Many machine learning and optimization algorithms solve hidden root-finding problems through the magic of stochastic approximation (SA). Unfortunately, these algorithms are slow to converge: the optimal convergence rate for the mean squared error (MSE) is of order O(n⁻¹) at iteration n.
Far faster convergence rates are possible by reconsidering the design of exploration signals used in these algorithms. In this lecture the focus is on quasi-stochastic approximation (QSA), in which a multi-dimensional clock process defines exploration. It is found that algorithms can be designed to achieve a MSE convergence rate approaching O(n⁻⁴).
Although the framework is entirely deterministic, this new theory leans heavily on concepts from the theory of Markov processes. Most critical is Poisson’s equation to transform the QSA equations into a mean flow with additive “noise” with attractive properties. Existence of solutions to Poisson’s equation is based on Baker’s Theorem from number theory---to the best of our knowledge, this is the first time this theorem has been applied to any topic in engineering!
The theory is illustrated with applications to gradient free optimization.
Joint research with Caio Lauand, current graduate student at UF.
References
[1] C. Kalil Lauand and S. Meyn. Approaching quartic convergence rates for quasi-stochastic approximation with application to gradient-free optimization. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems, volume 35, pages 15743–15756. Curran Associates, Inc., 2022.
[2] C. K. Lauand and S. Meyn. Quasi-stochastic approximation: Design principles with applications to extremum seeking control. IEEE Control Systems Magazine, 43(5):111–136, Oct 2023.
[3] C. K. Lauand and S. Meyn. The curse of memory in stochastic approximation. In Proc. IEEE Conference on Decision and Control, pages 7803–7809, 2023. Extended version. arXiv 2309.02944, 2023.
Quasi-Stochastic Approximation: Algorithm Design Principles with Applications to Machine Learning and Optimization.
1. Quasi-Stochastic Approximation
Algorithm Design Principles with Applications to Machine Learning and Optimization
Caio Kalil Lauand (Joint work with Sean Meyn)
Department of Electrical and Computer Engineering
University of Florida
Support from ARO award W911NF2010055 and NSF awards EPCN 1935389, CCF 2306023 is gratefully acknowledged.
2. Agenda
1 Root-Finding Under Presence of Noise
2 Quasi-Stochastic Approximation
3 A Return to Extremum Seeking Control
4 Conclusions
5 Appendices
6 References
1 / 40
3. Root-Finding Under Presence of Noise
Optimization
We have an objective Γ : Rd → R+ that we wish to minimize.
First order condition for optimality of θ∗ ∈ Rd:
∇Γ(θ∗
) = 0
2 / 40
4. Root-Finding Under Presence of Noise
Optimization
We have an objective Γ : Rd → R+ that we wish to minimize.
First order condition for optimality of θ∗ ∈ Rd:
s
f(θ∗
)
def
= −∇Γ(θ∗
) = 0
Our first root-finding problem
2 / 40
5. Root-Finding Under Presence of Noise
Optimization
We have an objective Γ : Rd → R+ that we wish to minimize.
First order condition for optimality of θ∗ ∈ Rd:
s
f(θ∗
)
def
= −∇Γ(θ∗
) = 0
Our first root-finding problem
Challenges we will address:
• In complex systems we might not have access to the gradient but only a
noisy measurement:
e
∇Γ(θn) = ∇Γ(θn) + Wn
first order optimization, stochastic gradient descent...
2 / 40
6. Root-Finding Under Presence of Noise
Optimization
We have an objective Γ : Rd → R+ that we wish to minimize.
First order condition for optimality of θ∗ ∈ Rd:
s
f(θ∗
)
def
= −∇Γ(θ∗
) = 0
Our first root-finding problem
Challenges we will address:
• In complex systems we might not have access to the gradient but only a
noisy measurement:
e
∇Γ(θn) = ∇Γ(θn) + Wn
first order optimization, stochastic gradient descent...
• We might only have noisy measurements of the objective:
Yn = Γ(θn) + Wn
zeroth order optimization, gradient free optimization, extremum seeking control...
