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%========================================

\setbeamercolor{block title}{fg=white,bg=red}

\title[\qquad FIDDLE for ATE] % (optional, use only with long paper titles)
{\black{\Large \bf
Factor Informed Double Deep Learning Estimation\\
For Average Treatment Effect
}}

\author[\bf Jianqing Fan (\ora{Princeton} {\bf University})] % (optional, use only with lots of authors)
{ \r{\Large Jianqing Fan}}

\date{}

%\institute%[\b{Princeton} University] % (optional, but mostly needed)
%{ \large
%  \ora{Princeton} University\\
%  \bigskip
%  \href{http://www.princeton.edu/~jqfan}{http://www.princeton.edu/$\sim$jqfan}
% }


\hypersetup{pdfpagemode=FullScreen}
\begin{document}
%\renewcommand{\pause}{}

%------------------------------------------------
\begin{frame}
  \titlepage
  \vspace*{-0.6 in}
  \begin{center}
    \ora{Princeton} \black{University}\\
    \href{https://fan.princeton.edu/}{https://fan.princeton.edu/}\\
    with \b{Soham Jana}, \b{Sanjeev Kulkarni} and \b{Qishuo Yin}
  \end{center}
  %\vspace*{0.3in}
  \centerline{
    \includegraphics[height=1.2in]{figs/firestone2}\;
    \includegraphics[height=1.2in]{figs/sherred1}\;
    \includegraphics[height=1.2in]{figs/Princeton01}
  }
\end{frame}



% outline page
\begin{frame}
\heading{Outlines}
\begin{enumerate}  \itemsep 0.15in
	\item Introduction
	\item FAST-NN (FAN-Lasso)
	\item <2-> FIDDLE
	\item <2-> Theoretical Guarantee 
	\item <3-> Numerical Results
	\item <3-> Conclusion Remarks
\end{enumerate}

\vspace{-1.5 in}
	\hfill	{
	\includegraphics[height=0.75in]{Figs/JanaSoham}\;
	\includegraphics[height=0.75in]{Figs/KulkarniSanjeev}; 
	\includegraphics[height=0.75in]{Figs/YinQishuo}}
\vspace*{-0.0 in}

\hfill \visible<1->{\scriptsize Soham Jana  \qquad  Sanjeev Kulkarni \qquad Qishuo Yin \quad}

\vspace*{.5 in}
\end{frame}


% introduction page
\section{Introduction}
\vfill
\begin{frame}
	\begin{center}
		\huge \bf \textsf{Introduction}
	\end{center}
		\addtocounter{framenumber}{-1}
\end{frame}






\begin{frame}
	\heading{Deep Learning}
	% \rst has achieved tremendous successes over the last decade.
	%Breakthrough examples include human-level skills
	
	\begin{itemize}
		\item \b{Games}: Atari, Go, Starcraft,  Texas hold'em,  \ldots \pause
		\smallskip
		\item \b{Control}: robotic hand, automatic driving, fleet management,  \ldots \pause
		\smallskip
		%\item \b{Interact}: auto-driving, recommend, personalized medicine, rideshare, \ldots
		\smallskip
		\item \b{Prediction}:  peotein folding, LLM, generative AI \ldots
	\end{itemize}
	\vspace*{-0.3in}
	
	\begin{center}
		\begin{tabular}{ccc}
			%	\includegraphics[width=0.26\textwidth]{\figsk/gym}
			\visible<1->{	\includegraphics[width=0.28\textwidth]{\figsk/alphago}}
			&
			\visible<1->{\includegraphics[width=0.28\textwidth]{\figsk/holdem}}
			%& \includegraphics[width=0.28\textwidth]{\figsk/rubic} 
			& \visible<3->{ \includegraphics[width=0.28\textwidth]{\figs/LLM-AI}}\\
			\visible<2->{	\includegraphics[width=0.28\textwidth]{\figsk/rubic} }
			%	\includegraphics[width=0.28\textwidth]{\figsk/drones} 
			&  \visible<2->{\includegraphics[width=0.28\textwidth]{\figs/Autodriving} }
			&  \visible<3->{\includegraphics[width=0.3\textwidth]{\figs/CausalAI2}}
		\end{tabular}
	\end{center}
	
	
\end{frame}





\begin{frame}
	\heading{Statistical Modeling}
\begin{itemize}
	\item [\rst] Nonparametric regression,  density estimation (ImageGen, DALLE-E)
	
	\item [\rst] Semiparametric, GLIM, Quantile regression, Survival Analysis \pause
	
	\item [\rst] Transfer and generative learning
	
	\item [\rst]  Causality Pursuit, Reinforcement Learning, and Fine Tuning\pause
	
\end{itemize}
\vfill 
\begin{center}
	\huge
	\r{Adapt to unknown low-d structure}  \pause
	\vfill
	
	\b{Causal Inference}
\end{center}
\end{frame}







\begin{frame}
\heading{Does aspirin relieves headaches?}
\begin{itemize}\itemsep 0.2 in
    \item [\rst] \r{Task}: study whether taking aspirin will help relieve headaches; 
    \item [\rst] \b{Unit}: the person;
    \pause

    \item [\rst] \b{Treatment T}: Aspirin / no Aspirin; 
    \item [\rst] \b{Outcome y}: headache / no headache; 
  
    \pause 
    \item [\rst] \r{Causal (Treatment) Effect}: improvement of headache relief by Aspirin;
    \item[\rst]  \b{Covariate x}: pretreatment information of the person.
\end{itemize}
\end{frame}

\begin{frame}
\heading{Potential outcome framework}
\nnu{Data} $\{(y_i, T_i, \boldsymbol{x}_i)\}_{i = 1}^n \overset{i.i.d.}{\sim} (y, T, \boldsymbol{x})$, \hfill Rubin (1974)
\vfill

\begin{itemize}\itemsep 0.2 in
    \item[\rst] \nnu{Observed covariates}: $\boldsymbol{x} \in \mathbb{R}^p$; \pause
    
