E. Gobet - UQ of Stochastic Approximation Limits

Uncertainty Quantification of Stochastic Approximation Limits [email protected] Centre de Mathématiques Appliquées, Ecole Polytechnique

Joint work with S. Crepey, G. Fort and U. Stazhynski. Preprint available on HAL. In revision for SIAM-ASA Uncertainty Quantification. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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Agenda 1

2

Financial context: risk measure calculations

4

1.1

Risk calculations in finance . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Few definitions and notations . . . . . . . . . . . . . . . . . . . . .

9

1.3

Computations of (VaRα (V ), CVaRα (V )) in practice . . . . . . . .

13

Stochastic approximation algorithm

15

2.1

Usual general setting (without parametric dependency) . . . . . .

15

2.2

Parametric Stochastic Approximation . . . . . . . . . . . . . . . .

16

2.3

Design of the algorithm . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4

Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . .

24

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3

4

Numerical results

30

3.1

Design Parameterization of the USA Algorithm . . . . . . . . . .

30

3.2

Benchmark Problems . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.3

Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.4

Impact of the Increasing Dimension . . . . . . . . . . . . . . . . .

33

3.5

Impact of a (step size of SA) . . . . . . . . . . . . . . . . . . . . .

37

3.6

Role of p (number of samples) . . . . . . . . . . . . . . . . . . . .

38

3.7

An example in risk measure calculation (Black-Scholes Call Option) 39

Conclusion

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1

1.1 Risk calculations in finance

Financial context: risk measure calculations Fatter Tails

The abnormal frequency of extreme currency market events since 2008: Causes and solutions

1.1

Risk calculations in finance by Brent Donnelly HSBC Foreign Exchange [email protected] October 6, 2017

Risks/volatilities get higher and higher... While it is well known that currency returns are not normally distributed, tail events in FX have been abnormally frequent in recent years. Take a look at the first chart which shows the largest range of the day for the US dollar vs. G7 currencies from 1990 until the end of September 2017:

1. Asia Crisis, LTCM unwind, October 8, 1998 2. Global Fin. Crisis, October 2008 3. US stock market flash crash, May 6, 2010 4. US downgraded by Standard & Poors, August 9, 2011 5. SNB puts in the floor, Sep 6, 2011 6. SNB removes the floor, January 15, 2015 Chart by AM/FX, data from Bloomberg Certainly not anti-fragile! A daily range of 6% or more in the dollar was nearly unheard of before 2008; since then, we have seen eight different 6%+ ranges, including three events that produced daily ranges in excess of 10%. The challenge in diagnosing the multitude of tail events is that when we look at the triggers and causes of the events individually, it is hard to find common ground. The numbers on the chart correspond to the events listed in this table:

7. China reval, August 24, 2015

1

9. Flash crash, October 7, 2016

USD versus G7 moves larger than 6%. Source: HSBC FX, 2017.

8. UK votes to leave EU, June 24, 2016

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1.1 Risk calculations in finance

Usually VaR − CVaR depend on time horizon, risk level, and losses under consideration . . . 1.1.1

Usual VaR − CVaR

The main financial risk measures in practice are VaR and CVaR (as an expected shortfall) of level α ∈ (0, 1) Insurance: Solvency capital requirement (SCR) determined as the 99.5%-value at risk for Solvency II, and the 99%-expected shortfall for the Swiss Solvency Test, of the one-year loss and profit of the company Banking: Basel II Pillar II defines economic capital as the 99% value-at-risk of the one-year depletion of core equity tier I capital (CET1), the regulatory metric that represents the wealth of the shareholders within the bank. Moreover, the FRTB (2019-...) requires a shift from 99% value-at-risk to 97.5% expected shortfall as the reference risk measure in capital calculations. Market risk: risk computations over 10 trading days. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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1.1.2

1.1 Risk calculations in finance

Initial-Margin Variation

Ref: Basel Committee on Banking Supervision. Margin requirements for non-centrally-cleared derivatives, 2015. http://www.bis.org/bcbs/publ/d317.pdf X Nowadays, banks and financial institutions have to post collateral to a central counterparty (CCP = clearing house) in order to secure their positions. X Variation margin= collateral to cover a new contract at inception and daily changes X Initial margin [Basel 2015, p.11 3(d)]: IM protects the transacting parties from the potential future exposure that could arise from future changes in the mark-to-market value of the contract during the time it takes to close out and replace the position in the event that one or more counterparties default.

