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Black Box Submodular Maximization: Discrete and Continuous
Settings
Lin Chen1† Mingrui Zhang1‡ Hamed Hassani] Amin Karbasi†
†
Department of Electrical Engineering, Yale University,
‡
Department of Statistics and Data Science, Yale University,
]
Department of Electrical and Systems Engineering, University of Pennsylvania
Abstract 2011, Rios and Sahinidis, 2013, Shahriari et al., 2016].
In this setting, we assume that the objective function
In this paper, we consider the problem of is unknown and we can only obtain zeroth-order infor-
black box continuous submodular maximiza- mation such as (stochastic) function evaluations.
tion where we only have access to the function Fueled by a growing number of machine learning ap-
values and no information about the deriva- plications, black-box optimization methods are usually
tives is provided. For a monotone and continu- considered in scenarios where gradients (i.e., first-order
ous DR-submodular function, and subject to a information) are 1) difficult or slow to compute, e.g.,
bounded convex body constraint, we propose graphical model inference [Wainwright et al., 2008],
Black-box Continuous Greedy, a derivative-free structure predictions [Taskar et al., 2005, Sokolov et al.,
algorithm that provably achieves the tight 2016], or 2) inaccessible, e.g., hyper-parameter turning
[(1 − 1/e)OP T − ] approximation guarantee for natural language processing or image classifications
with O(d/3 ) function evaluations. We then Snoek et al. [2012], Thornton et al. [2013], black-box
extend our result to the stochastic setting attacks for finding adversarial examples Chen et al.
where function values are subject to stochas- [2017c], Ilyas et al. [2018]. Even though heuristics such
tic zero-mean noise. It is through this stochas- as random or grid search, with undesirable dependen-
tic generalization that we revisit the discrete cies on the dimension, are still used in some applica-
submodular maximization problem and use tions (e.g., parameter tuning for deep networks), there
the multi-linear extension as a bridge between has been a growing number of rigorous methods to
discrete and continuous settings. Finally, we address the convergence rate of black-box optimiza-
extensively evaluate the performance of our tion in convex and non-convex settings [Wang et al.,
algorithm on continuous and discrete submod- 2017, Balasubramanian and Ghadimi, 2018, Sahu et al.,
ular objective functions using both synthetic 2018].
and real data.
The focus of this paper is the constrained continuous
DR-submodular maximization over a bounded convex
1 Introduction body. We aim to design an algorithm that uses only
zeroth-order information while avoiding expensive pro-
Black-box optimization, also known as zeroth-order or jection operations. Note that one way the optimization
derivative-free optimization2 , has been extensively stud- methods can deal with constraints is to apply the pro-
ied in the literature [Conn et al., 2009, Bergstra et al., jection oracle once the proposed iterates land outside
1
the feasibility region. However, computing the projec-
These authors contributed equally to this work.
2 tion in many constrained settings is computationally
We note that black-box optimization (BBO) and
derivative-free optimization (DFO) are not identical terms. prohibitive (e.g., projection over bounded trace norm
Audet and Hare [2017] defined DFO as “the mathematical matrices, flow polytope, matroid polytope, rotation ma-
study of optimization algorithms that do not use deriva- trices). In such scenarios, projection-free algorithms,
tives” and BBO as “the study of design and analysis of a.k.a., Frank-Wolfe [Frank and Wolfe, 1956], replace
algorithms that assume the objective and/or constraint
Proceedings of the 23rd International Conference on Artificial
functions are given by blackboxes”. However, as the differ-
Intelligence and Statistics (AISTATS) 2020, Palermo, Italy.
ences are nuanced in most scenarios, this paper uses them
PMLR: Volume 108. Copyright 2020 by the author(s).
interchangeably.Black Box Submodular Maximization: Discrete and Continuous Settings
Table 1: Number of function queries in different settings, where D1 is the diameter of K.
Function Additional Assumptions Function Queries
continuous DR-submodular monotone, G-Lipschitz, L-smooth O(max{G, LD1 }3 · d3 ) [Theorem 1]
3
stoch. continuous DR-submodular monotone, G-Lipschitz, L-smooth O(max{G, LD1 }3 · d5 ) [Theorem 2]
5
discrete submodular monotone O( d5 ) [Theorem 3]
the projection with a linear program. Indeed, our pro- [2008], a canonical example of DR-submodular func-
posed algorithm combines efficiently the zeroth-order tions, is only defined on a unit cube. Moreover, they
information with solving a series of linear programs to can only guarantee to reach a first-order stationary
ensure convergence to a near-optimal solution. point. However, Hassani et al. [2017] showed that for
a monotone DR-submodular function, the stationary
Continuous DR-submodular functions are an important
points can only guarantee 1/2 approximation to the
subset of non-convex functions that can be minimized
optimum. Therefore, if a state-of-the-art zeroth-order
exactly Bach [2016], Staib and Jegelka [2017] and maxi-
non-convex algorithm is used for maximizing a mono-
mized approximately Bian et al. [2017a,b], Hassani et al.
