x S ys)] x,y Ω = 1 E ˆ (x), (xs y S) x,y Ω R Y Taking the Eron-Stein decomposition o and applying linearity o expectation, we get In (S) = 1»i l E T (x), E [ U (y) y S = x S ] y Ω T,U [n] By the deinition o the Eron-Stein decomposition, E h U (y) ( i U (x), i U S, ys = x S = y Ω 0, otherwise So In (S) = 1 = 1 E T [n] U S T [n] U S hd T (x), U ERY i (x) D T, U E By the orthogonality o the Eron-Stein decomposition, when T U, T, U = 0 So D E T, U = D E T, T = T T [n] T S T S U S R Y By Parseval s identity, we also have that 1 = P T [n] T, so In (S) = T T = T T [n] T S T : S T A Proo o Proposition 3 Fischer et al [9] showed that in unctions with boolean ranges that are ar rom being juntas on a set J o coordinates, the set [n] \ J o coordinates has a signiicant amount o inluence A similar result was established by Diakonikolas et al [7] or unctions with non-boolean ranges, when a dierent notion o inluence ( binary variation ) is considered We use Hölder s Inequality to establish the analogous result with our notion o inluence Proposition 3 (Restated) I : Y is - ar rom being a k-junta, then or every set J [n] o size J k, In `[n] \ J Proo For a given set J o size J k, let h : Y be the unction deined by n o h(x) = argmax Pr [(x y Y z Jz J ) = y], where we break ties arbitrarily Then, Pr[(x) h(x)] = 1 E[ (x), h(x) x x R Y ]» D E = 1 E E[(x x z Jz J )], h(x) R» Y Ez = 1 E [(x x Jz J )] E [(x z Jz J )] 1» Ez 1 E [(x x Jz J )] = 1 S J S = S:S ([n]\j) S The irst equality ollows rom the act that and h are purevalued unctions The second ollows rom the act that h only depends on the coordinates in J and rom the linearity o expectation The third equality uses the two easilyveriied identities E z [(x Jz J )] = E z[(x Jz J )], h(x) R Y and E z [(x Jz J )] 1 = 1 The inequality is a special case o Hölder s Inequality The penultimate equality ollows rom the act that E z [(x Jz J )] = P S J S (x), and, inally the last equality ollows rom Parseval s Theorem The proposition then ollows rom Proposition 9