On Privacy in Smart Metering Systems with Periodically Time-Varying - - PowerPoint PPT Presentation

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On Privacy in Smart Metering Systems with Periodically Time-Varying Input Distribution

Yu Liua, Ashish Khistia, Aditya Mahajanb

a University of Toronto b McGill University

GlobalSIP Symposium on Privacy and Security 14 Nov, 2017

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Smart Meters empower smart grids

Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Smart Meters empower smart grids

Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Smart Meters empower smart grids

Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Smart Meters empower smart grids

Fine grained consumption measurements are needed for: Time-of-use pricing Demand response . . .

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What is the minimum information leakage rate if consumers obfuscate consumption using a rechargeable battery? What are privacy-optimal battery charging strategies?

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Home Applicances Power Grid Smart Meter Controller Demand: Xt Consumption: Yt

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Demand: Xt Consumption: Yt

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Why is the problem non-trivial?

𝒴 = 𝒵 = 𝒯 = {0, 1}, PX = [0.5, 0.5] (Binary model) Consv: St + Yt − Xt ∈ 𝒯

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Why is the problem non-trivial?

𝒴 = 𝒵 = 𝒯 = {0, 1}, PX = [0.5, 0.5] (Binary model) Consv: St + Yt − Xt ∈ 𝒯

Empty state St = 0

Xt = 0 ⟹ Yt ∈ {0, 1} Xt = 1 ⟹ Yt = 1

Full state St = 1

Xt = 0 ⟹ Yt = 0 Xt = 1 ⟹ Yt ∈ {0, 1}

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Why is the problem non-trivial?

𝒴 = 𝒵 = 𝒯 = {0, 1}, PX = [0.5, 0.5] (Binary model) Consv: St + Yt − Xt ∈ 𝒯

Empty state St = 0

Xt = 0 ⟹ Yt ∈ {0, 1} Xt = 1 ⟹ Yt = 1

Full state St = 1

Xt = 0 ⟹ Yt = 0 Xt = 1 ⟹ Yt ∈ {0, 1} Consider performance of memoryless policies

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Why is the problem non-trivial?

𝒴 = 𝒵 = 𝒯 = {0, 1}, PX = [0.5, 0.5] (Binary model) Consv: St + Yt − Xt ∈ 𝒯

Empty state St = 0

Xt = 0 ⟹ Yt ∈ {0, 1} Xt = 1 ⟹ Yt = 1

Full state St = 1

Xt = 0 ⟹ Yt = 0 Xt = 1 ⟹ Yt ∈ {0, 1} Consider performance of memoryless policies

Deterministic Memoryless Policy

P(Y|X = 0, S = 0) = [1 0]; P(Y|X = 1, S = 1) = [0 1]: Leakage = 1 (∵ Yt = Xt). P(Y|X = 0, S = 0) = [0 1]; P(Y|X = 1, S = 1) = [1 0]: Leakage ≈ 1 (∵ Yt = 1 − St).

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Why is the problem non-trivial?

𝒴 = 𝒵 = 𝒯 = {0, 1}, PX = [0.5, 0.5] (Binary model) Consv: St + Yt − Xt ∈ 𝒯

Empty state St = 0

Xt = 0 ⟹ Yt ∈ {0, 1} Xt = 1 ⟹ Yt = 1

Full state St = 1

Xt = 0 ⟹ Yt = 0 Xt = 1 ⟹ Yt ∈ {0, 1} Consider performance of memoryless policies

Deterministic Memoryless Policy

P(Y|X = 0, S = 0) = [1 0]; P(Y|X = 1, S = 1) = [0 1]: Leakage = 1 (∵ Yt = Xt). P(Y|X = 0, S = 0) = [0 1]; P(Y|X = 1, S = 1) = [1 0]: Leakage ≈ 1 (∵ Yt = 1 − St).

Randomized Memoryless Policy

P(Y|X = 0, S = 0) = [0.5 0.5]; P(Y|X = 1, S = 1) = [0.5 0.5]: Leakage = 0.5. Is this the best memoryless policy? Is this the optimal policy? How do we evaluate the performance of an arbitrary policy? Need ℙ(XT, YT)?

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Literature overview

Evaluate privacy of specific battery management policies

[Kalogridis et al., 2010] Monte-Carlo evaluation of best-efgort policy [Varodayan Khisti, 2011] Computing performance of battery conditioned stochastic charging policies using BCJR algorithm. [Tan Gündüz Poor, 2012] Generalized results of [Varodayan Khisti] to include models with energy harvesting. [Giulio Gündüz Poor, 2015] Bounds on performance of best-efgort and hide-and-store policies for infjnite battery size.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Literature overview

Evaluate privacy of specific battery management policies

[Kalogridis et al., 2010] Monte-Carlo evaluation of best-efgort policy [Varodayan Khisti, 2011] Computing performance of battery conditioned stochastic charging policies using BCJR algorithm. [Tan Gündüz Poor, 2012] Generalized results of [Varodayan Khisti] to include models with energy harvesting. [Giulio Gündüz Poor, 2015] Bounds on performance of best-efgort and hide-and-store policies for infjnite battery size.