tone DR-submodular function, it is likely to terminate
[2017], Mokhtari et al. [2018a], Hassani et al. [2019],
at a suboptimal stationary point whose approximation
Zhang et al. [2019b] This class of functions generalize
ratio is only 1/2.
the notion of diminishing returns, usually defined over
discrete set functions, to the continuous domains. They Our contributions: In this paper, we propose a
have found numerous applications in machine learn- derivative-free and projection-free algorithm Black-box
ing including MAP inference in determinantal point Continuous Greedy (BCG), that maximizes a monotone
processes (DPPs) Kulesza et al. [2012], experimental continuous DR-submodular function over a bounded
design Chen et al. [2018c], resource allocation Eghbali convex body K ⊆ Rd . We consider three scenarios:
and Fazel [2016], mean-field inference in probabilistic
(1) In the deterministic setting, where function eval-
models Bian et al. [2018], among many others.
uations can be obtained exactly, BCG achieves the
Motivation: Computing the gradient of a continu- tight [(1 − 1/e)OP T − ] approximation guarantee with
ous DR-submodular function has been shown to be O(d/3 ) function evaluations.
computationally prohibitive (or even intractable) in
(2) In the stochastic setting, where function evaluations
many applications. For example, the objective func-
are noisy, BCG achieves the tight [(1 − 1/e)OP T −
tion of influence maximization is defined via specific
] approximation guarantee with O(d3 /5 ) function
stochastic processes [Kempe et al., 2003, Rodriguez
evaluations.
and Schölkopf, 2012] and computing/estimating the
gradient of the mutliliear extension would require a (3) In the discrete setting, Discrete Black-box Greedy
relatively high computational complexity. In the prob- (DBG) achieves the tight [(1 − 1/e)OP T − ] approxi-
lem of D-optimal experimental design , the gradient mation guarantee with O(d5 /5 ) function evaluations.
of the objective function involves inversion of a po-
All the theoretical results are summarized in Table 1.
tentially large matrix [Chen et al., 2018c]. Moreover,
when one attacks a submodular recommender model, We would like to note that in discrete setting, due to
only black-box information is available and the service the conservative upper bounds for the Lipschitz and
provider is unlikely to provide additional first-order smooth parameters of general multilinear extensions,
information (this is known as the black-box adversarial and the variance of the gradient estimators subject to
attack model) [Lei et al., 2019]. noisy function evaluations, the required number of func-
tion queries in theory is larger than the best known re-
There has been very recent progress on developing
sult, O(d5/2 /3 ) in Mokhtari et al. [2018a,b]. However,
zeroth-order methods for constrained optimization
our experiments show that empirically, our proposed
problems in convex and non-convex settings Ghadimi
algorithm often requires significantly fewer function
and Lan [2013], Sahu et al. [2018]. Such methods typi-
evaluations and less running time, while achieving a
cally assume the objective function is defined on the
practically similar utility.
whole Rd so that they can sample points from a proper
distribution defined on Rd . For DR-submodular func- Novelty of our work: All the previous results in
tions, this assumption might be unrealistic, since many constrained DR-submodular maximization assume ac-
DR-submodular functions might be only defined on a cess to (stochastic) gradients. In this work, we address
subset of Rd , e.g., the multi-linear extension Vondrák a harder problem, i.e., we provide the first rigorousLin Chen1† , Mingrui Zhang1‡ , Hamed Hassani] , Amin Karbasi†
analysis when only (stochastic) function values can be maximization of monotone continuous DR-submodular
obtained. More specifically, with the smoothing trick functions Zhang et al. [2019a] is a closely related set-
[Flaxman et al., 2005], one can construct an unbiased ting to ours. However, to the best of our knowledge,
gradient estimator via function queries. However, this none of the existing work has developed a zeroth-order
estimator has a large O(d2 /δ 2 ) variance which may algorithm for maximizing a monotone continuous DR-
cause FW-type methods to diverge. To overcome this submodular function. For a detailed review of DFO
issue, we build on the momentum method proposed by and BBO, interested readers refer to book [Audet and
Mokhtari et al. [2018a] in which they assumed access Hare, 2017].
to the first-order information.
Given a point x, the smoothed version of F at x is de- 2 Preliminaries
fined as Ev∼B d [F (x + δv)]. If x is close to the boundary
of the domain D, (x + δv) may fall outside of D, leaving Submodular Functions We say a set function f :
the smoothed function undefined for many instances 2Ω → R is submodular, if it satisfies the diminishing
of DR-submodular functions (e.g., the multilinear ex- returns property: for any A ⊆ B ⊆ Ω and x ∈ Ω \ B,
tension is only defined over the unit cube). Thus the we have
vanilla smoothing trick will not work. To this end,
f (A ∪ {x}) − f (A) ≥ f (B ∪ {x}) − f (B). (1)
we transform the domain D and constraint set K in
a proper way and run our zeroth-order method on In words, the marginal gain of adding an element x
the transformed constraint set K0 . Importantly, we to a subset A is no less than that of adding x to its
retrieve the same convergence rate of O(T −1/3 ) as in superset B.
