From f20897a3160f1697c98738e577527798cd1ee63b Mon Sep 17 00:00:00 2001 From: Tue Herlau <tuhe@dtu.dk> Date: Fri, 18 Nov 2022 11:20:16 +0100 Subject: [PATCH] minor tex tweaks + references --- Latex/robustq.aux | 7 +- Latex/robustq.log | 99 +++++++++-------- Latex/robustq.out | 5 +- Latex/robustq.pdf | Bin 148360 -> 184305 bytes Latex/robustq.synctex.gz | Bin 14145 -> 53259 bytes Latex/robustq.tex | 227 +++++++++++++++++++++++++++++++++++++-- docs/1605.08803.pdf | Bin 0 -> 6681321 bytes docs/liu22a.pdf | Bin 0 -> 534732 bytes src/plots.py | 68 ++++++++++++ 9 files changed, 347 insertions(+), 59 deletions(-) create mode 100644 docs/1605.08803.pdf create mode 100644 docs/liu22a.pdf create mode 100644 src/plots.py diff --git a/Latex/robustq.aux b/Latex/robustq.aux index 52a7dba..604ca79 100644 --- a/Latex/robustq.aux +++ b/Latex/robustq.aux @@ -21,8 +21,11 @@ \citation{liu22a} \babel@aux{english}{} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Bellman operator}{2}{subsection.1.1}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2}Generalization:}{4}{section.2}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {3}Question: Can we use an embedding?}{5}{section.3}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.0.1}Defining objectives}{6}{subsubsection.3.0.1}\protected@file@percent } \bibstyle{alpha} \bibdata{pensum} \bibcite{liu22a}{LBB{$^{+}$}22} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Ideas:}{2}{subsection.1.1}\protected@file@percent } -\gdef \@abspage@last{2} +\gdef \@abspage@last{8} diff --git a/Latex/robustq.log b/Latex/robustq.log index e2dbfc7..bbcc4d4 100644 --- a/Latex/robustq.log +++ b/Latex/robustq.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=pdflatex 2022.8.30) 11 NOV 2022 11:04 +This is pdfTeX, Version 3.141592653-2.6-1.40.22 (TeX Live 2022/dev/Debian) (preloaded format=pdflatex 2022.8.30) 16 NOV 2022 14:30 entering extended mode \write18 enabled. %&-line parsing enabled. @@ -832,24 +832,24 @@ File: l3backend-pdftex.def 2022-01-12 L3 backend support: PDF output (pdfTeX) (./robustq.aux) \openout1 = `robustq.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. -LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 59. -LaTeX Font Info: ... okay on input line 59. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. +LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 62. +LaTeX Font Info: ... okay on input line 62. 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\newcommand{\std}{\operatorname{std} } +\newcommand{\rob}{\mathrm{rob} } +\newcommand{\KL}{\operatorname{KL} } +\newcommand{\Trob}{\mathcal{T}^{\text{rob}}_\delta } \title{Robust Q learning} \author{Tue Herlau} @@ -84,29 +87,231 @@ We also assume that the transition dynamics is jointly Gaussian, viz.: p_0(s'| s, a) = \mathcal{N}(x' |\mu^f(s,a), \sigma^f(s,a)^2 ) \end{align} +\subsection{Bellman operator} + +The bellman operator defined in the main reference is: +\begin{align} + \Trob(Q)(s,a) = r(s,a) + \gamma \sup_{\beta \geq 0}\left\{ + -\beta \log \left( \EE_{p^0_{s,a} } \left[ e^{ -\frac{1}{\beta}\max_b Q(s',b) } \right] \right) - \beta \delta + \right\} +\end{align} + This parametrization now allows us to solve the integral and get an explicit expression for the minimization problem. The integral is a standard product of two Gaussians and it becomes: \begin{align} \EE_{p_0}\left[ e^{ \frac{-V(s') }{\beta} } \right] -& = e^{a} \sqrt{ 2\pi \left( \frac{\sigma^v}{\sqrt{\beta} } \right)^2 } \left[ +& = e^{-\frac{a}{\beta}} \sqrt{ 2\pi \beta \left( \sigma^v \right)^2 } \left[ \frac{ e^{\frac{-1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{S_\beta^2} } }{ \sqrt{ 2\pi S_\beta^2 } } -\right], \quad S_\beta = \left( \sigma^f \right)^2 + \frac{ \left( \sigma^v \right)^2 }{\beta} +\right], \quad S_\beta^2 = \left( \sigma^f \right)^2 + \beta \left( \sigma^v \right)^2 \end{align} - This objective can then be maximized wrt. $\beta$ using a simple bisection algorithm or similar. - - -This leaves two things that need to be learnable. Firstly, the $s'$ distribution. This can be learned using standard techniques for learning a stochastic neural networks (i.e. minimize the log likelihood). Then, to learn the value function, knowing it represents +We plug it in and get that the quantity to be optimized is: +\begin{align} +& a -\beta \log( \sqrt{2\pi} \sigma^v) - \beta \frac{1}{2}\log \beta ++ \beta \frac{1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{S_\beta^2} ++ \beta \log \sqrt{2\pi} + \beta \log S_\beta - \beta \delta \\ +& = +a - \beta \frac{1}{2}\log \beta ++ \beta \frac{1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{S_\beta^2} + + \beta \log S_\beta - \beta (\delta + \log \sigma^v) \\ + & = a -\beta \log \left(\frac{ \sqrt{\beta} \sigma^v }{ S_\beta } \right) + + \beta \frac{1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{S_\beta^2} + - \beta \delta \\ + & = a -\beta \log \left(\frac{ \sqrt{\beta} \sigma^v }{ \sqrt{ (\sigma^2)^2 + \beta (\sigma^v)^2 } } \right) + + \beta \frac{1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{S_\beta^2} + - \beta \delta \\ + & = + a +\frac{1}{2}\beta \log \left( 1 + \frac{ (\sigma^f)^2 }{ \beta (\sigma^v)^2 } \right) + + \beta \frac{1}{2}\frac{ \left( \mu^f - \mu^v \right)^2}{(\sigma^f)^2 + \beta (\sigma^v)^2 } + - \beta \delta \\ +\end{align} +(TODO: simplify that ridiculous mess). +The questions is if we can help ourselves a bit in the optimization task by somehow showing this function has a single optimum or similar -- I am not sure that is possible, but you never know. A step towards this is to differentiate it. vlg., lets assume it has the form: +\begin{align} +F(x) & = x \log \left( 1+ \frac{\lambda}{x} \right) + \frac{A}{ \frac{\lambda}{x} +1 } - B x \\ +& = x \log h(x) + \frac{A}{h(x)} - Bx \\ +h(x) & = 1+ \frac{\lambda}{x} +\end{align} +We differentiate to get: +\begin{align} +F'(x) & = \log h(x) + x \frac{h'(x) }{h(x)} + +\frac{h'(x) }{h(x) }\left( \frac{-A}{h(x)} \right) +- B \\ +& = \log h(x) + \frac{h'(x) }{h(x) }\left[\frac{h(x) x - A}{h(x) } \right] - B \\ +\frac{h'(x) }{h(x)}& = \frac{ + -\frac{\lambda}{x^2} +}{ +1+ \frac{\lambda}{x} +} = \frac{1}{x}\frac{-\lambda}{x + \lambda} +%\log \left( 1+ \frac{\lambda}{x} \right) + \frac{A}{ \frac{\lambda}{x} +1 } - B x \\ +%x \log h(x) + \frac{A}{h(x)} - Bx \\ +%h(x) & = 1+ \frac{\lambda}{x} +\end{align} +So therefore +\begin{align} +F'(x) & = \log h(x)-B - \frac{\lambda}{x + \lambda} +\left[ \frac{x h(x) - A}{x h(x)} \right] \\ +& = \log h(x) - B - \frac{\lambda}{x + \lambda} +\left[ \frac{x + \lambda - A}{x + \lambda } \right] +\end{align} +So we can now start to bound this. We get that +\begin{align} +F'(x) \leq \frac{\lambda}{x} - B - \frac{\lambda}{x + \lambda} +\left[ \frac{x + \lambda - A}{x + \lambda } \right] +\end{align} +Solving for $F'(x) \leq 0$ can be done by reducing this to a 3rd degree polynomial and solve for the smallest root. Less exactly, we can also just ignore higher-order terms