Data-driven distributionally robust LQR with multiplicative noise

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Data-driven distributionally robust LQR

with multiplicative noise

Peter Coppens, Mathijs Schuurmans, Panagiotis Patrinos

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Control in uncertain environments model = reality generalize model ≠ reality safety conservative performance

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Uncertain systems financial (bio-)chemical climate control power distribution robotics/autonomous vehicles

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Learning to generalize

classical controlidentification and controller design are kept seperate.

Data identification Model synthesis Controller

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coarse id controlexplicitly include ambiguity in the model throughout the design1.

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Robustify — Stochastic and robust optimization • Robust optimization (𝑤 ∈ 𝒲) min 𝑥 max𝑤∈𝒲 𝑓(𝑥,𝑤) s. t. 𝑔(𝑥,𝑤) ≤ 0,∀𝑤 ∈ 𝒲 • Stochastic optimization (𝑤 ∼ IP𝑤) min 𝑥 IE[𝑓(𝑥,𝑤)] s. t. IP[𝑔(𝑥,𝑤) ≤ 0] ≥ 1 − 𝛿

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Robustify — Stochastic and robust optimization

• Robust optimization (𝑤 ∈ 𝒲)

min

𝑥 max𝑤∈𝒲 𝑓(𝑥,𝑤)

s. t. 𝑔(𝑥,𝑤) ≤ 0,∀𝑤 ∈ 𝒲

strong guarantees, conservative

• Stochastic optimization (𝑤 ∼ IP𝑤)

min

𝑥 IE[𝑓(𝑥,𝑤)]

s. t. IP[𝑔(𝑥,𝑤) ≤ 0] ≥ 1 − 𝛿

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Robustify — Stochastic and robust optimization • Robust optimization (𝑤 ∈ 𝒲) min 𝑥 max𝑤∈𝒲 𝑓(𝑥,𝑤) s. t. 𝑔(𝑥,𝑤) ≤ 0,∀𝑤 ∈ 𝒲 • Stochastic optimization (𝑤 ∼ IP𝑤) min 𝑥 IE[𝑓(𝑥,𝑤)] s. t. IP[𝑔(𝑥,𝑤) ≤ 0] ≥ 1 − 𝛿

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Robustify — Stochastic and robust optimization • Robust optimization (𝑤 ∈ 𝒲) min 𝑥 max𝑤∈𝒲 𝑓(𝑥,𝑤) s. t. 𝑔(𝑥,𝑤) ≤ 0,∀𝑤 ∈ 𝒲 • Stochastic optimization (𝑤 ∼ IP𝑤) min 𝑥 IE[𝑓(𝑥,𝑤)] s. t. IP[𝑔(𝑥,𝑤) ≤ 0] ≥ 1 − 𝛿 less guarantees, optimal

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Robustify — Distributionally robust optimization • Distributionally robust (𝜉 ∼ IP𝑤,IP𝑤∈ 𝒜) min 𝑥 IPmax𝑤∈𝒜 IE_𝑤[𝑓(𝑥,𝑤)] s. t. IP𝑤[𝑔(𝑥,𝑤) ≤ 0] ≥ 1 − 𝛿,∀IP𝑤∈ 𝒜

conservatism decreases with data akin to automatic regularization2 already applied in control before3-5

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conservatism decreases with data

akin to automatic regularization2 already applied in control before3-5

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conservatism decreases with data akin to automatic regularization2

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conservatism decreases with data akin to automatic regularization2 already applied in control before3-5

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DR-LQR — The nominal problem minimize 𝑢0,𝑢1,… IE [ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = (𝐴₀+ ∑𝑛𝑤 𝑖=1𝐴𝑖𝑤 (𝑖) 𝑘 ) 𝑥𝑘+ (𝐵0+ ∑ 𝑛𝑤 𝑖=0𝐵𝑖𝑤 (𝑖) 𝑘 ) 𝑢𝑘, ∀𝑘 ∈ IN 𝑥₀given

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DR-LQR — The nominal problem minimize 𝑢0,𝑢1,… IE [ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = (𝐴₀+ ∑𝑛𝑤 𝑖=1𝐴𝑖 ) 𝑥𝑘+ (𝐵0+ ∑ 𝑛𝑤 𝑖=0𝐵𝑖 ) 𝑢𝑘, ∀𝑘 ∈ IN 𝑥₀given 𝑤(𝑖)_𝑘 𝑤(𝑖)_𝑘

