Data-Driven Conditional Robust Optimization
Anonymous Author(s)
AfﬁliationAddress email
Abstract
In this paper, we study a novel approach for data-driven decision-making underuncertainty in the presence of contextual information.
Speciﬁcally, we address thisproblem using a new Conditional Robust Optimization (CRO) paradigm that seeksthe solution of a robust optimization problem where the uncertainty set accountsfor the most recent side information provided by a set of covariates.
We proposean integrated framework that designs the conditional uncertainty set by jointlylearning a partition in the covariate data space and simultaneously constructingregion speciﬁc deep uncertainty sets for the random vector that perturbs the CROproblem.
We also provide theoretical guarantees for the coverage provided byconditional uncertainty sets and for the value at risk performances obtained usingthe proposed CRO model.
Finally, we use simulated and real world data to illustratethe implementation of our approach and compare it against two non-contextualrobust optimization benchmark approaches to demonstrate the value of exploitingcontextual information in robust optimization.
Introduction
In most real world decision problems, the decision maker (DM) faces uncertainty either in theobjective function that he aims to optimize, or some of the constraints that he needs to satisfy.
Stochastic Programming and Robust Optimization (RO) are the most popular methods for addressingthis type of issue.
With the growing availability of data, there has recently been a surge of interest inmodeling optimization under uncertainty as contextual optimization problems that seek to leveragerich feature observations to make better decisions [Ban and Rudin, 2019, Bertsimas and Kallus, 2020].
In a simple cost minimization problem, where X ⊆ R n and c ( x, ξ ) respectively capture the feasibleset of actions and a cost that depends on both the action x and a random perturbation vector ξ ∈ R m ,the “contextual” DM has access to a vector of covariates ψ ∈ R m assumed to be correlated to ξ .
This
DM therefore traditionally wishes to identify an optimal policy, i.e. a functional x : R m → X thatsuggests an action in X adapted to the observed realization of ψ , with respect to his expected costover the joint distribution of ( ψ, ξ ) :
[Equation]
From a theoretical point of view, one can exploit the interchangeability property (see Theorem 14.60,[Rockafellar and Wets, 2009]) to identify an optimal policy for Problem (1) using the followingconditional stochastic optimization (CSO) problem:
[Equation]
While the literature that treats contextual optimization through the CSO problem is rich, much lessattention has been given to contextual optimization in the risk averse setting.
Namely, one can easilythink about replacing the risk neutral expected value operator in problem (2) with a risk measure such as value-at-risk or conditional value-at-risk in order to prevent the DM from being exposedto the possibility of large costs.
Moreover, while robust optimization is being used pervasively indisciplines that employ decision models, including chemical, civil, electrical engineering, medecine,and physics (see respectively [Bernardo and Saraiva, 1998, Bendsøe et al., 1994, Mani et al., 2006,
Chu et al., 2005, Bertsimas et al., 2007]) to name a few, the question of how to systematically integratecontextual information in this important class of decision models remains to this day unexplored.
In this work, we therefore tackle for the ﬁrst time the contextual optimization problem from the pointof view of robust optimization.
Namely, we will consider a contextual DM that wishes to exploit theside information in the design and solution of a robust optimization problem.
This naturally givesrise to the following conditional robust optimization (CRO) problem
[Equation]
where U ( ψ ) is an uncertainty set designed to contain with high probability the realization of ξconditionally on observing ψ .
Our proposed approach will be data-driven in the sense that the designof the CRO problem will make use of historical observations of joint realizations of ψ and ξ .
Our contribution can be summarized as follows.
• We propose for the ﬁrst time a framework for learning from data an uncertainty set for RO
that adapts to side information
The “training” of this conditional uncertainty set is done by
jointly learning a partition in the covariate data space using deep clustering methods, and
simultaneously constructing region speciﬁc deep uncertainty sets, using techniques from
one-class classiﬁcation, for the random vector that perturbs the CRO problem.