2 / 40
7. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
3 / 40
8. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
For any fixed θ and a small ε > 0, let
f(θ, ξ) = −
1
ε
ξΓ(θ + εξ)
where ξ is zero-mean.
=⇒ s
f(θ) := E[f(θ, ξ)] approximates −∇Γ (θ)
3 / 40
9. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
f(θ, ξ) = −1
ε ξΓ(θ + εξ) with ξ is zero-mean.
3 / 40
10. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
f(θ, ξ) = −1
ε ξΓ(θ + εξ) with ξ is zero-mean.
• A bit of Taylor series...
f(θ, ξ) = −
1
ε
ξΓ(θ) − ξξ⊺
∇Γ(θ) + O(ε)
3 / 40
11. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
f(θ, ξ) = −1
ε ξΓ(θ + εξ) with ξ is zero-mean.
• A bit of Taylor series...
f(θ, ξ) = −
1
ε
ξΓ(θ) − ξξ⊺
∇Γ(θ) + O(ε)
• Taking expectations of both sides yields
E[f(θ, ξ)] = −Cov(ξ)∇Γ(θ) + O(ε)
3 / 40
12. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
f(θ, ξ) = −1
ε ξΓ(θ + εξ) with ξ is zero-mean.
• A bit of Taylor series...
f(θ, ξ) = −
1
ε
ξΓ(θ) − ξξ⊺
∇Γ(θ) + O(ε)
• Taking expectations of both sides yields
s
f(θ) := E[f(θ, ξ)] = −Cov(ξ)∇Γ(θ) + O(ε)
3 / 40
13. Root-Finding Under Presence of Noise
Gradient-Free Optimization
How would we estimate θopt
∈ arg min
θ
Γ if we have access to Γ for any θ?
1SPSA: f(θ, ξ) = −1
ε ξΓ(θ + εξ) with ξ is zero-mean.
• A bit of Taylor series...
f(θ, ξ) = −
1
ε
ξΓ(θ) − ξξ⊺
∇Γ(θ) + O(ε)
• Taking expectations of both sides yields
s
f(θ) := E[f(θ, ξ)] = −Cov(ξ)∇Γ(θ) + O(ε)
s
f(θ∗
) = 0 , s
f(θopt
) = O(ε)
see Spall [67] and Ariyur & Krstić [3].
3 / 40
14. Root-Finding Under Presence of Noise
How do we implement this algorithm?
• The vector θ∗ can be estimated recursively through
θn+1 = θn − α
1
ε
ξn+1Γ(θn + εξn+1)
4 / 40
15. Root-Finding Under Presence of Noise
How do we implement this algorithm?
• The vector θ∗ can be estimated recursively through
θn+1 = θn − α
1
ε
ξn+1Γ(θn + εξn+1)
⋄ α > 0 is a constant independent of n.
4 / 40
16. Root-Finding Under Presence of Noise
How do we implement this algorithm?
• The vector θ∗ can be estimated recursively through
θn+1 = θn − α
1
ε
ξn+1Γ(θn + εξn+1)
⋄ α > 0 is a constant independent of n.
⋄ {ξn} is a zero-mean sequence.
Spall takes this to be i.i.d., entries ±1
4 / 40
17. Root-Finding Under Presence of Noise
How do we implement this algorithm?
• The vector θ∗ can be estimated recursively through
θn+1 = θn − α
1
ε
ξn+1Γ(θn + εξn+1)
• Deterministic counterpart, as the ODE
d
dt Θt = −α
1
ε
ξtΓ(Θt + εξt)
4 / 40
18. Root-Finding Under Presence of Noise
How do we implement this algorithm?
• The vector θ∗ can be estimated recursively through
θn+1 = θn − α
1
ε
ξn+1Γ(θn + εξn+1)
• Deterministic counterpart, as the ODE
ESC-0: d
dt Θt = −α
1
ε
ξtΓ(Θt + εξt)
This is the simplest Extremum Seeking Control (ESC) ODE
4 / 40
19. Root-Finding Under Presence of Noise
Extremum Seeking Control
• Being born in the 1920s, Extremum seeking control (ESC) is said to be
the oldest approach to gradient-free optimization.