The amount of initial margin reflects the size

of the potential future exposure. [...]

It depends on a variety of factors,

and can change over time [...].

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1.1.3

1.1 Risk calculations in finance

XVA

Ref: [Armenti, Crépey 2017; Abbas-Turki, Crépey, Diallo 2017] Cost of funding initial margin: hZ MVA0 = λE

T

IMα t dt

i .

0

Cost of capital (for remunerating shareholders for their capital at risk): hZ T i KVA0 = E e−ht CVaRα t dt . 0

à Nested computations. 1.1.4

Stress-test

Ref: European Banking Authority (EBA), Draft Guidelines on the revised common procedures and methodologies for the supervisory review and evaluation process (SREP) and supervisory stress testing, EBA/CP/2017/18, October 2017. Objective: to define good practices for all banks (including model risk). Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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1.1.5

1.1 Risk calculations in finance

To summarise

X Risk metrics may require nested evaluations à simulation-based methods X Risks involve complex dependencies à necessity of coupling random variables X Relatively few data for these dependent and high dimensional model à statistical error is an issue X Larger and larger pressure from the regulation to account for model risk 

Accounting for errors in metric risks w.r.t. estimation and modelling is nowadays a "must-have"

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1.2

1.2 Few basic definitions

Few basic definitions

Definition (VaR, CVaR). X Loss V (scalar random variable), assumed to be integrable X VaR (Value-at-risk) is the left quantile at level α of V , i.e. VaRα (V ) = inf{x ∈ R : P(V ≤ x) > α} I

“Loss that, with a probability α, will not be exceeded.”

I

Typically α = 99%

I

Ref: [Artzner, Delbaen, Eber, Heath, Math Fin 2002; Follmer et Schied, Stochastic finance, 2002]

I

For continuous c.d.f., z = VaRα solves E [H(z, V )] = 0,

1 with H(z, v) = 1 − 1v≥z . 1−α

This is the zero of some expectation (à Robbins-Monro algorithm later, [Bardou, Frikha, Pages MCMA ’09]). Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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1.2 Few basic definitions

X CVaR (Conditional Value-at-risk) accounts for losses exceeding VaR. Depends on the full distribution in the right tail. Several similar definitions: I

Averaged VaR: 1 AVaR (V) = (1 − α) α

I

Z

1

VaRβ (V)dβ.

α

Convex optimization point-of-view [Rockafellar-Uryasev 00]     1 AVaRα (V ) = inf x + E (V − x)+ x 1−α and the inf is attained for x = VaRα (V ). [à AVaRα is sub-additivity, homogeneous, convex. . . whereas VaRα is not] 

Note that the first-order optimality condition is the same as for the α-quantile. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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I

1.2 Few basic definitions

Lemma (The inf may be attained for several x). Assume a continuous c.d.f. for V : the arginf solve x : P(V ≤ x) = α, and any such solution leads to the same value for   1 x+ E (V − x)+ . 1−α à Selecting any α-quantile provides the same AVaR (useful remark for later use of Stochastic Approximation algorithm).

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I

1.2 Few basic definitions

Expected Shortfall: ESα (V) = E [V | V ≥ VaRα (V)] .

I

Worst case expectation: WCEα (V) :=

sup

E [V | A] .

A:P(A)≥1−α

X In full generality on V , AVaRα (V) ≥ WCEα (V) ≥ ESα (V). X When the c.d.f. of V is continuous at VaRα (V ), then these three quantities coincide. X In the following, we simply write CVaRα (V ) and assume implicitly no atoms in the tails (continuous c.d.f.).