    
    \item[\rst] \nnu{Treatment}: 
$
        T = \begin{cases}
            1, & \text{ if treated}\\
            0, & \text{ if control}
        \end{cases}
$
    \pause
    \item[\rst] \nnu{Outcomes}: $
        y = \begin{cases}
            y(1), & \text{ if } T = 1\\
            y(0), & \text{ if } T = 0
        \end{cases}
   $
\end{itemize}
\end{frame}

\begin{frame}
\heading{Average treatment effect Estimation}
\begin{itemize}
    \item[\rst] \nnu{Propensity score}: 
    \begin{align*}
        {\color{red} \pi^*(\mathbf{x})} \overset{\text{def}}{=} \mathbb{P}(T = 1 | \boldsymbol{X} = \boldsymbol{x}) 
    \end{align*}
    \item[\rst] \nnu{Outcome model}: 
    \begin{align*}
        y(t) \overset{\text{def}}{=} {\color{red} \mu_t^*(\boldsymbol{x}) } + \epsilon, \quad t \in \{0, 1\} 
    \end{align*}
     \pause \vspace*{-0.2 in}
     
     
    \item[\rst] \nnu{Average treatment effect (ATE)}: 
    \begin{align*}
       \fcolorbox{red}{blue!25} {${\color{red} \tau} \overset{\text{def}}{=} \EE [Y(1)- Y(0)] = \mathbb{E}[\mu_1^*(\boldsymbol{x}) - \mu_0^*(\boldsymbol{x})]$}
    \end{align*}
    
    \nnu{Conditional ATE}  $\r{\tau(\bx)} = \mu_1^*(\boldsymbol{x}) - \mu_0^*(\boldsymbol{x})$
\end{itemize}

\end{frame}


\begin{frame}
	\heading{Questions}
	\Large
	\begin{center}
	\b{How to model propensity and outcome functions?}
	
	\vfill \pause
	
	\r{Which variables affect propensity and outcome?}
	
	\vfill \pause
	
	\b{How to address dependence in high-d $\bx$?}
	\end{center}
	
\end{frame}






\begin{frame}
\heading{About this work}
\begin{enumerate}\itemsep 0.25in 
    \item Address dependence in high-d covariate  by\\
    { \small \rst Factor model $\boldsymbol{x} = \boldsymbol{B} \boldsymbol{f} + \bu$,\\
     \rdash learned via diversified projection}
     \vspace*{-.8 in}
     
     \hfill 	\includegraphics[width=0.28\textwidth]{figs/RisingTide}
     
    \pause
    \vspace*{-0.4 in}
    
    \item Model propensity and outcome functions by\\
    {\small \rst  Factor Augmented Sparse Throughput Neural network,  FAST-NN model\\
    \rdash Select variables in \r{adaptive}, \b{nonparametric} and \r{algorithmic} manner\\
    }
    \pause
    
    \item Ensure robust ATE estimator by\\
    {\small \rst  Augmented Inverse Propensity Weighted (AIPW) estimator\\
    \rdash Easily get semiparametric \r{efficiency}}
\end{enumerate}
\end{frame}

\begin{frame}
\heading{ATE estimation via factor decomposition}  
\begin{itemize} \itemsep 0.2in
    \item[\rst] \nnu{Model}:  For subsets  $\mathcal{J}_0, \mathcal{J}_1, \mathcal{J}_2$ subset of covariates
    \begin{align*}
        \boldsymbol{x} &= \boldsymbol{B} \boldsymbol{f} + \boldsymbol{u}\\
        \mathbb{E}[T \, | \, \boldsymbol{f}, \boldsymbol{u}] &= \pi^*(\boldsymbol{f}, \boldsymbol{u}_{\mathcal{J}_2}) 
       = \tilde \pi^*(\boldsymbol{f}, \boldsymbol{x}_{\mathcal{J}_2}) \\
        \mathbb{E}[y(t) \, | \, \boldsymbol{f}, \boldsymbol{u}] &= \mu_t^*(\boldsymbol{f}, \boldsymbol{u}_{\mathcal{J}_t}),  \, t \in \{0, 1\}
    \end{align*}
    
    
    \pause
    \item[\rst] \nnu{Data}: 
    $\{(y_i, T_i, \boldsymbol{x}_i)\}_{i = 1}^n \overset{i.i.d.}{\sim} (y, T, \boldsymbol{x})$
    \pause
    
    \item[\rst] \nnu{Goal}: estimate ATE
    \begin{align*}
        \tau = \mathbb{E} \left\{ \mathbb{E}[y(1) - y(0) \, | \, \boldsymbol{f}, \boldsymbol{u}] \right\} = \mathbb{E} [\mu_1^*(\boldsymbol{f}, \boldsymbol{u}_{\mathcal{J}_1}) - \mu_0^*(\boldsymbol{f}, \boldsymbol{u}_{\mathcal{J}_0})] 
    \end{align*}
\end{itemize}
\end{frame}

\begin{frame}
\heading{Related works}
\begin{itemize}\itemsep 0.15in
    \item[\rst] \nnu{Factor model}: {\footnotesize Forni et al. (2000); Stock \& Watson (2002); Bai (2003); Bai \& Ng (2006, 2008); Hallin \& Liska, (2007); Fan et al. (2013, 2021, 2022); Fan \& Liao (2022))};
    \pause  
    \item[\rst] \nnu{Sparse additive models}: {\footnotesize Ravikumar et al. (2009); Meier et al. (2009); Raskutti et al. (2012), Yuan \& Zhou (2016)};
    \pause 
    \item[\rst] \nnu{Neural network models}: {\footnotesize Feng \& Simon (2017), Chen et al. (2019); Gu et al. (2021); Schmidt-Hieber (2020); Kohler and Langer (2021); Shen et al. (2021), Fan, Gu, Zhou (2022), Ohn \& Kim (2022), Fan et al. (2022, ; Fan \& Gu (2024)};
    \pause
    \item[\rst] \nnu{ATE estimation}: 
    {\footnotesize Rosenbaum and Rubin (1983); Rosenbaum (1989); Abadie \& Imbens (2006); Wager \& Athey (2018); Chernozhukov et al. (2018); Yoon et al. (2018); Bang \& Robins (2005); Farrell (2015), Farrell et al. (2021)}
\end{itemize}
\end{frame}




\begin{frame}
	\vfill
	\begin{center}
		\huge \bf \textsf{FAST-NN}
		\vfill
		
		\large \visible<2->{{\rst Factor-Adjusted}} Nonparametric Lasso  \visible<2->{(FAN-Lasso)}
	\end{center}
	
