Scalable Distributional Robustness in a Class of Non Convex Optimization with Guarantees
Abstract
Distributionally robust optimization (DRO) has shown lot of promise in providingrobustness in learning as well as sample based optimization problems.
We endeavorto provide DRO solutions for a class of sum of fractionals, non-convex optimizationwhich is used for decision making in prominent areas such as facility location andsecurity games.
In contrast to previous work, we ﬁnd it more tractable to optimizethe equivalent variance regularized form of DRO rather than the minimax form.
We transform the variance regularized form to a mixed-integer second order coneprogram (MISOCP), which, while guaranteeing near global optimality , does notscale enough to solve problems with real world data-sets.
We further proposetwo abstraction approaches based on clustering and stratiﬁed sampling to increasescalability, which we then use for real world data-sets.
Importantly, we provide nearglobal optimality guarantees for our approach and show experimentally that oursolution quality is better than the locally optimal ones achieved by state-of-the-artgradient-based methods.
We experimentally compare our different approaches andbaselines, and reveal nuanced properties of a DRO solution.
Introduction
Distributionally robust optimization (DRO) is a popular approach employed in robust machinelearning.
Mostly, if not always, these works have focussed on the task of classiﬁcation or regression.
However, often in practical applications the end goal of learning is a decision output z , which requiresyet another complex optimization that uses the output (cid:98) x 1 , . . .
, (cid:98) x N of a regressor f ( · ) .
For example,in facility location problem the learning of facility values is followed by an optimization using thevalues predicted to decide where to locate facilities and in security games adversary behavior modelis learned and then an optimal defense allocation computed based on the learned model.
Often thelearning output is provided as public datasets with no access to the underlying private dataset used forsuch learning.
In this set-up, we aim to provide robustness at the decision making level with access toonly the non-robust learning output (cid:98) x 1 , . . .
, (cid:98) x N .
However, often the objective F ( z , x ) of decision optimization is non-convex in the learning systemoutput x unlike the convex objective of classiﬁcation or regression, presenting signiﬁcant scalabilitychallenges.
In general, for decision making and speciﬁcally for the problem domains we consider,global optimality is important as sub-optimal decisions can lead to large revenue loss or loss of life;thus, the local optimality provided by gradient based methods is not sufﬁcient.
As a consequence, inthis paper, we study the scenario of calculating DRO decisions using the given multi-dimensionalreal valued outputs (cid:98) x 1 , . . .
, (cid:98) x N of a non-robust learned f .
A ﬁrst result (Theorem 1) characterizes thequality of DRO decision output compared to the scenario where we know the true f ∗ .
Our main focusis on addressing the scalability issue for the DRO decision making problem for a particular, but widelyused, class of sum of fractionals non-convex objective .
This objective arises from the well-knowndiscrete choice models Train [2003] of human behavior, which is known to not have scalable globally optimal solutions [Schaible, 1995, Li et al., 2019]; we use this for tackling two different decisionoptimization problem in facility location and a robust version of Bayesian Stackelberg security gamesproblem with quantal response.
As far as we know, this is a ﬁrst attempt to solve the aforementionednon-convex problems in a DRO setting to near global optimality.
Our ﬁrst contribution is a modelling construct, where we reformualate the variance regularizedform Duchi and Namkoong [2019] of our non-convex sum of fractionals objective as a mixed integersecond order cone program (MISOCP).
While the MISOCP form provides more scalability thanthe original formulation and guaranteed solution quality (Theorem 2), it still does not scale to realworld sized datasets.
Our second contribution is a pair of approaches that achieves further scalabilityby splitting the problem space into sub-regions and solving a smaller MISOCP over representativesamples from the sub-regions.
Under mild conditions, both approaches provide global optimalityguarantees (Theorem 3, 4).
Our ﬁnal contribution is detailed experiments validating the scalability of our approaches on a simulated security game problem as well as two variants of location facility using park and ride data-setsfrom New York [Holguin-Veras et al., 2012].
We compare with two gradient-based approaches [Linet al., 2020] and show the superior solution quality achieved by our approach, which also reveals theneed for global optimality.
