Efficient Policy Gradient Optimization/Learning of Feedback Controllers

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Efficient Policy Gradient Optimization/Learning of Feedback Controllers. Chris Atkeson. Punchlines. Optimize and learn policies. Switch from “value iteration” to “policy iteration”. This is a big switch from optimizing and learning value functions. Use gradient-based policy optimization.

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### Efficient Policy GradientOptimization/Learning of Feedback Controllers

Chris Atkeson

Punchlines
• Optimize and learn policies.

Switch from “value iteration” to “policy iteration”.

• This is a big switch from optimizing and learning value functions.
• Use gradient-based policy optimization.
Motivations
• Efficiently design nonlinear policies
• Make policy-gradient reinforcement learning practical.
Model-Based Policy Optimization
• Simulate policy u = π(x,p) from some initial states x0 to find policy cost.
• Use favorite local or global optimizer to optimize simulated policy cost.
• If gradients are used, they are typically numerically estimated.
• Δp = -ε ∑x0w(x0)Vp 1st order gradient
• Δp = -(∑x0w(x0)Vpp)-1 ∑x0w(x0)Vp 2nd order
• Deterministic policy: u = π(x,p)
• Policy Iteration (Bellman Equation):

Vk-1(x,p) = L(x,π(x,p)) + V(f(x,π(x,p)),p)

• Linear models: f(x,u) = f0 + fxΔx + fuΔu

L(x,u) = L0 + LxΔx + LuΔu

π(x,p) = π0 + πxΔx + πpΔp

V(x,p) = V0 + VxΔx + VpΔp

Vxk-1 = Lx + Luπx + Vx(fx + fuπx)

Vpk-1 = (Lu + Vxfu)πp + Vp

Handling Constraints
• Lagrange multiplier approach, with constraint violation value function.
Antecedents
• Optimizing control “parameters” in DDP: Dyer and McReynolds 1970.
• Optimal output feedback design (1960s-1970s)
• Multiple model adaptive control (MMAC)
• Policy gradient reinforcement learning
When Will LQBR Work?
• Initial stabilizing policy is known (“output stabilizable”)
• Luu is positive definite.
• Lxx is positive semi-definite and (sqrt(Lxx),Fx) is detectable.
• Measurement matrix C has full row rank.
Other Issues
• Model Following
• Stochastic Plants
• Receding Horizon Control/MPC