Mokhtari et al. [2018a] with a minimum number of func-
tion queries in different settings (continuous, stochastic For the continuous analogue, consider a function F :
continuous, discrete). X → R+ , where X = Πni=1 Xi , and each Xi is a compact
subset of R+ . We define F to be continuous submodular
We further note that by using more recent variance if F is continuous and for all x, y ∈ X , we have
reduction techniques [Zhang et al., 2019b], one might
be able to reduce the required number of function F (x) + F (y) ≥ F (x ∨ y) + F (x ∧ y), (2)
evaluations. where ∨ and ∧ are the component-wise maximizing
and minimizing operators, respectively.
1.1 Further Related Work
The continuous function F is called DR-submodular
Submodular functions Nemhauser et al. [1978], that Bian et al. [2017b] if F is differentiable and ∀ x ≤ y :
capture the intuitive notion of diminishing returns, ∇F (x) ≥ ∇F (y). An important implication of DR-
have become increasingly important in various machine submodularity is that the function F is concave in any
learning applications. Examples include graph cuts non-negative directions, i.e., for x ≤ y, we have
in computer vision Jegelka and Bilmes [2011a,b], data
summarization Lin and Bilmes [2011b,a], Tschiatschek F (y) ≤ F (x) + h∇F (x), y − xi. (3)
et al. [2014], Chen et al. [2018a, 2017b], influence maxi- The function F is called monotone if for x ≤ y, we
mization Kempe et al. [2003], Rodriguez and Schölkopf have F (x) ≤ F (y).
[2012], Zhang et al. [2016], feature compression Bateni
et al. [2019], network inference Chen et al. [2017a], ac- Smoothing Trick For a function F defined on Rd ,
tive and semi-supervised learning Guillory and Bilmes its δ-smoothed version is given as
[2010], Golovin and Krause [2011], Wei et al. [2015],
crowd teaching Singla et al. [2014], dictionary learn- F̃δ (x) , Ev∼B d [F (x + δv)], (4)
ing Das and Kempe [2011], fMRI parcellation Salehi
where v is chosen uniformly at random from the d-
et al. [2017], compressed sensing and structured spar-
dimensional unit ball B d . In words, the function F̃δ at
sity Bach [2010], Bach et al. [2012], fairness in machine
any point x is obtained by “averaging” F over a ball of
learning Balkanski and Singer [2015], Celis et al. [2016],
radius δ around x. In the sequel, we omit the subscript
and learning causal structures Steudel et al. [2010],
δ for the sake of simplicity and use F̃ instead of F̃δ .
Zhou and Spanos [2016], to name a few. Continuous
DR-submodular functions naturally extend the notion Lemma 1 below shows that under the Lipschitz as-
of diminishing returns to the continuous domains Bian sumption for F , the smoothed version F̃ is a good ap-
et al. [2017b]. Monotone continuous DR-submodular proximation of F , and also inherits the key structural
functions can be (approximately) maximized over con- properties of F (such as monotonicity and submodu-
vex bodies using first-order methods Bian et al. [2017b], larity). Thus one can (approximately) optimize F via
Hassani et al. [2017], Mokhtari et al. [2018a]. Bandit optimizing F̃ .Black Box Submodular Maximization: Discrete and Continuous Settings
Lemma 1 (Proof in Appendix A). If F is monotone setting and using recently proposed variance reduction
continuous DR-submodular and G-Lipschitz continuous techniques, we can then optimize F̃ near-optimally. Fi-
on Rd , then so is F̃ and nally, by Lemma 1 we show that the obtained optimizer
also provides a good solution for F .
|F̃ (x) − F (x)|≤ δG. (5)
Recall that continuous DR-submodular functions are
defined on a box X = Πni=1 Xi . To simplify the expo-
An important property of F̃ is that one can obtain an
sition, we can assume, without loss of generality,
Qd that
unbiased estimation for its gradient ∇F̃ by a single
the objective function F is defined on D , i=1 [0, ai ]
query of F . This property plays a key role in our
Bian et al. [2017a]. Moreover, we note that since
proposed derivative-free algorithms.
F̃ = Ev∼B d [F (x+δv)], for x close to ∂D (the boundary
Lemma 2 (Lemma 6.5 in [Hazan, 2016]). Given a of D), the point x + δv may fall outside of D, leaving
function F on Rd , if we choose u uniformly at random the function F̃ undefined.
from the (d − 1)-dimensional unit sphere S d−1 , then
we have To circumvent this issue, we shrink the domain D by δ.