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Bellman equation:solution is still quadratic 𝑉 (𝑥) = inf

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Bellman equation:solution is still quadratic 𝑉 (𝑥) = inf 𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]} IE [𝑥⊤ +𝑃 𝑥+ ∣ (𝑥, 𝑢)] = (A0𝑥 + B0𝑢) ⊤ ([1 𝜇 ⊤ 𝜇 Σ + 𝜇𝜇⊤] ⊗ 𝑃) (A0𝑥 + B0𝑢) , whereA0= [𝐴⊤0 𝐴⊤1 … 𝐴⊤𝑛𝑤] ⊤ andB0= [𝐵⊤0 𝐵1⊤ … 𝐵𝑛⊤𝑤] ⊤

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

generalized Riccati equation1-3

properties of optimal controller:

• linear 𝑢 = 𝐾𝑥 (solve SDP)

• safe mean-square stabilizing IE[𝑥⊤

𝑘𝑥𝑘] → 0 as 𝑘 → ∞ • performant true optimum

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

𝑘𝑥𝑘] → 0 as 𝑘 → ∞

• performant true optimum • only depends on 𝜇 and Σ

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

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𝑢 {ℓ(𝑥, 𝑢) + IE [𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

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DR-LQR — Learning to generalize

Data identification Model Controller synthesis

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quantify robustify

assumptions about the data:

• access to samples {𝑤_𝑖}𝑀_𝑖=1 • Σ−1/2(𝑤 − 𝜇) is sub-Gaussian • 𝜇̂= 1 𝑀∑ 𝑀 𝑖=1𝑤𝑖 • Σ̂ = _𝑀1 ∑_𝑖=1𝑀 (𝑤_𝑖− ̂𝜇)(𝑤_𝑖− ̂𝜇)⊤

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quantify robustify

• access to samples {𝑤_𝑖}𝑀_𝑖=1 • Σ−1/2(𝑤 − 𝜇) is sub-Gaussian1 • 𝜇̂= 1 𝑀∑ 𝑀 𝑖=1𝑤𝑖 • Σ̂ = _𝑀1 ∑_𝑖=1𝑀 (𝑤_𝑖− ̂𝜇)(𝑤_𝑖− ̂𝜇)⊤

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quantify robustify

• access to samples {𝑤_𝑖}𝑀_𝑖=1 • Σ−1/2(𝑤 − 𝜇) is sub-Gaussian1 • 𝜇̂= 1 𝑀∑ 𝑀 𝑖=1𝑤𝑖 • Σ̂ = _𝑀1 ∑_𝑖=1𝑀 (𝑤_𝑖− ̂𝜇)(𝑤_𝑖− ̂𝜇)⊤

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DR-LQR — The ambiguity set

quantization: determine 𝑟_𝜇and 𝑟_Σsuch that IP [IP⋆_𝑤 ∈ 𝒜] ≥ 1 − 𝛽

𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

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𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤}

Σ⪯ Σ̂ }

𝑟_𝜇(𝛽) 𝑟_Σ(𝛽)

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quantization: determine 𝑟_𝜇and 𝑟_Σsuch thatIP [IP⋆_𝑤 ∈ 𝒜] ≥ 1 − 𝛽

𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

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𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

Σ⪯ 𝑟_Σ(𝛽)Σ̂ }

• radii decrease with 1/√# samples

• no knowledge of true distribution required • sufficient samples required

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𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

Σ⪯ 𝑟_Σ(𝛽)Σ̂ }

• no knowledge of true distribution required

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𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

Σ⪯ 𝑟_Σ(𝛽)Σ̂ }

• no knowledge of true distribution required • sufficient samples required

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𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

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quantization: determine 𝑟_𝜇and 𝑟_Σsuch that IP [IP⋆_𝑤 ∈ 𝒜] ≥ 1 − 𝛽 𝒜 ∶= {IP_𝑤∣ (𝜇−𝜇)̂

⊤_Σ̂−1₍_𝜇₋_𝜇_̂_{) ≤ 𝑟} 𝜇(𝛽)