• We establish theoretical connections between CRO and Contextual Value-at-Risk Optimiza-
tion (CVO):
[Equation]
where VaR 1 − ε ( Z
= inf { t | P ( Z ≤ t ) ≥ 1 − ε } refers to the value-at-risk of 1 − ε conﬁdencelevel of Z .•
We demonstrate empirically that contextual robust optimization can improve the performanceof robust optimization models in a data-driven portfolio optimization problem that employsreal world data from the US stock market.
In particular, we ﬁnd that in conditions whereside information carries a strong signal about future returns, the risk of the portfolio can bereduced by up to 15%.
The paper is organized as follows.
Section 2 surveys related work.
Section 3 summarizes theapproach discussed in [Goerigk and Kurtz, 2020].
Section 4 presents a Deep Cluster then Classify(DCC) scheme and our Integrated Deep Cluster then Classify (IDCC) scheme to generate conditionaluncertainty sets.
It also establishes the connections to CVO.
Our case study based on real worldportfolio optimization is presented in section 5 followed by conclusions in section 6.
Related work
Conditional Stochastic Optimization [Hannah et al., 2010] was possibly the earliest work on CSO,where a kernel density estimation approach is exploited to formulate and solve a CSO problem.[Ban and Rudin, 2019] apply CSO to a newsvendor optimization problem where the performanceof linear policies and kernel density estimation is explored and where generalization error can becontrolled using regularization.
[Kallus and Mao, 2020] studied methods to train forest decisionpolicies for CSO in a way that directly targets the optimization costs.
[Ban et al., 2019] use residualtree methods to solve general multi-stage stochastic programs where information about the underlyinguncertainty is available through covariate information.
[Kannan et al., 2020a] propose data-driven
SAA frameworks for approximating the solution to two-stage stochastic programs with access to aﬁnite number of samples of random variables and concurrently observed covariates.
Recently, Linet al. [2022] has applied a conditional VaR constrained CSO formulation to the newsvendor problem. While most of the related work focuses on an “estimate-then-optimize” approach (see also [Srivastavaet al., 2021b] and [Hu et al., 2022]), there has also been recent efforts in designing CSO models usingan end-to-end paradigm (see [Elmachtoub and Grigas, 2022] and [Donti et al., 2017]).
Distributionally robust CSO One common challenge with the applications of CSO is due to thefact that often there is only a few sample (if any at all) drawn from the conditional distribution of ξgiven ψ for each realization of ψ [Hu et al., 2020].
This in turns causes a poor approximation of thetrue conditional distribution resulting in poor out-of-sample performance.
Most proposed solutions tothis issue have relied on distributionally robust optimization (DRO).
For example, [Bertsimas and
Van Parys, 2021], [Bertsimas et al., 2022, Nguyen et al., 2021], and [Srivastava et al., 2021a] allpropose DRO approaches that employ a distribution sets that are centered at either the estimatedconditional distribution or joint joint empirical distribution of ( ψ, ξ ) .
[Kannan et al., 2020b] appliesdistributionally robust optimization to the residual-based CSO model proposed in [Kannan et al.
,2020a].
We ﬁnally note that none of these works have considered the problem of conditional
DRO where the distributional ambiguity set itself, namely its support or size, depend on contextualinformation.
Data-driven Robust Optimization and One-class Classiﬁcation There has been a growing set ofpapers (see [Ohmori, 2021, McCord, 2019, Wang and Jacquillat, 2020]) proposing various frameworksthat use both supervised and unsupervised one-class classiﬁcation techniques in designing theuncertainty sets which are further integrated in the RO problems.
Some approaches make use ofvariance and covariance of historical data [Natarajan et al., 2008] while others [Goerigk and Kurtz,2020, Wang et al., 2021] have exploited the representative power of deep neural networks to constructcompact uncertainty sets Up to this day, none of the data-driven robust optimization approaches haveconsidered accounting for contextual information.
Deep Clustering Methods Traditional clustering methods like Gaussian Mixture Models (GMM)and k -means clustering rely on the original data representations and suffer from the curse of dimensionality.
Recent developments in DNNs led to learning of high quality representations, especiallyauto-encoder(AE) and decoder systems are particularly appealing as they are able to learn the representations in a fully unsupervised fashion.