We further show a nuanced property of the DRO solution in providingbetter decisions for low probability scenarios over non-robust versions.
Overall, our work providesdesired robustness with globally optimal solution guarantees .
Related work: Our work is built on a recent line of research that connects the concepts of DRO andvariance regularization [Duchi and Namkoong, 2019, Duchi et al., 2021, Lam, 2016, Maurer and
Pontil, 2009, Staib et al., 2019].
While most the previous studies along this research line focus onconvex and continuous problems or problems with submodular objectives, our work concerns a classof DRO problems with fractional structures, which are highly non-convex and requires new technicaldevelopments for globally optimal solution.
Recent work Yan et al.
[2020], Qi et al.
[2021] hasaddressed non-convex objectives in DRO using gradient based methods that converge to stationarypoints, which is insufﬁcient for decision making as we experimentally show that stationary pointsand globally optimal points can yield very different decision utilities.
The literature on DRO is vast and we refer the reader to Rahimian and Mehrotra [2019] for areview.
DRO methods can be classiﬁed by different ways to deﬁne ambiguity sets of distributions,for instance, ambiguity sets based on φ -divergences [Ben-Tal et al., 2013, Duchi and Namkoong,2019, Staib et al., 2019] or
Wasserstein distances
[Pﬂug and Wozabal, 2007, Esfahani and Kuhn,2018, Shaﬁeezadeh-Abadeh et al., 2015, Blanchet and Murthy, 2019].
In this work, we focus onφ -divergence based models, motivated by their interesting connections with variance regularizationand the tractability of the resulting non-convex DRO models.
We show that our DRO methods can be used in some popular decision-making problems such as
Stackelberg security game (SSG) with Quantal Response [Tambe, 2011, Xu, 2016, Fang et al., 2017,
Sinha et al., 2018, Yang et al., 2012, Haghtalab et al., 2016] or competitive facility location underrandom utilities [Benati and Hansen, 2002, Freire et al., 2016, Mai and Lodi, 2020, Dam et al., 2021].
To the best of our knowledge, a DRO Bayesian model has not been studied in existing SSG works.
Inthe context of competitive facility location under random utilities, we seem to be the ﬁrst to bring
DRO as a consideration.
We handle a DRO version of a facility cost optimization problem, which hasalso never been studied in prior work.
Background, Preliminary Notation and Result
We use bold fonts for vectors and non-bold font for vector components and scalars, e.g., x j is acomponent of x .
[ N ] denotes { 1 , . .
. , N } .
A d -dimensional vector is written as x = ( x j ) j ∈
[ d ] oras ( x 1 , . . .
, x d ) .
The positive part of a vector x is x + =
(max(0 , x j ))
j ∈ [ d ] , and the negative part isx − =
(min(0 , x j ))
j ∈ [ d ] . 0 , 1 represent the all zero and all one vector.
Distributionally Robust Optimization : Consider a function F with inputs being a decision variablez and parameter x ∈ X .
Both z and x lie in an Euclidean space and both are constrained by linearconstraints ; for notational ease we skip writing the constraints in the general formulation.
We seek tomaximize the following objective function max z E P [ F ( z , x )] .
For many classes of distributions theabove is generally not tractable and one needs to sample x from P .
Let (cid:98) x 1 , . . .
, (cid:98) x N be N samples, we can solve the sample average approximation (SAA) problem instead max z (cid:80) n ∈ [ N ] F ( z , (cid:98) x n ) .
Let (cid:98) P
Nbe the empirical distribution induced by the samples.
The SAA above is same as max z E (cid:98) P N [ F ( z , x )] .
A distributionally robust version of the SAA problem is max z min (cid:101) P ∈P ξ,n (cid:8) E (cid:101) P
[ F ( z , x )] (cid:9) , where theambiguity set P ξ,n = (cid:110) (cid:101) P | D φ ( (cid:101) P || (cid:98) P N ) ≤ ξ/N (cid:111) , and D φ ( P || Q ) is the χ 2 divergence:
D φ ( P || Q ) =12 (cid:82) ( dP/dQ − 1) 2 dQ .
