Taming Reluctant Random Walks In The Positive Quadrant 2 , Marni - - PowerPoint PPT Presentation
Taming Reluctant Random Walks In The Positive Quadrant 2 , Marni Mishna 2 , and Yann Ponty 2 3 Jrmie Lumbroso 1 Department of Mathematics 1 Princeton University Department of Mathematics 2 Simon Fraser
Jérémie Lumbroso 1
2 , Marni Mishna 2 , and Yann Ponty 2 3
Princeton University 2
Simon Fraser University 3
Inria Saclay
June 3rd, 2016
✷ ✶
◆
❊
❙
❲
This is a unidimensional model. We could represent it using only {✷, −✶}.
◆
❊
❙
❲
This is a unidimensional model. We could represent it using only {✷, −✶}.
◆
❊
❙
❲
This is a unidimensional model. We could represent it using only {✷, −✶}.
◆
❊
❙
❲
❘ ❩ × ❩ ❩ × ❩≥✵ ❩≥✵ × ❩≥✵
plane half-plane quarter-plane
S = ✇♥ = ✹♥ ❤♥ ∼ ❝ ✹♥♥−✶/✷ q♥ ∼ ❝✹♥♥−✷/✸ S = ✇♥ = ✺♥ ❤♥ ∼ ❝ ✹.✹✻♥♥−✸/✷ q♥ ∼ ❝ ✹.✸♥θ(♥)
(✐,❥)∈S(✐, ❥)
Consider the following model. Asymptotically: q♥ ⊲ ⊳ ✻♥ The drift is (0,0), naive generation is feasible. A random walk:
Consider the following model. Asymptotically: q♥ ⊲ ⊳ ✻♥ The drift is (0,0), naive generation is feasible. A surprisingly efficient strategy is Anticipated rejection. Florentine Algorithm ❬❇❛r❝✉❝❝✐✴P✐♥③❛♥✐✴❙♣r✉❣♥♦❧✐✱ ✾✹✴✾✺❪ Two dimensional analogue ❬❇❛❝❤❡r✴❙♣♦rt✐❡❧❧♦ ✶✹❪ for walks with zero drift The average complexity probably linear.
Consider the following model. Asymptotically: q♥ ⊲ ⊳ ✺.✵✻♥ Naive rejection is too inefficient!
(♣r♦❜. ≪ (✺.✵✻/✻)♥ ≈ ✶✵−✼✹)
A random walk of a thousand steps.
Positive drift walks are mostly easy to handle
Reluctant (negative drift) walks are less cooperative
Reluctant (negative drift) walks are less cooperative... and require some taming!
Theorem ❬❋♦❧❦❧♦r❡❄❪
Random uniform generation of ❦ ❞-dimensional walks confined to a subset ❘ ⊂ ❩❞ is in Θ(❦ · ♥ + ♥❞+✶) arithmetic operations, using storage for Θ(♥❞+✶) (large) numbers.
Idea: Adapt trivial step-by-step generation into grammar having Θ(♥❞+✶) NTs
qS
♥ (①, ②) =
①+✐≥✵,②+❥≥✵
qS
♥−✶(① + ✐, ② + ❥)
if ♥ > ✵, ✶ if ♥ = ✵ (1) ⇒ Can we do better than Θ(♥✹) time?
Generate walks in an associated half plane, and wait for a 1/4-plane walk.
1
A walk in a 1/4-plane is also a walk in a half plane.
2
Convert to a unidimensional walk model. Steps may be in ❘. ② ≥ −♠①
3
Generate unidimensional walks (usually easy) and map to 2D
4
Reject walks that escape the quarter plane (“sub-exponential”?)
THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪
There is a half plane ② ≥ −♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. q♥ ✺ ✵✻♥ ❤♥ ❑ ♥ ♠ ❑ ♠
THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪
There is a half plane ② ≥ −♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. Asymptotically: q♥ ⊲ ⊳ ✺.✵✻♥ ❤♥ ❑ ♥ ♠ ❑ ♠
THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪
There is a half plane ② ≥ −♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. Asymptotically: q♥ ⊲ ⊳ ✺.✵✻♥ Exponential growth of 1/2-plane walks with these steps in various half-planes. (ie. ❤♥ ⊲ ⊳ ❑ ♥) ♠ 1/2 3/4 1 2 10 ∞ ❑ 5.219 5.075 5.064 5.073 5.156 5.376 5.464 ♠
THEOREM ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❀●❛r❜✐t✴❘❛s❝❤❡❧ ✶✺✰❪
There is a half plane ② ≥ −♠① such that (asymptotically) the half plane and quarterplane walks have the same exponential growth factor. This step set in the quarter plane. Asymptotically: q♥ ⊲ ⊳ ✺.✵✻♥ Exponential growth of 1/2-plane walks with these steps in various half-planes. (ie. ❤♥ ⊲ ⊳ ❑ ♥) ♠ 1/2 3/4 1 2 10 ∞ ❑ 5.219 5.075 5.064 5.073 5.156 5.376 5.464 The “best” 1/2-plane is obtained for a slope ♠ = 0.735
INPUT: Projected up {❛✐}, down {❜❥} and horizontal {❝❦} steps. Algebraic specification ❬▼❡r❧✐♥✐✴❘♦❣❡rs✴❙♣r✉❣♥♦❧✐✴❱❡rr✐ ✾✾❀ ❉✉❝❤♦♥ ✵✵❪
P = D × Paux L✐ =
✇(❛)=✐
❛ × D +
♠✐♥(¯ ❛,✐+¯ ❜)
L❦ × R❦−✐ Paux = ε +
¯ ❛
L❦ × Paux R❥ =
✇(❜)=−❥
❜ × D +
♠✐♥(❥+¯ ❛,¯ ❜)
L❦−❥ × R❦ D =
✇(❝)=✵
❝ × D +
♠❛①(¯ ❛,¯ ❜)
L❦ × R❦ + ε
Number of rules: Θ((♠❛① |❛✐| + ♠❛① |❜❥|)✷) – can be huge! Recursive generation ❬❋❧❛❥♦❧❡t✴❩✐♠♠❡r♠❛♥♥✴❱❛♥ ❈✉st❡♠ ✾✹✱ ●♦❧❞✇✉r♠ ✾✺❪, Boltzmann sampling ❬❉✉❝❤♦♥✴❋❧❛❥♦❧❡t✴▲♦✉❝❤❛r❞✴❙❝❤❛❡✛❡r ✵✹❪. . . Non-rational slopes... more on this later.