\vfill
\vfill

\small
\rst  Fan, J. and Gu, Y. (2024).  Factor Augmented Sparse Throughput Deep ReLU Neural Networks for High Dimensional Regression.  {\em JASA}, {\bf 119} (548), 2680-2694.

	\addtocounter{framenumber}{-1}
\end{frame}


\begin{frame}
	\heading{Versality of FAST-model}
	
	
	\centerline{\fcolorbox{red}{red!15}{
			$\bx = \bB\bff + \bu , \quad \quad \mathbb{E}[y|\bff, \bu] = m^*(\bff, \bu_{\r{\mathcal{J}}}) = g^*(\bff, \bx_{\r{\mathcal{J}}}) , 
			$}}

	
	\vfill
	
	\begin{figure}
		\begin{tikzpicture}[->, auto]
			\node [shape=rectangle,draw] (A) {{\bf FAST model}};
			%\node[shape=rectangle,draw] (B) [below left=of A]{\fcolorbox{red}{red!5}{Sparse model}};
			%\node[shape=rectangle,draw] (C) [below =of B]{Sparse linear model};
			\node (B) [below left=of A]{\fcolorbox{red}{red!5}{\footnotesize Sparse nonpara model}};
			\node (C) [below =of B]{\fcolorbox{red}{red!5}{\scriptsize Sparse linear}};
			\node (D) [below =of A]{\fcolorbox{red}{blue!15}{\footnotesize Nonpara factor reg}};
			\node (E) [below =of D] {\fcolorbox{red}{blue!15}{\scriptsize Pincipal Comp Reg}};
			\node (F) [below right=of A]{\fcolorbox{red}{green!15}{\footnotesize Sparse model w/ dep}};
			\node (G) [below =of F]{\fcolorbox{red}{green!15}{\scriptsize FarmPredict}};
			
			\scriptsize
			\path (A) edge    node{$r=0$} (B)
			(B) edge    node{linear} (C)
			(A) edge    node{$|\mathcal{J}|=0$} (D)
			(D) edge    node{linear} (E)
			(A) edge    node{$r\neq 0$} (F)
			(F) edge    node{linear} (G);
		\end{tikzpicture}
	\end{figure}
	
\end{frame}


\begin{frame}
	\heading{Archtechure of FAST-model}
	
	
\centerline{\fcolorbox{red}{red!15}{
		$\bx = \bB\bff + \bu , \quad \quad \mathbb{E}[y|\bff, \bu] = m^*(\bff, \bu_{\r{\mathcal{J}}}) = g^*(\bff, \bx_{\r{\mathcal{J}}}) , 
		$}}

	\vfill
	
	\begin{figure}
		\centering
		\includegraphics[scale=0.8]{\figss/fast-nn-arch.pdf}
	\end{figure}
	\vfill
	\centerline{\b{\huge Joint training vs Pre-training?}}
	
\end{frame}


\begin{frame}
	\heading{Diversified projections for factor learning}
	\small	
	\nnu{Challenges}: Learning \r{simultaneously} factors and reg function via DNN overfit. 
    \pause
	\vfill    
	\nnu{Factor learning}: For a given $\bW \in \mathbb{R}^{p\times \bar{r}}$ ($\bar{r} \geq r$, Fan \& Liao, 22),  \hfill \fcolorbox{red}{red!5}{\scriptsize \r{+} no need $n \to \infty$}
	\begin{align*}
		\fcolorbox{red}{red!5}{$\tilde{\bff} = p^{-1} \bW^\top \bx$} =  \underbrace{p^{-1} \bW^\top \bB}_{\bH} \r{\text{\huge $\bff$}} + \underbrace{\b{p^{-1} \bW^\top \bu}}_{O_p(p^{-1/2})} 
	\end{align*}
	\nnu{Requirements}:  $\nu_{\min}(\bH) \gg p^{-1/2}$,  \qquad  $\|\bW\|_\infty \leq C$
    \pause
	\vfill  
	\nnu{Examples}: \rds $\boldsymbol{W}^{(1)} = \bB$ \qquad \rds $\boldsymbol{W}^{(2)} = \left[ \bB,   \times  \right]$ \\
    \b{pre-training}: $\bW$ = top $\bar{r} $ PCs using $n_1$ data  
    \hfill 
    \fcolorbox{red}{red!5}{Thm: $n_1 \gtrsim r^2 \log p$ suffices}.  
\end{frame}