The above optimization can be written equivalently as ( ∆ ξ,n deﬁned below)
[Equation]
where ∆ ξ,n = (cid:110) p ∈
R N + (cid:12)(cid:12)(cid:12) (cid:80) i p i = 1;
||
p − 1 /N ||
22 ≤ 2 ξN 2 (cid:111) .
We have earlier stated that (cid:98) x
i isoutput by a regressor, say f ∈ F for some function class F trained using loss L with N T datapoints,but this implies that (cid:98) x i =
f
( b i ) might not exactly same as x ∗ i = f ∗ ( b i ) for some underlying featurevalues
b i and best function f ∗ ∈ F .
We assume f ∗ is deterministic and has zero Bayes risk.
Let P ∗be the true distribution induced by f ∗ from which the (unknown) samples x ∗ ’s are obtained; hence,the true utility of a decision z is E P ∗
[ F ( z , x )] .
We prove an end to end guarantee about the outputdecision (cid:98)
z
∗∗ using (cid:98) x i ’s, which reveals that (cid:98)
z ∗∗ is not much worse than the decision z ∗∗ learned if x ∗would be available and used, and larger training data N T helps.
Theorem 1.
If the optimal decision when solving DRO is z ∗∗ using x ∗ i ’s and (cid:98)
z ∗∗ using (cid:98) x i ’s, F is τLipschitz in x , X is bounded, and a scaled L upper bounds ||·||
(i.e., || x − x (cid:48)
|| 2 ≤ max( k L ( x, x (cid:48) ) , (cid:15) )for constants k, (cid:15) )
then, the following holds with probability 1 − 2 δ − 2 δ 1 : E P ∗ [ F ( (cid:98)
z ∗∗ , x )] ≥
[Equation]
− (cid:15) N − (cid:15) N T , where (cid:15) K
= C 1 R K ( L ◦ H )
+ C 2 / √ Kand R K is the Rademacher complexity with K samples and C, C 1 , C 2 are constants dependent onδ, δ 1 , ξ, k, τ .
Variance Regularizer :
As a large number of samples are needed for a low variance approximationof the true distribution, another proposed robust version of the SAA [Maurer and Pontil, 2009, Duchiand Namkoong, 2019] is to optimize the following variance-regularized (VR) objective function
[Equation]
The above allows to directly optimize the trade-off between bias and variance.
In a fundamentalresult, Duchi and Namkoong [2019, Theorem 1] show that, with high probability, problem ( VR ) isequivalent to the problem ( DRO ).
Further, Duchi and Namkoong [2019] argue for solving the DROversion of the problem for concave F (note we are solving a maximization SAA problem) sinceconcave F results in concavity of min p ∈ ∆ ξ,n (cid:80) i ∈
[ N ] p i F ( z , (cid:98) x i ) , thus, the overall DRO problem is aconcave maximization problem.
In contrast, the objective in ( VR ) is not concave.
In this paper, our focus is on F that is not concave , thus, the choice of DRO or variance regularizedform is not obvious.
For the class of functions F that we analyze, we argue the variance regularizedversion is more promising as far as scalability for global optimality is concerned.
We work with theassumption that the variance regularized form is equivalent to DRO, which holds under the mildcondition shown in Equation (9) in Duchi and Namkoong [2019].
Towards a Globally Optimal Solution
In this section, we present results for a general class of non-concave functions F that has a fractionalform with non-linear numerator and denominator and that can be approximated by a linear fractionalform with binary variables.
Then, we show three prominent applications of our approach.
Notation : For ease of notation, we use shorthand to denote F ( z , (cid:98) x i ) by F i and 2 ξ N 2 by ρ .
3.1 General Recipe to Form a MISOCP
We perform a sequence of variable and constraint transformations of ( VR ), leading to a MISOCP.
Mixed Integer Concave Program :
The variance regularized objective in shorthand notation is:
[Equation]
We substitute l i = (cid:80)
i F i N − F i and
q = (cid:80) i F i N for all i ∈
[ N ] , such that (cid:80) i l i = 0 and F i =
q − l i .
The objective in Equation 1 thus becomes q − (cid:112) ρ (cid:80) i
l 2 i which is concave in the variables q and{ l i } .
