A generalized probability kernel on discrete distributions and its application in two-sample test
Authors: Le Niu
Venue:  (2021)
Keywords: maximum mean discrepancy
RKHS
two-sample test
empirical estimator
discrete distributions

Abstract: We propose a generalized probability kernel(GPK) on discrete distributions with finite support. This probability kernel, defined as kernel between distributions instead of samples, generalizes the existing discrepancy statistics such as maximum mean discrepancy(MMD) as well as probability product kernels, and extends to more general cases. For both existing and newly proposed statistics, we estimate them through empirical frequency and illustrate the strategy to analyze the resulting bias and convergence bounds. We further propose power-MMD, a natural extension of MMD in the framework of GPK, illustrating its usage for the task of two-sample test. Our work connects the fields of discrete distribution-property estimation and kernel-based hypothesis test, which might shed light on more new possibilities.

Under review as a conference paper at ICLR 2021
A GENERALIZED PROBABILITY KERNEL ON DIS- CRETE DISTRIBUTIONS AND ITS APPLICATION IN TWO- SAMPLE TEST
Anonymous authors Paper under double-blind review
ABSTRACT
We propose a generalized probability kernel(GPK) discrete distributions with finite support. This probability kernel, defined as kernel between distributions in- stead of samples, generalizes the existing discrepancy statistics such maximum mean discrepancy(MMMD as well as probability product kernels, and extends to more general cases. For both existing and newly proposed statisties, we estimate them through empirical frequency and illustrate the strategy to analyze the re- sulting bias and convergence bounds. We further propose power-MMID, a natural extension of MMD in the framework of GPK, illustrating its usage for the task sam-le test. Our work connects the fields of discrete distribution propppery estimation and kernel-based hypothesis test, which might shed light more new possibilities.
INTRODUCTION
We focus on the two-sample problem. which is given (2... {yn, 1/....,, Vn), could we infer the discrepancy between underlying distribu- tions they are drawn from. For such problem, the option of hypothesis test(two-sample test) most popular, and variety of statistics in estimating the discrepancy is proposed. In recent years, RKHS based method such as maximum mean discrepancy has gained a lot of attention. (Gretton al. 2012) has shown that in universal-RKHS. F, MMD P. q) 0 if and only if = 9, thus could be used for the two-sa hypothesis test. (Gretton et al. (012) further provides unbiased estimator of MMD with fast asymptotic convergence rate, illustrating its advantages.
i.id samples
On the other hand, estimating distribution properties with plugin(empirical estimators on discrete setting is active research area in recent years, where people focus on problem settings with large support size but not so large sample size. The Bernstein polynomial technique is introduced to ana- lyze the bias of the plugin estimators in (Yi Alon 2020). which provides remarkable progress on bias-reduction methods of the plugin estimators. It is thus interesting to ask if the plugin estimators could motivate new results for the RKHS- based wwo-sample test.
Another interesting topic is about the probability kernel, defined as kernel function over probabil- ities, instead of over samples. As easily seen, any discrepancy measure of distribution p and q could potentially be valid probability kernels, not so much work focuses on this. While (Jebara et al.) 2004] introduced the so called probability product kernels which generalize variety of discrepancy measures, its properties remain further study.
Motivated by above observations, our work focuses on specialized probability kernel function which is direct generalization of sample-based RKHS methods such MMD. We focus on using as the default estimator of the kernel function we defined, and illustrate that with the help of Bernstein polynomial techniques, Our work thus connects the fields discrete cewiiuution--roppeer estimation and kernel-based hypothesis test, which brings interesting possibilities.
could analyze the bias and convergence bounds of these

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2 NOTATION
We use bold symbol p. q E IR to represent a probability function over a discrete support with support size k, and Pi; 9: represents the ith entry of p and q. We use {11, U2,,.., ,, E R to represent the support of p. q. [k] := 21.2,3..., k} represents the set of indices of elements in {UI, U2, .... J. - We use Φ o (p: q) to denote an element-wise function from R R to R* where (oo (p. 9)),
k...,k]
o o (ps, 9:) and @ o to denote an element-wise function from R$ to R where p), 0 poop, Pt- With a slight abuse of notation, we denote p". p q as element- function defined above. We use kernel q) to denote kernel function which maps from R$ x IR& to real value R. And kernel(x, (1, x. E Rd represents kernel function from R. x R2d to real value I. We use K to denote the gram matrix generated from karnel(æ, y) on finite support fFi, U2,,... U/ ). where Kij = kvr Uj). We use {x1, :22, .... un. p and {311 1/2, Un to denote the samples from distribution p and q, where n is the sample size.
GENERALIZED PROBABILITY KERNEL
Probability kernel function, defined as kernel function between distributions instead of samples, is natural extension of the idea of kernel function in sample space.
Definition 1. Given distribution and q belongs to a family of discrete distribution with the same finite support (VI, u2,..., Uk o2 E Rd, where k is the support size, we define the probability kernel function as PK(p, q), which is a kernel function maps from TR x Tekk to real value RR.
Many discrepancy measures, such as MMD. can serve as probability kernel functions, but peo- ple usually don't use the term of probability kernel function when describing them. The reason is that for most of the time, we only consider a limited number of distributions, and do not need or have the resources to navigate through all the distributions within the family, For example, when looking into the two-sample problem, we usually assume two samples { 22, ..., Fn Rd and {11, 3/2, .... Vn E IRd are i.idd drawn from two distributions p and q, and use the discrepancy mea- sure MMD[F. p. 4) to determine if p and q are indistinguishable in the RKHS F. We do not consider all other distributions in F that is irrelevant to samples! So far the idea of kernel func- tion between distributions is in practice not so much useful, however, here in this paper, we propose, when considering the plugin-estimator of many of the existing discrepancy measures, is beneficial to view them as probability kernel functions.
