Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy
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Optimal Quantisation of Probability Measures
Using Maximum Mean Discrepancy
Onur Teymur Jackson Gorham Marina Riabiz Chris. J. Oates
Newcastle University Whisper.ai, Inc. King’s College London Newcastle University
Alan Turing Institute Alan Turing Institute Alan Turing Institute
Abstract sation picks the xi to minimise a Wasserstein distance
between ν and µ, which leads to an elegant connection
with Voronoi partitions whose centres are the xi (Graf
Several researchers have proposed minimisa-
and Luschgy, 2007). Several other discrepancies exist
tion of maximum mean discrepancy (MMD)
but are less well-studied for the quantisation task. In
as a method to quantise probability mea-
this paper we study quantisation with maximum mean
sures, i.e., to approximate a distribution by a
discrepancy (MMD), as well as a specific version called
representative point set. We consider sequen-
kernel Stein discrepancy (KSD), each of which admit
tial algorithms that greedily minimise MMD
a closed-form expression for a wide class of target dis-
over a discrete candidate set. We propose a
tributions µ (e.g. Rustamov, 2019).
novel non-myopic algorithm and, in order to
both improve statistical efficiency and reduce Despite several interesting results, optimal quantisa-
computational cost, we investigate a variant tion with MMD remains largely unsolved. Quasi
that applies this technique to a mini-batch of Monte Carlo (QMC) provides representative point sets
the candidate set at each iteration. When the that asymptotically minimise MMD (Hickernell, 1998;
candidate points are sampled from the target, Dick and Pillichshammer, 2010); however, these re-
the consistency of these new algorithms—and sults are typically limited to specific instances of µ and
their mini-batch variants—is established. We MMD.1 The use of greedy sequential algorithms, in
demonstrate the algorithms on a range of im- which xn is selected conditional on the x1 , . . . , xn−1 al-
portant computational problems, including ready chosen, has received some attention in the MMD
optimisation of nodes in Bayesian cubature context—see Santin and Haasdonk (2017) and the sur-
and the thinning of Markov chain output. veys in Oettershagen (2017) and Pronzato and Zhigl-
javsky (2018). Greedy sequential algorithms have also
been proposed for KSD (Chen et al., 2018, 2019), as
1 Introduction well as a non-greedy sequential algorithm for minimis-
ing MMD, called kernel herding (Chen et al., 2010).
This paper considers the approximation of a proba- In certain situations,2 the greedy and herding algo-
!n on a set X , by a dis-
bility distribution µ, defined rithms produce the same sequence of points, with the
crete distribution ν = n1 i=1 δ(xi ), for some repre- latter theoretically understood due to its interpreta-
sentative points xi , where δ(x) denotes a point mass tion as a Frank–Wolfe algorithm (Bach et al., 2012;
located at x ∈ X . This quantisation task arises in Lacoste-Julien et al., 2015). Outside the translation-
many areas including numerical cubature (Karvonen, invariant context, empirical studies have shown that
2019), experimental design (Chaloner and Verdinelli, greedy algorithms tend to outperform kernel herd-
1995) and Bayesian computation (Riabiz et al., 2020). ing (Chen et al., 2018). Information-theoretic lower
To solve the quantisation task one first identifies an bounds on MMD have been derived in the liter-
optimality criterion, typically a notion of discrepancy ature on information-based complexity (Novak and
between µ and ν, and then develops an algorithm to Woźniakowski, 2008) and in Mak et al. (2018), who
approximately minimise it. Classical optimal quanti-
1
In Section 2.1 we explain how MMD is parametrised by
Proceedings of the 24th International Conference on Artifi- a kernel ; the QMC literature typically focuses on µ uniform
cial Intelligence and Statistics (AISTATS) 2021, San Diego, on [0, 1]d , and d-dim tensor products of kernels over [0, 1].
California, USA. PMLR: Volume 130. Copyright 2021 by 2
Specifically, the algorithms coincide when the kernel
the author(s).
on which they are based is translation-invariant.Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy
studied representative points that minimise an energy The remainder of the paper is structured thus. In Sec-
distance; the relationship between energy distances tion 2 we provide background on MMD and KSD. In
and MMD is clarified in Sejdinovic et al. (2013). Section 3 our novel methods for optimal quantisation
are presented. Our empirical assessment, including
The aforementioned sequential algorithms require
comparisons with existing methods, is in Section 4 and
that, to select the next point xn , one has to search
our theoretical assessment is in Section 5. The paper
over the whole set X . This is often impractical, since
concludes with a discussion in Section 6.