\begin{frame}
    \heading{Deep ReLU neural network}
	\nnu{Deep ReLU neural network with width $N$ and depth $L$} 
    \vspace{-5pt}
    \begin{align*}
		f: \mathbb{R}^d \to \mathbb{R}, \qquad f(\bx; \btheta) = \b{\mathcal{L}_{L+1}} \circ \r{\sigma} \circ \b{\mathcal{L}_{L}} \circ \cdots\r{ \sigma} \circ \b{\mathcal{L}_2 } \circ \r{\sigma} \circ \b{ \mathcal{L}_1}(\bx)
	\end{align*}
	\vspace{-25pt}
	\begin{figure}
		\centering
		\includegraphics[width=0.4\textwidth]{figs/relu-dnn.pdf}
	\end{figure}
	\vspace{-20pt}
	\begin{itemize}
		\item $\mathcal{L}_i(\bx) = \bW_i \bx + \bb_i$ with  weights $\bW_i, \bb_i$, \hfill \b{$\sigma(t) = t_+$ ReLU} activation
		\item Dim: $(d_0, d_1, \cdots, d_{L}, d_{L + 1}) = (d, N, \cdots, N, 1)$ 
		\item \r{Parameter $\btheta = \{\bW_i, \bb_i\}_{i = 1}^L$ } \hfill  \r{\small \# parameter  $\asymp  N^2 L$}
	\end{itemize}
	\vfill
	\nnu{Bounded ReLU-DNN functions} : {\footnotesize $\mathcal{G}_n  = \{ \blue{T_M g(\boldsymbol{x})}: \sup_{\ell \in [L + 1]} \|\boldsymbol{W}_{\ell}\|_{\max} \lor \|\boldsymbol{b}_{\ell}\|_\infty \le B\}$}
\end{frame}



\begin{frame}
	\heading{Estimation for FAST model}
	
	\begin{itemize} \small \itemsep 0.15in
		\item[\rst] Let diversified projection matrix \blue{$\boldsymbol{W}$}, variable selection matrix $\red{\boldsymbol{\Theta}} \in \mathbb{R}^{p\times N}$.
		\pause
		\item[\rst] With depth $L$ and width $N$ ReLU $\r{g}_{NN}$,  output the model
		\begin{align*}
			m(\boldsymbol{x}) = \red{\boldsymbol{g}}_{NN} \left(\left[ p^{-1} \blue{\boldsymbol{W}}^\top \boldsymbol{x}, \bar{T}_M(\red{\boldsymbol{\Theta}}^\top \boldsymbol{x})\right]\right) 
		\end{align*}
		\pause  \vspace*{-0.3 in}
		
		\item[\rst] Run penalized least squares with parameters \blue{$\lambda$} and \blue{$\tau$}: \hfill \fcolorbox{red}{red!5}{FAST-NN}
		{\footnotesize
			\begin{align*}
				\hat{\mathsf{R}}_{\mathtt{FAST}}(\red{\boldsymbol{\Theta}}, \red{\boldsymbol{g}_{NN}}) = \frac{1}{n} \sum_{i = 1}^n \{y_i - m(\boldsymbol{x}_i)\}^2 + \blue{\lambda} \sum_{j, k}  {|\boldsymbol{\Theta}_{j, k}|} \land {\blue{\tau}}
			\end{align*}
		}
	\end{itemize}
	\vspace{-10pt}
	\begin{figure}
		\centering
		\includegraphics[scale=0.8]{figs/fast-nn-arch.pdf}
	\end{figure}
\end{frame}








\begin{frame}
	\heading{Hierachial Composition Model}
	\vspace*{0.2 in}
	
	\begin{block}{\yellow{Hierachical Composition Model}:  $\text{HCM}(\cP) = $ Finite compositions  }
		$Q^* = Q_1 \circ \cdots \circ Q_q$ of $t$-variate function with smoothness $\beta$, for $(t, \beta) \in \cP$.
	\end{block}
	\rdash  $\gamma^*=\min_{(t, \beta) \in \cP } \frac{\beta}{t}$ is the hardest component.  \hfill \Ref{Kohler and Krzyzak, 2017}
	\pause
	\vfill \vfill
	
	\begin{enumerate}\itemsep 0.15in
		\item [\rst] e.g. $Q^* = f_1(x_1)+\cdots + f_p(x_p)$ \hfill $Q^* = f_1(x_1, f_2(x_2, x_3))+ f_3(x_2, x_5, x_9)$ \pause
		\item [\rst] Complexity determined by  $\gamma^* = \min_{(t, \beta)\in \cP} \frac{\beta}{t}$. %, \hfill denoted by $\cH(\gamma^*)$ 
		\pause
		\item [\rst] \nnu{Adaptively learned} by neural networks 
	\end{enumerate}
	\vspace*{-.7in}
	
	\hfill \visible<2>{                                   
		\hfill \includegraphics[width=1.4in]{figs/additive}
	}
	
	
\end{frame}



\begin{frame}
	\begin{center}
		\huge \bf \textsf{Factor Informed Double Deep Learning Estimator}
		
		\Large  (FIDDLE)
	\end{center}
		\addtocounter{framenumber}{-1}
\end{frame}


\begin{frame}
\heading{FIDDLE for ATE estimation}
\begin{itemize}
    \item[\rst] Outcome models \red{$\mu_t^*$} and propensity \blue{$\pi^*$} estimated by FAST-NN: 
    \begin{align*}
        \red{\hat{\mu}_t(\boldsymbol{x}_i)} &= m^{\text{FAST}}_t(\boldsymbol{x}_i), \quad t = 0, 1\\
        \blue{\hat{\pi}(\boldsymbol{x}_i)} &= m^{\text{FAST}}_2(\boldsymbol{x}_i)
    \end{align*}
    \pause
    
    
    \item[\rst] FIDDLE: ATE estimation via AIPW \Ref{Robins et al, 1994;  Bang \& Robins, 2005}
    \begin{align*}
        \hat{\tau} = \frac{1}{n} \sum_{i = 1}^n \left[ \red{\hat{\mu}_1(\boldsymbol{x}_i)} + \frac{T_i (y_i - \red{\hat{\mu}_1(\boldsymbol{x}_i)})}{\blue{\hat{\pi}(\boldsymbol{x}_i)}} - \red{\hat{\mu}_0(\boldsymbol{x}_i)} - \frac{(1 - T_i) (y_i - \red{\hat{\mu}_0(\boldsymbol{x}_i)})}{\blue{1 - \hat{\pi}(\boldsymbol{x}_i)}} \right]
    \end{align*}
     %\vspace*{-0.3 in}
    
