frame-training.tex

% SPDX-License-Identifier: CC-BY-4.0
% Copyright 2018 Toni Dietze
\documentclass[beamer]{standalone}
% \usepackage{amsmath}  % has to be loaded before unicode-math
% \usepackage[math-style=ISO]{unicode-math}
% \usepackage{tikz}
% \usetikzlibrary{calc}
% \usetikzlibrary{fit}
% \usetikzlibrary{positioning}
\input{preamble.tex}
\begin{document}
%         1 2 3 4 5 6 7 8 9
\timeline{- + + + - - - - -}{tlDefCorpus}
\timeline{- - + + - - - - -}{tlDefLklhd}
\timeline{- - - + - - - - -}{tlDefMle}
\timeline{- - - - + + + + +}{tlStack}
\timeline{- - - - + + + + +}{tlPlot}
\timeline{- - - - - + + + +}{tlPlotUnderFitting}
\timeline{- - - - - - + + +}{tlPlotOverFitting}
\timeline{- - - - - - - + +}{tlHoldOut}
\timeline{- - - - - - - - +}{tlPlotHoldOut}
\begin{standaloneframe}[t]{\jobname}
	\centering%
	\alt<\tlStack>{%
		%\newcommand{\stacklist}{8/0.390625, 7/0.78125, 6/1.5625, 5/3.125, 4/6.25, 3/12.5, 3/12.5, 2/25, 1/50, 0/100}%
		%
		% let n = 8; size = 1.25; dist = 10 in mapM_ (\ (a, b) -> putStrLn $ ", " ++ show a ++ "em/" ++ show b ++ "%") $ map (\i -> (size * (n + dist) * i / (n * (dist + i)), 100 * 2 ** (-i))) [n, n-1 .. 0]
		\newcommand{\stacklist}
		{ 1.25em/0.390625%
		, 1.1580882352941178em/0.78125%
		, 1.0546875em/1.5625%
		, 0.9375em/3.125%
		, 0.8035714285714286em/6.25%
		, 0.6490384615384616em/12.5%
		, 0.46875em/25.0%
		, 0.2556818181818182em/50.0%
		, 0.0em/100.0%
		}%
	}{%
		\newcommand{\stacklist}{0/100}%
	}%
	\begin{tikzpicture}
	[ data/.style = {align=center, draw, rounded corners}
	, func/.style = {align=center, draw}
	]
		\node[data] (training data) {\strut\color{HKS41K70} training data};
		\node[func, below = 1.5em of training data, align = center] (training) {training \\ (maximize {\color{HKS41K70}reward})};
		\foreach \s/\c in \stacklist {
			\node
				[ data
				, left = of training
				, align = center
				, color = black!\c
				, fill = white
				, xshift = \s
				, yshift = \s
				] (model) {model \(\mathbfit{M}\)\\ \((p_θ \mid θ ∈ Θ)\)};
			\node
				[ data
				, below = 1.5em of training
				, color = black!\c
				, fill = white
				, xshift = \s
				, yshift = \s
				] (param) {\strut \(\hat{θ} ∈ Θ\)};
		}

		\draw[->] (training data) -- (training);
		\draw[->] (model) -- (training);
		\draw[->] (training) -- (param);

		\visible<\tlHoldOut>{%
			\node[data, right = 0.3333em of training data] (hold-out data) {\strut\color{HKS57K100} hold-out data};
			\node[inner sep = 0] at ([yshift = 1.75em]{$(training data)!0.5!(hold-out data)$}) (data) {data};
			\node[data, fit=(data) (training data) (hold-out data)] {};
			\node[func, align = center] (validation) at (hold-out data |- param) {hold-out \\ validation};
			\foreach \s/\c in \stacklist
				\node
					[ data
					, right = of validation
					, color = black!\c
					, fill = white
					, xshift = \s
					, yshift = \s
					] (reward) {\strut\color{HKS57K100} reward};

			\draw[->] (hold-out data) -- (validation);
			\draw[->] (param) -- (validation);
			\draw[->] (validation) -- (reward);
		}%
		\path[use as bounding box] ([xshift = 1.25em]reward.east);
	\end{tikzpicture}%
	\only<\tlDefCorpus,\tlDefLklhd,\tlDefMle>{%
		\\[0.5em]
		\begin{columns}[t]
		\column<\tlDefCorpus>{0.19\linewidth}
			\centering
			training data
			\begin{block}{\strut Corpus}
				\(c\colon \trees{Σ} → ℝ_{≥ 0}\)
				\\[0.5em] \(\support{c}\) finite
			\end{block}
		\column<\tlDefLklhd>{0.35\linewidth}
			\centering
			reward\vphantom{g}
			\begin{block}{\strut Likelihood}
			\vspace{0.65em}%
			\(\displaystyle
				\lklhd{c}{p} = ∏_{t ∈ \support{c}} p(t)^{c(t)}
			\)
			\end{block}
		\column<\tlDefMle>{0.39\linewidth}
			\centering
			training
			\begin{block}{Maximum Likelihood Estimation (mle)}
			\(\displaystyle
				\mle[\mathbfit{M}]{c} = \argmax_{θ ∈ Θ} \lklhd{c}{p_θ}
			\)
			\end{block}
		\end{columns}
	}%
	\only<\tlPlot>{%
		\\[1.5em]
		\begin{tikzpicture}[anchor = base]
			\only<\tlPlotUnderFitting>{%
				\fill[fill=lightgray, path fading=east] (0, 0) rectangle (9em, 5em);
				\node[gray, anchor=base east] at ( 9em, 0.3em) {underfitting};
			}%
			\only<\tlPlotOverFitting>{%
				\fill[fill=lightgray, path fading=west] (11em, 0) rectangle (16em, 5em);
				\node[gray, anchor=base west] at (11em, 0.3em) {overfitting};
			}%

			% axes
			\draw[<->] (18em, 0)
				-- (0, 0  ) node[pos = 0, below right, xshift = -2em] (xlabel) {model complexity}
				-- (0, 5em) node[midway, above, rotate = 90] (ylabel) {reward};

			\draw[HKS41K70] (0, 1em)
				.. controls +(4em, 3em) and +(0, 0) .. (16em, 4.5em)
				node[right] {reward for training data};

			\visible<\tlPlotHoldOut>{%
				\draw[HKS57K100] (0, 0.5em)
					.. controls +(0  , 0) and +(-4em, 0) .. (10em, 3em)
					.. controls +(3em, 0) and +( 0  , 0) .. (16em, 2.5em)
					node[right] {reward for hold-out data};

				\draw[dashed] (10em, -0.05) -- ++(0, 5em);
				\node at ({10em, 0} |- xlabel.base) {chosen model};
			}%
		\end{tikzpicture}
	}%
\end{standaloneframe}
\end{document}