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1.3

1.3 Computations of (VaRα (V ), CVaRα (V )) in practice

Computations of (VaRα (V ), CVaRα (V )) in practice

a) Parametric method: fit a Gaussian distribution N (m, σ 2 ) to V and use an approximation ! 2 e−x /2 α α −1 √ (VaR (V ), CVaR (V )) ∼ m + σN (α), m + σ (1 − α) 2π x=N −1 (α) Simple but tightly related to Gaussian assumptions. b) Historical method: compute an empirical version of (VaRα (V ), CVaRα (V )) from historical data. Data driven method In practice, computed using too few data (because non-stationary data)

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1.3 Computations of (VaRα (V ), CVaRα (V )) in practice

c) Monte-Carlo method: fit a distribution µ to the data, and then perform a α Monte-Carlo approximation of (VaRα (V ), CVaR µ µ (V )) using samples under µ Approaches a) and c) are tightly related to a model. What if the model is incorrectly specified or estimated? B Our concerns: quantity the uncertainty of (VaRα (V ), CVaRα (V )) when the distribution of V depends on unknown/uncertain parameter V ∼ µ(θ, dv),

θ ∼ π(dθ)?

Bayesian point of view. B Our approach: since VaRα (V ) is the zero of some expectation (and CVaRα (V ) is some expectation at some critical parameter), we wish to design a Robbins-Monro type algorithm for this kind of problem.

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2 2.1

2.1 Usual general setting (without uncertain parameter)

Stochastic approximation algorithm Usual general setting (without uncertain parameter)

X For H : Rq × V → Rq , find the solution z ? ∈ Rq of E (H(z, V)) = 0.

X Strategy: use the iterative form of the Stochastic Approximation zk+1 = zk − γk+1 H(zk , Vk+1 ) where (Vk )k are i.i.d. Vk ∼ V . Under “good” hypotheses for (γk )k and H we indeed obtain lim zk = z? .

k→+∞

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2.2

2.2 Parametric Stochastic Approximation

Parametric Stochastic Approximation

X Variant: Given Θ ⊂ Rd , find z : Θ → Rq such that Eθ (H(z(θ), Vθ , θ)) = 0, where Eθ denotes expectation with respect to the random vector V θ ∼ µ(θ, dv). X Stochastic Approximation scheme for a fixed θ: θ φk+1 (θ) = φk (θ) − γk+1 H(φk (θ), Vk+1 , θ),

for an i.i.d. sequence {Vkθ , k ≥ 0} sampled from µ(θ, dv). X The value of z(θ) is given by z(θ) = lim φk (θ) k

a.s.

Crude method which does not allow to retrieve z(θ) for all θ ∈ Θ simultaneously. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.2 Parametric Stochastic Approximation

Outline of the Uncertainty Quantification methodology 1. Decomposition of z : Θ 7→ Rq on a L2 -basis w.r.t. π 2. Iterative computation of the coefficients (stochastic approximation): (a) which decreasing step, how many iterations? (b) how many coefficients? (c) stochastic approximation in increasing (or fixed large) dimensions? 3. Uncertainty quantification: (a) sample (θs : 1 ≤ s ≤ M ) (b) compute the empirical distribution of (ˆ z (θs ) : 1 ≤ s ≤ M ) or if (B0 = 1, B1 , . . . ) is an orthonormal basis in L2 (π) and zˆ(.) ≈

Pm

i=0

ui Bi (.)

(a) Eθ (z(θ)) = u0 Pm (b) Varθ (z(θ)) ≈ i=1 ui u> i (c) . . . Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.3 2.3.1

2.3 Design of the algorithm

Design of the algorithm Basis

X Let π(dθ) be a distribution on the set Θ ⊂ Rd X Let {θ 7→ Bi (θ), i ∈ N} be an orthonormal functional basis on Θ w.r.t. the scalar product Z hf ; gi = f (θ)g(θ)π(dθ). X We use the natural isomorphism between the sequences (ui , i ∈ N) ∈ l2 and P the functions ( i≥0 ui Bi , ui ∈ Rq ) ∈ L2 (π). X We approximate the function z(θ) using the first m + 1 elements of the basis: z(θ) ≈ zm (θ) =

m X

u?i Bi (θ).