3.1 DEFINATION OF GENERALIZED PROBABILITY KERNEL
Definition 2 (Generalized probability kernel). Given the family S of discrete distribution on sup- port (1i, u.,.., } where vs E Rdd Let F be unit ball in universal-RRe RKHSS H with associated continuous kernel RK(I, y). where for any x E IRd and y E Rdd, RK(z, y) maps from Id x Rd to RR We denote gram matrix Kūj RK(E,, vz).
The generalized probability kernel function on distribution p,q E S is GPKr, = oo (p. q) KΦ o (q.p)² EiEjk & (pr; 9:) Kijo (gj.P))
where ó a (p. q) is element-wise mapping function discrete distribution p.q S, which maps from R2 x R to &
Obviously, under this definition. the GPK a symmetric probability kernel function where GPKF GPKF,o(q.P
Mapping function é represent great amount of possibilities. For most cases, we need to narrow down the region and equipped it with some convenient properties so that the GPK measure could be useful.
One example is for the measurement of discrepancy, where we want GPK5 (p. & p q.
q) = 0 and only
Definition 3 (discrepancy probability kemnel.. Let S be family of discrete distribution p S A discrepancy probability kernel is a kernel function PK(p, q) that PK(p,q) = 0 if and only if q
on support (v,, U2..... Uv
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Theorem 1. ooKF.o(p.9) with the mapping function that satisfies:
symmetric or antisymmetric with respect to p and q: o (p. q) = o oro - (q.p)
16 o (p, g)]l2 = 1 p2 = if and only if = q, where || lll represents L2 nomm.
is discrepancy probability kernel.
Proof.
GPKF,0(P.0) = Σ
Σ
Φo (pi, K)) (gj.P,)
iE|M] jE[{]
o (p. KΦ (q.p)* to (p. q) KΦ (p.q)"
=
K is semipositive definite matrix, thus by definition of positive definite matrix, vKo 2 0, where equality holds if and only if v = 0, and since v = ó (p, q), this condition further means (p. q) which holds if and only if p - q.
Another example the polynomial GPK, which our main focus of this paper. Such subelass of GPK is interesting since we can build unbiased estimators of it using techniques of Bernstein poly- nomial (Qian et al., 2011). As we will show section 5., we also have analyzable convergence bounds for the resulting unbiased estimators, illustrating its potential usage for applications such as two-sample test.
Definition 4 (polynomial GPK). The polynomial GPK is the subset of GPK that equipped with the mapping function 6 that polynomial in p and q: 6o (p. q) = Ei- E. ai,pp'q q" where Z is the degree of the polynomial, and 01,a E IR is the coefficient
Below we give some examples of polynomial GPK, which include MMD proposed in (Gretton al. 2012), and the newly proposed power-MMD in this paper, which is a natural extension of MMD, from the view point of probability kernels.
3.1.1 EXAMPLE 1: MMD AS MEMBER OF POLYNOMIAL GPK
Given discrete distribution p, with support (vi, U2, v. we can rewrite MMD with distribution probability function Pi, qe:
MMD (p. q) = [xnpf(x) -Evi ( (2')1
Σ
[(vi)p: Σ [v:))q
Σ
[(vi)p: f(v))
(E(M]
KE(M]
KE|K
ΣΣ
Σ (p-- (v=))))(v))((()))) 4j)
(p. 9: Kij(pp -gj)
SE|R| JE
KE|M JE[
= - GPK F (p.q)
Where di (p, 4) = p - q, is the RKHS defined in MMD literature, and f is the function that maps 1 to H.
GPKF, (p, q) is a special case of polynomial GPK where 01,0 1, ap,1 = -1, and all other coefficients are 0.
3.1.2 EXAMPLE PRODUCT GPK AS MEMBERS OF POLYNOMIAL GPK
Definition 5 (product GPK). Let p and be probability distributions on support {1i U2, Uk and E & be nonnegative integer. The product GPK is subset polynomial GPK where 01,0 1, and all other coefficients are O. the corresponding mapping function is: ((p.q) p'
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The probability product kernel as in (Jebara et al. 2004) is special case of product GPK where K a identity matrix.
3.1.3 EXAMPLE 3: POWER-MMD AS MEMBERS OF POLYNOMIAL GPK
Another interesting subset of polynomial GPK is the one extends MMD case into power form and denote it power-MMD:
Definition 6 (power-MMD). Lei p and q be probability distributions on support {ur, V2,.... Uk) and p E Z be positive integer. then the power-MMD is a subset of polynomial GPK where 0.0 = 1, 0 = -1, and all other coefficients are O. the corresponding mapping function is: pp. 9) p" q
Apparently, MMD is a special case of power-MMD where = 1, and power-MMD satisfies the requirement in Theorem IT, thus has the potential usage of discrepancy measure. In section [ we will show that power-MMD has unbiased estimator with analyzable convergence bounds thus could be used for two-sample test.
3.2 DISCUSSION OF GPK IN DISCRETE SETTING
As one may easily notice, the definition of GPK includes gram matrix generated by the kernel function RK(u,, (y) which measures the discrepancy between us Uj E {U1, U2, U. While con- sidering the cases of categorical distribution, values of discrete variables does relate to any notion of distance, this raises the question: how the introduced gram matrix will be beneficial in any cases?