• A typical architecture for ESC for optimization is illustrated below:
5 / 40
20. Root-Finding Under Presence of Noise
Extremum Seeking Control
ESC-0
• The high-pass (HP) filter is removed entirely:
ξ̌t = ξt
Y̌n
t = Yn
t =
1
ε
Γ(Θt + εξt)
5 / 40
21. Root-Finding Under Presence of Noise
Extremum Seeking Control
ESC-0
• The low-pass (LP) filter is an integrator:
d
dt Θt = −αM e
∇tΓ = −αξt
1
ε
Γ(Θt + εξt)
5 / 40
23. Quasi-Stochastic Approximation
Zooming Out
• In quasi-stochastic approximation, ξ is a smooth deterministic process,
θn+1 = θn + αn+1f(θn, ξn+1)
The probing signal ξ is typically chosen as: ξt = G(Φt) where Φ ∈ CK
with entries
Φi
t = exp(2πj[ωit + ϕi])
and {ωi} distinct.
6 / 40
24. Quasi-Stochastic Approximation
Zooming Out
• In quasi-stochastic approximation, ξ is a smooth deterministic process,
θn+1 = θn + αn+1f(θn, ξn+1)
The probing signal ξ is typically chosen as: ξt = G(Φt) where Φ ∈ CK
with entries
Φi
t = exp(2πj[ωit + ϕi])
and {ωi} distinct.
• Expressed as ODEs for ease of analysis,
QSA ODE: d
dt Θt = atf(Θt, ξt)
Common choices for {at} include:
⋄ Vanishing gain: at = (t + 1)−ρ with ρ ∈ (1/2, 1)
⋄ Constant gain: at ≡ α > 0 for all t
6 / 40
26. Quasi-Stochastic Approximation
Quasi-Stochastic Approximation d
dt Θt = atf(Θt, ξt)
• Algorithm design and analysis are based upon another ODE,
Mean Flow: d
dt ϑt = s
f(ϑt)
s
f(θ) := lim
T→∞
1
T
Z T
0
f(θ, ξt) dt
If stable: ϑt → θ∗
and s
f(ϑt) → s
f(θ∗
) = 0
7 / 40
27. Quasi-Stochastic Approximation
Quasi-Stochastic Approximation d
dt Θt = atf(Θt, ξt)
• Algorithm design and analysis are based upon another ODE,
Mean Flow: d
dt ϑt = s
f(ϑt)
s
f(θ) := lim
T→∞
1
T
Z T
0
f(θ, ξt) dt
If stable: ϑt → θ∗
and s
f(ϑt) → s
f(θ∗
) = 0
• {Θt} couples with solutions of the mean flow under general conditions.
7 / 40
28. Quasi-Stochastic Approximation
Quasi-Stochastic Approximation d
dt Θt = atf(Θt, ξt)
• Algorithm design and analysis are based upon another ODE,
Mean Flow: d
dt ϑt = s
f(ϑt)
s
f(θ) := lim
T→∞
1
T
Z T
0
f(θ, ξt) dt
If stable: ϑt → θ∗
and s
f(ϑt) → s
f(θ∗
) = 0
• {Θt} couples with solutions of the mean flow under general conditions.
Caveat: Global Lipschitz continuity of f is crucial. Always assumed.
7 / 40
29. Quasi-Stochastic Approximation
Quasi-Stochastic Approximation d
dt Θt = atf(Θt, ξt)
• Algorithm design and analysis are based upon another ODE,
Mean Flow: d
dt ϑt = s
f(ϑt)
s
f(θ) := lim
T→∞
1
T
Z T
0
f(θ, ξt) dt
If stable: ϑt → θ∗
and s
f(ϑt) → s
f(θ∗
) = 0
• {Θt} couples with solutions of the mean flow under general conditions.
Caveat: Global Lipschitz continuity of f is crucial. Always assumed.
• Pertubative mean flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
7 / 40
30. Quasi-Stochastic Approximation
Quasi-Stochastic Approximation d
dt Θt = atf(Θt, ξt)
• Algorithm design and analysis are based upon another ODE,
Mean Flow: d
dt ϑt = s
f(ϑt)
s
f(θ) := lim
T→∞
1
T
Z T
0
f(θ, ξt) dt
If stable: ϑt → θ∗
and s
f(ϑt) → s
f(θ∗
) = 0
• {Θt} couples with solutions of the mean flow under general conditions.