Precisely, the shrunk domain is defined as
d
∇F̃ (x) = Eu∼S d−1 F (x + δu)u . (6) Dδ0 = {x ∈ D|d(x, ∂D) ≥ δ}. (10)
δ
Qd
Since we assume D = i=1 [0, ai ], the shrunk domain
3 DR-Submodular Maximization d
is Dδ0 = i=1 [δ, ai − δ]. Then for all x ∈ Dδ0 , we
Q
have x + δv ∈ D. So F̃ is well-defined on Dδ0 . By
In this paper, we mainly focus on the constrained
optimization problem: Lemma 1, the optimum of F̃ on the shrunk domain
Dδ0 will be close to that on the original domain D, if
max F (x), (7) δ is small enough. Therefore, we can first optimize
x∈K
F̃ on Dδ0 , then approximately optimize F̃ (and thus
where F is a monotone continuous DR-submodular F ) on D. For simplicity of analysis, we also translate
function on Rd , and the constraint set K ⊆ X ⊆ Rd is the shrunk domain Dδ0 by −δ, and denote it as Dδ =
Qd
convex and compact. i=1 [0, ai − 2δ].
For first-order monotone DR-submodular maximiza- Besides the domain D, we also need to consider the
tion, one can use Continuous Greedy Calinescu et al. transformation on constraint set K. Intuitively, if there
[2011], Bian et al. [2017b], a variant of Frank-Wolfe is no translation, we should consider the intersection of
Algorithm [Frank and Wolfe, 1956, Jaggi, 2013, Lacoste- K and the shrunk domain Dδ0 . But since we translate Dδ0
Julien and Jaggi, 2015], to achieve the [(1−1/e)OP T − by −δ, the same transformation should be performed
] approximation guarantee. At iteration t, the FW on K. Thus, we define the transformed constraint set
variant first maximizes the linearization of the objective as the translated intersection (by −δ) of Dδ0 and K:
function F :
K0 , (Dδ0 ∩ K) − δ1 = Dδ ∩ (K − δ1). (11)
vt = arg maxhv, ∇F (xt )i. (8)
v∈K
It is well known that the FW Algorithm is sensitive to
Then the current point xt moves in the direction of vt the accuracy of gradient, and may have arbitrarily poor
with a step size γt ∈ (0, 1]: performance with stochastic gradients Hazan and Luo
[2016], Mokhtari et al. [2018b]. Thus we incorporate
xt+1 = xt + γt vt . (9) two methods of variance reduction into our proposed al-
gorithm Black-box Continuous Greedy which correspond
Hence, by solving linear optimization problems, the to Step 7 and Step 8 in Algorithm 1, respectively.
iterates are updated without resorting to the projection First, instead of the one-point gradient estimation in
oracle. Lemma 2, we adopt the two-point estimator of ∇F̃ (x)
Here we introduce our main algorithm Black-box Con- [Agarwal et al., 2010, Shamir, 2017]:
tinuous Greedy which assumes access only to function
d
values (i.e., zeroth-order information). This algorithm (F (x + δu) − F (x − δu))u, (12)
2δ
is partially based on the idea of Continuous Greedy.
The basic idea is to utilize the function evaluations of where u is chosen uniformly at random from the unit
F at carefully selected points to obtain unbiased esti- sphere S d−1 .We note that (12) is an unbiased gradi-
mations of the gradient of the smoothed version, ∇F̃ . ent estimator with less variance w.r.t. the one-point
By extending Continuous Greedy to the derivative-free estimator. We also average over a mini-batch of BtLin Chen1† , Mingrui Zhang1‡ , Hamed Hassani] , Amin Karbasi†
Algorithm 1 Black-box Continuous Greedy ξ is stochastic zero-mean noise. In particular, in Step
1: Input: constraint set K, iteration number T , ra- 6 of Algorithm 1, we obtain independent stochastic
+ −
dius δ, step size ρt , batch size Bt function evaluations F̂ (yt,i ) and F̂ (yt,i ), instead of the
+ −
2: Output: xT +1 + δ1 exact function values F (yt,i ) and F (yt,i ). For unbiased
3: x1 ← 0, ḡ0 ← 0 function evaluation oracles with uniformly bounded
4: for t = 1 to T do variance, we have the following theorem.