We add the new constraints (cid:80) i l i = 0 and F i = q − l i for all i ∈
[ N ] .
Note that, while theobjective is now concave with above changes, we have pushed the non-convexity into the constraints
F i − q + l i =
for all i ∈ [ N ] that are added to the optimization.
If F i can be written (or approximated) as a fraction with afﬁne numerator and denominator, wecan convert the constraint F i − q + l i = 0 into a convex constraint, giving us an overall concave program.
The conversion is explained next.
Suppose F i can be written (or approximated)as a Ti v + b i a (cid:48)
Ti v + b (cid:48)
i where v represents binary variables after conversion ( v completely replaces z anda i , a (cid:48)
i , b i , b (cid:48)
i are dependent on (cid:98) x i ’s).
Typically, such a linear fractional form is constructed via apiece wise linear approximation of the original non-linear numerator and denominator of F .
Assume v is of dimension d ; typically d will depend on the number of pieces.
Deﬁne y
i
=
v t iwhere t
i =
[Equation]
Then, we can (re)write the fractional form for F i as F i = a Ti y i
+ b i t i .
This yields the linear constraints below with the non-linearity now restricted to y i =
v t i .
N (cid:88)
[Equation]
We handle y i =
v t i using McCormick relaxation technique [McCormick, 1976].
Typically, McCormick relaxation is applied for bilinear terms that are the product of two continuous variables,in which case, it is an approximation.
However, in our case since v is a binary vector variable, the
McCormick relaxation yields an exact reformulation of the bilinear term.
For applying McCormicktechnique, we need an upper and lower bound of v and t i .
Since v a vector of binary variables, wehave lower bound v L = 0 and upper bound v U = 1 .
Similarly, t Li =
[Equation]
(recall superscript + and − indicate positive and negative part of a vector respectively).
Further, it isassumed t Ui and t Li exist.
Note that these bounds are not variables but ﬁxed constants that depend onthe ﬁxed parameters a i , a (cid:48)
i , b i , b (cid:48)
i , hence these need to be computed just once.
Using the upper andlower bounds of v and t i in McCormick technique we get:
[Equation]
−
[Equation]
It is straightforward to check the above set of equations is equivalent to y i =
v t
i .
With the changes,we obtain a mixed integer concave program (with all constraints linear).
Next, while the above can besolved using branch and bound with general purpose convex solvers for intermediate problem, weshow that a further transformation to a MISOCP is possible.
Specialized SOCP’s solvers providemuch more scalability than a general purpose convex solvers [Bonami and Tramontani, 2015] andhence partially address the scalability challenge.
Mixed Integer SOCP : We transform further by introducing another variable s to stand for (cid:112) ρ (cid:80) i l 2 i .
We use r =
( s, q, ( l i ) i ∈
[ N ] , v , ( t i ) i ∈
[ N ] , y 1 , . . .
, y N ) to denote all the variables of the optimization.
Thus, the objective becomes the linear function q − s with an additional constraint that
[Equation]
The above is same as || A r || 2 ≤ c T r for the constant matrix A (with entries 0 or √ ρ ) and constantvector c (with 1 in the s component, rest 0’s) that picks the l i ’s and s respectively.
This is a SOCPform of constraint, and the linear objective q − s makes the problem after this transformation a
MISOCP.
In the above reformulation, the only approximation is introduced in writing F i as a linear fractional term.
One way of such approximation is via piecewise linear approximation (PWLA).
Suppose the fraction function F ( z , (cid:98) x i ) has a separable (in z ) numerator and denominator of the form(cid:80)
j n ( z j , (cid:98) x i ) (cid:80) j d ( z j , (cid:98) x i ) where j ranges over the components of z , and n ( z j , (cid:98) x i ) and d ( z j , (cid:98) x i ) are non-negativeand Lipschitz continuous.
In this case, a general PWLA is possible with the following guarantee:
Theorem 2.