S = {(✶, ✵), (✵, ✶), (−✶, ✵), (✶, −✶), (−✶, −✶), (−✷, −✶)} Using a generic Boltzmann implemention by A. Darrasse
INPUT: S ⊂ ❩✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S
1
Find optimal slope ♠ (explicit solution computable)
2
If optimal slope is rational, build associated grammar
3
Generate element α from grammar If α confined within quarter plane RETURN α ♥ ❈✶❑ ♥
♠♥ ✸ ✷
❈✷❑ ♥♥
r
♥r
✶ ✷
❑♠ ❑
❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪
r
INPUT: S ⊂ ❩✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S
1
Find optimal slope ♠ (explicit solution computable)
2
If optimal slope is rational, build associated grammar
3
Generate element α from grammar If α confined within quarter plane RETURN α Expected time= Time to generate a 1/2-plane walk
O(♥) × ❈✶❑ ♥
♠♥−✸/✷
❈✷❑ ♥♥−r = O(♥r−✶/✷) since ❑♠ = ❑
❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪
r
INPUT: S ⊂ ❩✷ (reluctant); Length ♥ OUPUT: Uniform random 1/4-plane walk with steps from S
1
Find optimal slope ♠ (explicit solution computable)
2
If optimal slope is rational, build associated grammar
3
Generate element α from grammar If α confined within quarter plane RETURN α Expected time= Time to generate a 1/2-plane walk
O(♥) × ❈✶❑ ♥
♠♥−✸/✷
❈✷❑ ♥♥−r = O(♥r−✶/✷) since ❑♠ = ❑ REMARK 1: Optimal slope computable ❬❏♦❤♥s♦♥✴▼✐s❤♥❛✴❨❡❛ts ✶✺✰❪ Bound r??
Let S ⊆ {±✶, ✵}✷ define a non-trivial quarter plane walk model. Consider the characteristic polynomial P(①, ②) =
①✐② ❥ and its critical point (α, β) such that ∂P(①,②)
∂①
②=β = ∂P(①,②)
∂②
②=β = ✵
Theorem 4 ❬❇♦st❛♥✴❘❛s❝❤❡❧✴❙❛❧✈② ✶✹❪
The number ❡♥ of excursions of length ♥ over S obeys ❡♥ ∼ ❈ P(α, β)♥♥−r❡ with r❡ = ✶ + π ❛r❝❝♦s
P①② (α,β)
√
P①① (α,β)P②② (α,β) .
Remark 1: For reluctant small steps, one has ✸.✸ < r❡ < ✼.✺ Remark 2: Since excursions ⊆ quadrant walks, thus r ≤ r❡. Work by Raschel/Garbit conjecture that r ≪ r❡, generically.
Asymptotically: q♥ ∼ ❈✺.✵✻♥♥−r From the excursion bound: r ≤ r❡ = ✷.✻✼✸ Empirically: r ∼ ✷.✸
Generating a walk of length ♥ = ✶✵✵✵ requires on average ♥(✷.✻✼−✶.✺) = ✶✵✵✵✶.✶✼ = ✸✵✵✵ trials.
INPUT: Fixed value of ♥, 1D step set A IDEA: Build grammar for rational approximation A∆θ of A ♥ ♥✷ ❦ ❦ ♥✷ ♥
INPUT: Fixed value of ♥, 1D step set A IDEA: Build grammar for rational approximation A∆θ of A ∆θ For sufficiently small values of ∆θ, the positive 1D walks of length ♥ generated by A∆θ and A are in bijection. ♥✷ ❦ ❦ ♥✷ ♥
INPUT: Fixed value of ♥, 1D step set A IDEA: Build grammar for rational approximation A∆θ of A ∆θ For sufficiently small values of ∆θ, the positive 1D walks of length ♥ generated by A∆θ and A are in bijection. Walks in A∆θ are generated by a grammar having Θ(♥✷) rules. Generation of ❦ walks in Θ(❦ · ♥✷) (Boltzmann) or worse. . . CONJECTURE: Grammar with ♥ rules may be sufficient
Random generation in higher di- mension can also be done efficiently through rejection from a 1D walks. Problem. How to find the optimal hyperplane? Problem: Generate reluctant 1D walks without a grammar? Problem: Characterize walks such that r < ✸ + ✸/✷
The strategy also works for non-reluctant (→ O(✶)) and non-trivial walks. Details are slightly different but essentially the same. Anticipated rejection is easily integrated, and seems to reduce the time by a linear factor in the zero drift case. Motivates more efficient samplers for 1D positive walks taking arbitrary numbers of steps.