in $x$ and solve to get: +\begin{align} +0 & < \frac{\lambda}{x} - B - \frac{\lambda}{x + \lambda} + \frac{A}{(x + \lambda)x} \\ +x & > \frac{1}{2B} \sqrt{ (\lambda B)^2 + 4 B (\lambda^2 + \lambda A) }-\frac{\lambda}{2} +\end{align} +This can be simplified obviously (TODO). The lower bound can be obtained by simply dropping terms to get: +\begin{align} + F'(x) & > \log h(x) - B - 1 \\ +x & \leq \frac{\lambda }{ e^{B + 1}- 1 } +\end{align} +(TODO: Much better lower bound can be obtained by back-substituting upper and lower bounds into expression for $F'$ and solve again for lower bound. This will give explicit $A$-dependency. + +This gets us an interval for $x$ within which to search for a root using the bisection method: \begin{align} -V(s) = \max Q(s,b) +x \in \left[ + \frac{\lambda }{ e^{B + 1}- 1 }\ ; \ \frac{1}{2B} \sqrt{ (\lambda B)^2 + 4 B (\lambda^2 + \lambda A) }-\frac{\lambda}{2} \right] \end{align} -This too can be learned using gradient descent. We can even let it share layers with the dynamics. Then finally the $Q$-function must be learned using the new (robust) optimization targets. -\subsection{Ideas:} +\section{Generalization:} +Suppose the dynamics of the model is given by: +\begin{align} +x_{k+1} & = f(x_k, a_k, L\epsilon_k) \\ +w_k & \sim P_W(\cdot) +\end{align} +Where $L$ is a linear operator (let's see). We then define the state as: +\begin{align} +s_k \equiv \begin{bmatrix} + x_k \\ x_{k-1} \\ a_{k-1} \\ L\epsilon_k +\end{bmatrix} +\end{align} +This definition contains redundant information. The update rule is then +\begin{align} +s_{k+1} = \bar f(s_k, a_k, L\epsilon_k) += \begin{bmatrix} + f(x_k, a_k, L\epsilon_k) \\ x_k \\ a_k \\ L \epsilon_k +\end{bmatrix} +\end{align} +Then the hard part of the update rule is: +\begin{align} +\EE_{p^0_{s,a} } \left[e^{-\frac{V(s')}{\beta} } \right] +\end{align} +Where $p_0$ is the dynamics of the (unperturbed) version of the system. +To compute the expectation, we will assume a special form of the value function namely: +\begin{align} +V(s_{k+1} ) \approx \hat V(s_{k+1} ) = V^{(1)}(x_k,a_k) + V^{(2)}(x_k,a_k)^{\top} w_k + \frac{1}{2}w_k^\top V^{(3)}(x_k,a_k)w_k +\end{align} +We will also assume the transition dynamics takes the form: +\begin{align} +p^0_{s,a} \approx \hat p^0_{s,a}(s' | s,a) = + V^{(1)}(x_k,a_k) + V^{(2)}(x_k,a_k)^{\top} w_k + \frac{1}{2}w_k^\top V^{(3)}(x_k,a_k)w_k\\ +\end{align} +Then the transition function is assumed to be: +\begin{align} +x_{k+1} \approx \bar{f}(x_k, a_k, w_k) = f^{(1)}(x_k, a_k) + +f^{(2)}(x_k, a_k)^\top w_k ++ \frac{1}{2} w_k^\top f^{(3)}(x_k, a_k) w_k +\end{align} +The expectation can then still be computed exactly, and the coefficients above can be learned using regression (even local linear?). + +\section{Question: Can we use an embedding?} +Assume a new generative model of the form: +\begin{align} +x_{k+1} = f(h(x_k, a_k) + w_k) +\end{align} +Where $h$ is an embedding and $w_k$ is assumed to be $\mathcal N(0,I)$. Let $g = f^{-1}$ so that +\begin{align} +g(x_{k+1} ) = h(x_k, a_k) + w_k. +\end{align} +Then it holds that +\begin{align} +p(x_{k+1} | x_k, a_k)dx_{k+1} = \mathcal N(w_k=g(x_{k+1} )-h(x_k, a_k); 0, I ) |J_g(x_{k+1} ) | dw_k +\end{align} +We also assume that $x_{k-1}$, $a_{k-1}$ and $w_{k-1}$ are part of the state $s_k$. Therefore we can introduce the approximation (assuming $W$ is a second degree polynomial): +\begin{align} +V(s_{k+1}) \approx \hat V(s_{k+1} ) = \hat