For F ( z , (cid:98) x i ) =
(cid:80) j n ( z j , (cid:98) x i ) (cid:80) j d
z j , (cid:98) x i ) as stated above and approximated as a Ti v
+ b i a (cid:48)
Ti v + b (cid:48)
i , a PWLAapproximation with K pieces yields |G
( z ∗ ) − G ( (cid:98) z
∗∗ ) | ≤ O (1 /K ) , where G ( z ∗ ) and G ( (cid:98) z
∗∗ ) are theoptimal objective values with approximation (MISOCP) and without the approximation respectively.
Next, we show instantiation of the just presented general recipe for three widely studied problems.
3.2 Applications
Notation :
In the SSG (facility location) application m resources (facility) are allocated to M targets(locations).
x maps to type θ a x of adversary, and type θ d x of defender in SSG, and type θ x of clients offacility or directly V x utility for each client type in facility location.
Bayesian Stackelberg Security Game with Quantal Response : A SSG models a Stackelberggame where a defender moves ﬁrst to allocate m security resources for protecting M targets.
Therandomized allocation is speciﬁed by decision variables z of dimension M with the constraintsthat (cid:80)
Mi
=
z i ≤ m ( z i ∈ [0 , 1] );
z i is interpreted as the protection probability of the target i .
Past works have used the model of a quantal responding adversary [Sinha et al., 2018]].
Wegeneralize this to a Bayesian game version where there is a continuum of attackers types with thetype speciﬁed by a parameter x and an unknown prior distribution over these types.
The attacker’sutility in attacking the target j is a function of the protection probability of target j and type:h ( z j , θ a x ) .
Similarly, the defender’s utility when target j is attacked is: u ( z j , θ d x ) for some playerspeciﬁc parameters θ that depend on x .
Following quantal response model (for attacker only), theattacker of type x attacks a target j with probability
[Equation]
x )) and the defender utilityis F ( z , x
[Equation]
=
[Equation]
Note that in case the defender’s utilities u ( z j , θ d x ) takenegative values and the assumptions of Theorem 2 will be violated.
This issue can be simplyﬁxed by choosing α such that α ≥ max z , x {− u ( z j , θ d x ) } and replacing F ( z , x ) by F ( z , x ) + α =(cid:80) j ∈
[Equation]
( u ( z
[Equation]
,θ
[Equation]
x )+ α )exp
( h ( z j ,θ
[Equation]
x )) (cid:80) j
[Equation] [Equation]
] exp( h ( x j ,θ
[Equation]
x )) .
This will make all the numerators and enumerators of the objectivefunction non-negative, while keeping the same optimization problem.
We also note that quantalresponse is also known as multinomial logit model in the discrete choice model literature [Train,2003].
Our generalization here to multiple types of adversary makes the problem akin to the mixedlogit model in discrete choice models, which is generally considered intractable.
As a consequence,our solution addresses a basic problem in discrete choice models also.
Following our set-up, we observe N samples of the types of attackers (cid:98) x 1 , . . .
, (cid:98) x N (which gives(cid:98) θ a 1 , . . .
, (cid:98) θ aN , (cid:98) θ d 1 , . . .
, (cid:98) θ dN ) and we solve a robust version of the problem.
Further, following our generalrecipe for solving the robust problem, we piecewise approximate the numerator and denominator of
F using K pieces, where the dimension of v is d = MK .
For this approximation, we require twoadditional linear constraints over the constraints in Equations (2-11).
The optimization then is:
[Equation]
subject to Constraints (2 - 11) ,
[Equation]
The overall additive solution bound of O (1 /K ) can be readily inferred from Theorem 2.
Max-Capture Competitive Facility Location (MC-FLP) :
In this problem [Mai and Lodi, 2020], aﬁrm has M locations ( [ M ] ) to set up at most m < M facilities.
The aim is to maximize the numberof clients using this ﬁrm’s facilities.
The competitor(s) already have facilities running at locations
Y ⊂
[ M ] .
There are different types of clients, where types are denoted by x .
The number of clientsof type x is known and equal to s x .
However, the distribution over types is unknown .
The ﬁrm’sdecision of which location to choose is given by binary variables z j ∈ { 0 , 1 } for j ∈ [ M ] .
A utilityof any client of type x for visiting location j is V x ,j .