X will typically be an infinite set and may not have
useful structure (e.g. a vector space) that can be ex-
ploited by a numerical optimisation method. 2 Background
Extended notions of greedy optimisation, where at Let X be a measurable space and let P(X ) denote
each step one seeks to add a point that is merely ‘suffi- the set of probability distributions on X . First we
ciently close’ to optimal, were studied for KSD in Chen introduce a notion of discrepancy between two mea-
et al. (2018). Chen et al. (2019) proposed a stochastic sures µ, ν ∈ P(X ), and then specialise this definition
optimisation approach for this purpose. However, the to MMD (Section 2.1) and KSD (Section 2.2).
global non-convex optimisation problem that must be
solved to find the next point xn becomes increasingly For any µ, ν ∈ P(X ) and set F consisting of real-
difficult as more points are selected. This manifests, valued measurable functions on X , we define a dis-
for example, in the increasing number of iterations re- crepancy to be a quantity of the form
quired in the approach of Chen et al. (2019). "# # "
DF (µ, ν) = supf ∈F " f dµ − f dν " , (1)
This paper studies sequential minimisation of MMD
over a finite candidate set, instead of over the whole assuming F was chosen so that all integrals in (1) exist.
of X . This obviates the need to use a numerical optimi- The set F is called measure-determining if DF (µ, ν) =
sation routine, requiring only that a suitable candidate 0 implies µ = ν, and in this case DF is called an
set can be produced. Such an approach was recently integral probability metric (Müller, 1997). An example
described in Riabiz et al. (2020), where an algorithm is the Wasserstein metric—induced by choosing F as
termed Stein thinning was proposed for greedy min- the set of 1-Lipschitz functions defined on X —that
imisation of KSD. Discrete candidate sets in the con- is used in classical quantisation (Dudley, 2018, Thm.
text of kernel herding were discussed in Chen et al. 11.8.2). Next we describe how MMD and KSD are
(2010) and Lacoste-Julien et al. (2015), and in Paige induced from specific choices of F.
et al. (2016) for the case that the subset is chosen from
the support of a discrete target. Mak et al. (2018) pro- 2.1 Maximum Mean Discrepancy
posed sequential selection from a discrete candidate set
to approximately minimise energy distance, but theo- Consider a symmetric and positive-definite function
retical analysis of this algorithm was not attempted. k : X × X → R, which we call a kernel. A kernel re-
produces a Hilbert space of functions H from X → R
The novel contributions of this paper are as follows: if (i) for all x ∈ X we have k(·, x) ∈ H, and (ii) for
all x ∈ X and f ∈ H we have 〈k(·, x), f 〉H = f (x),
• We study greedy algorithms for sequential min- where 〈·, ·〉H denotes the inner product in H. By the
imisation of MMD, including novel non-myopic al- Moore–Aronszajn theorem (Aronszajn, 1950), there is
gorithms in which multiple points are selected si- a one-to-one mapping between the kernel k and the
multaneously. These algorithms are also extended reproducing kernel Hilbert space (RKHS) H, which we
to allow for mini-batching of the candidate set. make explicit by writing H(k). A prototypical exam-
Consistency is established and a finite-sample-size ple of a kernel on X ⊆ Rd is the squared-exponential
error bound is provided. kernel k(x, y; ℓ) = exp(− 12 ℓ−2 (x − y(2 ), where ( · ( in
• We show how non-myopic algorithms can be cast this paper denotes the Euclidean norm and ℓ > 0 is a
as integer quadratic programmes (IQP) that can positive scaling constant.
be exactly solved using standard libraries. Choosing the set F in (1) to be the unit ball B(k) :=
• A detailed empirical assessment is presented, in- {f ∈ H(k) : 〈f, f 〉H(k) ≤ 1} of the RKHS H(k) enables
cluding a study varying the extent of non-myopic the supremum in (1) to be written in closed form and
selection, up to and including the limiting case in defines the MMD (Song, 2008):
which all points are selected simultaneously. Such
MMDµ,k (ν)2 := DB(k) (µ, ν)
non-sequential algorithms require high computa- ## ##
tional expenditure, and so a semi-definite relax- = k(x, y) dν(x) dν(y) − 2 k(x, y) dν(x) dµ(y)
##
ation of the IQP is considered in the supplement. + k(x, y) dµ(x) dµ(y) (2)Teymur, Gorham, Riabiz, Oates
Our notation emphasises ν as the variable of interest, where subscripts are used to denote the argument upon
since in this paper we aim to minimise MMD over pos- which a differential operator
# acts. Since kµ (x, ·) ∈
sible ν for a fixed kernel k and a fixed target µ. Under H(kµ ), it follows that kµ (x, ·) dµ(x) = 0 and from
suitable conditions on k and X it can be shown that (2) we arrive at the kernel Stein discrepancy (KSD)
MMD is a metric on P(X ) (in which case the kernel is ##
called characteristic); for sufficient conditions see Sec- MMDµ,kµ (ν)2 = kµ (x, y) dν(x) dν(y).