Several works like [Chang et al., 2017, Guo et al., 2017, Jiet al., 2017] combine variational AEs and GMMs to perform clustering and non-linearly map the inputdata into a latent space.
Few works like [Fard et al., 2020] try to jointly learn the representations andjointly cluster with k -means and learning representations.
We modify these algorithms to introducinga probability simplex which interact with the centroids and also the center of the uncertainty sets.
The Deep Data-Driven Robust Optimization (DDDRO) Approach
Focusing on a classical robust optimization model, i.e. min x ∈X max ξ ∈U c ( x, ξ ) , the authors of[Goerigk and Kurtz, 2020] propose to employ deep learning to characterize the uncertainty set U in adata-driven environment.
In particular, they consider describing the uncertainty set U in the form:
[Equation]
where f W : R m → R d is a deep neural network, parametrized using W , that projects the perturbationvector ξ to a new vector space where the uncertainty set can be more simply deﬁned as a sphere ofradius R centered at some ¯ f 0 .
Given a dataset D ξ = { ξ 1 , ξ 2 . . .
ξ N } , they propose discovering the underlying structure of U bytraining the NN using a method found in the one-class classiﬁcation literature, namely minimizingthe empirical centered total variation of the projected data points:
[Equation]
where ¯ f 0 :=
(1 /N )
(cid:80) i ∈
[ N ] f W 0 ( ξ i ) is the center of the projected points under some initial randomchoice of f W 0 .
Once the network is trained, they calibrate the radius R of U in order to reach atargeted coverage 1 − (cid:15) of the data set.
In terms of NN architecture, they favor a special class of fully connected neural network of depth L :
[Equation]
where each W (cid:96) captures a linear projection while each σ (cid:96) captures a term-wise piecewise linearactivation function (e.g. ReLU, Hardtanh, or hard sigmoid):
[Equation]
with { a (cid:96)k , b (cid:96)k , α (cid:96)k , α (cid:96)k } Kk =1 as the parameters that identiﬁes each of the K afﬁne pieces.
The motivation for such an architecture comes from the proposed solution scheme for the RO problem,which relies on a constraint generation approach (See Algorithm 3 in Appendix).
This scheme relieson progressively adding scenarios to a reduced set U (cid:48) ⊆
U until the worst-case cost of the solutionunder U (cid:48) is the same as under U .
Numerically, a critical step consists in identifying the worst-caserealization in U , which is shown to reduce to a mixed-integer linear program when c ( x, ξ ) is linear inξ under the selected NN architecture due to the following representation of U ( W, R ) :
[Equation]
 
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
where we assume for simplicity that each layer of the deep neural network has d neurons and φ (cid:96)is the output at l -th layer of the neural network.
We refer interested readers to [Goerigk and Kurtz,2020] for more details.
Deep Data-driven Conditional Robust Optimization
Let ( ψ, ξ ) be a pair of random vectors deﬁning respectively the side-information and randomperturbation vectors of a contextual optimization problem.
We can call our dataset D ψξ :={ ( ψ 1 , ξ 1 ) , . . .
, ( ψ N , ξ N ) } .
Our objective is to train a data-driven conditional uncertainty set U ( ψ )that will lead to robust solutions that are adapted to the type of perturbance that are experienced whenψ is observed.
In this section, we propose two algorithms, namely Deep cluster then classify (DCC)and the Integrated Deep cluster then classify (IDCC), to do so, and propose a calibration procedurethat offers some guaranties with respect to a contextual value-at-risk problem.
4.1 The Deep “Cluster then Classify” (DCC) Approach
A direct extension of G&K’s 3 DDDRO approach consists in reducing the side-information ψ to aset of K different clusters, which provides states of the environment in which one wishes to designcustomized data-driven uncertainty sets.
Mathematically, U ( ψ ) :=
U a ( ψ ) , where a : R m → [ K ] ,is a trained K -class cluster assignment function for ψ , and each U k , for k = 1 , . .
, K , is anuncertainty sets for ξ that is trained and sized using the procedure described in section 3 with thedataset D kξ := ∪ ( ψ,ξ ) ∈D ψξ : a ( ψ )=
k { ξ } .
