diff --git a/flux/nn_prior/linear_sys.jl b/flux/nn_prior/linear_sys.jl index b8fda731f813f2f64dc5ed421a689d058ef83ec8..367a3a97c75155ce0451a992984043ef9cd061fa 100644 --- a/flux/nn_prior/linear_sys.jl +++ b/flux/nn_prior/linear_sys.jl @@ -2,36 +2,27 @@ import Parameters -struct LinearSys <: AbstractSystem +@with_kw struct LinearSys <: AbstractSystem A B - N - n - ns - h - σ0 - sind - uind - s1ind + N = 1000 + n = 10 + ns = n + h = 0.02 + σ0 = 0 + sind = 1:ns + uind = ns+1:(ns+n) + s1ind = (ns+n+1):(ns+n+ns) end -function LinearSys(seed; - N = 1000, - n = 10, - ns = n, - h = 0.02, - σ0 = 0, - sind = 1:ns, - uind = ns+1:(ns+n), - s1ind = (ns+n+1):(ns+n+ns)) - +function LinearSys(seed; kwargs...) srand(seed) A = randn(n,n) A = A-A' # skew-symmetric = pure imaginary eigenvalues A = A - h*I # Make 'slightly' stable A = expm(h*A) # discrete time B = h*randn(n,n) - LinearSys(A,B,N, n, ns, h, σ0, sind, uind, s1ind) + LinearSys(A=A, B=B, kwargs...) end function generate_data(sys::LinearSys, systype, seed, validation=false) diff --git a/gridworld.jl b/gridworld.jl index d4344259b36900c41f5225483c59f3ca41b808ea..243e1c4c159112db0bf1b5a9d177370d3e29a03b 100644 --- a/gridworld.jl +++ b/gridworld.jl @@ -1,12 +1,13 @@ using Plots -function gridworld() - grid_size = (10,10) - iters = 10 - A = [1,2,3,4] - running_cost = 0.1 - grid = zeros(grid_size) - Q = 0*ones(length(A),grid_size...) - goal = (5,5) +function gridworld(; + grid_size = (10,10), + iters = 100, + A = [1,2,3,4], + running_cost = 0.1, + grid = zeros(grid_size), + Q = 0*ones(length(A),grid_size...), + goal = (5,5)) + grid[goal...] = 5 function transition(x,y,a) @@ -28,7 +29,7 @@ function gridworld() for y = 1:grid_size[2] for x = 1:grid_size[1] if (x,y) == goal - Q[:,x,y] = grid[goal...] + Q[:,x,y] .= grid[goal...] continue end for a = A @@ -38,17 +39,17 @@ function gridworld() end end end - heatmap(squeeze(maximum(Q,1),1)); gui() + heatmap(dropdims(maximum(Q,dims=1),dims=1)); gui() # println(i) # sleep(0.01) end println("done") Q end -@time Q = gridworld() +@time Q = gridworld(); -for y = 1:grid_size[2] - for x = 1:grid_size[1] +for y = 1:10#grid_size[2] + for x = 1:10#grid_size[1] a = indmax(Q[:,x,y]) sym = a == 1 ? :utriangle : a == 2 ? :rtriangle : a == 3 ? :dtriangle : :ltriangle scatter!([x],[y],m=(sym,5), lab="", legend=false) diff --git a/jump_lin_id/figs/force.tex b/jump_lin_id/figs/force.tex deleted file mode 100644 index 97a93acbcc692f174795a52bf0a80d4e6b0a3466..0000000000000000000000000000000000000000 --- a/jump_lin_id/figs/force.tex +++ /dev/null @@ -1,2298 +0,0 @@ -% This file was created by matplotlib2tikz v0.6.2. -\begin{tikzpicture} - -\definecolor{color1}{rgb}{0.88887350027252,0.43564919034819,0.278122936141944} -\definecolor{color0}{rgb}{0,0.605603161175225,0.978680117569607} -\definecolor{color3}{rgb}{0.764440175493436,0.444111779468777,0.824297535923276} -\definecolor{color2}{rgb}{0.242224297852199,0.64327509315763,0.304448651534115} 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a/jump_lin_id/pres/pres_idpaper.pdf b/jump_lin_id/pres/pres_idpaper.pdf index 164ca83d0f5e568dbceacf3acc12fba83f84ca9c..619843dad4fc1e05f09db6bc96df78cfd47fb21e 100644 Binary files a/jump_lin_id/pres/pres_idpaper.pdf and b/jump_lin_id/pres/pres_idpaper.pdf differ diff --git a/jump_lin_id/pres/pres_idpaper.tex b/jump_lin_id/pres/pres_idpaper.tex index dfae9ee9ca70fae5488d6d52b05de5ba4826bbd8..aadd70136c817d43930c8d6924299c661b3fc309 100644 --- a/jump_lin_id/pres/pres_idpaper.tex +++ b/jump_lin_id/pres/pres_idpaper.tex @@ -118,6 +118,44 @@ label=center:{{$\sum$}}, minimum width=2em}} \end{frame} +%==================================================================== +%==================================================================== +\begin{frame}{Outline}{} + \begin{itemize} + \item Introduce a number of LTV identification methods + \item Use of methods for reinforcement learning + \end{itemize} +\end{frame} + + +%==================================================================== +%==================================================================== +\begin{frame}{Motivating