Σ⪯ 𝑟_Σ(𝛽)Σ̂ }

synthesis: find the distributionally robust controller

min 𝑢₀,𝑢₁,…IPmax𝑤∈𝒜 IE𝑤[ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = 𝐴(𝑤_𝑘)𝑥_𝑘+ 𝐵(𝑤_𝑘)𝑢_𝑘, ∀𝑘 ∈ IN 𝑥₀given

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DR-LQR — Uncertain covariance min 𝑢₀,𝑢₁,…_Σ⪯𝑟max ΣΣ̂ IE𝑤[ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = 𝐴(𝑤_𝑘)𝑥_𝑘+ 𝐵(𝑤_𝑘)𝑢_𝑘, ∀𝑘 ∈ IN 𝑥₀given

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DR-LQR — Uncertain covariance min 𝑢₀,𝑢₁,… IE𝑤[ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = 𝐴(𝑤_𝑘)𝑥_𝑘+ 𝐵(𝑤_𝑘)𝑢_𝑘, ∀𝑘 ∈ IN 𝑥₀given max Σ⪯𝑟_ΣΣ̂

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𝑢 {ℓ(𝑥, 𝑢) +_Σ⪯𝑟max ΣΣ̂

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Bellman equation:solution is still quadratic 𝑉 (𝑥) = inf 𝑢 {ℓ(𝑥, 𝑢) + max_Σ⪯𝑟 ΣΣ̂ IE𝑤[𝑉 (𝑥+) ∣ (𝑥, 𝑢)]} max Σ⪯𝑟_ΣΣ̂ IE𝑤[𝑥⊤+𝑃 𝑥+ ∣ (𝑥, 𝑢)] = (A0𝑥 + B0𝑢) ⊤ ([1 𝜇 ⊤ 𝜇 𝑟_ΣΣ̂+ 𝜇𝜇⊤] ⊗ 𝑃) (A0𝑥 + B0𝑢)

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𝑢 {ℓ(𝑥, 𝑢) + max_Σ⪯𝑟 ΣΣ̂

IE𝑤[𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

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𝑢 {ℓ(𝑥, 𝑢) + max_Σ⪯𝑟 ΣΣ̂

IE𝑤[𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

generalized Riccati equation

• safe mean-square stabilizing w.h.p. IE[𝑥⊤

𝑘𝑥𝑘] → 0 as 𝑘 → ∞

• performant converges to true optimum • only depends on 𝜇 and𝑟_ΣΣ̂

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𝑢 {ℓ(𝑥, 𝑢) + max_Σ⪯𝑟 ΣΣ̂

IE𝑤[𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

𝑘𝑥𝑘] → 0 as 𝑘 → ∞ • performant converges to true optimum

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𝑢 {ℓ(𝑥, 𝑢) + max_Σ⪯𝑟 ΣΣ̂

IE𝑤[𝑉 (𝑥+) ∣ (𝑥, 𝑢)]}

𝑘𝑥𝑘] → 0 as 𝑘 → ∞ • performant converges to true optimum • only depends on 𝜇 and𝑟_ΣΣ̂

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DR-LQR — Full uncertainty min 𝑢0,𝑢1,… max IP𝑤∈𝒜 IE𝑤[ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = 𝐴(𝑤_𝑘)𝑥_𝑘+ 𝐵(𝑤_𝑘)𝑢_𝑘, ∀𝑘 ∈ IN 𝑥₀given

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DR-LQR — Full uncertainty min 𝑢0,𝑢1,… IE𝑤[ ∞ ∑ 𝑘=0 𝑥⊤ 𝑘𝑄𝑥𝑘+ 𝑢⊤𝑘𝑅𝑢𝑘] subj. to 𝑥_𝑘+1 = 𝐴(𝑤_𝑘)𝑥_𝑘+ 𝐵(𝑤_𝑘)𝑢_𝑘, ∀𝑘 ∈ IN 𝑥₀given max IP𝑤∈𝒜

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Bellman equation:solutionno longer quadratic 𝑉 (𝑥) = inf

𝑢 {ℓ(𝑥, 𝑢) +IPmax𝑤∈𝒜

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Bellman equation:approximate solution1

minimize 𝑃 𝑥 ⊤ 0𝑃 𝑥0 subj. to 𝑥⊤_{𝑃 𝑥 ≥ min} 𝑢 {ℓ(𝑥, 𝑢) + maxIP𝑤∈𝒜IE𝑤[𝑥 ⊤ +𝑃 𝑥+]} , ∀𝑥