tion 3 of Sriperumbudur et al. (2010). Furthermore,
Under stronger conditions on µ and k it can be shown
under stronger conditions on k, MMD metrises the
that KSD controls weak convergence to µ in P(Rd ),
weak topology on P(X ) (Sriperumbudur et al., 2010,
meaning that if MMDµ,kµ (ν) → 0 then ν ⇒ µ
Thms. 23, 24). This provides theoretical justification
(Gorham and Mackey, 2017, Thm. 8). The description
for minimisation of MMD: if MMDµ,k (ν) → 0 then
of KSD here is limited to Rd , but constructions also
ν ⇒ µ, where ⇒ denotes weak convergence in P(X ).
exist for discrete spaces (Yang et al., 2018) and more
Evaluation of MMD requires that µ and ν are either general Riemannian manifolds (Barp et al., 2018; Xu
explicit or can be easily approximated (e.g. by sam- and Matsuda, 2020; Le et al., 2020). Extensions that
pling), so as to compute the integrals appearing in use other operators Aµ (Gorham et al., 2019; Barp
(2). This is the case in many applications and MMD et al., 2019) have also been studied.
has been widely used (Arbel et al., 2019; Briol et al.,
2019a; Chérief-Abdellatif and Alquier, 2020). In cases
3 Methods
where µ is not explicit, such as when it arises as an
intractable posterior in a Bayesian context, KSD can
In this section we propose novel algorithms for minimi-
be a useful specialisation of MMD that circumvents
sation of MMD over a finite candidate set. The sim-
integration with respect to µ. We describe this next.
plest algorithm is described in Section 3.1, and from
this we generalise to consider both non-myopic selec-
2.2 Kernel Stein Discrepancy tion of representative points and mini-batching in Sec-
tion 3.2. A discussion of non-sequential algorithms,
While originally proposed as a means of proving dis-
as the limit of non-myopic algorithms where all points
tributional convergence, Stein’s method (Stein, 1972)
are simultaneously selected, is given in Section 3.3.
can be used to circumvent the integration against µ
required in (2) to calculate the MMD. Suppose we
have an operator Aµ defined on a set of functions G 3.1 A Simple Algorithm for Quantisation
#
such that Aµ g dµ = 0 holds for all g ∈ G. Choosing
In what follows we assume that we are provided with
F = Aµ G := {Aµ g : g "∈
# G} in (1),
" we would then have a finite candidate set {xi }ni=1 ⊂ X from which rep-
DAµ G (µ, ν) = supg∈G " Aµ g dν ", an expression which
resentative points are to be selected. Ideally, these
no longer involves integrals with respect to µ. Appro-
candidates should be in regions where µ is supported,
priate choices for Aµ and G were studied in Gorham
but we defer making any assumptions on this set un-
and Mackey (2015), who termed DAµ G the Stein dis-
til the theoretical analysis in Section 5. The simplest
crepancy, and these will now be described.
algorithm that we consider greedily minimises MMD
Assume µ admits a positive and continuously differen- over the candidate set; for each i, pick
tiable density pµ on X = Rd ; let ∇ and ∇· denote the $ ! %
i−1
gradient and divergence operators respectively; and let π(i) ∈ argmin MMDµ,k 1i i′ =1 δ(xπ(i′ ) ) + 1i δ(xj ) ,
k : Rd × Rd → R be a kernel that is continuously dif- j∈{1,...,n}
ferentiable in each argument. Then take to obtain, after m steps, an index sequence π ∈
Aµ g := ∇ · g + uµ · g,
uµ := ∇ log pµ , {1, . . . , n}m and associated empirical distribution ν =
1
!m !0
!d m i=1 δ(xπ(i) ). (The convention i=1 = 0 is used.)
G := {g : R → R : i=1 〈gi , gi 〉H(k) ≤ 1}.
d d
Explicit formulae are contained in Algorithm 1. The
computational complexity of selecting m points in this
Note that G is the unit ball in the d-dimensional tensor
manner is O(m2 n), provided that the integrals appear-
product of H(k). Under mild conditions on k and µ
ing in Algorithm 1 can be evaluated in O(1). Note that
(Gorham
# and Mackey, 2017, Prop. 1), it holds that
candidate points can be selected more than once.
