% \documentclass{uai2024} % for initial submission
\documentclass[accepted]{uai2024} % after acceptance, for a revised version; 
% also before submission to see how the non-anonymous paper would look like 
                        
%% There is a class option to choose the math font
% \documentclass[mathfont=ptmx]{uai2024} % ptmx math instead of Computer
                                         % Modern (has noticeable issues)
% \documentclass[mathfont=newtx]{uai2024} % newtx fonts (improves upon
                                          % ptmx; less tested, no support)
% NOTE: Only keep *one* line above as appropriate, as it will be replaced
%       automatically for papers to be published. Do not make any other
%       change above this note for an accepted version.

%% Choose your variant of English; be consistent
\input{math_commands.tex}

\usepackage[american]{babel}
% \usepackage[british]{babel}

%% Some suggested packages, as needed:
\usepackage{natbib} % has a nice set of citation styles and commands
    \bibliographystyle{plainnat}
    \renewcommand{\bibsection}{\subsubsection*{References}}
\usepackage{mathtools} % amsmath with fixes and additions
% \usepackage{siunitx} % for proper typesetting of numbers and units
\usepackage{booktabs} % commands to create good-looking tables
\usepackage{tikz} % nice language for creating drawings and diagrams

\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc}    % use 8-bit T1 fonts
\usepackage{hyperref}       % hyperlinks
\usepackage{url}            % simple URL typesetting
\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}         % colors
\usepackage{bbm}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{amsthm}
\usepackage[ruled,vlined,linesnumbered,algo2e,noend]{algorithm2e}
\usepackage{enumitem}
\usepackage{multirow}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{pifont}
\usepackage{float}
\usepackage{wrapfig}
\usepackage{arydshln}
\usepackage{enumitem}
\usepackage{bbm}
\usepackage{tablefootnote}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{remark}{Remark}

\DeclareMathOperator{\concat}{concat}
\DeclareMathOperator{\flatten}{flatten}
\DeclareMathOperator{\oodavg}{OOD_{Avg}}
\DeclareMathOperator{\idavg}{ID_{Avg}}
\DeclareMathOperator{\oodwrt}{OOD_{Wrt}}
\DeclareMathOperator{\acc}{acc}
\DeclareMathOperator{\enc}{Enc}
\DeclareMathOperator{\trans}{Trans}
\DeclareMathOperator{\lstm}{LSTM}
\DeclareMathOperator{\diam}{diam}

\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%

\newcommand{\hoang}[1]{\textcolor{red}{#1}}
\newcommand{\xr}[1]{\textcolor{blue}{#1}}
\newcommand{\xueru}[1]{\textcolor{purple}{#1}}
%% Provided macros
% \smaller: Because the class footnote size is essentially LaTeX's \small,
%           redefining \footnotesize, we provide the original \footnotesize
%           using this macro.
%           (Use only sparingly, e.g., in drawings, as it is quite small.)

%% Self-defined macros
\newcommand{\swap}[3][-]{#3#1#2} % just an example

\title{Non-stationary Domain Generalization: Theory and Algorithm}

% The standard author block has changed for UAI 2024 to provide
% more space for long author lists and allow for complex affiliations
%
% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
% information below.
%
% Add authors
% % \author[1]%{\href{mailto:<jj@example.edu>?Subject=Your UAI 2024 paper}{Jane~J.~von~O'L\'opez}{}}
% \author[1,2]{Thai-Hoang Pham}
% \author[1]{Xueru Zhang}
% \author[1,2]{Ping Zhang}
% % Add affiliations after the authors
% \affil[1]{%
%     Department of Computer Science and Engineering\\
%     The Ohio State University\\
%     USA
% }
% \affil[2]{%
%     Department of Biomedical Informatics\\
%     The Ohio State University\\
%     USA
% }

\newcommand*{\affaddr}[1]{#1}
\newcommand*{\affmark}[1][*]{\textsuperscript{#1}}
\newcommand*{\email}[1]{\texttt{#1}}

\author{%
\hfill Thai-Hoang Pham\affmark[1,2] \hfill Xueru Zhang\affmark[1] \hfill Ping Zhang\affmark[1,2] \hfill \phantom \\
\affaddr{\affmark[1]Department of Computer Science and Engineering, The Ohio State University, USA}\\
\affaddr{\affmark[2]Department of Biomedical Informatics, The Ohio State University, USA}\\
\email{\{pham.375,zhang.12807,zhang.10631\}@osu.edu}\\
}
  
\begin{document}
\maketitle

\begin{abstract}
Although recent advances in machine learning have shown its success to learn from independent and identically distributed (IID) data, it is vulnerable to out-of-distribution (OOD) data in an open world. Domain generalization (DG) deals with such an issue and it aims to learn a model from multiple source domains that can be generalized to unseen target domains. Existing studies on DG have largely focused on stationary settings with homogeneous source domains. However, in many applications,  domains may evolve along a specific direction (e.g., time, space). Without accounting for such non-stationary patterns, models trained with existing methods may fail to generalize on OOD data. In this paper, we study domain generalization in non-stationary environment. We first examine the impact of environmental non-stationarity on model performance and establish the theoretical upper bounds for the model error at target domains. Then, we propose a novel algorithm based on adaptive invariant representation learning, which leverages the non-stationary pattern to train a model that attains good performance on target domains. Experiments on both synthetic and real data validate the proposed algorithm.
\end{abstract}

\input{1_introduction}
\input{2_related_work}
\input{3_problem_formulation}
\input{4_theory}
\input{5_method}
\input{6_experiment}
\input{7_conclusion}
\input{11_ack}

% References

\bibliography{ref}

\newpage

\onecolumn

\title{Non-stationary Domain Generalization: Theory and Algorithm\\(Supplementary Material)}
\maketitle

\appendix

\input{8_app_proof.tex}
\newpage
\input{9_app_model.tex}
\newpage
\input{10_app_exp.tex}

\end{document}