Dynamic programming decomposition to identify optimal policies

[Yao Venkitasubramanian, 2013] Dynamic program, computable inner and upper bounds. Li Kshiti Mahajan, 2016 Dynamic program, closed form optimal strategy for i.i.d. case.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Inspired by the approach used for capacity of Markov channels with feedback (Goldsmith Varaiya 1996, Tatikonda Mitter 2009, Permuter et al 2008) The DP is similar to the DP for POMDPs but the per-step cost is concave rather than linear. v(π) is concave. So, computational approaches for POMDPs work.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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[LKM] Salient features of the solution

I(S − X; X) is concave in 𝒬𝒯

J∗ and θ∗ may be computed using Blahut-Arimoto algorithm.

Optimal policy is stationary and memoryless

q∗

t(y|xt, st) = b∗(y|xt, st)

(note b∗ does not depend on πt) If St ∼ θ∗, then St+1 ∼ θ∗ and St+1 ⊥ Yt.

Support of consumptions

Even if 𝒵 ⊃ 𝒴, under the optimal policy the support of PY is 𝒴.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

8

This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

Solution idea

We can use the qualitative properties of the i.i.d. solution to get achievable upper

• bounds. Compare them with non-achievable lower bounds.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

9

Achievable scheme and lower bound

Achievable scheme

Arbitrarily restrict attention to periodic policies: For odd time: q1(yt|xt, st) For even time: q2(yt|xt, st) Pick q1 and q2 to ensure invariance condition: St+1 ⊥ Yt. This induces ℙ(St) = PS1 for odd times and PS2 for even times. L∗ ≤ L∞(𝐫) = 1 2I(S1, X1; X1) + 1 2I(S2, X2; X2).

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Achievable scheme and lower bound

Achievable scheme

Arbitrarily restrict attention to periodic policies: For odd time: q1(yt|xt, st) For even time: q2(yt|xt, st) Pick q1 and q2 to ensure invariance condition: St+1 ⊥ Yt. This induces ℙ(St) = PS1 for odd times and PS2 for even times. L∗ ≤ L∞(𝐫) = 1 2I(S1, X1; X1) + 1 2I(S2, X2; X2).

Lower bound

L∗ ≥ 1 2 min

PS1

I(S1 − X1; X1) + 1 2 min

PS2

I(S2 − X2; X2) Same as assuming that the input distribution was Q1 for fjrst T/2 time steps and Q2 as last T/2 time steps.

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Numerical Results

Binary Model

𝒴 = 𝒵 = {0, 1}. Q1 = [0.7 0.3], Q2 = [0.3 0.7]. 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Battery size Leakage rate lower bound achievable policy

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Numerical Results

Ternary Model

𝒴 = 𝒵 = {0, 1, 2}. Q1 = [0.33 0.33 0.33], Q2 = [0.25 0.5 0.25]. 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Battery size Leakage rate lower bound achievable policy

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Numerical Results

Ternary Model

𝒴 = 𝒵 = {0, 1, 2}. Q1 = [0.33 0.33 0.33], Q2 = [0.25 0.5 0.25]. 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Battery size Leakage rate lower bound achievable policy

The performance of the proposed policy numerically matches that of the lower bound. Could we show optimality?

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

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Summary

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate)

SLIDE 35

Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

6

[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt)

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Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

6

[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

8

This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

Solution idea

We can use the qualitative properties of the i.i.d. solution to get achievable upper

• bounds. Compare them with non-achievable lower bounds.

SLIDE 38

Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

6

[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

8

This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

Solution idea

We can use the qualitative properties of the i.i.d. solution to get achievable upper

• bounds. Compare them with non-achievable lower bounds.

Smart-meter privacy–(Liu, Khisti, and Mahajan)

9

Achievable scheme and lower bound

Achievable scheme

Arbitrarily restrict attention to periodic policies: For odd time: q1(yt|xt, st) For even time: q2(yt|xt, st) Pick q1 and q2 to ensure invariance condition: St+1 ⊥ Yt. This induces ℙ(St) = PS1 for odd times and PS2 for even times. L∗ ≤ L∞(𝐫) = 1 2I(S1, X1; X1) + 1 2I(S2, X2; X2).

Lower bound

L∗ ≥ 1 2 min

PS1

I(S1 − X1; X1) + 1 2 min

PS2

I(S2 − X2; X2) Same as assuming that the input distribution was Q1 for fjrst T/2 time steps and Q2 as last T/2 time steps.

SLIDE 39

Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

6

[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

8

This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

Solution idea

We can use the qualitative properties of the i.i.d. solution to get achievable upper

• bounds. Compare them with non-achievable lower bounds.