Many natural processes produce discrete distributions where there possibly exists similarity measure in values which imply the similarity in frequencies of occur- rence(probability values). For example, in the field of natural language process(NI.P), one may treat words atomic units with no notion of similarity between words, as these are represented as indices in vocabulary. However, given large number of training samples, similarity measure between words could be made possible using techniques such as words2vec(Mikole) et al. 2013). Such techniques generally result better performance and have become the important preprocess- ing techniques for NLP tasks(Goodfellow et al. 2016). 2. As there are discrete variables are totally irrelevant, or people may use kernel function RK(U;, U2) which doesn't 't correctly imply the similarity in probability values, the GPK framework may still capture the simi- larity between distributions. One example is the case of MMD, which is, as we discussed above, an element of GPK family. As proved in (Gretton et al. 2012). MMD distribution free measure- ment between two samples, which means no matter what kind of and kernel y), we have, the MMD": 3 (p, q) measure will be 0 if and only if p does have a negative effect on convergence bounds of the empirical estimator proposed in (Gretton et al. 2012), and will influence the results of two- two-sample test. For this reason, we mainly focus on dataset with known relativity measures our experiment section.
The answer twofold: 1.
where the values of
However, the bad choice of kernel function
4 PLUGIN-ESTIMATOR FOR GPK
So far we have defined the GPK and discussed some subsets of GPK with potential usage of two-sample test. Next we discuss how build an estimator, given member of GPK. In this section, we propose the plugin-estimator, which based on the count of occurrence of each value is E (11, 02, ..., Uh samples {a1, .2,,.., Fn. E p or q. We illustrate that by doing so, the techniques of Bernstein polynomial in (Qian et al. 201T) could be used to help building unbiased estimators for any members of polynomial GPK. Furthermore, we provide analyzable convergence bounds of these estimators.
We begin with the definition of :etegnnestimatos::
Definition 7. Suppose we have i..d samples of distribution p Xn, 121, En, p and Xn, : {Ini+1, In, +2 u.. Fny } ~ p. And also the i.i.d samples of distribution q as Ymy (1/ 1/, .... yong } ~ q and Y lng {ymn +1, B.ny +2, Vmn +my] q.
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Let N/( denotes the number of occurrence of value vs {ui, U2,..., Vs} in sample Xup and Sin == (N(")) ni) .n) denotes the collection of ( and ny. The same follows for Xny Yms and Yma
We define the plugin-estimann of GPKF,o(P.. as
(PKK|F,,,X,Y)
Σ
Σ S (Sim St,ma) KigSo (5 Sj,mz)
(E[M]
where So is a function related to function é. and K is the gram matrix brought by F.
Here setting is different from the unbiased estimator MMD of Gretton et al. 2012). where in their setting X Xn, represent the same sample from and so do for Y Ym Y m2 from In- stead, we are using the same setting as the linear time statistic MMD proposed in (Getton et al., 2012). Another way of viewing this is that for our setting, given two samples 1 .... Fn 11,, y2,.... .... Un from p and we depart each sample of and into two parts, yielding 4 different samples with size n2, n2,, my, m2, and then calculate the empirical frequencies for glugin-estimane defined above.
4.1 POLYNOMIAL GPK WITH UNBIASED PLUGIN-ESTIMATORS
One of our main contributions of this paper is the proposal that we can always find an unbiased plugin-estimate for any members in polynomial GPK family. The basic idea is that analyze the expectation of plugin-estimators through Bernstein polynomial, and use the existing results of (Qian et al. 201T) Th to build the unbiased plugin estimators.
Theorem 2. Denote
(,(k, = ( k
) for j S k
93 (k, n)
for k
Then any member of polynomial GPK[F, D.P. q) equipped with polynomial mapping function Φ(p q) = Eiz Lezo Ont,p'' q" of degree &, has an unbiased plagin-estimator with mapping function to be:
So Stm,)
on.ssn(N mi)gg(N( (N(mm)) ,m))
1=0 am0
Proof. The basic idea is directly using the result of Bernstein polynomial QQan et al., 201 to build unbiased estimators. We put our formal proof in appendix
For notation simplicity, we define the plugin-estimator discussed above to be the default- default-plugin- estimator for polynomial GPK:
Definition 8 (default- (default-plugin plugin- esstimator for polynomial GPK). The plugin-estimator defined in The- oremer 2 is the platuuut-plug--eeee estimator for polynomial GPK.
This plugin-estimatoo according Theorem is unbiased estimator
4.2 DEVIATION BOUND OF PLUGIN-ESTIMATE
Another topic about default-plugin equality to derive the default
is its deviation bound. We directly use the McDiamid's in- plugin wwwg-p--stimaaanneenn estimator for polynomial GPK:
Theorem 3. The default-pluee estimator of GPK(F é. p.p.o g] equipped with polynomial mapping function o(p q) = Eizo Eizo 0n,pp' a.p qe of degree o E
has the convergence bound:
Va > 0, Pr[ o, X, Y| -F 2 S 2e -22
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where
=
((m)
[11)
(n.mm))
)
2
+ +m
)
(Tnj,mms) )²)
(2)
)
2
mg m2. n2
+
m2
((1)
+ n22
(T(2)
(1222112)
2
K2
mar
L n,m
(1 .n)gs(N, mm)
RE[W]
1=0 ==0
(1)
|on 9((N) ,n)g (N! m))
7n.m
[E[A]
sup
1=0 sm0
N
[nj
0
|a, (((2))) m))a((V m)
7..m
sup K
((m)
Kman is the largest value of entries K
Proof. The basic idea is to the McDiamid's inequality, and we put formal proof into the appendix.