Aµ g dµ = 0 for all g ∈ G. The set Aµ G can then be
shown (Oates et al., 2017) to be the unit ball B(kµ ) in Theorems 1, 2 and 4 in Section 5 provide novel finite-
a different RKHS H(kµ ) with reproducing kernel sample-size error bounds for Algorithm 1 (as a special
case of Algorithm 2). The two main shortcomings of
kµ (x, y) := ∇x · ∇y k(x, y) + ∇x k(x, y) · uµ (y) Algorithm 1 are that (i) the myopic nature of the op-
(3)
+ ∇y k(x, y) · uµ (x) + k(x, y)uµ (x) · uµ (y), timisation may be statistically inefficient, and (ii) theOptimal Quantisation of Probability Measures Using Maximum Mean Discrepancy
Algorithm 1: Myopic minimisation of MMD Algorithm 2: Non-myopic minimisation of MMD
Data: A set {xi }ni=1 , a distribution µ, a kernel k and Data: A set {xi }ni=1 , a distribution µ, a kernel k, a
a number m ∈ N of output points number of points to select per iteration s ∈ N
Result: An index sequence π ∈ {1, . . . , n}m and a total number of iterations m ∈ N
for i = 1, . . . , m do& Result: An index sequence π ∈ {1, . . . , n}m×s
!i−1 for i = 1, . . . , m do &
π(i) ∈ argmin 12 k(xj , xj ) + i′ =1 k(xπ(i′ ) , xj ) !
j∈{1,...,n} # ' π(i, ·) ∈ argmin 12 j,j ′ ∈S k(xj , xj ′ )
−i k(x, xj )dµ(x) S∈{1,...,n}s
end !i−1 !s !
+ i′ =1 j=1 j ′ ∈S k(xπ(i′ ,j) , xj ′ )
! # '
requirement to scan through a large candidate set dur- −is j∈S k(x, xj )dµ(x)
end
ing each iteration may lead to unacceptable computa-
tional cost. In Section 3.2 we propose non-myopic and
argmin 12 v ⊤ Kv + ci⊤ v s.t. 1⊤ v = s (4)
mini-batch extensions to address these issues. v∈Ns0
Kj,j ′ := k(xj , xj ′ ), 1j := 1 for j = 1, . . . , n,
3.2 Generalised Sequential Algorithms ! ! #
i−1 s
cij := i′ =1 j ′ =1 k(xπ(i′ ,j ′ ) , xj ) − is k(x, xj ) dµ(x)
In Section 3.2.1 we describe a non-myopic extension of
Algorithm 1, where multiple points are simultaneously Remark 1. If one further imposes the constraint
selected at each step. The use of non-myopic optimisa- vi ∈ {0, 1} for all i, so that each candidate may be se-
tion is impractical when a large candidate set is used, lected at most once, then the resulting binary quadratic
and therefore we explain how mini-batches from the programme (BQP) is equivalent to the cardinality con-
candidate set can be employed in Section 3.2.2. strained k-partition problem from discrete optimisa-
tion, which is known to be NP-hard (Rendl, 2016).
3.2.1 Non-Myopic Minimisation (The results we present do not impose this constraint.)
Now we consider the simultaneous selection of s > 1
representative points from the candidate set at each 3.2.2 Mini-Batching
step. This leads to the non-myopic algorithm
The exact solution of (4) is practical only for moderate
$ !
1 i−1 !s values of s and n. This motivates the idea of consider-
π(i, ·) ∈ argmin MMDµ,k is i′ =1 j=1 δ(xπ(i′ ,j) )
S∈{1,...,n}s ! % ing only a subset of the n candidates at each iteration,
1
+ is j∈S δ(xj ) , a procedure we call mini-batching and inspired by the
similar idea from stochastic optimisation. There are
whose output is a bivariate index π ∈ {1, . . . , n}m×s , several ways that mini-batching can be performed, but
together
!m with
!s the associated empirical distribution ν = here we simply state that candidates denoted {xij }bj=1
1
ms i=1 j=1 δ(xπ(i,j) ). Explicit formulae are con- are considered during the ith iteration, with the mini-
tained in Algorithm 2. The computational complexity batch size denoted by b ∈ N. The non-myopic algo-
of selecting ms points in this manner is O(m2 s2 ns ), rithm for minimisation of MMD with mini-batching is
which is larger than Algorithm 1 when s > 1. The- then
orems 1, 2 and 4 in Section 5 provide novel finite- $ !