5: Sample ut,1 , . . . , ut,Bt i.i.d. from S d−1 Theorem 2 (Proof in Appendix C). Under the
+ −
6: For i = 1 to Bt , let yt,i ← δ1 + xt + δut,i , yt,i ← condition of Theorem 1, if we further assume that
+ −
δ1 + xt − δut,i and evaluate F (yt,i ), F (yt,i ) for all x, E[F̂ (x)] = F (x) and E[|F̂ (x)−F (x)|2 ] ≤
PBt d + −
7: gt ← B1t i=1 2δ [F (yt,i ) − F (yt,i )]ut,i
σ02 , then we have
8: ḡt ← (1 − ρt )ḡt−1 + ρt gt
9: vt ← arg maxv∈K0 hv, ḡt i (1 − 1/e)F (x∗ ) − E[F (xT +1 + δ1)]
10: xt+1 ← xt + vTt 3D1 Q1/2 LD12 √
11: end for ≤ + + δG(1 + ( d + 1)(1 − 1/e)),
T 1/3 2T
12: Output xT +1 + δ1
where D1 = diam(K0 ), Q = max{42/3 G2 , 6L2 D12 +
(4cdG2 + 2d2 σ02 /δ 2 )/Bt }, c is a constant, and x∗
independently sampled two-point estimators for fur- is the global maximizer of F on K.
ther variance reduction. The second variance-reduction 3 3 2
technique is the momentum method used in [Mokhtari Remark √ 2. By setting T = O(1/ ), Bt = d / , and
et al., 2018a] to estimate the gradient by a vector ḡt δ = / d, the error term (RHS) is at most O(). The
which is updated at each iteration as follows: total number of evaluations is at most O(d3 /5 ).
ḡt = (1 − ρt )ḡt−1 + ρt gt . (13) 4 Discrete Submodular Maximization
Here ρt is a given step size, ḡ0 is initialized as an all zero
In this section, we describe how Black-box Continuous
vector 0, and gt is an unbiased estimate of the gradient
Greedy can be used to solve a discrete submodular max-
at iterate xt . As ḡt is a weighted average of previous
imization problem with a general matroid constraint,
gradient approximation ḡt−1 and the newly updated
i.e., maxS∈I f (S), where f is a monotone submodular
stochastic gradient gt , it has a lower variance compared
set function and I is a matroid.
with gt . Although ḡt is not an unbiased estimation of
the true gradient, the error of it will approach zero as For any monotone submodular set function f : 2Ω →
time proceeds. The detailed description of Black-box R≥0 , its multilinear extension F : [0, 1]d → R≥0 , de-
Continuous Greedy is provided in Algorithm 1. fined as
Theorem 1 (Proof in Appendix B). For a mono-
X Y Y
F (x) = f (S) xi (1 − xj ), (14)
tone continuous DR-submodular function F , which S⊆Ω i∈S j ∈S
/
is also G-Lipschitz continuous and L-smooth on
a convex and compact constraint set K, if we set is monotone and DR-submodular [Calinescu et al.,
ρt = 2/(t + 3)2/3 in Algorithm 1, then we have 2011]. Here, d = |Ω| is the size of the ground set
Ω. Equivalently, we have F (x) = ES∼x [f (S)], where
(1 − 1/e)F (x∗ ) − E[F (xT +1 + δ1)] S ∼ x means that the each element i ∈ Ω is included
3D1 Q1/2 LD12 √ in S with probability xi independently.
≤ 1/3
+ + δG(1 + ( d + 1)(1 − 1/e)).
T 2T It can be shown that in lieu of solving the discrete
optimization problem one can solve the continuous op-
where Q = max{42/3 G2 , 4cdG2 /Bt + 6L2 D12 }, c is
timization problem maxx∈K F (x), where K = conv{1I :
a constant, D1 = diam(K0 ), and x∗ is the global
I ∈ I} is the matroid polytope [Calinescu et al., 2011].
maximizer of F on K.
This equivalence is obtained by showing that (i) the
Remark 1. By setting T = O(1/3 ), Bt = d, and δ = optimal values of the two problems are the same, and
√
/ d, the error term (RHS) is guaranteed to be at most (ii) for any fractional vector x ∈ K we can deploy effi-
O(). Also, the total number of function evaluations is cient, lossless rounding procedures that produce a set
at most O(d/3 ). S ∈ I such that E[f (S)] ≥ F (x) (e.g., pipage rounding
[Ageev and Sviridenko, 2004, Calinescu et al., 2011]
We can also extend Algorithm 1 to the stochastic case and contention resolution [Chekuri et al., 2014]). So we
in which we obtain information about F only through can view F̃ as the underlying function that we intend to
its noisy function evaluations F̂ (x) = F (x) + ξ, where optimize, and invoke Black-box Continuous Greedy. AsBlack Box Submodular Maximization: Discrete and Continuous Settings
Algorithm 2 Discrete Black-box Greedy 3)2/3 , St,i = l in Algorithm 2, then we have
1: Input: matroid constraint I, transformed con-
straint set K0 = Dδ ∩ (K − δ1) where K = conv{1I : (1 − 1/e)f (X ∗ ) − E[f (XT +1 )]
I ∈ I}, number of iterations T , radius δ, step size
p
3D1 Q1/2 2M d(d − 1)D12
ρt , batch size Bt , sample size St,i ≤ +
T 1/3 √ √T
2: Output: XT +1
+ 2M δ d(1 + ( d + 1)(1 − 1/e)).