i=0

2dθ √ 1 − θ2 on Θ = [−1, 1] then Example. If π(dθ) has the density π {Bi (θ), i ∈ N} is given by the Chebyshev polynomials of the second kind. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.3.2

2.3 Design of the algorithm

Naive algorithm

X Let us approximate the k-th term of the stochastic algorithm: θ φk+1 (θ) = φk (θ) − γk+1 H(φk (θ), Vk+1 , θ) 

k P k k k by φ (θ) ≈ i≤m ui Bi (θ) where ui = φ (θ); Bi (θ) .

X Naive algorithm: taking the scalar product of the stochastic algorithm with each of the basis function B0 (θ), ..., Bm (θ) we obtain  E      D P k θ k H u B (θ), V , θ ; B0 (θ) uk+1 u j j k+1 j≤m 0 0       ..  ..   ..     .  =  .  − γk+1   .     D   E P k θ uk+1 ukm H m j≤m uj Bj (θ), Vk+1 , θ ; Bm (θ) ⇒ The algorithm requires the simulation of V θ for all θ at each step, as well as the calculation of the scalar product. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.3.3

2.3 Design of the algorithm

Modified algorithm

X Note that *       + X k X k θ θ       E H uj Bj (θ), Vk+1 , θ ; Bi (θ) =E H uj Bj (θ), Vk+1 , θ Bi (θ) j≤m

j≤m

where the second expectation is w.r.t. joint distribution of the pair (V θ , θ) ∼ π(dθ)µ(θ, dv). X This gives us a new stochastic approximation procedure with the following s s k update rule: for an i.i.d. sample (Vk+1 , θk+1 )M s=1 ∼ π(dθ)µ(θ, dv) and compute       k+1   B0 (θs ) u0 uk0 k+1  Mk      X X 1 ..  ..   ..   s s s  H ukj Bj (θk+1 ), Vk+1 , θk+1  . = .  − γk+1   . Mk s=1       j≤m s uk+1 ukm Bm (θk+1 ) m So far, the number m of coefficients is fixed. . . Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.3.4

2.3 Design of the algorithm

Number of coefficients m → +∞

X We want to increase the number of coefficients u?i under approximation as k → +∞. X Let mk be the number of "active" coefficient at the iteration k, typically mk = k β , β < 1. X Now we are working in an infinite-dimensional space l2 . X In order to establish the convergence we apply projection at each iteration. X Assume that we know a convex subset A of l2 such that u? ∈ A.

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2.3.5

2.3 Design of the algorithm

The main algorithm

USA Algorithm (Uncertainty for Stochastic Approximation) Input: Sequences {γk , k ≥ 1}, {mk , k ≥ 1}, {Mk , k ≥ 1}, K ∈ N, {u0i , i = 0, . . . , m0 }, a convex set A ⊆ l2 . for k = 0, . . . , K − 1 do for s = 1 . . . , Mk+1 do s s Sample independently (θk+1 , Vk+1 ) under the distribution π(dθ)µ(θ, dv) end for For all i > mk+1 define uki = 0 for i = 0, . . . , mk+1 do P  P M m −1 k+1 k k s s s s u ˆk+1 = uki − γk+1 Mk+1 i s=1 H j=0 uj Bj (θk+1 ), Vk+1 , θk+1 Bi (θk+1 ) end for uk+1 = ΠA (ˆ uk+1 ) end for return The vector {uK i , i = 0, . . . , mK }. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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2.3.6

2.3 Design of the algorithm

Literature background

˜ X SA in Hilbert space: in the case H(z, V ) = H(z) + V , see [Walk, ZWVG ’79; Yin, Zhu, JMA ’90; Berman, Shwartz, SPA ’89]

X Increasing the dimension aka Sieve method [Golstein, JTP’88; Kiefer, Wolfowizz, AMS ’52; Nixdorf JMA ’84; Yin, SAA’ 92 ]. These references do not account for the uncertainty, and for the real computational cost of Hilbert space approximations. X Projection in Hilbert spaces: [Chen, White, SNDE ’02]. As a difference, our scalar product is not explicit and our assumptions are weaker.