Caveat: Global Lipschitz continuity of f is crucial. Always assumed.
• Pertubative mean flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
Can we do any better?
7 / 40
31. Quasi-Stochastic Approximation
Métivier and Priouret To The Rescue!
• Pertubative mean (p-mean) flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
• Representation for e
Ξ based on solutions to Poisson’s equation.
First instance, solution ˆ
f with forcing function f:
d
dt
ˆ
f(θ, Φt) = −[f(θ, ξt) − s
f(θ)] , θ ∈ Rd
8 / 40
32. Quasi-Stochastic Approximation
Métivier and Priouret To The Rescue!
• Pertubative mean (p-mean) flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
• Representation for e
Ξ based on solutions to Poisson’s equation.
First instance, solution ˆ
f with forcing function f:
d
dt
ˆ
f(θ, Φt) = −[f(θ, ξt) − s
f(θ)] , θ ∈ Rd
=⇒ d
dt
ˆ
f(Θt, Φt) = −e
Ξt + ∂θ
ˆ
f(Θt, Φt) · d
dt Θt
8 / 40
33. Quasi-Stochastic Approximation
Métivier and Priouret To The Rescue!
• Pertubative mean (p-mean) flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
• Representation for e
Ξ based on solutions to Poisson’s equation.
First instance, solution ˆ
f with forcing function f:
d
dt
ˆ
f(θ, Φt) = −[f(θ, ξt) − s
f(θ)] , θ ∈ Rd
=⇒ d
dt
ˆ
f(Θt, Φt) = −e
Ξt + ∂θ
ˆ
f(Θt, Φt)[atf(Θt, ξt)]
e
Ξt = zero mean + small
8 / 40
34. Quasi-Stochastic Approximation
Métivier and Priouret To The Rescue!
• Pertubative mean (p-mean) flow #1:
d
dt Θt = at[ s
f(Θt) + e
Ξt] , e
Ξt := f(Θt, ξt) − s
f(Θt)
• Representation for e
Ξ based on solutions to Poisson’s equation.
First instance, solution ˆ
f with forcing function f:
d
dt
ˆ
f(θ, Φt) = −[f(θ, ξt) − s
f(θ)] , θ ∈ Rd
=⇒ d
dt
ˆ
f(Θt, Φt) = −e
Ξt + ∂θ
ˆ
f(Θt, Φt)[atf(Θt, ξt)]
e
Ξt = zero mean + small
• Borrowed from the stochastic approximation literature:
disturbance decomposition introduced by Métivier and Priouret.
8 / 40
35. Quasi-Stochastic Approximation
Perturbative Mean Flow
The perturbative mean (p-mean) flow representation
d
dt Θt = at[ s
f(Θt) + e
Ξt]
e
Ξt = −at
s
Υ(Θt) +
2
X
i=0
a2−i
t
di
dti
Wi
t
where {s
Υt, Wi
t : i = 0, 1, 2} are smooth deterministic functions of (Θt, Φt)
admitting representations in terms of solutions to Poisson’s equation.
• Opens doors for analysis: transient bounds and filter design.
9 / 40
36. Quasi-Stochastic Approximation
Perturbative Mean Flow
The perturbative mean (p-mean) flow representation
d
dt Θt = at[ s
f(Θt) + e
Ξt]
e
Ξt = −at
s
Υ(Θt) +
2
X
i=0
a2−i
t
di
dti
Wi
t
where {s
Υt, Wi
t : i = 0, 1, 2} are smooth deterministic functions of (Θt, Φt)
admitting representations in terms of solutions to Poisson’s equation.
• Opens doors for analysis: transient bounds and filter design.