3: x1 ← 0, ḡ0 ← 0,
4: for t = 1 to T do 2d2 M 2 ( 1
+8c)
5: Sample ut,1 , . . . , ut,Bt i.i.d. from S d−1 where D1 = diam(K0 ), Q = max{ lδ 2
Bt +
+ 2 2 5/3 2 ∗
6: For i = 1 to Bt , let yt,i ← δ1 + xt + 96d(d − 1)M D1 , 4 dM }, c is a constant, X is
− the global maximizer of f under matroid constraint
δut,i , yt,i ← δ1 + xt − δut,i , independently
sample subsets Yt,i +
and Yt,i −
for St,i times I.
+ −
according to yt,i , yt,i , get sampled subsets Remark 3. By setting T = O(d3 /3 ), Bt = 1, l =
+
Yt,i,j −
, Yt,i,j , ∀j ∈ [St,i ], evaluate the function d2 /2 , and δ = /d, the error term (RHS) is at most
values f (Yt,i,j + −
), f (Yt,i,j ), ∀j ∈ [St,i ], and calcu- O(). The total number of evaluations is at most
PSt,i +
f (Yt,i,j )
O(d5 /5 ).
late the averages f¯t,i
+
← j=1
St,i , f¯t,i
−
←
PSt,i − We note that in Algorithm 2, f¯t,i +
is the unbiased estima-
j=1 f (Yt,i,j )
St,i
PBt d ¯+ tion of F (yt,i ), and the same holds for f¯t,i
+ −
and F (yt,i−
).
gt ← B1t i=1 ¯−
2δ (ft,i − ft,i )ut,i
7: As a result, we can analyze the algorithm under the
8: ḡt ← (1 − ρt )ḡt−1 + ρt gt framework of stochastic continuous submodular maxi-
9: vt ← arg maxv∈K0 hv, ḡt i mization. By applying Theorem 2, Lemma 3, and the
10: xt+1 ← xt + vTt facts E[|f¯t,i
+ + 2
−F (yt,i )| ] ≤ M 2 /St,i , E[|f¯t,i
− − 2
−F (yt,i )| ] ≤
11: end for M 2 /St,i directly, we can also attain Theorem 3.
12: Output XT +1 = round(xT +1 + δ1)
5 Experiments
a result, we want that F is G-Lipschitz and L-smooth
as in Theorem 1. The following lemma shows these In this section, we will compare Black-box Continuous
properties are satisfied automatically if f is bounded. Greedy (BCG) and Discrete Black-box Greedy (DBG)
with the following baselines:
Lemma 3. For a submodular set function f defined on
Ω with supX⊆Ω |f (X)|≤ M , its multilinear extension (1) Zeroth-Order Gradient Ascent (ZGA) is the projected
√
gradient ascent algorithm equipped with the same two-
p
F is 2M d-Lipschitz and 4M d(d − 1)-smooth.
point gradient estimator as BCG uses. Therefore, it is
We note that the bounds for Lipschitz and smooth- a zeroth-order projected algorithm.
ness parameters actually depend on the norms that we
(2) Stochastic Continuous Greedy (SCG) is the state-of-
consider. However, different norms are equivalent up
the-art first-order algorithm for maximizing continuous
to a factor that may depend on the dimension. If we
DR-submodular functions Mokhtari et al. [2018a,b].
consider another norm, some dimension factors may be
Note that it is a projection-free algorithm.
absorbed into the norm. Therefore, we only study the
Euclidean norm in Lemma 3. (3) Gradient Ascent (GA) is the first-order projected
gradient ascent algorithm Hassani et al. [2017].
We further note that computing the exact value of F is
difficult as it requires evaluating f over all the subsets The stopping criterion for the algorithms is whenever a
S ∈ Ω. However, one can construct an unbiased esti- given number of iterations is achieved. Moreover, the
mate for the value F (x) by simply sampling a random batch sizes St,i in Algorithm 1 and Bt in Algorithm 2
set S ∼ x and returning f (S) as the estimate. We are both 1. Therefore, in the experiments, DBG uses 1
present our algorithm in detail in Algorithm 2, where query per iteration while SCG uses O(d) queries.
we have D = [0, 1]d , since F is defined on [0, 1]d , and
We perform four sets of experiments which are de-
thus Dδ = [0, 1 − 2δ]d . We state the theoretical result
scribed in detail in the following. The first two sets
formally in Theorem 3.