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2.4

2.4 Convergence analysis

Convergence analysis

2.4.1

Assumptions     Z   X ? ? 2 ? Let T := u ∈ l ; H ui Bi (θ), v, θ µ(θ, dv) = 0, π-a.s. .   V i≥0

C1. The set T ? is compact and non-empty. C2. {Mk , k ≥ 1} and {mk , k ≥ 1} are deterministic sequences of positive integers; {γk , k ≥ 1} is a deterministic sequence of positive real numbers such that, for some κ > 0, X X X Qm X 1+κ k 2 γk = +∞, γk < +∞, γk < +∞, γk1−κ qmk < +∞, Mk k≥1

k≥1

k≥1

k≥1

where the sequences {qm , m ∈ N} and {Qm , m ∈ N} are defined by X X ? 2 |Bi (θ)|2 . qm := sup |ui | , Qm := sup u? ∈T ?

i>m

θ∈Θ

i≤m

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2.4 Convergence analysis

C3. For any z ∈ Rq , Z |H(z, v, θ)| µ(θ, dv)π(dθ) < ∞; Θ×V

For any z ∈ Rq and θ ∈ Θ, Z h(z, θ) =

H(z, v, θ) µ(θ, dv) V

exists; For any φ ∈ L2π , the mapping h(φ(·), ·) : θ 7→ h(φ(θ), θ) is in L2π ; The mapping φ 7→ h(φ(·), ·) from L2π into itself is continuous. C4. For π-almost every θ, for any zθ , zθ? ∈ Rq such that h(zθ , θ) 6= 0 and h(zθ? , θ) = 0, (zθ − zθ? ) · h(zθ , θ) > 0.

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2.4 Convergence analysis

C5. a) There exists a constant CH such that, for any z ∈ Rq , Z sup |H(z, v, θ)|2 µ(θ, dv) ≤ CH (1 + |z|2 ). θ∈Θ

L2π

V

b) The map from into R defined by φ 7→ V×Θ |H(φ(θ), v, θ)|2 π(dθ)µ(θ, dv) is bounded, i.e. it maps bounded sets into bounded sets. R

Note that C5-b implies that φ 7→ h(φ(·), ·) is a bounded map from L2π into itself. C6. For any B > 0, there exists a constant CB > 0 such that, for any (φ, φ? ) ∈ L2π × Is(T ? ) with kφ − φ? kπ ≤ B, Z

2 ? ¯

(φ − φ ) (θ) · h(φ(θ), θ) π(dθ) ≥ CB min φ − φ π . ?) ¯ φ∈Is(T

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2.4.2

2.4 Convergence analysis

Projection sets

We address the convergence of the USA algorithm for three possible choices regarding the projection set A (which always includes T ? ). Case 1) A := l2 . No projection needed. Case 2) A is a closed ball of l2 containing T ? . A is bounded but projection easy given by   B u 7→ min 1, u. kukl2 Case 3) A is a closed convex set of l2 containing T ? , with compact intersections to closed balls of l2 . A may be unbounded but projection may be difficult.

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2.4 Convergence analysis

Example (for Case 3). Given a positive sequence {an , n ∈ N} such that P 2 i≥0 ai < ∞ and an increasing sequence of non-negative integers {dn , n ∈ N}, define the closed convex set A:     X A := u ∈ l2 : |ui |2 ≤ a2n ∀n ∈ N .   dn ≤i 0. X The orthogonal projection on A consists in projecting (udn , . . . , udn+1 −1 ) on the ball of radius an for all n ∈ N.

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2.4.3

2.4 Convergence analysis

Main Result

Theorem. Assume C1 to C4 and C5-a if A is unbounded or C5-b if A is bounded. Let there be given i.i.d. random variables {(θks , Vks ), 1 ≤ s ≤ Mk , k ≥ 1} with distribution π(dθ)µ(θ, dv). Let uK and φK be the outputs of the USA Algorithm.

k ? ? ?