What is s
Υ? It appears with multiplicative noise:
s
Υ(θ) := − lim
T→∞
1
T
Z T
0
∂θ
ˆ
f(θ, Φt)f(θ, ξt) dt
9 / 40
37. Quasi-Stochastic Approximation
Convergence and Acceleration d
dt Θt = atf(Θt, ξt)
• When at = (1 + t)−ρ with ρ ∈ (1/2, 1),
Θt = θ∗
+ at[A∗
]−1 s
Υ∗
+ nicet
o
⇒ ∥Θt − θ∗
∥2
= O(a2
t )
where s
Υ∗ = s
Υ(θ∗) and A∗ = ∂θ
s
f(θ∗).
10 / 40
38. Quasi-Stochastic Approximation
Convergence and Acceleration d
dt Θt = atf(Θt, ξt)
• When at = (1 + t)−ρ with ρ ∈ (1/2, 1),
Θt = θ∗
+ at[A∗
]−1 s
Υ∗
+ nicet
o
⇒ ∥Θt − θ∗
∥2
= O(a2
t )
where s
Υ∗ = s
Υ(θ∗) and A∗ = ∂θ
s
f(θ∗).
• Convergence is accelerated through Polyak-Ruppert (PR) averaging
ΘPR
T =
1
T − δT
Z T
δT
Θt dt , δ ∈ (0, 1)
10 / 40
39. Quasi-Stochastic Approximation
Convergence and Acceleration d
dt Θt = atf(Θt, ξt)
• When at = (1 + t)−ρ with ρ ∈ (1/2, 1),
Θt = θ∗
+ at[A∗
]−1 s
Υ∗
+ nicet
o
⇒ ∥Θt − θ∗
∥2
= O(a2
t )
where s
Υ∗ = s
Υ(θ∗) and A∗ = ∂θ
s
f(θ∗).
• Convergence is accelerated through Polyak-Ruppert (PR) averaging
ΘPR
T =
1
T − δT
Z T
δT
Θt dt , δ ∈ (0, 1)
• Extremely fast rates are obtained:
ΘPR
T = θ∗
+ O(aT ∥s
Υ∗
∥) + O(a2
T ) ⇒ ∥ΘPR
T − θ∗
∥2
= O(a4
T )
| {z }
If s
Υ∗=0
10 / 40
40. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
11 / 40
41. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
Cleverness # 1: existence of solutions to Poisson’s equation.
• Solutions can be represented as sums of integrals
Z t
0
exp(2πj[ω◦
t + ϕ◦
]) dt
ω◦ = n1ω1 + · · · nKωK.
=⇒ Require bounds on 1/ω◦
11 / 40
42. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
Cleverness # 1: existence of solutions to Poisson’s equation.
• Solutions can be represented as sums of integrals
Z t
0
exp(2πj[ω◦
t + ϕ◦
]) dt
ω◦ = n1ω1 + · · · nKωK.
=⇒ Require bounds on 1/ω◦
Great lower bounds on |ω◦| from Baker’s Theorem.
11 / 40
43. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
Cleverness # 1: existence of solutions to Poisson’s equation.
• Solutions can be represented as sums of integrals
=⇒ Require bounds on 1/ω◦
Great lower bounds on |ω◦| from Baker’s Theorem.
Cleverness # 2: ĝ ⊥ h for smooth functions g, h of the probing signal.
11 / 40
44. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
Cleverness # 1: existence of solutions to Poisson’s equation.
• Solutions can be represented as sums of integrals
=⇒ Require bounds on 1/ω◦
Great lower bounds on |ω◦| from Baker’s Theorem.
Cleverness # 2: ĝ ⊥ h for smooth functions g, h of the probing signal.
s
Υi(θ) =
X
j
⟨ĝi,j, hj⟩, with g = ∂θf and h = f.
11 / 40
45. Quasi-Stochastic Approximation
Killing s
Υ∗
Φi
t = exp(2πj[ωit + ϕi])
Clever Probing design
⋄ Design ξ so that ξt = G(Φt) with G analytic and choose frequencies
{ω1 , . . . , ωK} satisfying,
ωi = log(ai/bi) > 0 , {ai, bi} positive integers.
Cleverness # 1: existence of solutions to Poisson’s equation.
• Solutions can be represented as sums of integrals
=⇒ Require bounds on 1/ω◦
Great lower bounds on |ω◦| from Baker’s Theorem.