5 EXAMPLE: POWER-MMD AS NATURAL EXTENSION TO MMD FROM GPK VIEWPOINT
In this section, we mainly discuss power-MMD as defined in 3.1.3 We analyze the bias and con- vergence bound of its plugin-estimaten using the techniques introduced so far, illustrating that such a natural extension to MMD from GPK viewpoint could be beneficial for two-sample test.
5.1 PLUGIN-ESTTMMTORSS OF POWER-MMID
As we already discussed section 4.1.3, power-MMD is a subset of polynomial GPK. According Theorem 2 any member GPKF (p, q) in power-MMD has defaallt-lugin-essiiatt with the mapping function ( SLm,) = (N,, (M, m))
Remark 3.1. When = 1, the power-MMD return to the original MMD case. Remarkably, the default-llugin-essimaae this case is equivalent to the linear time statistic MMD} proposed in Gretton al. 2012): GPKR[F, [KKK[F,r,,, X,Y] FM, XY For details of the derivation, see appendix
5.2 DEVIATION BOUND OF PLUGIN-ESTIMATORR OF POWER-MMD
Corollary 3.1. Denote Tn supie[t] (NfA) .n)
The default- teittt--ppiii--taaaee of power-MMD GPK5, oo (p,q) has uniform convergence bound de- fined in Theorem 3 with (1) and Tm.m Tm
(2)
Corollary 3.2. Consider the case where = n2 = = = n The default-plugin-ssimee
of poW GPKr, (p.q) has uniform convergence bound:
Pr[[GPKg[ Y E|GPK& op. X, X,YIII 2 a) < 2e 10 2 miging 1K < 2e Spr man
Proof. The first inequality above comes from:
P N (N, p(N: - 1)(N - 2)... - -p+ 1)
P NP-1
1P < P
<
n(n - 1)... +

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where supte[k] ( (N(,,))) = a only stands for extreme case such that there exist N, = n, i.. all the samples belongs the same value 2s E {ui, V2, uk And the second inequality above comes from:
o p2,m=
Σ (E(A)
9r (N(
..) ,n)
(o( m))
S
Σ
gp(N
m))
Σ sE(Ai
9 ( m)
sc(A]
NZ(1))
N:(ma)
NZ(1)
N/(m)
+
S
Σ KE[K]
n
m
n
m
KE[A]
EE kg]
(E|M
Remark 3.2. recall
(Gretton et al. 2012 the deviation bound for linear time estimator
MMD x, Y,
Va > O, M.PP((MMD)FF...Y) - EF, 2
2e
SK
Treus
Interestingly, this bound is the same as the case p = in Corollary 3.2 section 6.1, the eeffaul-plugiinesttnnee MMD case in (Gretton et al. 2012). Our bound generalize the bound in (Gretton et al. 2012 and provide lighter version. Note that the bounds for special case of p has simpler derivation, and the reader may refer to appendix for more details.
Note that according to
of power-MMD with p 1 is actually in equivalent to
5.3 TWO-SAMPLE TEST USING POWER- -MMD
Corollary 3.3, A hypothesis test of level for the null hypothesis p has the acceptance region [F (log((()) "1 Where Z is defined in Corollary 3.1
V2
The two-sample test for power-MMDD then follows this procedure: 1. (F... check if < (log((()) ()) if so, accept the null hypothesis, otherwise
calculate
VZ
reject the null hypothesis.
Next we analyze the performance of our proposed two-sample test under two cases: p and p>1
5.3.
CASE
For case, since GPK [F, X, is equivalent to MMD & Y, the only difference between our proposal and that of Gretton et al.] (2012) is the convergence bound. According to Remark 3.2 provide tighter bound for the test statistic, thus will certainly have better performance using power-MMD.
5.3.2 p >>> CASE
We need to answer two questions for the > 1 case: 1. when applying MMD in practice, the proposed statistics numerical stable? 2. will the performance of two-sample test gets better when / gets larger?
For the question of numerical stability, since (N,,n) < NN:) (N:)², the term will exponentially de- crease with the increase of p. This effect will cause numerical problem when N; & n and p is large. which is numerical stable.
(
One solution is to find upper-bound of GPKg[F.
Corollary 3.4. Consider the simplest case where m n2 mh m2 Define
CN (N(())) (()))) ...., ((1)), N((m)), (m2) ...., (122) (m(m)) ( (mm)
(ma) N (m(2) (ma) ..... (m22))

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be the set of all counts of occurrence in the four samples Xm1 Xx2 Xm1: Xmp- Denote SN
suNN,ecN (N₂) to be the maximium value in the set CN We have:
GPKE|F, pp. X,Y] <
SN GPK's
VZ
Kmaupy 2n 40
where
GPK's = Σ
(go(N(")), ,,) (N( ,Sx) Kis (go(N ,Sx
Sxx) -go(N sN
[JE(M]
and
Φ' Σ 9o(NN( SN) - -p(()(m) SN)]
(E(R)
For cases when N2 are not far less from SN. GPK's will be much more numerical stable than GPK=[F. op. X, Y].