Stability. For any φ ∈ Is(T ), limk→+∞ φ − φ π exists, is finite a.s., and we have h i

2 sup E φk − φ? < +∞. k≥0

π

Convergence. In addition, in Case 3, and in Cases 1 and 2 under the additional assumption C6, there exists a random variable φ∞ taking values in Is(T ? ) such that h

k

p i ∞ k ∞ lim φ − φ π = 0 a.s. and, for any p ∈ (0, 2), lim E φ − φ π = 0. k→∞

k→∞

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3 3.1

3.1 Design Parameterization of the USA Algorithm

Numerical results Design Parameterization of the USA Algorithm

We consider sequences of parameters the form γk = k −a ,

mk = bk b c + 1,

Mk = bk p c + 1,

for a, p ≥ 0 and b > 0, and we discuss how to choose these constants assuming that

−δ ∆ qm = O m , Qm = O m . C2 is satisfied if 0 < a ≤ 1,

2 − δb < 2a,

b∆ + 1 < 2a + p.

In the case p = 0, the colored area is the admissible set of points (a, b). From left to right: (δ, ∆) = (2, 1), (0.5, 1), (4, 3), and (0.5, 5). Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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3.2

3.2 Benchmark Problems

Benchmark Problems

Here 1 1[−π,π] dθ, µ(θ, dv) = N (0, θ2 ), 2π  cos(v) ? H1 (z, v, θ) := (z − φ (θ)) 1 + sin(z − φ? (θ)) + v, 2   cos(v) H2 (z, v, θ) := (z − φ? (θ)) 1 + sin(z − φ? (θ)) , 2 4 1 ? 2 φ (θ) := + exp(sin(θ)) − cosh(sin(θ) ) (1 + sin(2θ)). 5 4

Θ = [−π, π], and

π(dθ) =

We have q = 1 and Is(T ? ) = {φ? }. For any z ∈ R, θ ∈ Θ we have Z 1 |Hi (z, v, θ)|2 µ(θ, dv) ≤ 8|z − φ? (θ)|2 + 2θ2 , (z − φ? (θ)) · hi (z, θ) ≥ (z − φ? (θ))2 , 2 V R where hi (θ) = V Hi (z, v, θ)µ(θ, dv) à assumptions C3, C4, C5, and C6 are satisfied (we can take A = l2 , no projection needed). Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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3.3 Performance Criteria

Orthonormal basis {Bi , i ∈ N}: normalized trigonometric basis, thus Qm = O(m) à ∆ = 1. Truncation error: φ? is piecewise C 1 à δ = 2 (at least).

3.3

Performance Criteria

We test algorithms X with increasing mk and with fixed m X different choices of (a, b) with p = 0 X output statistics computed over 50 independent runs of the algorithm r h i 2 X After K iterations we estimate the RMSE E = E kuK − u? kl2 with 2 2 E 2 = Esa + Etr ,

2 Esa

"m # K X ? 2 =E (uK , i − ui )

2 Etr =

i=0

+∞ X

(u?i )2 .

i=mK +1

X u?i are pre-calculated by high-precision numerical integration. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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3.4

3.4 Impact of the Increasing Dimension (mk )k

Impact of the Increasing Dimension (mk )k

Heuristics for tuning parameters: X (δ, ∆) = (2, 1) X a = 0.875 (see the previous graph) X For Model 1, we expect Esa = O(mk γk ) = O(k b−a ) and Etr = O(k −bδ ) à we choose b such that b − a = −bδ (i.e. b ≈ 0.3) X For Model 2 (zero variance at the limit), we simply take b = 0.45

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3.4 Impact of the Increasing Dimension (mk )k

The total error E as a function of the number of iterations, for different choices of the sequence {mk , k ∈ N}: mk increasing (solid line) and: [left] for Model 1 with mk = m = 8, 12, 16, 20 (other lines); [right] for Model 2 with mk = m = 10, 20, 30, 40, 50 (other lines). X Seemingly fixing the dimension is never the better solution X For Model 2, the impact is even stronger (variance reduction effect) Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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E. Gobet - UQ of Stochastic Approximation Limits

3.4 Impact of the Increasing Dimension (mk )k

For Model 2, φ? and φK are displayed in respective solid line and dashed lines, as a function of θ ∈ [−π, π]. At the top, increasing {mk , k ∈ N}. At the bottom, constant mk = 30. From left to right, K ∈ {128, 256, 512}. 

