Multilevel Monte Carlo
Metamodeling
Imry Rosenbaum
Jeremy Staum
Outline
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What is simulation metamodeling?
Metamodeling approaches
Why use function approximation?
Multilevel Monte Carlo
MLMC in metamodeling
Simulation Metamodelling
• Simulation
–
–
–
–
Given input 𝜃 we observe 𝑌 𝜃 .
Each observation is noisy.
Effort is measured by number of observations, 𝑚.
We use simulation output 𝑌 𝜃; 𝑚 to estimate the response
surface 𝜇 𝜃 = 𝐸 𝑌 𝜃 .
• Simulation Metamodelling
– Fast estimate of 𝜇 𝜃 given any 𝜃.
– “what does the response surface look like?”
Why do we need Metamodeling
• What-if analysis
– How things will change for different scenarios .
– Applicable in financial, business and military settings.
• For example
– Multi-product asset portfolios.
– How product mix will change our business profit.
Approaches
• Regression
• Interpolation
• Kriging
– Stochastic Kriging
• Kernel Smoothing
Metamodeling as Function
Approximation
• Metamodeling is essentially function approximation
under uncertainty.
• Information Based Complexity has answers for such
settings.
• One of those answers is Multilevel Monte Carlo.
Multilevel Monte Carlo
• Multilevel Monte Carlo has been suggested as a
numerical method for parametric integration.
• Later the notion was extended to SDEs.
• In our work we extend the multilevel notion to
stochastic simulation metamodeling.
Multilevel Monte Carlo
• In 1998 Stefan Heinrich introduced the notion of
multilevel MC.
• The scheme reduces the computational cost of
estimating a family of integrals.
• We use the smoothness of the underlying function
in order to enhance our estimate of the integral.
Example
• Let us consider 𝑓 ∈ 𝐶 0,1
compute
2
and we want to
1
𝑢 𝜃 =
𝑓 𝜃, 𝑡 𝑑𝑡
0
For all 𝜃 ∈ Θ = 0,1 .
• We will fix a grid 𝜃𝑖 =
𝑖
,
𝑛
𝑖 = 0, … , 𝑛 , estimate
the respective integrals and interpolate.
Example Continued
We will use piecewise linear approximation
𝑛
𝜂 𝜃 =
𝑢 𝜃𝑖 𝜑𝑖 𝜃
𝑖=0
Where 𝜑𝑖 are the respective hat functions and 𝑢 𝜃𝑖
are Monte Carlo estimate, i.e,
𝑢 𝜃𝑖 =
1
𝑀
𝑀
𝑗=1 𝑓
𝜃𝑖 , 𝜉𝑗 .
𝜉𝑗 are iid uniform random variables.
Example Continued
• Let us use the root mean square norm as metric
for error
1
𝑒 𝜂 =
𝜂 𝜃 −𝑢 𝜃
2
1/2
𝑑𝜃
.
0
• It can be shown that under our assumption of
smoothness that 𝑒 𝜂 = 𝑂 𝑀−1/2 at the cost of
𝑂 𝑀3/2 .
Example Continued
• Let us consider a sequence of grids
𝜃ℓ𝑖 =
𝑖
,
2ℓ
𝑖 = 0, … , 2ℓ .
• We could represent our estimator as
𝜂 𝜃 =
𝐿
ℓ=0 𝜂ℓ
𝜃 − 𝜂ℓ−1 𝜃 , ( 𝜂−1 = 0).
Where, 𝜂ℓ 𝜃 is the estimation using the ℓ𝑡ℎ grid.
• We define each one of our decision variables in terms
of M, as to keep a fair comparison.
Example Continued
level
Square root of
variance
0
ℓ
L
𝑀−1/2
2−ℓ 𝑀−1/2
2−𝐿 𝑀−1/2
Cost
𝑀
2ℓ 𝑀
2𝐿 𝑀
• The variance reaches its maximum in the first level
but the cost reaches its maximum in the last level.