example}{} + \begin{block}{LQR} + \begin{itemize} + \item Control signal trajectory + \item Feedback gain + \item Dynamic programming + \item Linear systems + \item Quadratic cost + \end{itemize} + \end{block} + \pause + + \begin{block}{Iterative LQR} + \begin{itemize} + \item Linearize system + \item Differential Dynamic programming\footfullcite{levine2013guided} + \item Nonlinear systems + \item Stepsize or constraint + \end{itemize} + + \end{block} + +\end{frame} + + + %==================================================================== %==================================================================== \begin{frame}{Outline}{} @@ -150,32 +188,32 @@ label=center:{{$\sum$}}, minimum width=2em}} -\begin{frame}{LTI identification}{} - Linear in the parameters, can be written $\y = \A\w$ - \begin{align*} - \y &= - \begin{bmatrix} - {x_1} \\ \vdots \\ {x_T} - \end{bmatrix} - & &\inspace{Tn} \\ - \w &= \vec{\bmatrixx{A\T & B\T}} & &\inspace{K}\\[0.2em] - \A &= - \begin{bmatrix} - I_n \otimes x_0\T & I_n \otimes u_0\T \\ - \vdots & \vdots\\ - I_n \otimes x_{T-1}\T & I_n \otimes u_{T-1}\T - \end{bmatrix} - & &\in \mathbb{R}^{Tn\times K} - \end{align*} - \pause - - Closed form solution - \begin{align} - \w^* &= \argmin_{\w} \normt{\y - \A \w}^2 \label{eq:lscost}\\ - ~ &= \PI \y \label{eq:ls} - \end{align} - -\end{frame} +% \begin{frame}{LTI identification}{} +% Linear in the parameters, can be written $\y = \A\w$ +% \begin{align*} +% \y &= +% \begin{bmatrix} +% {x_1} \\ \vdots \\ {x_T} +% \end{bmatrix} +% & &\inspace{Tn} \\ +% \w &= \vec{\bmatrixx{A\T & B\T}} & &\inspace{K}\\[0.2em] +% \A &= +% \begin{bmatrix} +% I_n \otimes x_0\T & I_n \otimes u_0\T \\ +% \vdots & \vdots\\ +% I_n \otimes x_{T-1}\T & I_n \otimes u_{T-1}\T +% \end{bmatrix} +% & &\in \mathbb{R}^{Tn\times K} +% \end{align*} +% \pause +% +% Closed form solution +% \begin{align} +% \w^* &= \argmin_{\w} \normt{\y - \A \w}^2 \label{eq:lscost}\\ +% ~ &= \PI \y \label{eq:ls} +% \end{align} +% +% \end{frame} @@ -195,17 +233,17 @@ label=center:{{$\sum$}}, minimum width=2em}} \end{block} \pause - Parameters $\w$ are assumed to evolve according to the dynamical system - \begin{equation} - \label{eq:dynsys} - \begin{split} - k_{t+1} &= H_t k_t + w_t\\ - y_t &= \big(I_n \otimes \bmatrixx{x_t\T & u_t\T}\big) \w_t - \end{split} - \end{equation} - \pause - - Free to choose $H$, e.g., $H = I$ + % Parameters $\w$ are assumed to evolve according to the dynamical system + % \begin{equation} + % \label{eq:dynsys} + % \begin{split} + % k_{t+1} &= H_t k_t + w_t\\ + % y_t &= \big(I_n \otimes \bmatrixx{x_t\T & u_t\T}\big) \w_t + % \end{split} + % \end{equation} + % \pause + % + % Free to choose $H$, e.g., $H = I$ \end{frame} @@ -233,14 +271,16 @@ label=center:{{$\sum$}}, minimum width=2em}} %==================================================================== %==================================================================== \begin{frame}{Trend filtering} - Class of identification methods for signal reconstruction, \emph{trend filtering}~\footfullcite{kim2009ell_1, tibshirani2014adaptive}. + Class of identification methods for signal reconstruction~\footfullcite{kim2009ell_1, tibshirani2014adaptive}. \pause - As a simple example, consider the reconstruction $\hat y$ of a noisy signal $y = \{y_t\inspace{}\}_{t=1}^T$ with piecewise constant segments. - \begin{equation*} \label{eq:tf} - \minimize{\hat{y}} \normt{y-\hat{y}}^2 + \lambda\sum_t |\hat{y}_{t+1} - \hat{y}_t| - \end{equation*} - \pause + \begin{block}{Example} + Reconstruction $\hat y$ of a noisy signal $y = \{y_t\inspace{}\}_{t=1}^T$ with piecewise constant segments. + \begin{equation*} \label{eq:tf} + \minimize{\hat{y}} \normt{y-\hat{y}}^2 + \lambda\sum_t |\hat{y}_{t+1} - \hat{y}_t| + \end{equation*} + \end{block} + \begin{itemize} \item Fitness function \item (Sparsity promoting) Regularization @@ -253,33 +293,35 @@ label=center:{{$\sum$}}, minimum width=2em}} %==================================================================== \begin{frame}{Norms for Regularization}{} \begin{description} - \item[$\norm{k}_1$ 1-norm] \emph{Sparsity-promoting} penalty.