To answer the question related to the performance of two-sample test when P get larger, we need to increase with the increase of p. Unfortunately,
analyze the case when p # q. if (FK,, there is no clear answer to this.
vZ
SUMMARY
To summarize, we introduce the framework of generalized probability kerkel(GPK). While GPK represents large family of probability kemels. we focus polynomial GPK since all members of such subset of GPK have unbiased plugin-estimaten from the viewpoint of polynomial GPK, which call power-MMID, could be used for two-sample test. Theoretical study shows that for proposed in Gretton et al. (2012), and the performance of p > members of GPK which do not belong to polynomial GPK, it is not easy to design an unbiased estimators. However, bais reduction techniqes proposed in (Yi al., 2018) and (Yi Alon 2020) could be used, and we still have the chance to apply two-sample test with the resulting estimators. Such possibility is also left for future work.
Remarkably, a natural extension of MMD
case, power-MMDD outperforms linear time MMD case is left for future work. For
REFERENCES
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Hao Yi and Orlitsky Alon. Data amplification: Instance-optimal property estimation. International Conference on Machine Learning, 2020.
Hao Yi, Orlitsky Alon, Theertha S. Ananda, and Wu Yihong. Data amplification: unified and
competitiveapproach to property estimation. Neural Information Processing Systems, 2018.
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A APPENDIX
A.1 BERNSTEIN POLYNOMIAL
Drawing i.i.d. samples ym from any distribution p the expected value of the empirical estimator for a distribution property is
M
IB ME (X
= Σ
Mimbin(m.)))
M.
iE(M)
Note that for any function f, m N, and x E [0, 1], the degree-m Bernstein polynomial of
Bm =
x' -
(1-1=))"- =
j
i=0
Therefore, express the expectation of the empirical property estimator
E Ymup
Me (YT))
Σ kE(K]
Bm (h, Pt)
A.2 PROOF OF THEOREM 2
Proof. Recall the definition of polynomial GPK:
= Σ
(pr. = Σ
Σ
Kis Σ a.
RE TA jE[
JE(K
recall in QQan et 2011
Σ
(k, n)bx,. = ( (k,
where gf (k, n) defined in the beginning of the theorem
GPK(F, (PK(F, = Σ
Σ
Kij Σ
on Cr.
kETRi JE(A]
Σ
Σ
Kiy
Kij
01, (in m1)
min(pp..
SETR JE(K]
Larie0
E Ny wbin(g, m2) I (o(
n2 Ny -bin b
E
(N (N nn nn2)E x wbin( ffmru g (aa(i (aa
= E ΣΣ Ki. Σ
a )g, (N(mi) m. (N) n2)g (mm .2m)
NE(R] JELN
Lante0
= E E F,
A.3 PROOF THEOREM 3
Lemma 4. Let SN.m. == (N n1) denotes the collection of N: (m1) and n1. The same follows for Xng, Ym, and Y Ymg. Also for notation simplicity, let Si,z == SN
For the plugin-estimator
GPK X, - fo (S SN., Kiffo (SN SNy SE(A]
with mapping function (S SN having the following properties:

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Jo(SN,, SN,m1)) monotonic function related ((mi) and N/ma); VN, N or < N¹ fa(SN., SN.m.) > fo(Sw, SN,m..)
The same follows for ((m)
If No SN...)) fS(S Sw..m S Tm
and
(S m,,+1,,;) (S(SN.m Sw. Tms
where Tay is constant related sample size m. Tiyy is a constant related sample size mi, the same follows for Tag and Tin
have:
2a) S. 2e -22
Va
where
Z ((1IT n22T2)))) $2 + (mITm, -2 mmam)) $2} K2 max
Φ = Σ
= Σ
[ S ma))
JE[R]
jE(M]
Kmav is the largest value of entries in K
Proof. recall McDiamid's incquality
Theorem 5. Let YI.................. ....... ., Ym be independent random variables taking values in ranges Ri...., Rm Rm - C. with the property that if one freezes all but the w co- ordinate of F (yI...., 3m) for some thus F (y1,... :.nb-...-........ 1,,-1, Mw. ywtt....., Vmm) F (Vi...., tw- Vw: bottl.... ,9m) I Cw for all Yj Rí and Vna Rw for Si m
and let F Rı x
w m, then F only fluctuates by most Cu 0,
2u2
Then for any O, one has Pr([F(Y)- - E[F 2 a) S 2e
ME
considering the plugin-estimator GPK family:
GPKg|F.6.X.y1 Σ Σ fo (S St.,,) Kijfo (Sima Sp.az)
jE[A
Without loss of generality, we rewrite the function fo as:
fo (SN SN,m, = F(21, 22, Ear Iny N ((1)) m1) = FN.
Assume freeze all but one element in X,,, (x1, T2, ... Eay..n Fny }. and only my is allowed to change its value.
obviously, matter how this element change, it always lies in the finite set of support {U,, u2,..., ) without loss of generality, we assume x, changes its value from U; to U.s thus the corresponding count of occurrence N, changes N, 1 and Na changes to N₂
10

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Under review as a conference paper at ICLR 2021
have for 2i Xn1
sups,
GPKg (2 Fn) - GPKg(21, F2, Fs x..)
Σ
To: Ini No N/mi), mm)) F(x1, 22, ..... Egg. Dny N/(ma), ma)) Kiffo (S)m2 Sp,22)
sups.