1 i−1 !s i′
sample-size error bounds for Algorithm 2. π(i, ·) ∈ argmin MMDµ,k is i′ =1 j=1 δ(xπ(i′ ,j) )
S∈{1,...,b}s ! %
1 i
Despite its daunting computational complexity, we + is j∈S δ(x j )
have found that it is practical to exactly implement
Algorithm 2 for moderate values of s and n by cast- Explicit formulae are contained in Algorithm 3. The
ing each iteration of the algorithm as an instance of a complexity of selecting ms points in this manner is
constrained integer quadratic programme (IQP) (e.g. O(m2 s2 bs ), which is smaller than Algorithm 2 when
Wolsey, 2020), so that state-of-the-art discrete opti- b < n. As with Algorithm 2, an exact IQP formulation
misation methods can be employed. To this end, we can be employed. Theorem 3 provides a novel finite-
represent the indices S ⊂ {1, . . . , n}s of the s points sample-size error bound for Algorithm 3.
to be selected at iteration i as a vector v ∈ {0, . . . , s}n
whose jth element indicates the number of copies of 3.3 Non-Sequential Algorithms
xj that!are selected, and where v is constrained to
n
satisfy j=1 vj = s. It is then an algebraic exercise Finally we consider the limit of the non-myopic Al-
to recast an optimal subset π(i, ·) as the solution to a gorithm 2, in which all m representative points are
constrained IQP: simultaneously selected in a single step:Teymur, Gorham, Riabiz, Oates
Figure 1: Quantisation of a Gaussian mixture model using MMD. A candidate set of 1000 independent samples
(left), from which 12 representative points were selected using: the myopic method (centre-left); non-myopic
selection, picking 4 points at a time (centre-right), and by simultaneous selection of all 12 points (right). Simu-
lations were conducted using Algorithms 1 and 2, with a Gaussian kernel whose length-scale was ℓ = 0.25.
Algorithm 3: Non-myopic minimisation of MMD 4 Empirical Assessment
with mini-batching
This section presents an empirical assessment3 of Algo-
Data: A set {{xij }bj=1 }m i=1 of mini-batches, each of
rithms 1–3. Two regimes are considered, correspond-
size b ∈ N, a distribution µ, a kernel k, a
ing to high compression (small sm/n; Section 4.1)
number of points to select per iteration s ∈ N,
and low compression (large sm/n; Section 4.2) of the
and a number of iterations m ∈ N
target. These occur, respectively, in applications to
Result: An index sequence π ∈ {1, . . . , n}m×s
Bayesian cubature and thinning of Markov chain out-
for i = 1, . . . , m do &
! put. In Section 4.3, we compare our method to a
π(i, ·) ∈ argmin 12 j,j ′ ∈S k(xij , xij ′ ) variety of others based on optimisation in continuous
S∈{1,...,b}s
!i−1 !s ! ′ spaces, augmenting a study in Chen et al. (2019). For
+ i′ =1 j=1 j ′ ∈S k(xiπ(i′ ,j) , xij ′ ) details of the kernels used, and a sensitivity analysis
! # '
−is j∈S k(x, xij )dµ(x) for the kernel parameters, see Appendix C.
end
Figure 1 illustrates how a non-myopic algorithm may
$ % outperform a myopic one. A candidate set was con-
1
! structed using 1000 independent samples from a test
π∈ argmin MMDµ,k m i∈S δ(xπ(i) ) (5)
S∈{1,...,n}m measure, and 12 representative points selected using
the myopic (Alg. 1 with m = 12), non-myopic (Alg.
The index set π can again be recast as the solu- 2 with m = 3 and s = 4), and non-sequential (Alg. 2
tion to !
an IQP and the associated empirical measure with m = 1 and s = 12) approaches. After choosing
1 m
ν = m i=1 δ(xπ(i) ) provides, by definition, a value the first three samples close to the three modes, the
for MMDµ,k (ν) that is at least as small as any of the myopic method then selects points that temporarily
methods so far described (thus satisfying the same er- worsen the overall approximation; note in particular
ror bounds derived in Theorems 1–4). the placement of the fourth point. The non-myopic
However, it is only practical to exactly solve (5) for methods do not suffer to the same extent: choosing 4
small m and thus, to arrive at a practical algorithm, points together gives better approximations after each
we consider approximation of (5). There are at least of 4, 8 and 12 samples have been chosen (s = 4 was
two natural ways to do this. Firstly, one could run a chosen deliberately so as to be co-prime to the number
numerical solver for the IQP formulation of (5) and of mixture components, 3). Choosing all 12 points at
terminate after a fixed computational limit is reached; once gives an even better approximation.