Cleverness # 2: ĝ ⊥ h for smooth functions g, h of the probing signal.
s
Υi(θ) =
X
j
⟨ĝi,j, hj⟩, with g = ∂θf and h = f.
= 0
11 / 40
47. Quasi-Stochastic Approximation
Fixed Gain Algorithms for QSA
• The QSA ODE with fixed gain is old news! (recall the averaging principle)
d
dt Θt = αf(Θt, ξt) , α > 0
see Khalil [24].
13 / 40
48. Quasi-Stochastic Approximation
Fixed Gain Algorithms for QSA
• The QSA ODE with fixed gain is old news! (recall the averaging principle)
d
dt Θt = αf(Θt, ξt) , α > 0
see Khalil [24].
• Motivation is tracking: { s
ft} ⇒ {θ∗
t }
13 / 40
49. Quasi-Stochastic Approximation
Fixed Gain Algorithms for QSA
• The QSA ODE with fixed gain is old news! (recall the averaging principle)
d
dt Θt = αf(Θt, ξt) , α > 0
see Khalil [24].
• Motivation is tracking: { s
ft} ⇒ {θ∗
t }
• Without averaging, MSE is
lim sup
t→∞
∥Θt − θ∗
∥2
= O(α2
)
13 / 40
50. Quasi-Stochastic Approximation
Fixed Gain Algorithms for QSA
• The QSA ODE with fixed gain is old news! (recall the averaging principle)
d
dt Θt = αf(Θt, ξt) , α > 0
see Khalil [24].
• Motivation is tracking: { s
ft} ⇒ {θ∗
t }
• Without averaging, MSE is
lim sup
t→∞
∥Θt − θ∗
∥2
= O(α2
)
• A p-mean flow representation inspires a low pass filter to obtain
lim sup
t→∞
∥ΘF
t − θ∗
∥2
= O(α4
)
13 / 40
53. Quasi-Stochastic Approximation
Filtering d
dt Θt = αf(Θt, ξt)
Second order filter is required
d2
dt2
ΘF
t + 2γζ
d
dt
ΘF
t + γ2
ΘF
t = γ2
Θt
with γ = O(α) and ζ ∈ (0, 1).
Obtain from p-mean flow
lim sup
t→∞
∥ΘF
t − θ∗
∥2
= O(α2
∥s
Υ∗
∥) + O(α4
)
14 / 40
54. Quasi-Stochastic Approximation
Filtering d
dt Θt = αf(Θt, ξt)
Second order filter is required
d2
dt2
ΘF
t + 2γζ
d
dt
ΘF
t + γ2
ΘF
t = γ2
Θt
with γ = O(α) and ζ ∈ (0, 1).
Obtain from p-mean flow
lim sup
t→∞
∥ΘF
t − θ∗
∥2
= O(α2
∥s
Υ∗
∥) + O(α4
)
Recall s
Υ can be eliminated with careful design of ξ.
14 / 40
55. Quasi-Stochastic Approximation
Error Attenuation in Tracking
ESC-0: f(Θt, ξt) = −
1
ε
ξtΓt(Θt + εξt)
ΘF
t
t
θopt
Θt
Γ(θ −
− θopt
t )
Traveling Camel
Transients: First 20% Final 80%
ESC-0
with
and
without
filtering
15 / 40
56. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = −1
ε ξΓ(θ + εξ)
Recall: global Lipschitz continuity of f is always assumed
16 / 40
57. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = −1
ε ξΓ(θ + εξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
16 / 40
58. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = −1
ε ξΓ(θ + εξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
Finite escape time is typical
16 / 40
59. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = −1
ε ξΓ(θ + εξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
Finite escape time is typical
State-dependent probing gain:
f(θ, ξ) = −
1
ϵ(θ)
ξΓ(θ + ϵ(θ)ξ)
16 / 40
60. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = − 1
ϵ(θ) ξΓ(θ + ϵ(θ)ξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
Finite escape time is typical