(EE(K]
suPi,
Σ
((FN--1- (N,)Ki, + (FN++1 PNaJ) Kay) fo(SN, m2 S2,
]E(R]
Tmi (- (-Ki, + ( SNy,
I s/n K Kmar
Σ jEll
[S((N SN,,ma)
JE[K]
=Tu, KmanD D22
where $2 = Ejeik [o(SNg SNy.ma)
Note that Kij == Kij
Similarly, for I, X22
Co S Tny Kma L
where 1 =jE(N (P1))111)
for v. Ym
Cg S Tmy K Kmaz $2
for v3 Ymz
C. S Tio Kmax $1
thus according to McDiamid's inequality, we have
Pr([GPKg[F. - -E GPK &, 2
We set
C Tny Kmaz $2
for H, E {Xnr).
Tm, Kman P
for F, E (.
C = Tng Kmaz $1
for v, E (X
G: Tiy Kmaz $1
for H, E (Yma}
and get
mitn +ma
of = ni (Trs Kmaz 22)² + n22 (Tnz Kmaz Di D1)²
+ m. (Tm, K max 22)² m2 (5mz (K Kmnn 22)
=((117) + mm m $3 (122 m 2
mi b7) K2
maz

---

Under review as a conference paper at ICLR 2021
We ready to proof theorem
Proof. Define
Div n.
!gr( 1, n)g. (NV (NV!T)) .m) (N,
(N, .m)] mm]
(N; (NN 1)... (N:: -O0N---40 -p+ 2)
Ni (N. - (o.(N/ m))
mi 1)... (2. - -p p
ni mi(mi - (11. - -
91 (N( m)) | m))
NT(1) 1++ +1
1
(((( (N( m)
N(()) -1 +
(2) Dir
(N (m) 1, (N m) (N( (N, mm)
(N; - 1)(N: - 2)... -
N.(N, 1004-11)..(0 1)...(N,, pp
- m1(m1 1m1 11+ (o m))
mi(m - (11 - p 1)
N (( .n) [ m))
9( (N/")) .m)
(1) Divn, max(Dio
Diu p (( (N) m)
Recall the mapping function of default-llugin of polynomial GPK f ( Eis Ei=o (N ,n) m)
Apparently fo is monotonic function with respect to N, and M,, thus condition 1, for Theorem satisfied
Since also have:
So(S N(())()) (y(m) mn SM. 5
or Döun,t.o
I=0 s=u
(: (u (a) (n) .m)
I
a
1=0 ==0
thus condition 2. for Theorem
3 is satisfied
12

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Under review as a conference paper at ICLR 2021
A.4 PROOF OF COROLLERY 3.4
Proof. Define T = SuPN,ECx A. Syy (1),,. Pa,m = ( P Pmzznz
Since we have 9o(N.n) 9o(N, M), we could get
(M(Maa)
EnjERa (gp(N)!m), n) - (p(N!(ts) .ni) Kis (gp(N) .n) - g .n)
GPK F, Op. X.
VZ
Kmar 4 nm 2n
EnJelh (go(N) m)) (o( m)) Kij 1pp (N .)) 9p ( .
Kmaz Sw (o(SN,,) EiElh 9pl (2(NV ..) (p(N ni) 22n
SN EnjelH (Dp(N(rr) .xx) - (N( ,Sw)) Kij (g,(A) ,Sx) 9p (p( SNN)
KmaxpV 2n ΣiE(H) (N .5x) (N, .SN)
A.5 DETAILS OF REMARK 3.1
Σ
(N)(N) mn) ( mi) Kij (g)(Nr m2)) m2)) 91 (.( m2)
iE(R]
JE(K]
N (m1)
N (m1)
Ny(ma)
N/maa
EΣ
Kig
11
mi
122
m22
KE(A)
1
1
k 25) +
l (yt. By)
k (04,81)
k (Ix, 81)
m2m2
n1112
12
m2n
1=1 ==1
1=1
(=1 j=1
MMD X,y
A.6 THE CONVERGENCE BOUNDS OF POWER FOWER-MMD WITH p=
When P 1, Tai gN
For simplicity consider the case where m
n1
n2 m1 =m2 n. from Corollary
we have
Pr([GPKE[F 0i, X,
EE G or, X, X,Y]] 2 a) < 2e 2C2
(m))
((
(y
Since 4- = (f
ΣjE(K)
< Σ jE(M]
E ΣjE(N)
jE(NN]
2
we have
Pr[[GPKe[F Pi, X, Y] E[GPKK[[F Pi, X, X,Y]JI 2 a) 2e Kmag
S 2c
sic Faz
recall in Gretton al. 2012). the deviation bound for linear time estimator MMD pP
Pr( PM - [F, [,, 2 a) S 2e SK
Thus our bound generalize the bound in (Gretton et al. 2012) and provide tighter version.
A.7 IS BOUNDS GET TIGHTER WHEN P GETTING LARGER?
As we've already known, p = 1 case is equivalent to MMD, in (Gretton et al. 2012). one question rises: Would the performance of cases p > better than widely used case?
13

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Under review as a conference paper at ICLR 2021
moo? e Mux the convergence bounds for p > cases
According to Corollary 3.2. since e SK moe
seem looser than p case, and this may give negative answer to the question above.
However, the bound above is based on the worst cases where sup; (N₂) n. such that Tn S and + s 2. In practice, we are less likely to come across such phenomena, instead, may assume the (N; to be far smaller.
(upp( Without loss of generality, assume we have mar (up((()())) easily seen:
S a where 2 1, it is
In S P No
p
n no-1
an-In n
and
!!