  
\end{itemize}
\end{frame}

\begin{frame}
\heading{Augmented inverse propensity weighted estimator}

\begin{itemize}
    \item[\rst] Why Augmented Inverse Propensity Weighted \textbf{\blue{(AIPW)}} Estimator?
    \pause  \vfill
  \item[ \rst] \small  $1^{st}\mbox{-term } \approx \EE \left[  \hat{\mu}_1(\bX) + \frac{T (Y - \hat{\mu}_1(\bX))}{\hat{\pi}(\bX)} \right ] =
 \EE \left[  \hat{\mu}_1(\bX) + \frac{\pi^*(\bX) ( \mu_1^*(\bX) - \hat{\mu}_1(\bX))}{\hat{\pi}(\bX)} \right ]$  \pause \\
 $$= \EE  \left[   \mu^*_1(\bX)   + \frac{( \pi^*(\bX) - \hat{\pi} (\bX)) ( \mu_1^*(\bX) - \hat{\mu}_1(\bX))}{\hat{\pi}(\bX)} \right ]
 \qquad \Longleftarrow \qquad \mbox{\r{double-robustness}}
 $$
 \vfill
 
 
  \item[\rst] 
    \textbf{\red{either}} propensity \textbf{\red{or}} outcome models \textbf{\red{correctly specified}} \\
    $\Longrightarrow$  \textbf{\red{consistent}} ATE estimation    $\Longrightarrow$  \b{error multiplied}
\end{itemize}
\end{frame}


\begin{comment}
\begin{frame}
\heading{Proof sketch}
\begin{itemize}
    \item[\rst] Define the RW of \textbf{\blue{doubly robust outcome}} estimation
    \begin{align*}
        \mu_1^{\text{dr}}(\boldsymbol{x}) &= \hat{\mu}_1(\boldsymbol{x}) + \frac{T \left( y(1) - \hat{\mu}_1 ( \boldsymbol{x}) \right) }{\hat{\pi}(\boldsymbol{x})} \\
        \mu_0^{\text{dr}}(\boldsymbol{x}) &= \hat{\mu}_0 (\boldsymbol{x}) + \frac{\left( 1 - T \right) \left( y(0) - \hat{\mu}_0 (\boldsymbol{x}) \right) }{1 - \hat{\pi} (\boldsymbol{x})}
    \end{align*}
    \pause
    \item[\rst] Check \textbf{\blue{unbiasedness}} of doubly robust outcome estimation
    \begin{align*}
         \mathbb{E} \left[ \mu_1^{\text{dr}}(\boldsymbol{x}) - y(1) \right] 
         &= \mathbb{E} \left\{ \mathbb{E} \left[ \frac{T - \hat{\pi}(\boldsymbol{x})}{\hat{\pi}(\boldsymbol{x})} \left( y(1) - \hat{\mu}_1(\boldsymbol{x}) \right) \Big| \boldsymbol{x} \right] \right\} \\
         &= \mathbb{E} \left[ \frac{\pi^* - \hat{\pi}(\boldsymbol{x})}{\hat{\pi}(\boldsymbol{x})} \left( \mu_1^* - \hat{\mu}_1(\boldsymbol{x}) \right) \right]
    \end{align*}
    only require either $\hat{\pi}(\boldsymbol{x}) = \pi^* $ or $ \hat{\mu}_1(\boldsymbol{x}) = \mu_1^*$.
\end{itemize}
\end{frame}

\end{comment}

% theory page
\section{Nonasymptotic Theory}
\vfill
\begin{frame}
	\begin{center}
		\huge \bf \textsf{Theory}
	\end{center}
	
		\addtocounter{framenumber}{-1}
\end{frame}

\begin{frame}\heading{Technical assumptions}
	\nnu{Estimator:}  Approximate ERM $\hat{m}$ with $\hat{\mathsf{R}} (\hat{m}) \le \inf_{m} \hat{\mathsf{R}} (m) + \delta_{\mathtt{opt}}$.
		\begin{itemize} \footnotesize
	    \item[\rst]  Truncation level $M$ is a large constant.
	    %chosen to be large enough but of order $\mathcal{O}(1)$.
	\end{itemize}
    \pause 
    \vfill
    \nnu{Assump 1}: $\|\boldsymbol{B}\|_{\max} \le b$, $\boldsymbol{f}$ and $\boldsymbol{u}$ zero mean, $\|\boldsymbol{f}\|_\infty, \|\boldsymbol{u}\|_\infty \le b$.
    \vfill
    \nnu{Assump 2}: Weak dependence of $\boldsymbol{u}$: $\sum_{j, k} |\mathbb{E} [u_j u_k]| \lesssim p$.
    \pause 
    \vfill  
    \nnu{Assump 3}: Sub-Gaussian noise: $\mathbb{P}[|\varepsilon|>t | \boldsymbol{f}, \boldsymbol{u}] \le 2e^{-c_1 t^2} ~ a.s.$
    \vfill
    \nnu{Assump 4}: $\pi^*, \mu_0^*, \mu_1^*$ is $C$-Lipschitz and $\|\pi^*\|_\infty, \|\mu_0^*\|_\infty, \|\mu_1^*\|_\infty \lesssim 1$.
\end{frame}