Smart-meter privacy–(Liu, Khisti, and Mahajan)

9

Achievable scheme and lower bound

Achievable scheme

Arbitrarily restrict attention to periodic policies: For odd time: q1(yt|xt, st) For even time: q2(yt|xt, st) Pick q1 and q2 to ensure invariance condition: St+1 ⊥ Yt. This induces ℙ(St) = PS1 for odd times and PS2 for even times. L∗ ≤ L∞(𝐫) = 1 2I(S1, X1; X1) + 1 2I(S2, X2; X2).

Lower bound

L∗ ≥ 1 2 min

PS1

I(S1 − X1; X1) + 1 2 min

PS2

I(S2 − X2; X2) Same as assuming that the input distribution was Q1 for fjrst T/2 time steps and Q2 as last T/2 time steps. Smart-meter privacy–(Liu, Khisti, and Mahajan)

10

Numerical Results

Binary Model

𝒴 = 𝒵 = {0, 1}. Q1 = [0.7 0.3], Q2 = [0.3 0.7]. 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Battery size Leakage rate lower bound achievable policy

SLIDE 40

Smart-meter privacy–(Liu, Khisti, and Mahajan)

12

Summary

Smart-meter privacy–(Liu, Khisti, and Mahajan)

2

Home Applicances Power Grid Smart Meter Controller Evesdropper/ Adversory Battery ( State St) Yt − Xt Demand: Xt Consumption: Yt

Energy conservation

St+1 = St + Yt − Xt, St ∈ 𝒯 (Size of battery)

Randomized charging strategy

qt(yt | xt, st, yt−1): Choose consumption given history . . .

Objective

Choose battery charging strategy 𝐫 = {qt}t≥1 to min lim

T→∞

1 T I

𝐫(XT; YT)

(mutual information rate) Smart-meter privacy–(Liu, Khisti, and Mahajan)

5

[LKM] Main results: Markovian demand

Structure of optimal strategies

Defjne belief state πt(x, s) = ℙ(Xt = x, St = s|Yt−1) Charging strategies of the form qt(yt|xt, st, πt) are optimal.

Dynamic programming decomposition

Let 𝒝 denote the class of conditional distributions on 𝒵 given (𝒴, 𝒯). Suppose there exists a J ∈ ℝ and v∶ 𝒬X,S → ℝ that satisfjes the following: J∗ + v(π) = inf

a∈𝒝 {I(a; π) + ∑ x,s,y

π(x, s)a(y|x, s)v(φ(π, y, a))} Then, J∗ is the minimum leakage rate Let f∗(π) denote the arg min of the RHS and a∗ = f∗(π). Then, J∗ is achieved by the charging policy q∗(y|xt, st, πt) = a∗(y|xt, st) (note a∗ depends on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

6

[LKM] Main results: i.i.d. demand

Solution of the dynamic program

J∗ ∶= min

θ∈𝒬S I(S − X; X)

where X ∼ PX and S ∼ θ. Let θ∗ denote the arg min of the RHS. Then, J∗ is the minimum leakage rate

Optimal strategies

Defjne b∗(y|x, s) = ⎧ ⎨ ⎩ PX(y)θ∗(y + x − s) Normalize if y ∈ 𝒴 and y is feasible 0,

• therwise

. Then, J∗ is achieved by time-invariant action q∗

t(y|xt, st, πt) = b∗(y|xt, st)

(note b∗ does not depend on πt) Smart-meter privacy–(Liu, Khisti, and Mahajan)

8

This paper: Periodic Input Distribution

Periodic input

Xodd ∼ Q1(⋅) and Xeven ∼ Q2(⋅). We assume that the input cycles between two distributions (each of length one). Results easily generalize to a larger cycle or staying at each distribution for a difgerent amount of time.

Conceptual diff.

Same as before. The leakage rate is a multi-letter mutual information expression that depends on ℙ(XT, YT).

Solution idea

We can use the qualitative properties of the i.i.d. solution to get achievable upper

• bounds. Compare them with non-achievable lower bounds.

Smart-meter privacy–(Liu, Khisti, and Mahajan)

9

Achievable scheme and lower bound

Achievable scheme

Arbitrarily restrict attention to periodic policies: For odd time: q1(yt|xt, st) For even time: q2(yt|xt, st) Pick q1 and q2 to ensure invariance condition: St+1 ⊥ Yt. This induces ℙ(St) = PS1 for odd times and PS2 for even times. L∗ ≤ L∞(𝐫) = 1 2I(S1, X1; X1) + 1 2I(S2, X2; X2).

Lower bound

L∗ ≥ 1 2 min

PS1

I(S1 − X1; X1) + 1 2 min

PS2

I(S2 − X2; X2) Same as assuming that the input distribution was Q1 for fjrst T/2 time steps and Q2 as last T/2 time steps. Smart-meter privacy–(Liu, Khisti, and Mahajan)

10

Numerical Results

Binary Model

𝒴 = 𝒵 = {0, 1}. Q1 = [0.7 0.3], Q2 = [0.3 0.7]. 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Battery size Leakage rate lower bound achievable policy