The choice probability of a client of type x choosing any of this ﬁrm’s location is given as a quantal response model (cid:80) j ∈ [ M ] z j e V x ,j (cid:80) j ∈
[ M ] z j e V x ,j + (cid:80) j ∈ Y e V x ,j .
For shorthand, we abuse notation and use V x ,j to replace e V x ,j and U x ,Y to replace (cid:80) j ∈ Y e V x ,j .
Thisgives F ( z , x ) =
s x (cid:80) j ∈
[ M ] z j V x ,j (cid:80) j ∈ [ M ] z j V x ,j + U x ,Y , which is interpreted as the expected number of clients of type xchoosing this ﬁrm’s facilities.
Following our set-up, we observe N samples of the types of clients samples( (cid:98) V 1 ,j ) j ∈
[ M ] , . . .
, ( (cid:98) V N,j ) j ∈ [ M ] and we solve a robust version of the problem.
Here, we get
F i =
[Equation]
Y .
Next, following our general recipe for solving the robust problem, we notethat F
i is already in the form a Ti v + b i a [Equation]
Ti v + b (cid:48)
i where z plays the role of v .
Thus, the dimension of v is Mand no approximation is needed here for F i . Thus, by Theorem 2, we achieve the global optimalsolution by solving the MISOCP optimally.
The full MISOCP with an additional number of locationconstraint is (note v directly replaces z ):
[Equation]
to Constraints (2 - 11) , (cid:88)
[Equation]
Max-Capture Facility Cost Optimization (MC-FCP) :
In the previous MC-FLP problem, thebudget was speciﬁed as a constraint on the number of facilities.
However, often a more realistic set-upis where there is a monetary constraint and the attractiveness of a facility depends on the investmentinto the facility.
Thus, modifying the previous problem slightly, z j takes a different meaning ofthe amount of investment into facility at location j (zero investment indicates no facility).
Giventhis investment, the attractiveness of a facility j for the client of type x is given as h (
z j , θ x ,j ) forsome parameter θ dependent on x and j .
And the choice probability of a client of type x choosingany of this ﬁrm’s location is given as a quantal response model
[Equation]
x ,Y .
This gives
F ( z , x ) =
[Equation]
x
[Equation]
Y , which is interpreted similar to MC-FLP .
As stated, we observe
N samples of the types of clients which gives ( (cid:98) θ 1 ,j ) j ∈ [ M ] , . .
, ( (cid:98) θ N,j ) j ∈
[ M ] and we solve a robustversion of the problem.
Here, we get F i =
[Equation]
U
i
[Equation]
Y .
Next, as can be seen from theform, this is similar to the SSG problem and, following our general recipe, with two additional linearconstraints the optimization formulation is exactly same as Equation ( SSG ).
Scaling up in Number of Samples
The transformation to a MISOCP helps in scalability over a general mixed integer concave program,but for real world dataset sizes (e.g., 80 , 000 data points in our experiments) we need further scalability.
We explore two related techniques towards this end: clustering and stratiﬁed sampling.
For bothapproaches, we obtain a representative subset of S data points ( S << N ) and a modiﬁed weightedobjective, which converts to a much smaller tractable MISOCP compared to the original problem.
For solution guarantees, we need mild assumptions: in particular, for the rest of this section weassume a bounded F , i.e., for some ﬁxed ψ max z { F ( z , x ) } − min z { F ( z , x ) } ≤ ψ∀ (cid:98) x 1 , . . .
, (cid:98) x
Nand τ -lipschitzness of F in the argument x : | F ( z , x (cid:48) ) − F ( z , x ) | ≤ τ || x (cid:48) − x ||∀ z .
Clustering Approach : We cluster the N points x 1 , ...
x N into S groups and for each group s wehave
||
x
i − x s || ≤ (cid:15) , where x s is the cluster center of cluster s .
We call (cid:15) the clustering radius.
Let C sbe the number of points in the cluster s , hence (cid:80) s ∈ [ S ] C s = N .
We use a shorthand for the originalobjective function of the MISOCP G ( z ) :(cid:88)
[Equation]
where (cid:91) Mean is empirical mean and (cid:99) Var is unnormalized variance .