The progressive dimension growth plays a key role in the USA algorithm performance. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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3.4 Impact of the Increasing Dimension (mk )k

Model 2: [left] in the case mk → ∞ (solid line) and mk = m = 3 (dotted line),  P 1/2 3 ? 2 the SA error E i=0 (uK as a function of the number of iterations i − ui ) K; [right] in the case mk = m = 3, the truncation error Etr (dashed line) and the total error E (dash-dot line) displayed as a function of K. 

The algorithm with fixed m does typically not converge to the first (m + 1) coefficients of the decomposition of φ? . In the USA algorithm, having mk → +∞ is a key point to ensure convergence to the right limit.

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3.5 Impact of a (step size of SA)

E. Gobet - UQ of Stochastic Approximation Limits

3.5

Impact of a (step size of SA)

Here p = 0, with b = 0.3 for Model 1 and b = 0.45 for Model 2.

The total error E as a function of the number of iterations, for different values of a in {0.75, 0.80, 0.85, 0.9, 0.95, 1.0}: [left] Model 1 and [right] Model 2. 1/2

As known for standard SA for which RM SE = O(γk ) ∼ k −a/2 , the convergence rate is better for larger values of a. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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3.6

3.6 Role of p (number of samples)

Role of p (number of samples)

Model 1, total error E of the USA algorithm for different values of p ∈ {0, 0.1, 0.2, 0.3} as a function of the number of iterations [left] and of the total number of Monte Carlo draws [right]. Here a = 0.875 and b = 0.3(p + 1). No real difference in terms of convergence speed when the latter is assessed with respect to the total number of simulations. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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E. Gobet - UQ of Stochastic Approximation 3.7 Limits Another example in risk measure calculation (Black-Scholes Call Option)

3.7

Another example in risk measure calculation (Black-Scholes Call Option)

X Call option:   + Ct = E (ST − K) | St 2

where K is the strike, T is the maturity, St = S0 e

− σ2 t+σWt

.

X Recall the close-form formula of Ct : Ct = St Φ(d1 (St , σ)) − KΦ(d2 (St , σ)) where d1 (s, σ) :=

√1 σ T −t



s log( K )

+

2

σ 2

 (T − t) ,

√

d2 (s, σ) := d1 (s, σ) − σ T − t.

Uncertainty of (VaRα (C), CVaR(C)) w.r.t. θ = (S0 , σ), S0 ∈ [50, 250], σ ∈ [0.2, 0.6]. X We take α = 0.95, K = 100, t = 1, T = 5. Workshop Optimization and Learning - IMT, Toulouse - September 10th/14th, 2018

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E. Gobet - UQ of Stochastic Approximation 3.7 Limits Another example in risk measure calculation (Black-Scholes Call Option)

Figure 1: L2 -error convergence of the PCE coefficients.

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4

4 Conclusion

Conclusion

X Design of Algorithm for computing the Polynomial Chaos Expansion of SA limits X Relevant for Uncertainty Quantification of (VaRα , CVaRα ) w.r.t. incertain model parameters X SA in Hilbert space with projections. Weaker conditions than previously stated in the literature. X Rates of convergence available: see the Phd thesis of Uladzislau Stazhinsky

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4 Conclusion

Advertisement X From September 2018, starting Chaire on Stress Tests. X Research program between Ecole Polytechnique and BNP Paribas X Several lines of research: I

Rare-event simulation and meta-modeling using machine-learning

I

Uncertainty quantification of risk metrics

I

Modeling and estimation of multidimensional dependencies

X Several PostDoc positions and one Assistant Professor position. X If you are interested, contact me at [email protected]

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