This process implicitly involves multiple sequential stepsof training of deep neural networks.
Following [Moradi Fard et al., 2020], when performing deep
K -mean clustering to obtain a ( ψ ) , training can take the form of Algorithm 5, where the deep Kmeans algorithm trains simultaneously a representation g V E : R m → R d , using an encoder andg V D : R d → R m , using a decoder network, and a K-mean classiﬁer ¯ a θ ( φ ) :
= argmin k ∈ [ K ] (cid:107) φ − θ k (cid:107) 2by minimizing, using stochastic gradient descent in a coordinate descent scheme, a trade-off (usingα K ) between reconstruction error and the within cluster centered total variation in the encoded space:
[Equation]
where a ( ψ ) :
= ¯ a θ ( g V E ( ψ )) .
To solve this problem, we iterate between improving V :=
( V E , V D )while keeping θ ﬁxed, and improving θ while preserving V ﬁxed.
Once the K -mean and one-class classiﬁers are trained, we correct for a deﬁciency of DDDROapproach, who assume wrongfully that the projected f W k ( ξ ) are normalized for each D kξ .
Namely,we replace U ( W, R ) with a set that employs an ellipsoid in the projected space according to thestatistics of D kξ :
[Equation]
where S k is short for ( µ kf , Σ kf ) with
[Equation]
The calibration of each R k can ﬁnally be done using the same procedure as in [Goerigk and Kurtz,2020]but using the reduced dataset D kξ .
4.2 The Integrated Deep Cluster-Classify (IDCC) Approach
While the simplicity of the approach presented in section 4.1 makes it appealing, we identify twoimportant weaknesses.
First, by separating the training in multiple step, it omits to tackle theconditional uncertainty set learning problem as a whole.
Namely, that low total variation in the ψspace (or a projection of it) does not necessarily imply that low total variation can easily be achievedin a projection of the ξ space.
Second, it is unclear how to adapt the approach to a context where aclear separation of the clusters is impossible and where the notion of partial membership to a clusteris more appropriate.
To address the ﬁrst problem, we propose an integrated framework for performing deep clusteringand deep uncertainty set design jointly.
Namely, we propose to optimize all of V , θ , and { W k } Kk =1jointly using a loss function that trades-off between the objectives used for clustering and each ofthe K versions of one-class classiﬁers.
We also tackle the issue of hard assignments by training aparameterized random assignment policy π : R m → ∆ K , where ∆ K is the probability simplex in R K ,and θ the parameters that deﬁne the policy space.
In the context of employing a soft version of deep
K -means [Fard et al., 2020]; this random assignment policy takes the form of π ( ψ ) :
= ¯ π θ ( g V ( ψ )) ,where
[Equation]
With these adjustments, our proposed loss function takes the form of:
[Equation]
where ˜ a ( ψ ) ∼ ¯ π θ ( g V E ( ψ )) is the randomized assignment based on ψ , TotalVar π D ( φ, θ | ˜ a ( ψ )) :
= 185 (cid:80) dj
=1 E π D [( φ j − θ j ) 2 | ˜ a ( ψ )] is the conditional centered total variation of given ˜ a ( ψ ) .
In fact, allstatistics are measured using the empirical distribution expressed in D ψξ and the conditional distribution produced by the randomized assignment policy ¯ π θ ( g V ( ψ )) , i.e. P π D (( ψ, ξ, ˜ a ) ∈ E ) =(1 /N )
(cid:80) Ni =1 (cid:80)
Kk =1 1 { ( ψ i , ξ i , k ) ∈
E} ¯ π θk ( g V ( ψ i )) .
The explicit form of equation (11) can be foundin Appendix B.1.
Overall, L 3 α trades off (using α S ) between the reconstruction error of the encoder-decoder networkson ξ , the expected recognizability of the K clusters, i.e. the fact that the observed features g V E ( ψ )form distinct clusters of points, and the average compactness of the produced conditional uncertaintysets.
In particular, as α S → 1 , we can expect the minimizer of L 3 α to converge to the minimizerof the cluster then classify approach.