\begin{frame}
\heading{A data-driven method for $\boldsymbol{W}$}
\nnu{PCA} for other $n_1$ data $\boldsymbol{x}_1, \cdots, \boldsymbol{x}_{n_1}$ with a large $\overline{r}$.
    \begin{itemize} \small
        \item[\rst] Apply PCA to sample covariance matrix $\widehat{\boldsymbol{\Sigma}} = \frac{1}{n_1} \sum_{i = 1}^{n_1} \boldsymbol{x}_i \boldsymbol{x}_i^\top$. 
        \item[\rst] $\widetilde{\boldsymbol{W}} = [\sqrt{\widehat{\lambda}_1} \widehat{\boldsymbol{v}}_1, \cdots, \sqrt{\widehat{\lambda}_{\bar{r}}} \widehat{\boldsymbol{v}}_{\overline{r}}]$ for top-$\overline{r}$ eigenvalues and eigenvectors of $\widehat{\boldsymbol{\Sigma}}$.\\
        \hfill {\r{$\leadsto$} remove the incoherence assump  in Fan and Gu (24).}
    \end{itemize}
    \pause
    \vfill
    \begin{block}{\small Theorem 1. Under assumptions 1-2,}
        \small
        if $\lambda(\boldsymbol{B}^\top \boldsymbol{B}) \asymp p$ then w.h.p., 
        \vspace{-5pt}
        \begin{align*}
            \nu_{\min}(p^{-1} \widetilde{\boldsymbol{W}}^\top \boldsymbol{B}) \geq C \left(1 - r \sqrt{\frac{\log p}{n_1}} + r^2 \sqrt{\frac{\log r}{n_1}} + \sqrt{\frac{1}{p}} \right).
        \end{align*}
        \vspace{-10pt}
    \end{block}
    \pause 
    \fcolorbox{red}{red!5}{$\leadsto$ only requires $n_1 \gg r^2 \log p$}. \hfill {\rst \footnotesize negligible}
\end{frame}

\begin{frame}
\heading{Oracle-type inequality for FAST-NN estimator}
\begin{block}{\small Theroem 2.  Under assumptions 1-4, \hfill [Fan \& Gu, 2024] }
    {if $N \geq 2(r + |\mathcal{J}|)$, $B \geq c_1 r |\mathcal{J}|$ and $\lambda \geq c_1 \frac{\log(p)}{n}$ and $\eta^{-1} \geq c_2 n$, then for each estimator $\hat{g}$ (outcome and propensity functions), we have
    \begin{align*}
        \mathbb{P} \Big( \| \hat{g}_j - g_j^* \|_2^2 & \leq c_3 \, ( \delta_{\text{opt}} + {\color{blue} \delta_{j,a}} + {\color{red} \delta_s} + {\color{teal} \delta_f} + \frac{t}{n} )  \Big) \geq 1 - 3e^{-t}
    \end{align*}}
\end{block}
\pause 
\vspace*{-0.1 in}

\begin{align*}
    \color{blue} \text{approx.  error: } & \color{blue} \delta_{j,a} = \inf_{g_j \in \mathcal{G}_{\tiny NN}} \lVert m^{\text{FAST}}_j(\cdot) - g_j^*\rVert_\infty^2\\
    \color{red} \text{stochastic error: } & \color{red} \delta_s = \lambda + \frac{(LN)^2}{n}\\
    \color{teal} \text{factor est error: } & \color{teal} \delta_f = \frac{|\mathcal{J}| \, r \cdot \bar r}{\nu_{\min}^2(\boldsymbol{H}) \cdot p},  \qquad
    \bH = p^{-1} \widetilde{\boldsymbol{W}}^\top \boldsymbol{B}
\end{align*}  
\pause 
\vspace*{-.3 in}

\rst Need modifications of the proofs to random samples with $T=0$ and $T=1$.

\end{frame}

\begin{frame}
\heading{Asymptotic normality}
\begin{block}{\small Theorem 3. Under assumptions 1-4, $r + |\mathcal{J}| \leq C$, $\|\boldsymbol{u}\|_\infty \leq b$.} 
\begin{itemize}
	\item $n^{c_1} < p <  n^{100}$ for some constant $c_1 > \frac{1}{2}$,
    \item $\mu_0^*, \mu_1^*, \pi^* \in \mathcal{H}(r + |\mathcal{J}|, l, \mathcal{P})$, \hfill $\gamma^* > \frac{1}{2}$
	\item  estimated by the FAST-NN estimators with $\delta_{opt} < (n / \log n)^{-\frac{\gamma^*}{2\gamma^* + 1}}$. \pause
\end{itemize}

\begin{itemize}
    \item[\rst] asymptotic normality: 
    \begin{align*}
        \sqrt{n} \left( \hat{\tau}^{\text{FIDDLE}} - \tau \right) \xrightarrow{d} N(0, \sigma^2),
    \end{align*}
    \item[\rst] semiparametric efficiency:  \hfill {\scriptsize Hahn (1998)}
    \begin{align*}
        \sigma^2 = \mathbb{E} \left[ (\mu_1^*(\bx) - \mu_0^*(\bx)-\mu)^2  + \frac{\Var[y(1) | \bx]}{\pi^*(\bx)}+ \frac{\Var[y(0) | \bx]}{1 - \pi^*(\bx)} \right]
    \end{align*}
\end{itemize}
\end{block}

\vfill \pause
\rst  Required $\gamma_0^* \gamma_2^* > \frac14$ and $\gamma_1^* \gamma_2^* > \frac 14$. \hspace{1 in} \rst hold for $r = 0$ 
\end{frame}



% empirical page
\section{Empirical Results}
\vfill
\begin{frame}
	\begin{center}
		\huge \bf \textsf{Numerical Results}
	\end{center}
		\addtocounter{framenumber}{-1}
\end{frame}

\begin{frame}
\heading{Simulations: data generating process}
\begin{itemize}
    \item[\rst] $r = 4$, $\mathcal{J} = \{1, \cdots, 5\}$, $n = 5000$, varying $p \in \{10, \cdots, 10000\}$; 
    \bigskip
    \pause
    \item[\rst] \b{Propensity model}:  \hfill $f_i, u_i \sim_{iid} U(-1, 1)$, $b_{i,j} \sim_{iid} U(-\sqrt{3}, \sqrt{3})$.
    \begin{align*}
        \pi^*(\boldsymbol{f}, \boldsymbol{u}) = 0.8 \r{\sigma} \Bigl (\sin(f_1) + \tan(f_2) + f_3 + f_4 + \sum_{j=1}^{5} u_j \Bigr )+0.1;
    \end{align*}
    \pause
    