Example Continued
• Let us now use a different number of observations in
each level, 𝑚ℓ , thus the estimator will be
𝐿
𝜂𝑀𝐿𝑀𝐶
𝜃 =
ℓ=1
1
𝑚ℓ
𝑚ℓ
2ℓ
2ℓ−1
𝑢 𝜃𝑖ℓ 𝜑𝑖ℓ 𝜃 −
𝑗=1
𝑖=0
𝑢 𝜃𝑖ℓ−1 𝜑𝑖ℓ−1 𝜃
𝑖=0
• We will use 𝑚ℓ = 2−3ℓ/2 𝑀 to balance between cost
and variance.
Example Continued
level
Square root of
variance
0
ℓ
L
𝑀−1/2
2−ℓ/4 𝑀−1/2
2−𝐿/4 𝑀−1/2
Cost
𝑀
2−ℓ/2 𝑀
2−𝐿/2 𝑀
• It follows that the square root of the variance is
𝑂 𝑀−1/2 while the cost is 𝑂 𝑀 .
• Previously, same variance at the cost of 𝑂 𝑀3/2 .
Generalization
• Let Θ ⊂ 𝑅 𝑑1 and 𝐺 ⊂ 𝑅 𝑑2 be bounded open sets
with Lipschitz boundary.
• We assume the Sobolev embedding condition
•
•
𝑊𝑞𝑟,0
𝑓
Θ × 𝐺 = 𝑓 ∈ 𝐿𝑞 Θ × 𝐺
𝑊𝑞𝑟,0
=
𝛼 ≤𝑟
𝜕𝛼 𝑓
𝑞
𝜕𝜃 𝛼
𝐿𝑞
𝜕𝛼 𝑓
: 𝛼
𝜕𝜃
1/𝑞
.
𝑟
𝑑1
>
1
.
𝑞
∈ 𝐿𝑞 , 𝛼 ≤ 𝑟 .
General Thm
Theorem 1 (Heinrich). Let 1 < 𝑞 < ∞, 𝑝 = 𝑚𝑖𝑛 2, 𝑞 . Then
there exist constants 𝑐1 , 𝑐2 > 0 such that for each integer 𝑀 > 1
there is a choice of parameters 𝐿, 𝑚ℓ 𝐿ℓ=1 such that the cost of
computing 𝜂 𝑀𝐿𝑀𝐶 is bounded by 𝑐1 𝑀 and for each 𝑓 ∈ 𝑊𝑞𝑟,0 (Λ
Issues
• MLMC requires smoothness to work, but can we
guarantee such smoothness?
• Moreover, the more dimensions we have the more
smoothness that we will require.
• Is there a setting that will help with alleviating these
concerns?
Answer
• The answer to our question came from the derivative
estimation setting in Monte Carlo simulation.
• Derivative Estimation is mainly used in finance to
estimate the Greeks of financial derivatives.
• Glasserman and Broadie presented a framework
under which a pathwise estimator is unbiased.
• This framework will be suitable as well in our case.
Simulation MLMC
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Goal
Framework
Multi Level Monte Carlo Method
Computational Complexity
Algorithm
Results
Goal
• Our goal is to estimate the response surface 𝜇 𝜃
=𝐸 𝑌 𝜃 .
• The aim is to minimize the total number of
observations used for the estimator.
• Effort is relative to amount of precision we require.
Elements We will Need for the MLMC
• Smoothness provided us with the information how
adjacent points behave.
• Our assumptions on the function will provide the
same information.
• The choice of approximation and grid will allow to
preserve this properties in the estimator.
The framework
• First we assume that our simulation output is a
Holder continuous function of a random vector 𝑋,
𝑌 𝜃, 𝜔 = 𝑓 𝑋 𝜃, 𝜔 .
• Therefore, there exist 𝑐 ∈ ℜ and 𝜁 ∈ (0,1] such that
𝑓 𝑈 − 𝑓 𝑉 < 𝑐 𝑈 − 𝑉 𝜁 for all 𝑈, 𝑉 in ℜ𝑑
Framework Continued…
• Next we assume that there exist a random variable, 𝜅
with a finite second moment such that
𝑋 𝜃1 − 𝑋 𝜃2 ≤ 𝜅 𝜃1 − 𝜃2 for all 𝜃1 , 𝜃2 ∈ Θ, a.s.
• Furthermore, we assume that Θ = 1 and that it is
compact.
