Confidence regions and intervals for the expected value formulation
of stochastic variational inequalities
Shu Lu
Based on joint work with Amarjit Budhiraja and Michael Lamm
Department of Statistics and Operations Research
The University of North Carolina at Chapel Hill
XIV International Conference on Stochastic Programming
Búzios, June 25-July 1, 2016
Outline
1 Stochastic optimization and stochastic variational inequalities
2 Confidence regions and confidence intervals
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Outline
1 Stochastic optimization and stochastic variational inequalities
2 Confidence regions and confidence intervals
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
A stochastic optimization problem
min E [Φ(x, ξ)]
x∈S
where
x is the variable, which is constrained in the set S ⊂ Rn
ξ(ω) is a random vector taking values in a set Ξ ⊂ Rd
Φ is a function from S × Ξ to R
If E [Φ(x, ξ)] is differentiable at a local minimizer x0 , then x0 satisfies the
first-order necessary condition
−∇x E [Φ(x0 , ξ)] ∈ NS (x0 )
(FONC-SO)
where NS (x0 ) is the normal cone to S at x0 , and for convex S is given by
NS (x0 ) = {v ∈ Rn | hv , s − x0 i ≤ 0 ∀s ∈ S}
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
The expected-value formulation of stochastic variational inequalities1
−E [F (x, ξ)] ∈ NS (x)
(SVI)
where
x is the variable, which is constrained in the set S ⊂ Rn
ξ(ω) is a random vector taking values in a set Ξ ⊂ Rd
F is a function from S × Ξ to Rn
(FONC-SO) fits in (SVI) when
∇x E [Φ(x0 , ξ)] = E [∇x Φ(x0 , ξ)]
1
See [Chen, Wets and Zhang 2012], [Rockafellar and Wets 2016] and their
references for alternative SVI formulations
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
The normal map formulation of variational inequalities2
The normal map associated with a function f : S → Rn and a set S ⊂ Rn is a
function fS : Rn → Rn , defined as
fS (z) = f (ΠS (z)) + z − ΠS (z) for each z ∈ Rn
where ΠS (z) is the Euclidean projection of z on S
z=x−f (x)
−f (x) ∈ NS (x)
−
−−−−−
→
←−−−−−
fS (z)
=
0
x=ΠS (z)
ΠS is piecewise affine
The normal manifold of S: the
polyhedral subdivision of Rn
corresponding to ΠS
fS is piecewise smooth if f is smooth,
and is piecewise affine if f is affine
2
Details about normal maps can be found in [Robinson 1992], [Ralph 1993],
[Facchinei and Pang 2003], [Scholtes 2012] and references therein
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
The sample average approximation (SAA)
Define the true function as f0 (x) = E [F (x, ξ)], which often does not
have a closed form expression. Consider the SAA function
fN (x) = −N −1
N
X
F (x, ξ i (ω))
i=1
1
where ξ , · · · , ξ
N
are i.i.d. r.v. with distribution same as ξ
Suppose we have found xN , a solution to
−fN (x) ∈ NS (x)
(SAA-VI)
We aim to give confidence regions for x0 , the true SVI solution
−f0 (x) ∈ NS (x)
(SVI)
Let z0 = x0 − f0 (x0 ) and zN = xN − fN (xN ) be the corresponding
solutions to the normal map formulations
(f0 )S (z) = 0
(SVI-NM)
and
(fN )S (z) = 0
(SAA-NM)
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
From SAA solutions to the true solution3
Suppose S is polyhedral convex. Under certain
Assumptions
For a.e. ω, (SAA-VI) has a locally unique solution xN for N large
enough, with limN→∞ xN = x0
The corresponding solution zN to (SAA-NM) is also locally unique,
with limN→∞ zN = z0 almost surely
Let Σ0 be the covariance matrix of F (x0 , ξ), and N (0, Σ0 ) be a
normal r.v. in Rn with zero mean and covariance matrix Σ0 . Then,
√
NLK (zN − z0 ) ⇒ N (0, Σ0 )
(Conv-Dist-z)
√
N(xN − x0 ) ⇒ ΠK ◦ (LK )−1 (N (0, Σ0 ))
(Conv-Dist-x)
where L = ∇x E [F (x0 , ξ)], K = TS (x0 ) ∩ E [F (x0 , ξ)]⊥ , LK is the
normal map associated with L and K , and (LK )−1 is its inverse
LK is a piecewise linear approximation of the normal map (f0 )S
around z0
3
See related results in [King and Rockafellar 1993], [Gürkan, Özge and
Robinson 1999], [Demir 2000], [Shapiro, Dentcheva and Ruszczyński 2009],
[Xu 2010] etc.