No(m)
+ 11 )
s n.m
+
$ 2
m
RE[A]
define r = max (Tai) Tag: Timg: Tmg)) and +h = max ( m1 1711 $ m2g
We have
p² 2 (1 P + ( 41-
S 2 max S 16
K2 mar Zb
Plotting the Zo value with respect variety value of and in Fig. the bound will be looser given larger However, for a larger than around 1.25, which means the (N,) is slightly smaller than the sample size, the bound will become tighter when p is large. This illustrate the benefit of using power-MMD with larger p in practice.
see that for 1,
We could also get tighter bound according to Corollary 3.1
Practically, it will be m) and Pn,m. =
much more beneficial to calculate the
suPse(k)
(N(
Eiel (N(" m) (p(")"") .m) on the fly. That is to say, we do not estimate the convergence bounds before we receive the samples, instead, the calculation of the bounds is carried out together with the calculation of default- defaull-plugin plugin-estimators. Remarkably this is still distribution-fiet bounds, since we make no assumptions on the probability functions we apply hypothesis test upon.
However, the issue is although Z decreases when p increases,
GPKg Ap. X, Y also decreases when o increases. It is not clear how p will influence the value of (T.,. v2
n1 = 1000, Koar
(ag(2
235
Figure 1: log (o)( with respect to variety of and
14


Reviews:
[{"review_id": "xn8vr0P4wAI", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper tries to propose a kernel-based discrepancy measure called generalised probability kernel that can unify MMD and KSD which is an interesting topic of discussion. The paper applies the new discrepancy to perform two-sample tests. \nThe new kernel proposed, unlike the previous RKHS kernels that only depend on data-points, incorporate the notion of probability. e.g. kernel K_{p,q} depends on density p and q. also a symmetric version on discrete KSD is discussed.\nDespite the idea is interesting, there are several flaws which can be reviewed.\n\nFirstly, I think the paper is not clearly presented, with some confusing notation.\n--in Definition 1, you defined a kernel, on distributions p and q, that is a k x k matrix; while in definition 2, the notion of K, are on samples and is a scalar output.\nit is unclear of how \\phi is defined in general; only examples are given later for specific cases so that we got an conjuncture.\n--in Definition 5, why is it different from stein operator of KDSD? or it is supposed to say difference operator?\n\nIn addition I have several confusions:\n1. why is MMD_E^2 an unbiased estimator? what happened to k(x_i, x_i)? it is not clear from the Bernstein polynomial introduced in appendix. \n2. in abstract, it claims that the kernels are between distributions instead of samples, but in the main text it is still evaluation at p_i=p(x_i) on samples; I m confused of the difference and novelty claimed.\n3. The above concern brings up the question while applying on two sample test. \n--When the MMD is used to perform two-sample test, it is assumed that both p and q are unknown. however, to my understanding, we need to know p and q to define k_{prob}; how is this going to be applied to two-sample test? \n--for KSD setting, when the symmetric KDSD is introduced, it also seems to require p and q to known for two-sample testing. In the Liu2016 setting, where goodness-of-fit test is proposed with KSD, q is known (up to normalization) while p is unknown with samples; that is a key point why KSD is useful for goodness-of-fit test.\nIn addition, is there any argument on why the symmetric-KDSD might be better than KDSD Yang et.al 2018?\n\nAn additional point is regarding literature review, which is yet throughout  to check; e.g. as\nChwialkowski, et. al  \"A kernel test of goodness of fit.\" proposed independently as Liu et.al for KSD goodness-of-fit test, that might be useful to cite.\n\nIn my point of view, ICLR may not be a venue of fit either. More reviews and clarifications may be required, for both kernel construction and application. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "SZK27KWfx2Z", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary.\nThe authors describe a family of kernel functions on discrete probability measures. The kernel generalizes existing discrepancies such as the MMD and the KSD. The authors further provide plugin estimates based on empirical frequencies and some arguments for unbiased-ness.\n\nWhile it is interesting to think about alternative estimators for comparing probability distributions, this paper falls short on the execution. I would recommend a major work-over before considering a submission again. There are many missing points in theory, experiments (there are none), and presentation. See below.\n\nIt is unclear to me why we would care about the proposed estimators\n* There is no analysis showing that the presented kernels are useful in any way.\n* I appreciate that the authors show that existing discrepancies are special cases of the proposed one, but I wonder again what that is useful for?\n* This is in particular as the authors do not provide any sort of asymptotic analysis of the presented estimators. How can we use them for two-sample testing without that? Answering this question is one of the major parts of the kernel two-sample testing literature.\n\nTheory.\n* The presented theory consists of elementary manipulations that mostly follow existing literature, so there is very little actual innovation. For example of of page 5.\n\nExperimental evaluation\n* There is *no* experimental evaluation of the proposed estimators.\n* It would have been interesting to compare the variances as a function of dataset characteristics.\n\nPresentation\n* There many grammar glitches, spelling mistages, missing articles, etc, to a point that it is hard to follow.\n* There is no overview of the series of arguments in the later part of the paper.\n* There is a lot of re-cited derivations from existing papers.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "6KYySP4YZT5", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper under review proposes to generalize MMD for discrete random variables whose labels take value in $\\mathbb{R}^k$. They propose to estimate these generalized probability kernel distance using empirical estimators. Their properties are studied for two particular examples, namely a kernelized Stein discrenpancy and polynomials versions. Consistency and bias of both estimators are studied and bias corrected.