of experiments are maximization of continuous DR-
Theorem 3 (Proof in Appendix E). For submodular functions, which Black-box Continuous
a monotone submodular set function f with Greedy is designed to solve. The last two are sub-
supX⊆Ω |f (X)|≤ M , if we set ρt = 2/(t + modular set maximization problems. We will apply
Discrete Black-box Greedy to solve these problems. TheLin Chen1† , Mingrui Zhang1‡ , Hamed Hassani] , Amin Karbasi†
function values at different rounds and the execution problem is defined by f (S) = log det(I + ΣS,S ), where
times are presented in Fig. 1 and Section 5. The first- S ⊆ [d] and ΣS,S is the principal submatrix indexed
order algorithms (SCG and GA) are marked in orange, by S. The total number of 22 attributes are parti-
and the zeroth-order algorithms are marked in blue. tioned into 5 disjoint subsets with sizes 4, 4, 4, 5 and
5, respectively. The problem is subject to a partition
Non-convex/non-concave Quadratic Program-
matroid requiring that at most one attribute should be
ming (NQP): In this set of experiments, we apply our
active within each subset. Since this is a submodular
proposed algorithm and the baselines to the problem of
set maximization problem, in order to evaluate the
non-convex/non-concave quadratic programming. The
gradient (i.e., obtain an unbiased estimate of gradi-
objective function is of the form F (x) = 12 x> Hx + b> x,
ent) required by first-order algorithms SCG and GA,
where x is a 100-dimensional vector, H is a 100-by-100
it needs 2d function value queries. To be precise, the
matrix, and every component of H is an i.i.d. ran-
i-th component of gradient is ES∼x [f (S ∪ {i}) − f (S)]
dom variable whose distribution is equal to that of the
and requires two function value queries. It can be ob-
negated absolute value ofPa standard normal
P60 distribu-
30 served from Fig. 1c that DBG outperforms the other
tion. The constraints are i=1 xi ≤ 30, i=31 xi ≤ 20,
P100 zeroth-order algorithm ZGA. Although its performance
and i=61 xi ≤ 20. To guarantee that the gradient is slightly worse than the two first-order algorithms
is non-negative, we set bt = −H > 1. One can observe SCG and GA, it require significantly less number of
from Fig. 1a that the function value that BCG attains function value queries than the other two first-order
is only slightly lower than that of the first-order algo- methods (as discussed above).
rithm SCG. The final function value that BCG attains
is similar to that of ZGA. Influence Maximization In the influence maximiza-
tion problem, we assume that every node in the network
Topic Summarization: Next, we consider the topic is able to influence all of its one-hop neighbors. The ob-
summarization problem [El-Arini et al., 2009, Yue and jective of influence maximization is to select a subset of
Guestrin, 2011], which is to maximize the probabilistic nodes in the network, called the seed set (and denoted
coverage of selected articles on news topics. Each news by S), so that the total number of influenced nodes,
article is characterized by its topic distribution, which including the seed nodes, is maximized. We choose
is obtained by applying latent Dirichlet allocation to the social network of Zachary’s karate club Zachary
the corpus of Reuters-21578, Distribution 1.0. The [1977] in this study. The subjects in this social net-
number of topics is set to 10. We will choose from 120 work are partitioned into three disjoint groups, whose
news articles. The probabilistic coverage of a subset sizes are 10, 14, and 10 respectively. The chosen seed
of news articles (denoted by X) is defined by f (X) = nodes should be subject to a partition matroid; i.e., We
1
P10 Q
10 j=1 [1 − a∈X (1 − pa (j))], where pa (·) is the topic will select at most two subjects from each of the three
distribution of article a. PThe multilinear extension groups. Note that this problem is also a submodular
1 10 Q
function of f is F (x) = 10 j=1 [1− a∈Ω (1−p a (j)xa )], set maximization problem. Similar to the situation in
120
where x ∈ [0, 1] Iyer et al. [2014]. The constraint the active set selection problem, first-order algorithms
P40 P80 P120
is i=1 xi ≤ 25, i=41 xi ≤ 30, i=81 xi ≤ 35. It need function value queries to obtain an unbiased es-
can be observed from Fig. 1b that the proposed BCG timate of gradient. We can observe from Fig. 1d that
algorithm achieves the same function value as the first- DBG attains a better influence coverage than the other
ordered algorithm SCG and outperforms the other two. zeroth-order algorithm ZGA. Again, even though SCG
As shown in Fig. 2a, BCG is the most efficient method. and GA achieve a slightly better coverage, due to their
The two projection-free algorithms BCG and SCG run first-order nature, they require a significantly larger
faster than the projected methods ZGA and GA. We number of function value queries.
will elaborate on the running time later in this section.