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Assumptions
Assumption 1: Implies the continuous differentiability of f0 on O, the
1
n
almost sure convergence fN → f0 as an element of C
√ (X , R ) for any
compact set X ⊂ O, and the weak convergence of N(fN − f0 )
(a) E kF (x, ξ)k2 < ∞ for all x ∈ O, where O is an open set in Rn .
(b) The map x 7→ F (x, ξ(ω)) is cont diff on O for a.e. ω ∈ Ω.
(c) There exists a square integrable random variable C such that
kF (x, ξ(ω)) − F (x 0 , ξ(ω))k + kdF (x, ξ(ω)) − dF (x 0 , ξ(ω))k ≤ C (ω)kx − x 0 k,
for all x 0 , x ∈ O and a.e. ω ∈ Ω.
Assumption 2: Guarantees the existence, local uniqueness, and stability
of the true solution under small perturbation of f0
Suppose that x0 ∈ O solves (SVI). Let z0 = x0 − f0 (x0 ), L = df0 (x0 ),
K = TS (x0 ) ∩ {z0 − x0 }⊥ , and assume that the normal map LK induced by
L and K is a homeomorphism from Rn to Rn
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Properties of the limiting distributions
(LK )−1 (N √
(0, Σ0 ))
limiting dist of
and
N(zN −z0 )
ΠK ◦ (LK )−1 (N
(0, Σ0 ))
√
limiting dist of
N(xN −x0 )
For a given q ∈ Rn , ΠK ◦ (LK )−1 (q) is the solution h of a linear VI:
−Lh + q ∈ NK (h)
and when L is symmetric it is the unique solution of the QP
1
min hT Lh − q T h
h∈K 2
(LK )−1 (q) = h − Lh + q
If K is a subspace, ΠK ◦ (LK )−1 (q) and (LK )−1 (q) are linear
functions of q, and xN and zN are asymptotically normal
If K is a polyhedral convex cone but not a subspace, then
ΠK ◦ (LK )−1 (q) and (LK )−1 (q) are piecewise linear functions with
multiple pieces, and xN and zN are not asymptotically normal
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Example: a linear complementarity problem
F : R2 × R6 → R2 given by F (x, ξ) =
ξ1
ξ3
ξ2
ξ4
x1
ξ
+ 5
x2
ξ6
ξ uniformly distributed on [0, 2] × [0, 1] × [0, 2] × [0, 4] × [−1, 1] × [−1, 1]
1 1/2
Then f0 (x) = E [F (x, ξ)] =
x
1
2
Let S = R2+ . The SVI is an LCP:
−f0 (x) ∈ NR2+ (x),
which has a unique solution x0 = 0, and
z0 = x0 − f0 (x0 ) = 0 is the unique
solution for (SVI-NM)
1 1/2
Here, L =
and K = S = R2+
1 2
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
In the example: scatter plots for zN
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Left: solutions to 200 SAA problems with N = 10; Right: N = 30
Curves are boundaries of sets
T
{z ∈ R2 | N LK (z − z0 ) Σ−1
LK (z − z0 )] ≤ χ2N (α)}
0
which contain zN with (approximately) probability 1 − α for
α = 0.1, · · · , 0.9
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
0.3
0.4
Outline
1 Stochastic optimization and stochastic variational inequalities
2 Confidence regions and confidence intervals
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Two approaches on confidence regions and intervals
1
Obtain C.R. for z0 first, then compute C.I.’s for z0 and x0
√
Recall NLK (zN − z0 ) ⇒ N (0, Σ0 )
(Conv-Dist-z)
LK = d(f
√0 )S (z0 ), the B-derivative of the normal map (f0 )S at z0 . One can
show − Nd(fN )S (zN )(z0 − zN ) ⇒ N (0, Σ0 )
An asymptotically exact (1 − α)100% C.R. for z0 :
T
2
{z ∈ Rn | N d(fN )S (zN )(z − zN ) Σ−1
N d(fN )S (zN )(z − zN )] ≤ χn (α)}
where ΣN = Sample Cov of F (xN , ξ)
w.h.p. d(fN )S (zN ) is linear, and this confidence region is an
2