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minimize 𝑃 𝑥 ⊤ 0𝑃 𝑥0 subj. to 𝑥⊤_{𝑃 𝑥 ≥ min} 𝑢 {ℓ(𝑥, 𝑢) + maxIP𝑤∈𝒜IE𝑤[𝑥 ⊤ +𝑃 𝑥+]} , ∀𝑥 solve approximate SDP2

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Bellman equation:approximatesolution1

minimize 𝑃 𝑥 ⊤ 0𝑃 𝑥0 subj. to 𝑥⊤_{𝑃 𝑥 ≥ min} 𝑢 {ℓ(𝑥, 𝑢) + maxIP𝑤∈𝒜 IE_𝑤[𝑥⊤ +𝑃 𝑥+]} , ∀𝑥 solveapproximateSDP2

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properties of approximateoptimal controller:

𝑘𝑥𝑘] → 0 as 𝑘 → ∞

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properties of approximateoptimal controller:

𝑘𝑥𝑘] → 0 as 𝑘 → ∞ • performant converges to true optimum

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DR-LQR — Numerical result 103 ₁₀4 ₁₀5 ₁₀6 10−6 10−5 10−4 10−3 10−2 # samples R el. Subop t. uncertain covariance full uncertainty

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DR-LQR — Numerical result 103 ₁₀4 ₁₀5 ₁₀6 10−6 10−5 10−4 10−3 10−2 # samples R el. Subop t. uncertain covariance full uncertainty 1/# samples

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DR-LQR — Future work

• what to do when not enough samples?

• what to do when only state measurements? • constraints, state observers, …

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• what to do when not enough samples? • what to do when only state measurements?

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• what to do when not enough samples? • what to do when only state measurements? • constraints, state observers, …

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References i

Abadeh, Soroosh Shafieezadeh, Peyman Mohajerin Esfahani, and Daniel Kuhn.

“Distributionally Robust Logistic Regression”.In: Advances in Neural Information Processing Systems 28. 2015, pp. 1576–1584.

Abadeh, Soroosh Shafieezadeh et al.“Wasserstein distributionally robust Kalman filtering”.In: Advances in Neural Information Processing Systems. 2018, pp. 8474–8483. Ben-Tal, Aharon, Laurent El Ghaoui, and Arkadi Nemirovski.“Robustness”.In: Handbook

of Semidefinite Programming. Boston, MA: Springer US, 2000, pp. 139–162.

Boyd, Stephen et al.Linear Matrix Inequalities in System and Control Theory.Society for Industrial and Applied Mathematics, 1994.

Coppens, P. and P. Patrinos.Sample Complexity of Data-Driven Stochastic LQR with Multiplicative Uncertainty.2020. arXiv: 2005.12167.

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References ii

Costa, Oswaldo L.V. and Carlos S. Kubrusly.“Quadratic optimal control for discrete-time infinite-dimensional stochastic bilinear systems”.In: IMA Journal of Mathematical Control and Information 14.4 (1997), pp. 385–399.

Delage, Erick and Yinyu Ye.“Distributionally robust optimization under moment uncertainty with application to data-driven problems”.In: Operations Research 58.3 (2010), pp. 595–612.

Gravell, Benjamin, Peyman Mohajerin Esfahani, and Tyler Summers.“Learning robust control for LQR systems with multiplicative noise via policy gradient”.In: arXiv preprint arXiv:1905.13547 (2019). arXiv: 1905.13547.

Hsu, Daniel, Sham M. Kakade, and Tong Zhang.“A tail inequality for quadratic forms of subgaussian random vectors”.In: Electronic Communications in Probability 17.52 (2012), pp. 1–6.

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References iii

Hsu, Daniel, Sham M. Kakade, and Tong Zhang.“Tail inequalities for sums of random matrices that depend on the intrinsic dimension”.In: Electronic Communications in Probability 17.52 (2012), pp. 1–13.

Rahimian, H. and S. Mehrotra.Distributionally Robust Optimization: A Review.2019. arXiv: 1908.05659.