Behavior of Adjacnt Points
• bias of estimating 𝑌 𝜃 + ℎ using 𝑌 𝜃 is
𝐸 𝑌 𝜃+ℎ −𝑌 𝜃
≤𝐸 𝑌 𝜃+ℎ −𝑌 𝜃
≤𝐸 𝑐 𝑋 𝜃+ℎ −𝑋 𝜃 𝜁 =𝑂 ℎ 𝜁
• It follows immediately that,
𝑉𝑎𝑟 𝑌 𝜃 + ℎ − 𝑌 𝜃 ≤ 𝐸 𝑌 𝜃 + ℎ − 𝑌 𝜃
= 𝑂 ℎ 2𝜁
2
Multi Level Monte Carlo
• Let us assume that we have a sequence of grids Δℓ
with increasing number of points 𝑘 𝑑 ℓ+1 .
• The experiment designed are structured such that the
maximum distance between a point 𝜃 and point in the
experiment design is 𝑂 𝑘 −ℓ𝜏 , denoted by Δℓ .
• Let 𝑌ℓ 𝜃, 𝜔 denote an approximation of 𝑌 𝜃 using
the same 𝜔 at each design point.
Approximating the Response
MLMC Decomposition
• Let us rewrite the expectation of our approximation in
the multilevel way
𝐸 𝑌ℓ 𝜃, 𝜔
= 𝐸 𝑌0 𝜃, 𝜔
+
ℓ
𝑖=1 𝐸
𝑌𝑖 𝜃, 𝜔
− 𝐸 𝑌𝑖−1 𝜃, 𝜔 .
• Let us define the estimator of 𝐸 𝑌ℓ 𝜃, 𝜔
observations, 𝜇ℓ 𝜃, 𝑚, 𝜔𝑖
=
1
𝑚
𝑚
𝑖=1 𝑌ℓ
using m
𝜃, 𝜔𝑖 .
MLMC Decomposition Continued
• Next we can write the estimator in the multilevel
decomposition,
𝜇ℓ 𝜃, 𝑚, 𝜔𝑖
ℓ
= 𝜇0 𝜃, 𝑚, 𝜔𝑖
ℓ
ℓ
+
𝜇𝑗 𝜃, 𝑚, 𝜔𝑖
ℓ
− 𝜇𝑗−1 𝜃, 𝑚, 𝜔𝑖
ℓ
𝑗=1
• Do we really have to use the same 𝑚 for all levels?
The MLMC estimator
• We will denote the MLMC estimator as
𝑍ℓ 𝜃,
𝜔𝑖0
,…,
𝜔𝑖ℓ
=
ℓ
𝑖=0 Δ𝑍𝑗
𝜃,
𝑗
𝜔𝑖
• Where
𝑗
Δ𝑍𝑗 𝜃, 𝜔𝑖
= 𝜇𝑗 𝜃, 𝑀𝑗 , 𝜔𝑖
𝑗
− 𝜇𝑗−1 𝜃, 𝑀𝑗 , 𝜔𝑖
𝑗
Multilevel Illustration
Δ
Multi Level MC estimators
• Let us denote
𝑟
𝑟
𝐵𝑜𝑥𝑟 𝜃 = 𝑥: ∀𝑖 = 1, … , 𝑑, 𝜃𝑖 − ≤ 𝑥𝑖 ≤ 𝜃𝑖 +
2
2
• We want to consider approximation of the form of
𝑘 𝑑ℓ
𝑔ℓ 𝜃 − 𝜃𝑖ℓ ⋅ 𝑌 𝜃𝑖ℓ , 𝜔
𝑌ℓ 𝜃, 𝜔 =
𝑖=0
Approximation Reqierments
• We assume that for each 𝜉 > 0 there exist a window
size 𝑟 ℓ, 𝜉 > Δℓ ) which is 𝑂 𝑘 −𝜈ℓ . Such that for
each, 𝑠 ≥ 𝑟 ℓ, 𝜉 we have 0 ≤ 𝑔ℓ 𝑠 ≤
for each 𝜃 ∈ Θ we have
𝜉
1 − ℓ𝜈 ≤
𝑘
ℓ
𝜃𝑖 ∈𝐵𝑜𝑥𝑟 ℓ,𝜉 𝜃
𝑔ℓ 𝜃 −
𝜃𝑖ℓ
𝜈
𝑘 ℓ 𝑑+𝜈
and
𝜉
≤ 1 + ℓ𝜈
𝑘
Bias and Variance of the
Approximation
• Under these assumptions we can show that
– 𝜇 𝜃 − 𝐸 𝜇ℓ 𝜃
– 𝑉𝑎𝑟 Δ𝑍ℓ 𝜃
= 𝑂 𝑘 −ℓ𝜈𝜁
= 𝑂 𝑘 −2ℓ𝜈𝜁
• Our measure of error is Mean Integrated Square Error
𝑀𝐼𝑆𝐸 𝜇 𝜃
=𝐸
Θ
𝜇 𝜃 −𝜇 𝜃
2
𝑑𝜃
• Next, we can use a theorem provided by Cliffe et al.