    \item[\rst] \b{Outcome model}: 
    $y = \mu^*(\boldsymbol{f}, \boldsymbol{u}) + T \, \tau^*(\boldsymbol{f}, \boldsymbol{u_{\mathcal{J}}}) + \varepsilon$ with $\varepsilon \sim \mathcal{N}(0, 1/4)$ %independent of $\boldsymbol{f}, \boldsymbol{u}$
    \begin{align*}
		\mu^*(\boldsymbol{f}, \boldsymbol{u}) &= 10 + f_1 + f_2 f_3 + \sin(f_4) + \log(5 + u_1 + u_2 u_3) + \tan(u_4) + u_5;\\
		\tau^*(\boldsymbol{f}, \boldsymbol{u}) &= 5 + f_1 + f_2 + \sin(f_3) + \tan(f_4) + u_1 + u_2 + \sin(u_3 + u_4) + \tan(u_5).
	\end{align*}
    \pause
    \item[\rst] ground truth $\text{ATE}^* = \mathbb{E} [ \tau^*(\boldsymbol{f}, \boldsymbol{u}) ] = 5$  \hfill  $N_{sim} = 100$
\end{itemize}
\end{frame}

\begin{frame}
\heading{Comparison of Methods}
	\begin{center}
		\includegraphics[height=6.7cm]{figs/method_compare_rmse_plot.png}
		
	\footnotesize	n = 5000 \qquad $N_{sim} = 100$
	  % \rsq RMSE for different numbers of covariates
	\end{center}
\footnotesize

\rst FIDDLE close to oracle  \qquad \rst Vanilla-NN suffers COD. \qquad \rst outperforms other methods \\


\end{frame}

\begin{frame}
\heading{Performance accross $n$ and $p$}
\begin{center}
		\includegraphics[height=7.5 cm]{figs/size_compare_mse_heatmap.png}
		
	%RMSE of FIDDLE for ATE estimation across different  $n$ and $p$
	%sample sizes and covariate dimensions. Dark regions and light regions indicate high and low RMSE, respectively.
\end{center}
\end{frame}



\begin{frame}
	\heading{Rates of Convergence}
	\begin{center}
		\includegraphics[height=7.5 cm]{figs/size_compare_rmse_vs_n}

	\end{center}
\end{frame}


\begin{frame}
\heading{Semi-synthetic image data}
\begin{itemize}
    \item[\rst] \nnu{CIFAR-10 dataset} (Canadian Institute For Advanced Research)
    \item[\rst] benchmark in computer vision: $n = 60,000$ and $p = 3,072$ \hfill {\scriptsize $32\times 32\times 3$}
    \pause 
    %semi-synthetic data by solution of covariate factor decomposition: 
    %\begin{align*}
     %   \underset{\boldsymbol{B} \in \mathbb{R}^{p \times r}, \boldsymbol{F} \in \mathbb{R}^{n \times r}}{\mathrm{minimize}} \sum_{i = 1}^n \lVert \boldsymbol{X}_i - \boldsymbol{B} \boldsymbol{f}_i \rVert^2 = \lVert \boldsymbol{X} - \boldsymbol{F} \boldsymbol{B}^T \rVert^2_F.
    %\end{align*}
    %\pause
	\item[\rst] \b{synthetic propensity model}:  \hfill {\scriptsize $\widetilde{f}_{ij}, \widetilde{u}_i$ taken from fitting factor model $r=4$}
    \begin{align*}
        \pi^*(\widetilde{\boldsymbol{f}}, \widetilde{\boldsymbol{u}}) = 0.8 \sigma(\sin(\widetilde{f}_1) + \sum_{i = 2}^4 \widetilde{f}_i + \sin (\widetilde{u}_1) + \sum_{j = 2}^5 \widetilde{u}_j) + 0.1
    \end{align*}
    \pause
    
    \item[\rst] \b{synthetic outcome model}: {\small 
    $y = \mu^*(\boldsymbol{f}, \boldsymbol{u}) + T \, \tau^*(\boldsymbol{f}, \boldsymbol{u_{\mathcal{J}}}) + \varepsilon$ with $\varepsilon \sim \mathcal{N}(0, 1/4)$}
    \begin{align*}
		\mu^*(\widetilde{\boldsymbol{f}}, \widetilde{\boldsymbol{u}}) &= 10 + \widetilde{f}_1 + \sin(\widetilde{f}_2) + \widetilde{f}_3 \widetilde{f}_4 + \widetilde{u}_1 (\widetilde{u}_2 + \sin(\widetilde{u}_3)) + \widetilde{u}_4 + \widetilde{u}_5, \\
		\tau^*(\widetilde{\boldsymbol{f}}, \widetilde{\boldsymbol{u}}) &= \widetilde{f}_1 (\widetilde{f}_2 + 3) + \widetilde{f}_3 + \sin(\widetilde{f}_4) + \sin(\widetilde{u}_1) + \widetilde{u}_2 + \widetilde{u}_3 \widetilde{u}_4 \widetilde{u}_5.
	\end{align*}
\end{itemize}
\end{frame}

\begin{frame}
\heading{CIFAR-10 semi-synthetic dataset results}
\begin{table}[H]
    \centering
  %  \small
    \begin{tabular}{l c c}
        \hline
        \textbf{Method} & \textbf{RMSE} & \textbf{(SE)} \\
        \hline
        $\text{Oracle AIPW}$ & 0.009 & (0.001) \\
        \textcolor{red}{$\text{FIDDLE}$} & \color{red}{0.030} & \color{red}{(0.003)} \\
        $\text{Vanilla NN}$ & 0.282 & (0.012) \\
        \text{GANITE} & 1.389 & (0.032) \\
        \text{DR} & 1.664 & (0.007) \\
        \text{DML} & 1.427 & (0.007) \\
        \text{CF} & 1.878 & (0.007) \\
        $\text{Factor CF}$ & 1.990 & (0.006) \\
        $\text{Oracle IPW}$ & 0.448 & (0.030) \\
        \hline
    \end{tabular}
  %  \caption{RMSE and its standard error (in parentheses) of candidate ATE estimation methods on the semi-synthetic dataset based on CIFAR-10 over $100$ replications.}
%\label{tab:CIFAR-10}

