[PPT] - Synthesizing Probabilistic Invariants via Doobs decomposition G. PowerPoint Presentation

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Synthesizing Probabilistic Invariants via Doob’s decomposition

G. Barthe, T. Espitau, L.M.F Fioriti, J. Hsu

CAV, 2016

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Introduction

1

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Introduction

1 Probabilistic Computations 2

Ubiquitous in many fields (ML, Crypto, Privacy,…) But… Difficult to prove Termination ? (Certainly, almost sure, non terminating)

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Introduction

1 Probabilistic Computations 2 Martingales? 3 Doob’s Decomposition

Ubiquitous in many fields (ML, Crypto, Privacy,…) But… Difficult to prove Termination ? (Certainly, almost sure, non terminating) Difficulty to transfer local to end of program Reason on average values Martingales have the required transfer property

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Introduction

1 Probabilistic Computations 2 Martingales? 3 Doob’s Decomposition

Ubiquitous in many fields (ML, Crypto, Privacy,…) But… Difficult to prove Termination ? (Certainly, almost sure, non terminating) Difficulty to transfer local to end of program Reason on average values Martingales have the required transfer property But (again)… Difficult to find good ones Automated generation? Doob’s decomposition Formal method to generate martingales from a seed.

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Martingale theory 101 (I)

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Martingale theory 101 (I)

Step 1: Some probabilities

Ω set of outcomes.
Sigma algebra:

Set F of subsets of Ω Closed under complements, countable unions, countable intersections.

Probability measure:

Countably additive mapping P : F → [0, 1] P(Ω) = 1. Probability space

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Martingale theory 101 (II)

Step 2: Stochastic process

Random variable:

X : Ω → R measurable ( X-1 ( (a,b] ) ∈F )

Filtration: ( Fi ) ⊂ F s.t:

Fi-1 ⊂ Fi

Process wrt filtration Fi:

Sequence (Xi) s.t: Xi is Fi measurable

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Martingale theory 101 (II)

Interlude: PL setting Ω: Element = Possible outcome of samples Fi: Events sampled at iteration i or before Process (Xi) is adapted to the filtration iff: Xi is defined in term of elements sampled at step i or before

i = 0 While b do z[i] ←$ Samplings... x[i] ← f(x[i-1], ... , f[0], z[i], … ,z[0]) i++ end

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Martingale theory 101 (III)

Step 3: Expectations & Moments

Expectation:

E[X] = ∑u∈Ω X(u) P(u)

Conditional expectation wrt G ⊂ F: E[X|G]

Y G-mesurable st E[X.1A ] = E[Y.1A] for A∈G

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Martingale theory 101 (IV)

Step 4 ( Final! ): Martingales

Martingale:

E [ Xi | Fi-1 ] = Xi-1 Average value of the current step is equal to the value of the previous step

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Playing with martingales

Doob’s decomposition (Xi) stochastic process → (Mi) martingale M0 = X0 Mi = X0 + ∑i

j=1 Xj - E [Xj | Fj-1]

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Black Magic of martingales

Optional Stopping theorem (Mi) martingale → Expectations are invariants E[Mj] = E[M0]

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Black Magic of martingales

Optional Stopping theorem E[Mj] = E[M0]

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Black Magic of martingales

Optional Stopping theorem E[MT] = E[M0] For T a stopping time : T : Ω → R { w ∈ Ω | T(w) ≤ i } ⊂ Fi

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Black Magic of martingales

Optional Stopping theorem E[MT] = E[M0] For T a stopping time : T : Ω → R { w ∈ Ω | T(w) ≤ i } ⊂ Fi

and...

|Mi - Mi-1 | ≤ C E[T] < ∞

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