to bound the computational complexity of the
MLMC.
Computational Complexity Theorem
Theorem. Let 𝜇 𝜃 denote a simulation response surface and
𝜇ℓ 𝜃 , an estimator of it using 𝑀ℓ replications for each design
point. Suppose there exist Δ𝑍ℓ 𝜃 , 𝑐1 , 𝑐2 , 𝑐3 , 𝛼, 𝛽, 𝛾, 𝑘 such that
𝑘 > 1, 𝛼 > 0.5𝑚𝑖𝑛 𝛽, 𝛾 and
1. 𝐸 𝜇 𝜃 − 𝜇ℓ 𝜃 ≤ 𝑐1 𝑘 −𝛼ℓ
2. 𝐸 Δ𝑍ℓ 𝜃
3. 𝑉𝑎𝑟 Δ𝑍ℓ 𝜃
𝐸 𝜇0 𝜃 , ℓ = 0
=
𝐸 𝜇ℓ 𝜃 − 𝜇ℓ−1 𝜃 , ℓ > 0
≤
𝑐2 −𝛽ℓ
𝑘
𝑀ℓ
4. The computational cost of Δ𝑍ℓ 𝜃 is bounded by 𝑐3 𝑀ℓ 𝑘 𝛾ℓ
Theorem Continued…
Then for every 𝜀 there exist values of 𝐿 and 𝑀ℓ for which the
MSE of the MLMC estimator 𝐿ℓ=0 Δ𝑍ℓ 𝜃 is bounded by 𝜀 2
with a total computation cost of
𝑂 𝜀 −2 ,
−2 𝑙𝑜𝑔𝜀
𝑂
𝜀
𝐶=
𝑂 𝜀
−2−
𝛾−𝛽
𝛼
2
,
𝛽>𝛾
,
𝛽=𝛾
𝛽<𝛾
Multilevel Monte Carlo Algorithm
• The theoretical results need translation into practical
settings.
• Out of simplicity we consider only the Lipschitz
continuous setting.
Simplifying Assumptions
• The constants 𝑐1 , 𝑐2 and 𝑐3 stated in the theorem are
crucial in deciding when to stop. However, in practice
they will not be known to us.
• If 𝐸 𝜇 𝜃 − 𝜇ℓ 𝜃 = 𝑐1 𝑘 −𝛼ℓ we can deduce that
𝐸 Δ𝑍ℓ 𝜃 ≈ 𝑘 − 1 𝐸 𝜇 𝜃 − 𝜇ℓ 𝜃 .
Simplifying Assumptions Continued
• Hence, we can use Δ𝑍ℓ 𝜃 as a pessimistic estimate
of the bias at level ℓ. Thus, we will continue adding
level until the following criterion is met
𝜀
Δ𝑍ℓ 𝜃 ≤ 𝑘 − 1
2
• However, due to its inherent variance we would
recommend using the following stopping criteria
1
𝜀
𝑀𝑎𝑥 Δ𝑍ℓ 𝜃 , Δ𝑍ℓ−1 𝜃
≤ 𝑘−1
𝑘
2
The algorithm
Black-Scholes
Black-Scholes continued
Conclusion
• Multilevel Monte Carlo provides an efficient
metamodeling scheme.
• We eliminated the necessity for increased smoothness
when dimension increase.
• Introduced a practical MLMC algorithm for
stochastic simulation metamodeling.
Questions?