the solver will return a feasible, but not necessarily
optimal, solution to the IQP. An advantage of this 4.1 Bayesian Cubature
approach is that no further methodological work is re-
quired. Alternatively, one could employ a convex re- Larkin (1972) and subsequent authors proposed to cast
laxation of the intractable IQP, which introduces an numerical cubature in the Bayesian framework, such
approximation error that is hard to quantify but leads that an integrand f is a priori modelled as a Gaussian
to a convex problem that may be exactly soluble at process with covariance k, then conditioned on data
reasonable cost. We expand on the latter approach in 3
Our code is written in Python and is available at
Appendix B, with preliminary empirical comparisons. https://github.com/oteym/OptQuantMMDOptimal Quantisation of Probability Measures Using Maximum Mean Discrepancy
Figure 2: Synthetic data model Figure 3: KSD vs. wall-clock time, and KSD × time vs. number of
formed of a mixture of 20 bivariate selected samples, shown for the 38-dim calcium signalling model (top two
Gaussians (top). Effect of varying panes) and 4-dim Lotka–Volterra model (bottom two panes). The kernel
the number s of simultaneously- length-scale in each case was set using the median heuristic (Garreau
chosen points on MMD when et al., 2017), and estimated in practice using a uniform subsample of
choosing 60 from 1000 indepen- 1000 points for each model. The myopic algorithm of Riabiz et al. (2020)
dently sampled points (bottom). is included in the Lotka–Volterra plots—see main text for details.
D = {f (xi )}ni=1 as the integrand is evaluated. The selections are seen to outperform more myopic ones.
posterior standard deviation is (Briol et al., 2019b) Note in particular that MMD of the myopic method
(# ) $! m % (s = 1) is observed to decrease non-monotonically.
Std f dµ|D = min MMDµ,k wi δ(xi ) . (6) This is a manifestation of the phenomenon also seen in
w1 ,...,wm ∈R i=1
Figure 1, where a particular selection may temporarily
The selection of xi to minimise (6) is impractical since worsen the quality of the overall approximation.
evaluation of (6) has complexity O(m3 ). Huszár and
Duvenaud (2012) and Briol et al. (2015) noted that Next we consider applications in Bayesian statistics,
(6) can be bounded above by fixing wi = m 1
, and that where both approximation quality and computation
quantisation methods give a practical means to min- time are important. In what follows the density pµ
imise this bound. All results in Section 5 can therefore will be available only up to an unknown normalisa-
be applied and, moreover, the bound is expected to be tion constant and thus KSD—which requires only that
quite tight—wi ≪ m 1
implies that xi was not optimally uµ = ∇ log pµ can be evaluated—will be used.
1
placed, thus for optimal xi we anticipate wi ≈ m .
BC is most often used when evaluation of f has a high 4.2 Thinning of Markov Chain Output
computational cost, and one is prepared to expend re-
sources in the optimisation of the point set {xi }m i=1 . The use of quantisation to ‘thin’ Markov chain out-
Our focus in this section is therefore on the quality of put was proposed in Riabiz et al. (2020), who studied
the point set obtained, irrespective of computational greedy myopic algorithms based on KSD. We revisit
cost. Figure 1 suggested that the approximation qual- the applications from that work to determine whether
ity of non-myopic methods depends on s. Figure 2 our methods offer a performance improvement. Unlike
compares the selection of 60 from 1000 independently Section 4.1, the cost of our algorithms must now be as-
sampled points from a mixture of 20 Gaussians, vary- sessed, since their runtime may be comparable to the
ing s. This gives a set of step functions. Less myopic time required to produce Markov chain output itself.Teymur, Gorham, Riabiz, Oates
The datasets4 consist of (i) 4 × 106 samples from the 10
MALA SP SP-RWM INFL
38-parameter intracellular calcium signalling model of 9
RWM SP-MALA LAST OPT MB 100-10
SVGD SP-MALA INFL OPT MB 10-1
Hinch et al. (2004), and (ii) 2 × 106 samples from a 4- MED SP-RWM LAST B-UNIF 100-10
8
parameter Lotka–Volterra predator-prey model. The
MCMC chains are both highly auto-correlated and 7
start far from any mode. The greedy KSD approach 6
log KSD
of Riabiz et al. (2020) was found to slow down dra- 5
matically after selecting 103 samples, due to the need
4
to compute the kernel between selected points and all
points in the candidate set at each iteration. We em- 3
ploy mini-batching (b < n) to ameliorate this, and also 2
investigate the effectiveness of non-myopic selection. 1
Figure 3 plots KSD against time, with the number of 4 6 8 10 12
log n eval
collected samples m fixed at 1000, as well as time-
adjusted KSD against m for both models. This ac-
Figure 4: Comparison of quality of approximation for
knowledges that both approximation quality and com-
various methods, adjusted by the number of evalua-
putational time are important. In both cases, larger
tions neval of µ or uµ . For details of line labels, refer
mini-batches were able to perform better provided that
to the main text, and to Fig. 4 of Chen et al. (2019).