At the other end of the spectrum, when α S → 0 , the modelwill produce more self contained conditional uncertainty sets but at the price of less distinguishableclusters (in terms of ψ ) that might poorly exploit the side-information.
Algorithm 1 presents ourproposed training scheme for the IDCC approach.
Given that we employ a random assignment policy, we propose replacing the deterministic CROproblem with its randomized version:
[Equation]
where ˜ U ( ψ ) :
=
U ( W ˜ a ( ψ ) , R ˜ a ( ψ ) , S ˜ a ( ψ ) ) 1 is a random uncertainty set, and where we express the factthat, conditionally on ψ , ˜ x ( ψ ) is a random policy that depends on the realization of ˜ a .
Given therandomness of ˜ U ( ψ ) , one needs to be more careful in deﬁning a calibration scheme for each R k .
Ourproposed scheme is motivated by the following Lemma, which proof can be found in appendix A.
Lemma 4.1.
Let the random uncertainty set ˜ U ( ψ ) satisfy:
[Equation]
[Equation]
then it satisﬁes:
[Equation]
In particular, this lemma suggests calibrating each R k using the bisection to solve:
[Equation]
given that the resulting R k are the smallest that satisfy (12).
Algorithm
Input: Data-set D ψξ , Number of clusters K , hyper-parameters α K , α S , β
Randomly initialize θ 0 , V 0 , and W
Let π 0 :
=
¯ π θ
( g V E 0 ( ψ )) and W k 0 :
= W 0 for all k ’s
Set t :=repeat
Set t :=
t + 1 .
Update θ kt :=
E π D [ g V Et − 1 ( ψ ) | ˜ a ( ψ ) =
k ] using π t −
Update ( V t , { W kt } Kk =1 ) using gradient descent on (11) with θ t
Get π t := ¯
π θ t ( g V Et ( ψ ))
until t ≥ T or convergence
Let π ( · ) :
= π t ( · ) and W k :=
W kt for all k ’s for k = 1 , . . .
K do
Calibrate R k using (14)
Let U k :
= U ( W k , R k , S k ) end for
Return π ( · ) and {U k } Kk =
4.3 Connections to Contextual Value-at-Risk Optimization
In the previous subsections, we proposed two different schemes to produce a possibly randomizeduncertainty set ˜ U ( ψ ) that can be employed in a randomized CRO problem.
We also proposed ascheme for radii calibration so that they would satisfy the coverage property in (13).
Hence, one canderive the following connection between conditional robust optimization and the CVO problem (1).
The proof is pushed to Appendix A.
Lemma 4.2.
When ˜ U satisﬁes (13) , the random policy ˜ x ( · ) to the randomized CRO problem togetherwith
[Equation]
[Equation]
[Equation]
provide a conservative approximate solution to the CVO problem under the empirical measure P π D .
Namely,
[Equation]
In particular, in the case of the proposed DCC and IDCC approaches we have that
[Equation]
As the robust optimization paradigm traditionally aims at offering statistical guarantees on the out-ofsample performance of the prescribed solutions, we describe below how a bootstrap method can beused to estimate the radii R k ’s.
Remark 4.1.
Using bootstrapping methods, we can get a conservative approximation of each R k as:
[Equation]
where P ˜ D measures the probabibility when resampling a new dataset of size N with replacementfrom D ψξ .
When N is large enough and assuming that each data point is drawn i.i.d.
according tosome unknown probability measure P , we asymptotically get the guarantee that P ( ξ ∈ ˜
U
( ψ ))
≥ 1 − (cid:15)with probability higher than approximately 1 − Kδ .
Experiments
In this section, we illustrate the coverage aspect of the IDCC approach using simulated data.
We willfurther demonstrate the advantage of the CRO problem using a standard risk minimizing portfoliooptimization problem.
We compare the performance of IDCC with that of DCC, DDDRO (withellipsoidal correction in (9)), and the classical ellipsoidal uncertainty approach (i.e. DCC with K = 1and f W 1 ( ξ ) := ξ ).