V(x_k, a_k, w_k) = W_{h(x_k, a_k)}(w_k; \Phi). +\end{align} + +\subsubsection{Defining objectives} +When learning the objectives, we train $Q$ by minimizing the specific objective given in the paper (viz., using the simulator) as: +\begin{align} +Q(s,a) = ... +\end{align} +So you could just as well train $V$ directly and use the simulator to define $Q$... but this way is more elegant (or is it?). Robust learning requires $V$ internally (to get 2nd degree polyal). Or does it? + +The $Q$-learning objective satisfy the Bellman equation +\begin{align} +Q(s_k,a_k) = \inf_{\KL(p|p_0)<\delta }\left\{ + r(s_{k}, a_{k} ) + \gamma \max_b Q(s_{k+1}, b) + \right\} +\end{align} +Re-writing this, and assuming for simplicity reward only depends on state/action, we get that for any constant $C$: +\begin{align} +Q^\rob(s_k,a_k) & = r(s_{k}, a_k) + \sup_{\beta >0}\left\{ -\beta \log \EE\left[ e^{- \frac{\max_b Q^\rob(s_{k+1}, b) }{\beta} }\right] + \delta \beta + \right\} \\ + & = r(s_{k}, a_k) + C + \sup_{\beta >0}\left\{ -\beta \log \EE\left[ e^{- \frac{V^\rob(s_{k+1} ) - C }{\beta} }\right] + \delta \beta + \right\} \\ +V^\rob(s_k) & = \max_b Q^{\rob}(s_k, b) +\end{align} +for any constant $C$ (probably a bad idea to have such a constant). Then to proceed, we need to find a way to compute the above expectation exactly and update. Let's say we divide this into two parts. The first computes this expectation (and updates). The second considers how to fit $V^\rob$. For each $V^\rob$, we can compute the target easily. Then perform the update as a regression problem (batched) which is trained jointly with $o$. IOW, we get that: +\begin{align} +V^\rob(s_k) & = \max_b Q^\rob(s_k, b) \\ +& \approx \hat V^\rob(s_k) \\ +& = C_0( h(x_{k-1}, a_{k-1}) ) ++ C_1( h(x_{k-1}, a_{k-1}) ) w_k ++ \frac{1}{2} w_k^\top C_2( h(x_{k-1}, a_{k-1}) ) w_k \\ +w_k & = g(x_k) - h(x_{k-1}, a_{k-1} ) +\end{align} +Train this using gradient descent, probably keeping $C_2$ diagonal. + +What happens if you use the $Q$-learning function and try to integrate? Let each coordinate be a 2nd degree polynomial. + + + +$V$ to minimize the recursion: + + + +is a second-order approximation of $V$ only depending on the latent state $h(x_k, a_k)$. We can learn the latent embedding by minimizing a loss over both $V$ and over $g$ (issue: $g$ most well-defined. Possibly fix $h$ insofar as second-order approx is concerned and use that $\epsilon$ automatically normal? Alternatively use that $V$ must satisfy, before optimization, that: +\begin{align} +V(s_{k+1}) = \max a_k \EE\left[ r(x_k, a_k) + V(x_{k+1} ) \right] +\end{align} + + +Then it holds that +\begin{align} +J_g(x_{k+1}) +\end{align} + + + + +that $h_k = h(x_k, a_k, w_k)$ is a latent embedding of some sort and the next state is given by: +\begin{align} +x_{k+1} = o(h_k) +\end{align} +Given embedding, we can write the value function as a quadratic in embedding parameter? + + + In other word the expectation is with respect to +\begin{align} +0 +\end{align} + +However, we can re-write this to be: + $$ -s' = h(s, \epsilon) +p(s' | s,a) $$ -I.e. let $s'$ be a function of something that looks Gaussian. Idea is that $V(s')$ should look +We will then assume a special parametrization of the value function in the form that: +\begin{align} +V(s') \approx V^{(1)}(s) + V^{(2)}(s) L \epsilon + (L\epsilon)^\top V^{(3)} (s)(L\epsilon) +\end{align} +There are some questions about whether this is a rigorous construction or not. 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