s was large enough to realise their potential. Non-
myopic selection shows a significant improvement over shown as a black solid line (OPT MB 100-10; b = 100,
batch-myopic in the Lotka–Volterra model, and a less s = 10) and dashed line (OPT MB 10-1; b = 10,
significant (though still visible) improvement in the s = 1), was applied to select 100 states from the first
calcium signalling model. The practical upper limit mb states visited in the RWM sample path, with m
on b for non-myopic methods (due to the requirement increasing and b fixed. The resulting quantisations are
to optimise over all b points) may make performance competitive with those produced by existing methods,
for larger s poorer relatively; in a larger and more com- at comparable computational cost. We additionally
plex model, there may be fewer than s ‘good’ samples include a comparison with batch-uniform selection (B-
to choose from given moderate b. This suggests that UNIF 100-10; b = 100, s = 10, drawn uniformly from
control of the ratio s/b may be important; the best the RWM output) in light grey.
results we observed occurred when s/b = 10−1 .
A comparison to the original myopic algorithm (i.e. 5 Theoretical Assessment
b = n) of Riabiz et al. (2020) is incuded for the Lotka–
Volterra model. This is implemented using the same This section presents a theoretical assessment of Al-
code and machine as the other simulations. The cyan gorithms 1–3. Once stated, a standing assumption is
line shown in the bottom two panes of Figure 3 rep- understood to hold for the remainder of the main text.
resents only 50 points (not 1000); collecting just these
Standing Assumption 1. Let X be a measurable
took 42 minutes. This algorithm is slower still for the
space equipped with a probability measure µ. Let k :
calcium signalling model, so it was omitted.
X ×X →
2
##R be symmetric positive definite and satisfy
Cµ,k := k(x, y)dµ(x)dµ(y) < ∞.
4.3 Comparison with Previous Approaches
For the kernels kµ described in Section 2.2, Cµ,kµ = 0
Here we compare against approaches based on continu- and the assumption is trivially satisfied. Our first re-
ous optimisation, reproducing a 10-dimensional ODE sult is a finite-sample-size error bound for non-myopic
inference task due to Chen et al. (2019). The aim algorithms when the candidate set is fixed:
is to minimise KSD whilst controlling the number of Theorem 1. Let {xi }ni=1 ⊂ X be fixed and let Cn,k 2
:=
evaluations neval of either the (un-normalised) target maxi=1,...,n k(xi , xi ). Consider an index sequence π of
µ or its log-gradient uµ . Figure 4 reports results length m and with selection size s produced by Algo-
for random walk Metropolis (RWM), the Metropolis- rithm 2. Then for all m ≥ 1 it holds that
adjusted Langevin algorithm (MALA), Stein varia-
$ ! %2
tional gradient descent (SVGD), minimum energy de- 1 m !s
MMDµ,k ms i=1 j=1 δ(xπ(i,j) )
signs (MED), Stein points (SP), and Stein point $ %
* !n +2 2 1+log m
MCMC (four flavours, denoted SP-∗, described in ≤ min MMDµ,k w δ(x ) + C ,
i=1 i i m
Chen et al., 2019). The method from Algorithm 3, 1⊤w=1
wi ≥0
4
Available at https://doi.org/10.7910/DVN/MDKNWM. with C := Cµ,k + Cn,k an m-independent constant.Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy
The proof is provided in Appendix A.1. Aside from The proof is provided in Appendix A.3. The mini-
providing an explicit error bound, we see that the out- batch size b plays an analogous role to n in Theorem 2
put of Algorithm 2 converges in MMD to the optimal and must be asymptotically increased with m in order
(weighted) quantisation of µ that is achievable using for Algorithm 3 to be consistent.
the candidate point set. Interestingly, all bounds we
In our second randomised setting the candidate set
present are independent of s.
arises as a Markov chain sample path. Let V be a
Remark 2. Theorems 1–4 are stated for general function V : X → [1, ∞) and, for a function f : X → R
MMD and apply in particular to KSD, for which we and a measure µ on X , let
set k = kµ . These results extend the work of Chen "# "
et al. (2018, 2019) and Riabiz et al. (2020), who con- (f (V := supx∈X |fV (x)|
(x) , (µ(V := sup&f &V ≤1
" f dµ" .
sidered only myopic algorithms (i.e., s = 1). A ψ-irreducible and aperiodic Markov chain with nth
Remark 3. The theoretical bounds being independent step transition kernel Pn is V-uniformly ergodic if and
of s do not necessarily imply that s = 1 is optimal only if there exists R ∈ [0, ∞) and ρ ∈ (0, 1) such that
in practice; indeed our empirical results in Section 4
suggest that it is not. (Pn (x, ·) − P (V ≤ RV (x)ρn (7)
for all initial states x ∈ X and all n ∈ N (see Thm.