Active Set Selection We study the active set selec- Running Time The running times of the our pro-
tion problem that arises in Gaussian process regres- posed algorithms and the baselines are presented in
sion Mirzasoleiman et al. [2013]. We use the Parkin- Section 5 for the above-mentioned experimental set-
sons Telemonitoring dataset, which is composed of ups. There are two main conclusions. First, the two
biomedical voice measurements from people with early- projection-based algorithms (ZGA and GA) require
stage Parkinson’s disease [Tsanas et al., 2010]. Let significantly higher time complexity compared to the
X ∈ Rn×d denote the data matrix. Each row X[i, :] projection-free algorithms (BCG, DBG, and SCG),
is a voice recording while each column X[:, j] denotes as the projection-based algorithms require solving
an attribute. The covariance matrix Σ is defined by quadratic optimization problems whereas projection-
Σij = exp(−kX[:, i] − X[:, j]k2 )/h2 , where h is set to free ones require solving linear optimization problems
0.75. The objective function of the active set selection which can be solved more efficiently. Second, when we
compare first-order and zeroth-order algorithms, weBlack Box Submodular Maximization: Discrete and Continuous Settings
Number of first-order oracle queries Number of queries to compute gradients
Number of first-order oracle queries Number of queries to compute gradients
×104 0 10 20 30
0 10 20 30 0 500 1000
0 1000 2000
1.00 6
3 1.5
Function values
Function values
Function values
Function values
0.75 4
2 BCG BCG 1.0 DBG
0.50 DBG
ZGA ZGA ZGA ZGA
1 2
SCG 0.25 SCG 0.5 SCG SCG
GA GA GA GA
0 0.00 0.0 0
0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60
Number of zeroth-order oracle queries Number of zeroth-order oracle queries Number of zeroth-order oracle queries Number of zeroth-order oracle queries
(a) NQP (b) Topic summarization (c) Active set selection (d) Influence maximization
Figure 1: Function value vs. number of oracle queries. Note that every chart has dual horizontal axes. Orange
lines use the orange horizontal axes above while blue lines use the blue ones below.
Rel. running time (DBG=1)
Rel. running time (DBG=1)
Rel. running time (BCG=1)
Rel. running time (BCG=1)
6
4 10 8
4 3 6
2 5 4
2
1 2
0 0 0 0
BCG SCG ZGA GA BCG SCG ZGA GA DBG SCG ZGA GA DBG SCG ZGA GA
(a) NQP (b) Topic summarization (c) Active set selection (d) Influence maximization
Figure 2: Relative running time normalized with respect to BCG (for continuous DR-submodular maximization in
the first two sets of experiments) and DBG (for submodular set maximization in the last two sets of experiments).
can observe that zeroth-order algorithms (BCG, DBG, is particularly preferable in a large-scale machine learn-
and ZGA) run faster than their first-order counterparts ing task and an application where the total number of
(SCG and GA). function evaluations or the running time is subject to
a budget.
Summary The above experiment results show the
following major advantages of our method over the 6 Conclusion
baselines including SCG and ZGA.
In this paper, we presented Black-box Continuous
1. BCG/DBG is at least twice faster than SCG and Greedy, a derivative-free and projection-free algo-
ZGA in all tasks in terms of running time (Figs. 2a rithm for maximizing a monotone and continuous
to 2d) DR-submodular function subject to a general convex
2. DBG requires remarkably fewer function evalua- body constraint. We showed that Black-box Continuous
tions in the discrete setting (Figs. 1c and 1d) Greedy achieves the tight [(1−1/e)OP T −] approxima-
tion guarantee with O(d/3 ) function evaluations. We
3. In addition to saving function evaluations, then extended the algorithm to the stochastic continu-
BCG/DBG achieves an objective function value ous setting and the discrete submodular maximization
comparable to that of the first-order baselines SCG problem. Our experiments on both synthetic and real
and GA. data validated the performance of our proposed al-
gorithms. In particular, we observed that Black-box
Furthermore, we note that the number of first-order Continuous Greedy practically achieves the same utility
queries required by SCG is only half the number re- as Continuous Greedy while being way more efficient in
quired by BCG. However, as is shown in Figs. 2a and 2b, terms of number of function evaluations.
BCG runs significantly faster than SCG since a zeroth-
order evaluation is faster than a first-order one.
Acknowledgements
In the topic summarization task (Fig. 1b), BCG ex-
hibits a similar performance to that of the first-order LC is supported by the Google PhD Fellowship. HH
baselines SCG and GA, in terms of the attained is supported by AFOSR Award 19RT0726, NSF HDR
objective function value. In the other three tasks, TRIPODS award 1934876, NSF award CPS-1837253,
BCG/DBG runs notably faster while achieving an only NSF award CIF-1910056, and NSF CAREER award
slightly inferior function value. Therefore, BCG/DBG CIF-1943064. AK is partially supported by NSFLin Chen1† , Mingrui Zhang1‡ , Hamed Hassani] , Amin Karbasi†
(IIS-1845032), ONR (N00014-19-1-2406), and AFOSR Andrew An Bian, Baharan Mirzasoleiman, Joachim
(FA9550-18-1-0160). Buhmann, and Andreas Krause. Guaranteed non-
convex optimization: Submodular maximization over
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