The IDCC and DCC methods incorporate the covariate information whereas
DDDRO and ellipsoid approach ignore this information.
The neural network architecture and othermodeling information is available in the AppendixB.
The code can be found on github 3 .
Our codeuses the Pytorch implementation from[Goerigk and Kurtz, 2020], which is available online 4 .
5.1 Conditional uncertainty set illustration using simulated data
For the ease of illustration, we consider a simulation environment where [ ψ T ξ T ] ∈ R 4 is a randomvector which distribution is an equal weighted mixture of two 4 -d multi-variate normal distributions.
We consider N = 500 and train IDCC (with K = 2 ), DDDRO, and the ellipsoid and calibrate theuncertainty sets for a probability coverage of 90%, 99% (i.e. (cid:15) ∈ { 1% , 10% } ).
As a result, DDDROand IDCC, which use deep neural networks, identify non-convex uncertainty sets, whose convexhulls are presented in Figure 1 together with the calibrated ellipsoid.
The ﬁgure also presents theconditional distribution of ξ according to P π D ( ·| ˜ a ( ψ ) = k ) , using IDCC’s randomized assignment,and the training dataset.
One can remark that the conditional sets produced by IDCC exploit theside information by concentrating the uncertainty set on the region that has the most mass accordingto P π D ( ·| ˜ a ( ψ ) = k ) thus leading to a less conservative RO problem then DDDRO and the ellipsoid,which are oblivious to ψ .
In fact, it appears to have successfully learned to at least partially recognizethe mixture membership using ψ and exploit this information to adapt the uncertainty set.
5.2 Robust portfolio optimization
We further investigate the empirical out of sample performance of the proposed uncertainty setson a classical robust portfolio optimization problem.
Namely, we consider a situation where aninvestor is trying to minimize the worst-case return based on an uncertainty set that provides 1 − (cid:15)probabilistic coverage of the uncertain future return vector.
In particula, given that x captures a vectorof investment in n = m different assets which return are captured using ξ , we let c ( x, ξ ) := − ξ (cid:124) x tocapture the return on investment, and let X :=
{ x ∈ R n | (cid:80) ni
=1 x i = 1 , x ≥ 0 } to capture the need
[Figure]
to invest one unit of wealth among
Following Lemma 4.2, this model can in turnbe interpreted as conservatively approximating a min x ∈X VaR 1 − (cid:15) ( ξ (cid:124) x ) , where the objective is a riskaverse value-at-risk metric.
Dataset
Our experiments make use of historical data from the US stock market.
We collect theadjusted daily closing prices for 70 stocks (as used in [Xu and Cohen, 2018]) coming from 8 differentsectors from January 1, 2012 to December 31, 2020 using the Y!Finance’s API.
Each year has 252data points and we compute the percentage gain/loss w.r.t the previous day to create our dataset forξ .
As for side information, we use trading volume of individual stocks, and other market indices 5over the same period as covariates.
Our algorithm gives the ﬂexibility to use any number of suchmetrics as contextual information.
Given the time series nature of the data, at a given instance, weuse 3 years of data to train and the following year as validation to pick the hyper parameters of ourmodel such as learning rate, weight decay, optimal number of clusters.
We then retrain the modelusing the 4 years of data to build the ﬁnal model.
Upon calibrating the uncertainty set, we use it tosolve the robust portfolio optimization problem.
We then apply this policy on the next 1 year dataand compute the performance metric, namely Value at risk (VaR) for different conﬁdence levels tocompare the performances.
VaR quantiﬁes the level of risk of a portfolio over a speciﬁed time frame.
Here, it gives an estimate of the maximum % loss the decision maker can incur over a period of1 year when he uses the policy from the RO model.
Intuitively, lower the VaR, less riskier is thegenerated policy.
Many ﬁnancial institutions use VaR to determine the amount of collateral neededwhen trading ﬁnancial products so lowering VaR for high conﬁdence levels is crucial.
Experiment Design To test for robustness of the IDCC algorithm, we experiment on variousrandomly sampled stock combinations across different time periods.
We randomly sampled a subsetof 15 stocks in a time window and repeated the experiment for 10 runs on 3 moving time frames.