\rst $n = 5000$
\end{table}


\end{frame}

\begin{frame}
\heading{Application: bariatric surgery}
\begin{itemize}
    \item[\rst] \nnu{MBSAQIP dataset} (Metabolic and Bariatric Surgery Accreditation and Quality Improvement Program): weight loss surgery
    \item[\rst] $174,013$ records with $42$ pretreatment covariates from year $2017$
    \pause
    \item[\rst] outcome: \r{30-day BMI reduction}; treatment: surgery type
    \begin{itemize}
        \item \nnu{Sleeve} Gastrectomy: most widely performed as \red{control};
        \item \nnu{RYGB} (Roux-en-Y Gastric Bypass);
        \item \nnu{AGB} (Adjustable Gastric Band);
        \item \nnu{BPD/DS} (Biliopancreatic Diversion with Duodenal Switch);
        \item \nnu{SADI-S} (Single Anastomosis Duodeno-Ileal Bypass with Sleeve Gastrectomy)
    \end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\heading{Bariatric surgery type comparison}
\begin{table}[t]
	\centering
	\footnotesize
	\setlength{\tabcolsep}{3pt}
	\begin{tabular}{l|cccc}
		\hline
		Surgery & RYGB & AGB & BPD/DS & SADI-S \\
		\hline
		\color{red}{FIDDLE} & \color{red}{-0.011} & \color{red}{-1.039} & \color{red}{0.336} & \color{red}{-0.502}\\
		& \color{red}{(-0.012, -0.011)} & \color{red}{(-1.052, -1.027)} & \color{red}{(0.311, 0.362)} & \color{red}{(-0.519, -0.485)} \\
		\hline
		Vanilla NN & -0.381 & -1.681 & -0.367 & -1.544\\
		& (-0.388, -0.375) & (-1.691, -1.670) & (-0.386, -0.348) & (-1.557, -1.530) \\
		\hline
		GANITE & -0.451 & -2.465 & -1.502 & -2.045\\
		& (-0.516, -0.386) & (-2.659, -2.271) & (-1.655, -1.348) & (-2.169, -1.922) \\
		\hline
		DR & -0.031 & -0.734 & 0.031 & -0.512\\
		& (-0.031, -0.031) & (-0.741, -0.727) & (0.031, 0.031) & (-0.518, -0.506) \\
		\hline
		DML & -0.015 & -0.240& 0.631& -0.546\\
		& (-0.016, -0.015) & (-0.273, -0.208) & (0.606, 0.655) & (-0.550, -0.542) \\
		\hline
		CF & -0.023 & -1.080 & 0.214 & -0.647\\
		& (-0.023, -0.022) & (-1.082, -1.079) & (0.211, 0.216) & (-0.649, -0.645) \\
		\hline
		Factor-CF & -0.057 & -1.092 & 0.175 & -0.7773\\
		& (-0.057, -0.056) & (-1.094, -1.091) & (0.173, 0.177) & (-0.779, -0.776)\\
		\hline
	\end{tabular}
%	\caption{Estimated ATE and 95\% confidence intervals for 30-day BMI reduction by different procedures, compared with Sleeve as the control.}
%	\label{tab:surgery}
\end{table}
\end{frame}

\begin{frame}
\heading{Bariatric surgery type comparison}
\begin{itemize}\itemsep 0.15 in
    \item[\rst] FIDDLE yields robust results as the comparing methods
    \item[\rst] \nnu{Sleeve} - removes part of the stomach - \red{as control}
    \pause
    \item[\rst] \nnu{RYGB} - create new stomach pouch - \red{small negative ATE}
    \item[\rst] \nnu{AGB} - restricts stomach with an adjustable band - \red{large negative ATE}
    \pause
    \item[\rst] \nnu{BPD/DS} - bypass surgery after sleeve gastrectomy - \red{positive ATE}
    \item[\rst] \nnu{SADI-S} - simplified BPD/DS similar to Sleeve - \red{moderate negative ATE}
\end{itemize}
\end{frame}

\begin{frame}
\heading{Concluding Remarks}
\begin{enumerate}\itemsep 0.15in
    \item Introduce FIDDLE for ATE estimation with high-dimensional covariate.
     \item Leverage \rdt FAST-NN for non-param propensity and outcome modeling. \pause
    \item Encompass most statistical models, \rdt factor models, \rdt sparse models.
   
    \item Learning attribution variables for propensity and outcome model \pause
    \item Guarantee doubly-robustness by \rdt AIPW estimator
    \item Demonstrate asymptotic normality and semiparametric efficiency \pause
    \item Verify results on both generated and semi-synthetic datasets
    \item Apply to bariatric surgery data for surgical type-weight loss analysis
\end{enumerate}
\end{frame}

\begin{frame}
\heading{The End}
\begin{figure}[tb]
	\centering
	\includegraphics[scale=1]{figs/thankyou.pdf}
\end{figure}
\vfill
\rst \footnotesize Fan, J., S. Jana, S. Kulkarni, and Q. Yin (2025). ``Factor Informed Double Deep Learning For Average Treatment Effect Estimation." arXiv preprint arXiv:2508.17136 .
\end{frame}


\end{document}