\n\nThe paper has numerous typos, with approximate English and lacks of rigorousness when introducing mathematical concepts; multiple notations are never defined. On the theoretical side, the setting on which the contribution relies on is quite strange: in general, labels of discrete variables does not relate to any notion of distance/ordering as in a classical RKHS setting making the relevance of the methodology quite questionable. These point with other technical issues are summarized in following remarks:\n\n## Major points\n* As mentioned above, it is quite rare in a discrete setting that the labels lies in $\\mathbb{R}$ and satisfies a notion of distance. The authors should better motivate this setting by giving at least on relevant example, either theoretical or practical, where such structure is relevant.\n* What does a Stein operator in a discrete setting means? There is a diffenretial operator in Definition 5 that is difficult to generalize and apply in a discrete context.\n* The symmetric KDSD introduced in Definition 6 is claimed to be a probability kernel, but the proof that is satisfies Definition 2 is not given.\n* The so-called polynomial probability kernel seems to obviously require $l=k$ to satisfy conditions of Definition 3, i.e., that $|\\phi(q,p)\\| = 0$ implies that $p=q$. It can be called a probability kernel only in such condition.\n\\end{enumerate}\n\n\n\n## Minor points\nThe paper has numerous typos and imprecisions; a subset of them are listed here.\n* p.1: 'underline' should be 'underlying'.\n* p.1 when using 'i.e.', always write ',e.i.,'.\n* p.1 and onward: there is always a space missing before each parenthesis.\n* p.1: Yi & Along (2020) should be a citep and not citet.\n* p.1: 'remain futher study' should be for instance 'is left for future work'.\n* p.1: KSD is not defined yet.\n* p.2: 'the introducting' is `the introduction'.\n* p.2 'in representing; is not right.\n* p.2 Is $[k]$ the sample space? If yes what is $\\{x_1,\\dots,x_n\\}$? A sample? What is the probability measure $v_i$? Do you mean the probability that $X$ falls in $v_i$?\n* p.2 Definition 1: 'Given that distributionS p and q belong ... distributionS with...'. Also 'map' is singular. What is the 'function space' that you refer to? Also where does this definition comes from? Please give proper referencing.\n* p.2: Why is there a line break right at the start of 4.1?\n* p.2: what is an 'instance of integral probability metric'?\n* p.2: last equation $\\mu_p$ is not defined, the product opertor $<.,.>_{\\mathcal{H}}$ is not defined. $\\mathcal{H}$ is not defined.\n* p.3: what these 'embedding functions'?\n* p.3 the RBH kernel is not defined.\n* p.3 second equation: what is $\\phi$?\n* p.3 Definition 2: the index the sample space should be k, i.e., $y_1,\\dots,y_k$ if it refers to the distributions and $n$ for a sample. Here it should be $k$ as it is written distribution.\n* p.3: 'examINE', 'members'.\n* p.4: the 'brief' proof provided here is only working for discrete variables, while the proof in Gretton et. al deals with continuous variables.\n* p.4: what is the 'term above'?\n* p.5 'illustrate'\n* p.5: there should not be such a thing as an 'art' in science. If you raise that question, then you should formally discuss this topic (choice of optimal $\\phi$).\n* p.5 Second equation: what are $x_s$ and $x_t$? Notations between this equation and the next are not consistent ($n$ is paired with $x$ and then with $y$ in the next equation).\n* p.6: what is this so-called 'same property'?\n* p.6: pmfs is never defined.\n* p.6 Definition 5: notation $\\mathcal{A}$ is never used. $s_p$ is not defined. $\\Delta^*$ is not defined. If the latter is a differential operator, what does it means in the context of discrete random variables?\n* p.6: what is 'form 5'?\n* p.7: what forms 5 and 6?\n* p.7: Theorem 5: operator $L$ is not defined. A dot is missing. Are $p$ and$q$ density functions of pmfs?\n* p.7: 'preliminary results'.\n* p.7: what does 'justing' mean?\n* p.8: what is requirement $2$?", "extra_scores": {}, "extra_text": {}}, {"review_id": "51mRl3J2PK", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The works proposes a generalization of MMD-squared distance. However, the submission seems to be an incomplete one.\n\nMajor comments:\n1. Definition 2 seems to be the key definition in the work. However, there are multiple issues:\n       a. It is not clear why it is called a kernel? Should not it be called distance? After all, it generalizes MMD-squared!\n       b. \"$K$\" seems to be mixed up with \"$k$\". \"$K$\" seems to be the gram matrix and not the kernel.\n       c. $k(y_i, y_j)$ is from $Y \\times Y\\rightarrow R$, and not from $R^n \\times R^n \\rightarrow R$.\n       d. why should \"\\phi\" belong to RKHS of k? Recall that RKHS of k will contain functions from R^n\\rightarrow R.\n2. I agree with the write-up which states  that results in section 4.2 are trivial.\n3. section 4.3.1 are known results and need to be skipped.\n4. Proof of theorem 5 seems to be completely missing. How is sup over f removed?\n5. In Definition 6, what is $f$? Is a sup over f missing? Because of this and the previous issue, section 5.1 seems very incomplete.\n6. Simulation section seems to be completely missing.\n7. Connection with Bernstien polynomials highlighted in intro etc. seems to be missing.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The focus of the submission is to define divergences on discrete probability measures. Particularly, the authors propose a common generalization of the well-known concept of maximum mean discrepancy and kernel Stein discrepancy.

As summarized by the reviewers the submission is in a rather preliminary form:
1)The work lacks motivation.
2)Literature review (there are 4 references in total) and numerical illustrations are missing.
3)The submission lacks proper mathematical formulation/rigor.
I highly recommend the authors to not submit similar draft manuscripts in the future.
Decision: Reject