\\ A solution with a small number of non-zero entries in $k$ is favored.\vspace{2mm} - \item[$\norm{k}_2$ 2-norm] Penalizes large entries in $k$. \\Does not care about small, non-zero entries. + \item[$\norm{k}_1$] \emph{Sparsity-promoting} penalty.\\ A solution with a small number of non-zero entries in $k$ is favored.\vspace{2mm} + \item[$\sum_t\norm{k_t}_2^2$] Penalizes large entries in $k_t$. \\Does not care about small, non-zero entries. + \item[$\sum_t\norm{k_t}_2$] Penalizes non-zero $k_t$. \\ Grouped lasso. \end{description} \end{frame} + + + %==================================================================== %==================================================================== -\begin{frame}{Identification methods}{} +\begin{frame}{Low-frequency time evolution} + System with slowly varying dynamics \begin{itemize} - \item Introduce a number of optimization problems - \item The difference lies in the regularization term - \item The regularization term can be given statistical interpretation + \item Temperature change + \item Smooth nonlinearities + \item Implicit assumption: parameters $k$ evolve as Brownian motion %($H = I$) \end{itemize} + \centering + \includegraphics[width=0.5\linewidth]{figs/kalman-0.png} \end{frame} - -%==================================================================== -%==================================================================== \begin{frame}{Low-frequency time evolution} A slowly varying signal is characterized by \emph{small first-order time differences}. \pause - Assume parameters $k$ evolve according to Brownian motion ($H = I$) - \begin{equation} \label{eq:slow} \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+1} - \w_{t}}^2 \end{equation} @@ -292,7 +334,19 @@ label=center:{{$\sum$}}, minimum width=2em}} \end{align} \pause - Parameters found efficiently by Dynamic programming! + \begin{itemize} + \item Parameters found efficiently by Dynamic programming! + \item Generalization in paper. + \end{itemize} +\end{frame} + + +\begin{frame}{Low-frequency time evolution} + \centering + \only<1>{\includegraphics[width=0.8\linewidth]{figs/kalman-1.png}} + \only<2>{\includegraphics[width=0.8\linewidth]{figs/kalman-4.png}} + \only<3>{\includegraphics[width=0.8\linewidth]{figs/kalman-7.png}} + \only<4>{\includegraphics[width=0.8\linewidth]{figs/kalman-9.png}} \end{frame} @@ -300,25 +354,35 @@ label=center:{{$\sum$}}, minimum width=2em}} %==================================================================== %==================================================================== -\begin{frame}{Smooth time evolution} - A smoothly varying signal is characterized by \emph{small second-order time differences}. - \pause +% \begin{frame}{Smooth time evolution} +% A smoothly varying signal is characterized by \emph{small second-order time differences}. +% \pause +% +% Parameters $k$ are integrated twice (inertia) +% \begin{equation} \label{eq:smooth} +% \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+2} -2 \w_{t+1} + \w_t}^2 +% \end{equation} +% \pause +% +% \begin{itemize} +% \item Closed-form solution and solution (intractable) +% \item Dynamic programming solution (efficient) +% \end{itemize} +% \end{frame} - Parameters $k$ are integrated twice (inertia) - \begin{equation} \label{eq:smooth} - \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+2} -2 \w_{t+1} + \w_t}^2 - \end{equation} - \pause +%==================================================================== +%==================================================================== +\begin{frame}{Piecewise constant time evolution} + Piecewise constant dynamics \begin{itemize} - \item Closed-form solution and solution (intractable) - \item Dynamic programming solution (efficient) + \item Abrupt failure + \item Change of operation mode + \item Non-holonomic constraints + \item Number of change points known / unknown \end{itemize} \end{frame} - -%==================================================================== -%==================================================================== \begin{frame}{Piecewise