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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. 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 Can we make model-based policy gradient more efficient? Analytic Gradients • 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 • Policy Gradient: Vxk-1 = Lx + Luπx + Vx(fx + fuπx) Vpk-1 = (Lu + Vxfu)πp + Vp Handling Constraints • Lagrange multiplier approach, with constraint violation value function. Vpp: Second Order Models Regularization LQBR: Linear (dynamics) Quadratic (cost) Bilinear (policy) Regulator Timing Test 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 • Adaptive critics, Werbos: HDP, DHP, GDHP, ADHDP, ADDHP 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. Locally Linear Policies Local Policies GOAL Cost Of One Gradient Calculation Continuous Time Other Issues • Model Following • Stochastic Plants • Receding Horizon Control/MPC • Adaptive RHC/MPC • Combine with Dynamic Programming • Dynamic Policies -> Learn State Estimator Optimize Policies • Policy Iteration, with gradient-based policy improvement step. • Analytic gradients are easy. • Non-overlapping sub-policies make second order gradient calculations fast. • Big problem: How choose policy structure?
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