After clustering, we solve for thesame problem but only with cluster centers and appropriate weighing , to get modiﬁed objective (cid:98) G ( z ) :
[Equation]
The conversion to MISOCP is exactly the same, except for F i ’s being weighted as shown above;details of conversion are in the appendix.
We bound the approximation incurred by the two termsabove (weighted mean and unnormalized weighted variance) separately below
Lemma 1.
Under assumptions stated above, we have (cid:12)(cid:12)(cid:12) (cid:91)
Mean ( F ( z , x )) − (cid:91) Mean S
( F ( z , x )) (cid:12)(cid:12)(cid:12) ≤ τ(cid:15) and
[Equation]
The next result is obtained by using the lemma above
Theorem 3.
Given the assumptions stated above, and (cid:98) z an optimal solution for max z (cid:98) G ( z ) and z ∗optimal for MISOCP max z G ( z ) , the following holds: [Equation]
Stratiﬁed Sampling :
Similar in spirit to clustering, the space of x space is divided into T strata.
Each strata has C t samples, such that (cid:80) t ∈
[ T ] C t = N .
Next, distinct from the clustering approach,we draw N t samples randomly from the t th stratum with a total of (cid:80) t N t =
S samples (note samenumber of total samples S as in clustering).
For each stratum t we have || x
i − x j || ≤ d t for anyx i , x j in stratum t .
We denote a random sample in stratum t as (cid:98) x j where j ∈ [ N t ] (note superscript isto distinguish from the subscript used to index all the (cid:98) x ’s).
This approach is the preferred one if theclustering approach results in cluster centers that are not allowed as parameter values (e.g., clustercenter may be fractional where x ’s can only be integral).
Let l t =
C t N t .
Use (cid:92) Mean T
( F ( z , x )) to stand for 1 N (cid:80) t ∈ [ T ] l t (cid:80) j ∈ [ N t ] F ( z , (cid:98) x j ) and (cid:100) V ar T ( z , (cid:98) x ) for(cid:80) t ∈
[ T ] l t (cid:80) j ∈ [ N t ] (cid:16) (cid:92) Mean ( F ( z , x ))
− F ( z , (cid:98) x j ) (cid:17) .
After stratiﬁed sampling our modiﬁed weightedobjective (cid:98) G ( z ) is (cid:92) Mean T ( F ( z , x )) − (cid:113) ρ (cid:100) V ar T ( z , (cid:98) x ) .
Next, similar to clustering, bounds for (cid:92) Mean Tand (cid:100) V ar T but with high probability (Lemma 2,3 in appendix) lead to the main result:
Theorem 4.
Let D = max z , x | F ( z , x ) | for bounded function F .
Given the assumptions stated above,and (cid:98) z an optimal solution for max z (cid:98) G ( z ) and z ∗ optimal for MISOCP max z G ( z ) , and N ∗ =
min t N t ,the following statement holds with probability ≥ 1 − 2 (cid:80) t exp
[Equation]
[Equation]
[Equation]
Thus, with increasing samples in all strata, optimality gap approaches 0 with prob.
approaching 1 .
Experiments
We evaluate our methods on (a) Stackleberg Security Games ( SSG ) with Quantal Response (syntheticdata), (b) Maximum capture Facility Location Planning ( MC-FLP ) and (c) Maximum capture Facility
Cost Planning ( MC-FCP ).
Empirically we demonstrate (i) better solution quality of our methodcompared to baselines, (ii) practical scalability of our method, and (iii) improvement over nonrobust optimization on those data points that contribute least to the objective (akin to rare classes inclassiﬁcation) while not sacriﬁcing average performance.
We ﬁx K = 10 in PWLA as we ﬁnd thatobjective increase saturates for this K (see Appendix K).
We use a 2.1 GHz CPU with 128GB RAM.
Baselines: We use the following two methods as baselines: (i) [Table]
[Table]
5.1 SSG with Quantal Response (Synthetic Data)
[Table]
We generate attacker and defender utilities following Yang et al.