Recht, Benjamin.“A Tour of Reinforcement Learning: The View from Continuous Control”.In: Annual Review of Control, Robotics, and Autonomous Systems 2.1 (2019), pp. 253–279.

Schuurmans, Mathijs, Pantelis Sopasakis, and Panagiotis Patrinos.“Safe

Learning-Based Control of Stochastic Jump Linear Systems: a Distributionally Robust Approach”.In: 2019 IEEE Conference on Decision and Control (CDC). IEEE. 2019,

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References iv

Vershynin, Roman.High-Dimensional Probability: An Introduction with Applications in Data Science.2018.

Wainwright, Martin J.High-Dimensional Statistics.Cambridge University Press, 2019. Wonham, W. Murray.“Optimal Stationary Control of a Linear System with

State-Dependent Noise”.In: SIAM Journal on Control 5.3 (1967), pp. 486–500. Yang, Insoon.Wasserstein distributionally robust stochastic control: A data-driven

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Assets

Images‘control in uncertain environments’: https://flic.kr/p/<e6tYCK,939fWN,2iUoC88> Images‘uncertain systems’:

https://flic.kr/p/<k283C7,xz8h9,2hRaa8G,r3LtVQ,azfYAW,n9mn42> Matlab code available on request

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Quantify – Sub-Gaussian

Definition (Sub-Gaussian)

Random variable 𝑋 is sub-Gaussian1if, for some 𝜎 ≥ 0, IE[𝑒𝜆(𝑋−IE[𝑋])] ≤ 𝑒𝜎2_𝜆2_/2

, ∀𝜆 ∈ IR

examples2_:

• Gaussian random variables 𝜎2= IE[(𝑋 − IE[𝑋])2_],

• Bernoulli random variable 𝜎 = 1/√log(2),

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, ∀𝜆 ∈ IR

examples2_:

• random variables with bounded support 𝜎 = ess sup |𝑋|/√log(2). Sub-Gaussian vector𝑍: for every 𝑢 ∈ 𝒮_𝑑we have 𝑢⊤_{𝑍 sub-Gaussian.}

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, ∀𝜆 ∈ IR

examples2_:

• random variables with bounded support 𝜎 = ess sup |𝑋|/√log(2). Sub-Exponential: |𝜆| < 1/𝛼.

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Quantify — Sum of random vectors/matrices

𝒜 = {IP𝑤 ∣ (𝜇̂−𝜇)Σ̂−1(𝜇̂−𝜇) ≤𝑟𝜇,Σ⪯𝑟ΣΣ̂} ,

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first consider isotropic case: IE[𝑤𝐼] = 0, IE[𝑤𝐼𝑤⊤𝐼] = 𝐼.

mean uncertainty: 𝑤_𝐼∈ IR𝑑∼ subG(𝜎) 𝜇̂_𝐼∼ subG𝑑(𝜎/

√ 𝑀). IP [‖𝜇̂_𝐼‖₂≤ 𝜎

2

𝑀(𝑛𝑤+ 2√𝑛𝑤log (1/𝛽) + 2log(1/𝛽))] ≥ 1 − 𝛽

Daniel Hsu, Sham M. Kakade, and Tong Zhang. “A tail inequality for quadratic forms of subgaussian random vectors”. In: Electronic Commu-nications in Probability 17.52 (2012), pp. 1–6

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first consider isotropic case: IE[𝑤𝐼] = 0, IE[𝑤𝐼𝑤⊤𝐼] = 𝐼. covariance uncertainty: 𝑤_𝐼∈ IR𝑑∼ subG(𝜎).

IP [‖Σ̂_𝐼− 𝐼‖₂≤ 𝜎 2 1 − 2𝜖(√ 32𝑞(𝛽, 𝜖, 𝑛_𝑤) 𝑀 + 2𝑞(𝛽, 𝜖, 𝑛_𝑤) 𝑀 , )] ≥ 1 − 𝛽, where 𝑞(𝛽, 𝜖, 𝑛𝑤) = 𝑛𝑤log(1 +1/𝜖) + log(2/𝛽).

Daniel Hsu, Sham M. Kakade, and Tong Zhang. “Tail inequalities for sums of random matrices that depend on the intrinsic dimension”. In: Electronic Communications in Probability 17.52 (2012), pp. 1–13

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Quantify — Final ambiguity set