ellipsoid
Obtain a consistent estimate for LK to compute C.R. and C.I.’s
Under additional assumptions, zN converges to z0 at an exponential rate
At zN we can define a function ΦN (zN ) : Rn × Rn so that
kΦN (zN )(h) − LK (h)k
φ
lim Prob sup
< 1/3 = 1
N→∞
khk
N
h∈Rn
Replacing LK by ΦN (zN ) in the weak convergence results gives a basis for
computing C.R. and C.I.’s
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Confidence regions for z0 computed from z10
0.93
−
0.75
An SAA for the LCP (N=10):
0.54
0.13
x+
∈ NR2+ (x)
2.11
0.29
A unique solution x10 = (0.08, 0.11) = z10 marked as ×
0.42
0.01
Σ10
0.01
0.19
0.93
0.75
0.54
2.11
T 4.88
z 10(z − z10 )
9.34
+: z0 = 0
9.34
2
(z − z10 ) ≤ χ2 (α)
24.26
(1−α)100% confidence region for z0
d(f10 )R2 (z10 )
+
0.5
Shown in the figure:
confidence regions for z0
at levels 0.1, · · · , 0.9
0.4
0.3
0.2
90% simultaneous
confidence intervals:
(z0 )1 : [-0.52, 0.68]
(z0 )2 : [-0.16, 0.38]
(x0 )1 : [0, 0.68]
(x0 )2 : [0, 0.38]
0.1
0
−0.1
−0.2
−0.6
−0.4
−0.2
0
0.2
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
0.4
0.6
Individual confidence intervals for z0 and x0 (target level: 90%)
Consider cells in the normal manifold of R2+ :
{0}, {0} × R+ , R+ × {0}, {0} × R− , R− × {0}, R2+ , R2− , R+ × R− , R− × R+ ,
CiN : the cell with the smallest dimension, among all cells that intersect
the 95% region. Here it is {0}
PN : the 2-dim cell that contains zN in its interior. Here it is R2+
Let z̃iN be any point in ri CiN , and KN = cone(PN − z̃iN ). Here it is R2+
With limiting probability ≥ 95%, KN gives the cone that contains zN in
the polyhedral subdivision of R2 corresponding to LK
1/2
Let M = (d(fN )S (zN ))−1 ΣN , and compute an number `N such that
Pr (|(MZ )j | ≤ `N , and MZ ∈ KN )
= 0.95,
Pr (MZ ∈ KN )
lim inf N→∞ Prob
√
N|(zN − z0 )j | ≤ `N
where Z ∼ N (0, I )
≥ 0.90
90% individual confidence intervals for z0 and x0 (computation of
intervals for x0 is analogous)
(z0 )1 : [-0.16, 0.32], (z0 )2 : [0, 0.22], (x0 )1 : [0, 0.32], (x0 )2 : [0, 0.22]
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities
Summary
Development and justification of methods to build computable
confidence regions and intervals for the true solutions of the
expected-value formulation of stochastic variational inequalities
Applied to stochastic Cournot-Nash production/distribution
problems and sparse-penalized statistical regression
This presentation is mainly based on the following papers:
Lamm, Lu. 2016. Generalized conditioning based approaches to compute
confidence intervals for stochastic variational inequalities. Working paper
Lamm, Lu and Budhiraja. 2016. Individual confidence intervals for solutions to
expected value formulations of stochastic variational inequalities. Mathematical
Programming B, accepted
Lu. 2014. Symmetric confidence regions and confidence intervals for normal
map formulations of stochastic variational inequalities. SIAM Journal on
Optimization. Vol. 24, No. 3, pp. 1458-1484
Lu and Budhiraja. Confidence regions for stochastic variational inequalities.
Mathematics of Operations Research, 2013, Vol. 38, No. 3, pp. 545-568
Confidence regions and intervals for the expected value formulation of stochastic variational inequalities