[2012]; a complete description ofdata generation and choice of f ∗ is in Appendix I.
We generate ﬁve datasets of size N = 500 each inorder to observe the variance of every result reported in this sub-section; this is also the largest sizethat we could solve exactly optimally within an hour using all the data points.
First, we empiricallyvalidate Theorem 1 by plotting in Figure 1(left) the relative gap between the true utility E P ∗ [ F ( · , x )]of the decisions output by running DRO on the output ( x ∗ ) i ∈
[ N ] of the true f ∗ (assumed ﬁxed linearfunction) versus on ( (cid:98) x ) i ∈
[ N ] from learned f , as the training data size N T for learning f is varied.
Next, we focus on only using ( (cid:98) x ) i ∈
[ N ] and the optimal solution for ( (cid:98) x ) i ∈
[ N ] is named as OPT .
Figure 1 (middle) demonstrates empirically that the solution of the optimization problem on clustercenters converges to OPT with only a few number of clusters and the time for the optimizationshown in Figure 1 (right) is reasonable.
Next, the results in Table 1 show a comparison of theclustering and stratiﬁed sampling approach using the metric of how close they get to OPT .
We ﬁndstratiﬁed sampling to be better than clustering in almost all cases.
Table 2 (left) demonstrates thatthe baselines struggle to reach the optimal value objective as the magnitude of regularization ( ξ )increases.
Intuitively, as ξ increases the variance term (which is highly non-convex) contributes moreto the objective and stationary points reached by the baselines are quite sub-optimal compared to theglobal optimal.
In addition, with increasing ξ the ambiguity set becomes larger possibly containingmore local optimal solutions.
We also study varying the parameter M ( m ﬁxed) and Table 2 (right)shows that our approaches outperform the gradient-based baselines across different values of M . [Figure] [Table] [Figure]
[Table]
[Figure]
5.2 MC-FLP and MC-FCP (Real Data)
P&R-NYC Dataset : We use a large and challenging Park-and-ride (P&R) dataset collected in New
York City, which provides utilities for 82341 clients ( N ) for 59 park and ride locations ( M ), alongwith their incumbent utilities for competing facilities [Holguin-Veras et al., 2012]; this data wasdirectly used for MC-FLP .
For MC-FCP we additionally use generated costs, which are not presentin the P&R data.
A complete description of data generation is in Appendix I.
Both these problemscould not be solved at all with our MISOCP alone (no clustering) as the optimization did not ﬁnish in24 hours.
Hence, we use our clustering approach with 50 clusters.
We compare to a baseline solution of the non-robust empirical risk minimization (ERM) (also calledthe sample average approximation or SAA).
We split the data (randomly) into training and test (80:20)and then obtain the decision (cid:98) z using the training data.
Then, we obtain the choice probability (recallthis as probability of a client choosing any of the ﬁrm’s facility) for every client in the test data for thedecision (cid:98) z .
We compare the performance of ERM and our method for clients (in test set) bucketed bychoice probabilities in Figure 2 and Table 3.
The buckets are made by sorting all the clients in the testset by ascending choice probabilities and then considering cumulative buckets as the ﬁrst 5%, the ﬁrst10% and so on.
In Fig.
2, we show that the average percentage improvement in choice probabilitiesof our robust approach over ERM is considerably higher for clients with lower choice probability(these clients contribute least to the objective) and the over all average over all clients (rightmost onx-axis) is slightly better than ERM.
In Table 3, note the signiﬁcantly increased probabilities for lowchoice probability clients (low choice prob.
using (cid:98) z ) without compromising the average performanceacross all clients for varying M .
Additional results are in Appendix J.
Conclusion
We presented an approach for a distributionally robust solution to a class of non-convex sum offractional solutions, with guaranteed near global optimality.
We presented application to three prominent practical problems and the connection to discrete choice models likely opens up possibilities ofapplying our approach to even more problems.
Our scalability is limited to hundred thousand datapoints, further investigation on how to cluster or stratify more effectively (than k-means) to achieveeven more scalability is a possible future research direction.
We hope that our work inspires tacklingrobust formulation of more classes of non-convex problems, with guarantees for global optimality.
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