Weused learning rate = 0.01, α K = 0 .
5 , α S = 0 .
5 , β = 0 .
1 for all the experiments.
We use a cold start
K-means approach to determine K for each run.
We do this across all these experiments as it willbe computationally expensive to tune the parameters through grid search for each run and also ourintention is to show the learning capability of our algorithm even with minimal tuning.
The parametertuning and implementation details can be found in appendix B.3.
Results Fig.
2 shows the avg.
VaR across the runs at different conﬁdence levels.
It is evident that
IDCC generally performs better than the baseline models.
This difference is especially noticeableat higher conﬁdence level, and vanishes as we move to lower conﬁdence levels.
Table 1 providesmore details by comparing the overall and conditional cluster level VaR with the baseline models.
Speciﬁcally, in each run we identify each cluster as either the “majority” or “minority” clusterdepending on its frequency and report averages of VaR (among the 10 runs) for each of these labels.
The average frequencies for each label is also reported in the table.
In particular, one can observe thatthe improvement on average overall VaR can reach up to ∼ 15% (see in 2019 at a 0.99 conﬁdencelevel).
This advantage is even more clearly visible when we look at the individual cluster level
[Table]
[Table] [Table]
0.99 Confidence level 0.
A v g .
V a
R (a)0.
0.99 Confidence level 0.
A v g .
V a
R (b)0.
0.99 Confidence level 0.
A v g . V a R
[Figure]
DCC
DDDRO
Ellipsoid conditional VaR.
For instance, in the year 2018 for the 0.99 conﬁdence level, the majority cluster( ∼ 68% data) provides an improvement of 19% and an overall improvement of 9% compared to thesecond best baseline model.
A similar pattern is observed for the year 2019 as well.
In year 2017,the overall performance of IDCC is close and for some conﬁdence levels slightly above the baselinemodels.
However, we see that the majority cluster ( ∼ 80% data) is performing better than the baselinemodels while the minority cluster has slightly higher risk.
We attribute this loss in performance tothe fact that the minority clusters is much less frequent ( ∼ 20% data) and therefore has less dataavailable to properly learn its conditional uncertainty set.
This large difference in frequencies mightalso indicate that the side information does not have a strong signal for the behavior of the returnsduring this period of time.
Conf.
1 − (cid:15) 0.
IDCC 0.
[Table]
overall VaR and conditional VaR given the membership to the majority/minority clusters are presented.
Conclusion and Future Work
In this work, we introduced a new approach, Conditional Robust Optimization, for solving contextualoptimization problems in a risk averse setting.
We proposed a novel integrated approach to designuncertainty sets that adapts to revealed covariate information.
We identiﬁed connections to contextualvalue-at-risk optimization and showed empirically that our method reduces the out-of-sample VaRconsiderably compared to non-contextual RO schemes when the level of protection needed is high.
As future work, we ﬁnd that it should be interesting to integrate data-driven conditional uncertaintysets in the context of multi-stage robust optimization models.
Given that clustering techniques areoften prone to learning correlations from the data that do not reﬂect true causal relation, there mightbe a need to integrate causal inference methods to our approach.
One might also be concernedregarding fairness considerations in contexts where side information might allow to treat certainclass of individuals differently from others.
This last issue might be addressed by adding fairnessconsideration in our integrated loss function.
Checklist
For all authors...
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contributions and scope?
[Yes]
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[Yes]
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[Yes]
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[Yes]
A
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
[Yes] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
[Yes] B
(c) Did you report error bars (e
g., with respect to the random seed after running experiments multiple times)?
[Yes] (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[TODO]
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(a) If your work uses existing assets, did you cite the creators?
[Yes] https://github.com/goerigk/RO-DNN
(b) Did you mention the license of the assets?
[N/A]
(c) Did you include any new assets either in the supplemental mate- rial or as a URL?
[Yes] https://anonymous.4open.science/r/ Data Driven [Table]
Conditional
[Table]
Robust
[Table]
Optimization
[Table]
E
[Table]
README.md
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identiﬁable
information or offensive content?
[N/A]
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]