constant time evolution}\label{sec:pwconstant} A signal which is mostly flat, with a small number of distinct level changes, is characterized by a \emph{sparse first-order time difference}. \pause @@ -331,8 +395,6 @@ label=center:{{$\sum$}}, minimum width=2em}} We can give \labelcref{eq:pwconstant} an interpretation as a \emph{grouped-lasso} cost function. - Penalty on the 1-norm on the \emph{length} of the difference vectors $\w_{t+1} - \w_t$ since $\norm{\normt{\cdot}}_1 = \normt{\cdot}$. - \pause \end{frame} @@ -351,38 +413,13 @@ label=center:{{$\sum$}}, minimum width=2em}} %==================================================================== %==================================================================== \begin{frame}{Piecewise constant time evolution -- Implementation} - Due to the non-squared norm penalty $$\sum_t \normt{ \w_{t+1} - \w_t}$$ problem \labelcref{eq:pwconstant} is significantly harder to solve than \labelcref{eq:smooth}. + Due to the non-squared norm penalty $$\sum_t \normt{ \w_{t+1} - \w_t}$$ problem \labelcref{eq:pwconstant} is significantly harder to solve than \labelcref{eq:slow}. An efficient implementation using the linearized ADMM algorithm \footfullcite{parikh2014proximal} is made available in the accompanying repository. \url{https://github.com/baggepinnen/LTVModels.jl} \end{frame} - - -%==================================================================== -%==================================================================== -\begin{frame}{Summary} - \begin{table}[] - \centering - \caption{Summary of optimization problem formulations. \hspace{\textwidth} $D_n$ refers to parameter vector time-differentiation of order $n$.} - \label{tab:opts} - \begin{tabular}{@{}lll@{}} - \toprule - Norm & $D_n$ & Result \\ \midrule - 1 & 1 & Small number of steps (piecewise constant) \\ - 1 & 2 & Small number of bends (piecewise affine) \\ - 2 & 1 & Small steps (slowly varying) \\ - 2 & 2 & Small bends (smooth) \\ \bottomrule - \end{tabular} - \end{table} - -\end{frame} - - - - - %==================================================================== %==================================================================== \begin{frame}{Example -- Jump-linear system} @@ -439,6 +476,80 @@ label=center:{{$\sum$}}, minimum width=2em}} + +% %==================================================================== +% %==================================================================== +% \begin{frame}{Summary} +% \begin{table}[] +% \centering +% \caption{Summary of optimization problem formulations. \hspace{\textwidth} $D_n$ refers to parameter vector time-differentiation of order $n$.} +% \label{tab:opts} +% \begin{tabular}{@{}lll@{}} +% \toprule +% Norm & $D_n$ & Result \\ \midrule +% 1 & 1 & Small number of steps (piecewise constant) \\ +% 1 & 2 & Small number of bends (piecewise affine) \\ +% 2 & 1 & Small steps (slowly varying) \\ +% 2 & 2 & Small bends (smooth) \\ \bottomrule +% \end{tabular} +% \end{table} +% +% \end{frame} + + + +%==================================================================== +%==================================================================== +\begin{frame}{Example -- Reinforcement learning} \label{sec:rl} + \begin{itemize} + \item Identify LTV models for reinforcement learning + \item Dampen oscillations of a pendulum on a cart + \item Quadratic cost on states and control + \begin{enumerate} + \item Fit model along trajectory + \item Optimize the cost function under + model using iterative LQG (DDP)\footnote{Implementation made available at + \href{github.com/baggepinnen/DifferentialDynamicProgramming.jl}{github.com/baggepinnen/DifferentialDynamicProgramming.jl}} + \item Bounds on the deviation between optimized trajectory and previous trajectory. + \end{enumerate} + \end{itemize} + \centering + \vspace{-5mm}\hspace{5cm}\includegraphics[width=0.3\linewidth]{figs/pendcart_draw.jpg} +\end{frame} +%==================================================================== +%==================================================================== +\begin{frame}{Example -- Reinforcement learning}{} + \footnotetext{$\normt{\w_{t+1} - \w_t}^2$} + \small + \begin{columns} + \begin{column}{0.5\textwidth} + We compare three different models + \begin{itemize} + \item The ground truth system model + \item LTV model \labelcref{eq:slow}\footnotemark + \item LTI model + \end{itemize} + \end{column} + \pause + \begin{column}{0.6\textwidth} + \begin{figure}[htp] + \centering + \setlength{\figurewidth}{0.99\linewidth} + \setlength{\figureheight }{5cm} + \pgfplotsset{every axis/.append style={ + legend style={draw=black!20!white, yshift=1mm}, + title style={yshift=1.3mm}, + }} + \input{../figs/ilc.tex} + \end{figure} + \end{column} + \end{columns} + +\end{frame} + + + + %==================================================================== %==================================================================== \begin{frame}{Example -- Robot arm} @@ -506,57 +617,6 @@ label=center:{{$\sum$}}, minimum width=2em}} \end{frame} - -%==================================================================== -%==================================================================== -\begin{frame}{Example -- Reinforcement learning} \label{sec:rl} - \begin{itemize} - \item Identify LTV models for reinforcement learning - \item Dampen oscillations of a pendulum on a cart - \item Quadratic cost on states and control - \begin{enumerate} - \item Fit model along trajectory - \item Optimize the cost function under - model using iterative LQG (DDP)\footnote{Implementation made available at - \href{github.com/baggepinnen/DifferentialDynamicProgramming.jl}{github.com/baggepinnen/DifferentialDynamicProgramming.jl}} - \item Bounds on the deviation between optimized trajectory and previous trajectory. - \end{enumerate} - \end{itemize} - \centering - \vspace{-5mm}\hspace{5cm}\includegraphics[width=0.3\linewidth]{figs/pendcart_draw.jpg} -\end{frame} -%==================================================================== -%==================================================================== -\begin{frame}{Example -- Reinforcement learning}{} - \footnotetext{$\normt{\w_{t+1} - \w_t}^2$} - \small - \begin{columns} - \begin{column}{0.5\textwidth} - We compare three different models - \begin{itemize} - \item The ground truth system model - \item LTV model \labelcref{eq:slow}\footnotemark - \item LTI model - \end{itemize} - \end{column} - \pause - \begin{column}{0.6\textwidth} - \begin{figure}[htp] - \centering - \setlength{\figurewidth}{0.99\linewidth} - \setlength{\figureheight }{5cm} - \pgfplotsset{every axis/.append style={ - legend style={draw=black!20!white, yshift=1mm}, - title style={yshift=1.3mm}, - }} - \input{../figs/ilc.tex} - \end{figure} - \end{column} - \end{columns} - -\end{frame} - - %==================================================================== %==================================================================== \begin{frame}{Discussion}{} @@ -574,8 +634,8 @@ label=center:{{$\sum$}}, minimum width=2em}} \begin{frame}{Discussion -- Reinforcement learning}{} \begin{itemize} \item In ILC and trajectory centric reinforcement learning, a first-order approximation to the dynamics is used for efficient optimization - \item Validity of the approximation ensured by constraints between two consecutive trajectories. - \item[\nice{+}] Proposed identification methods attractive in applications such as guided policy search (GPS)~\footfullcite{levine2013guided, levine2015learning} and non-linear ILC (ILC)~\footfullcite{bristow2006survey}. \\ + \item Validity of approximation ensured by constraints between two consecutive trajectories. + \item[\nice{+}] Useful in applications such as guided policy search (GPS)~\footfullcite{levine2013guided, levine2015learning} and non-linear ILC (ILC)~\footfullcite{bristow2006survey}. \\ Can lead to dramatically decreased sample complexity. \end{itemize} \end{frame} @@ -600,6 +660,7 @@ label=center:{{$\sum$}}, minimum width=2em}} \begin{itemize} \item Analysis of identifyability \item Kalman smoother for efficient identification + \item More identification methods \end{itemize} \end{frame}