Our remaining results explore the cases where the can- 16.0.1 of Meyn and Tweedie, 2012).
didate points are randomly sampled. Independent and #
Theorem 4. Assume that k(x, ·)dµ(x) = 0 for all
dependent sampling is considered and, in each case,
x ∈ X . Consider a µ-invariant, time-homogeneous,
the following moment bound will be assumed:
reversible Markov chain {xi }i∈N ⊂ X generated us-
Standing( Assumption ) 2. For some γ > 0, C1 := ing a V-uniformly ergodic transition kernel, such that
,
supi∈N E eγk(xi ,xi ) < ∞, where the expectation is (7) is satisfied with V (x) ≥ ,k(x, x) for all x ∈ X .
taken over x1 , . . . , xn . Suppose that C2 := supi∈N E[ k(xi , xi )V (xi )] < ∞.
In the first randomised setting, the xi are indepen- Consider an index sequence π of length m and selec-
dently sampled from µ, as would typically be possible tion subset size s produced by Algorithm 2. Then, with
2Rρ
when µ is explicit: C3 = 1−ρ , we have that
& $ ! %2 '
Theorem 2. Let {xi }ni=1 ⊂ X be independently sam- m !s
E MMDµ,k ms 1
i=1 j=1 δ(x π(i,j) )
pled from µ. Consider an index sequence π of length $ %$ %
m produced by Algorithm 2. Then for all s ∈ N and log(C1 ) C2 C3
≤ nγ + n + 2 Cµ,k 2
+ log(nC 1) 1+log m
.
γ m
all m, n ≥ 1 it holds that
& $ ! %2 '
m !s The proof is provided in Appendix A.4. Analysis of
E MMDµ,k ms 1
i=1 j=1 δ(xπ(i,j) ) mini-batching in the dependent sampling context ap-
$ %$ %
≤ log(C 1)
+ 2 C 2
+ log(nC1 ) 1+log m
. pears to be more challenging and was not attempted.
nγ µ,k γ m
The proof is provided in Appendix A.2. It is seen that 6 Discussion
n must be asymptotically increased with m in order
for the approximation provided by Algorithm 2 to be This paper focused on quantisation using MMD,
consistent. No smoothness assumptions were placed proposing and analysing novel algorithms for this task,
on k; such assumptions can be used to improve the but other integral probability metrics could be con-
O(n−1 ) term (as in Thm. 1 of Ehler et al., 2019), but sidered. More generally, if one is interested in com-
we did not consider this useful since the O(m−1 ) term pression by means other than quantisation then other
is the bottleneck in the bound. approaches may be useful, such as Gaussian mixture
An analogous (but more technically involved) argu- models and related approaches from the literature on
ment leads to a finite-sample-size error bound when density estimation (Silverman, 1986).
mini-batches are used: Some avenues for further research include: (i) extend-
Theorem 3. Let each mini-batch {xij }bj=1 ⊂ X be ing symmetric structure in µ to the set of representa-
independently sampled from µ. Consider an index se- tive points (Karvonen et al., 2019); (ii) characterising
quence π of length m produced by Algorithm 3. Then an optimal sampling distribution from which elements
∀ m, n ≥ 1 of the candidate set can be obtained (Bach, 2017);
& $ ! %2 ' (iii) further applications of our method, for example
m !s
E MMDµ,k ms 1
i=1 j=1 δ(x i
π(i,j) ) to Bayesian neural networks, where quantisation of the
$ %$ % posterior provides a promising route to reduce the cost
log(C1 ) 2 log(bC 1) 1+log m
≤ bγ + 2 Cµ,k + γ m . of predicting each label in the test dataset. •Teymur, Gorham, Riabiz, Oates
Acknowledgments Chérief-Abdellatif, B.-E. and Alquier, P. (2020).
MMD-Bayes: robust Bayesian estimation via max-
The authors are grateful to Lester Mackey and Luc imum mean discrepancy. Proc. Mach. Learn. Res.,
Pronzato for helpful feedback on an earlier draft, and 18:1–21.
to Wilson Chen for sharing the code used to produce
Figure 4. This work was supported by the Lloyd’s Dick, J. and Pillichshammer, F. (2010). Digital nets
Register Foundation programme on data-centric en- and sequences: discrepancy theory and quasi–Monte
gineering at the Alan Turing Institute, UK. MR Carlo integration. Cambridge University Press.
was supported by the British Heart Foundation–Alan Dudley, R. M. (2018). Real analysis and probability.
Turing Institute cardiovascular data science award CRC Press.
(BHF; SP/18/6/33805). Ehler, M., Gräf, M., and Oates, C. J. (2019). Optimal
Monte Carlo integration on closed manifolds. Stat.
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