State-dependent probing gain:
f(θ, ξ) = −
1
ϵ(θ)
ξΓ(θ + ϵ(θ)ξ)
Examples: ϵ(θ) = ε
p
1 + Γ(θ) [WLOG Γ ≥ 0]
16 / 40
61. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = − 1
ϵ(θ) ξΓ(θ + ϵ(θ)ξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
Finite escape time is typical
State-dependent probing gain:
f(θ, ξ) = −
1
ϵ(θ)
ξΓ(θ + ϵ(θ)ξ)
Examples: ϵ(θ) = ε
p
1 + Γ(θ) [WLOG Γ ≥ 0]
ϵ(θ) = ε
q
1 + ∥θ − θctr∥2/σ2
p
16 / 40
62. A Return to Extremum Seeking Control
Lipschitz Continuity Matters! f(θ, ξ) = − 1
ϵ(θ) ξΓ(θ + ϵ(θ)ξ)
Recall: global Lipschitz continuity of f is always assumed
This is rarely the case in pratice ! (consider Γ quadratic)
Finite escape time is typical
State-dependent probing gain:
f(θ, ξ) = −
1
ϵ(θ)
ξΓ(θ + ϵ(θ)ξ)
Examples: ϵ(θ) = ε
p
1 + Γ(θ) [WLOG Γ ≥ 0]
ϵ(θ) = ε
q
1 + ∥θ − θctr∥2/σ2
p
• The algorithm is globally stable
subject to coercivity of Γ ⊕ Lipschitz gradient
• It makes sense to explore more when Γ(θ) is big!
16 / 40
63. A Return to Extremum Seeking Control
Finite Escape Time for ESC f(θ, ξ) = − 1
ϵ(θ) ξΓ(θ + ϵ(θ)ξ)
• Consider the scalar-valued objective Γ(θ) = θ2
• ESC-0 was applied with ξt = 2 cos( t
4) and at = (t + 1)−0.8
a) ϵ = 0.5 , b) ϵ(θ) = 0.5
p
1 + ∥θ∥2
10
10
2
8
6
4
2
0
8
4
0
-4
-8
0 2 4 6 8 10
-140
-100
-60
-20
20
10
-1
17 / 40
64. A Return to Extremum Seeking Control
Vanishing vs Fixed Gain Algorithms for Optimization
a) at = 0.1(1 + t)−0.65 , b) αb = 3 × 10−3 , c) αs = 7 × 10−4
100
10−10
10−5
100
10−10
10−5
θ1
θ2
Γ(Θt)
Vanishing Gain Constant Gain
Θt
Θt
PR
Θt
Θt
F1
Θt
F2
at = a0(1 + t)−ρ αb αs
t × 104
Evolution
of
Estimates
Evolution
of
Cost
18 / 40
65. Conclusions
Steps to a Succesful Design:
1) Design s
f so the mean flow ϑ̇ = s
f(ϑ) is GAS to θ∗ and ∂ s
f(θ∗) is
Hurwitz.
2) Design step-size: αn = n−ρ with 1
2 ≤ ρ < 1 for optimization.
αn ≡ α > 0 for tracking.
3) Use “clever” exploration when applicable.
4) Perform PR Averaging or filtering with bandwidth O(α).
5) Repeat! Obtain histograms for M runs
{TN θPR
N
(m)
: 1 ≤ m ≤ M} TN = (N − N0)2ρ
with θPR
0
(m)
widely dispersed.
19 / 40
66. Conclusions
Final Thoughts
• A simple averaging (or filtering) trick can greatly improve algorithmic
performance:
⋄ Vanishing gain: MSE vanishes at rates arbitrarily close to O(n−4).
⋄ Fixed gain: In general, MSE bounds are O(α4).
• The incorporation of a state-dependent probing gain leads to ESC-0
algorithms that are globally stable (and more).
20 / 40
67. Conclusions
Final Thoughts
• A simple averaging (or filtering) trick can greatly improve algorithmic
performance:
⋄ Vanishing gain: MSE vanishes at rates arbitrarily close to O(n−4).
⋄ Fixed gain: In general, MSE bounds are O(α4).
• The incorporation of a state-dependent probing gain leads to ESC-0
algorithms that are globally stable (and more).
• Work in progress: two time-scale QSA comes up for ESC and elsewhere.
The p-mean flow representation should be done by today’s CDC deadline!
Thank you!
Financial support from ARO award W911NF2010055 and NSF award EPCN 1935389 is gratefully acknowledged.
20 / 40
68. Appendices
Assumptions for QSA
(A1)
VG: The process a is non-negative, monotonically decreasing, and
lim
t→∞
at = 0,
Z ∞
0
ar dr = ∞.
BG: For all t, the gain process satisfies at ≡ α > 0 for some
0 < α < α0 < 1.
(A2) The functions s
f and f are Lipschitz continuous: for a constant
Lf < ∞,
∥ s
f(θ′
) − s
f(θ)∥ ≤ Lf ∥θ′
− θ∥,
∥f(θ′
, ξ) − f(θ, ξ)∥ + ∥f(θ, ξ′
) − f(θ, ξ)∥ ≤ Lf [∥θ′
− θ∥ + ∥ξ′
− ξ∥] ,
θ′
, θ ∈ Rd
, ξ, ξ′
∈ Rm
21 / 40
69. Appendices
Assumptions for QSA
(A3) The ODE d
dtϑt = s
f(ϑt) is globally asymptotically stable with unique
equilibrium θ∗. Moreover, one of the following conditions holds:
(a) There is a Lipschitz continuous Lyapunov function V : Rd → R+, a
constant δ0 > 0 and a compact set S such that ∇V (ϑt) · s
f(ϑt) ≤
−δ0∥ϑt∥ whenever ϑt /
∈ S.
(b) The scaled vector field s
f∞ : Rd → Rd defined by s
f∞(θ) :=
limc→∞
s
f(cθ)/c, θ ∈ Rd, exists as a continuous function. Moreover,
the ODE@∞ defined by d
dt xt = s
f∞(xt) is globally asymptotically sta-
ble [48, §4.8.4].
(A4) The vector field s
f is differentiable, with derivative denoted Ā(θ) =
∂θ
s
f (θ).
That is, Ā(θ) is a d × d matrix for each θ ∈ Rd, with Āi,j(θ) =
∂
∂θj
s
fi (θ).
Moreover, the derivative Ā is Lipschitz continuous, and Ā∗ = Ā(θ∗) is
Hurwitz.
22 / 40
70. Appendices
Assumptions for QSA
(A5) Φ is the state process for a dynamical system d
dt Φt = H(Φt), H :
Ω → Ω with unique invariant measure π. It satisfies the following ergodic
theorems for the functions of interest, for each initial condition Φ0 ∈ Ω:
(i) For each θ there exists a solution ˆ
f(θ, · ) to Poisson’s equation with
forcing function f. That is,
ˆ
f(θ, Φt0 ) =
Z t1
t0
[f(θ, ξt) − s
f(θ)] dt + ˆ
f(θ, Φt1 ) , 0 ≤ t0 ≤ t1
and for each θ,
R
Ω
ˆ
f(θ, z) π(dz) = 0. Finally, ˆ
f is continuously
differentiable (C1) on Rd × Ω. Its Jacobian with respect to θ is
denoted
b
A(θ, z) := ∂θ
ˆ
f(θ, z)
where
Z
Ω
b
A(θ, z) π(dz) = 0 for each θ ∈ Rd
23 / 40
71. Appendices
Assumptions for QSA
(ii) For each θ, there are C1 solutions to Poisson’s equation with forcing
functions ˆ
f and Υ. They are denoted
ˆ
ˆ
f and b
Υ, respectively, satisfying
ˆ
ˆ
f(θ, Φt0 ) =
Z t1
t0
ˆ
f(θ, ξt) dt +
ˆ
ˆ
f(θ, Φt1 )
b
Υ(θ, Φt0 ) =
Z t1
t0
[Υ(θ, Φt) − s
Υ(θ)] dt + b
Υ(θ, Φt1 ) , 0 ≤ t0 ≤ t1
with s
Υ(θ) = −
Z
Ω
b
A(θ, z)f(θ, G(z)) π(dz)
Moreover, for each θ,
Z
Ω
ˆ
ˆ
f(θ, z) π(dz) =
Z
Ω
b
Υ(θ, z) π(dz) = 0
24 / 40
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