\documentclass[accepted]{uai2022} % for initial submission
% \documentclass[accepted]{uai2022} % 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]{uai2022} % ptmx math instead of Computer
                                         % Modern (has noticable issues)
% \documentclass[mathfont=newtx]{uai2022} % 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.

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\usepackage[american]{babel}
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%% 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

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%           redefining \footnotesize, we provide the original \footnotesize
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%% Self-defined macros
\newcommand{\swap}[3][-]{#3#1#2} % just an example

\input{math_commands.tex}

\usepackage{url}

\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography

\usepackage{geometry}
\usepackage{amsmath}
\usepackage{amssymb}

\usepackage{graphicx}
\usepackage{subfigure}
\usepackage{multirow}
\usepackage{makecell}
\usepackage{wrapfig}

\usepackage{caption}
%\usepackage{subcaption}
\usepackage{wrapfig}

\usepackage{algorithm,algpseudocode}
\usepackage{qiangstyle}


% \newcommand{\theHalgorithm}{\arabic{algorithm}}

% \title{Pareto Navigation Gradient Descent:
% a First-Order Algorithm for Optimization in Pareto Set}
\title{Pareto Navigation Gradient Descent: a First-Order Algorithm for Optimization in Pareto Set}

% The standard author block has changed for UAI 2022 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:<maoye21@utexas.edu>?Subject=Your UAI 2022 paper}{Mao Ye}{}}
\author[1]{Qiang Liu}
% Add affiliations after the authors
\affil[1]{%
    Computer Science Dept.\\
    The University of Texas at Austin.
}

  
\begin{document}

\global\long\def\th{\theta}%
\global\long\def\E{\mathrm{E}}%
\global\long\def\L{\vv{\ell}}%\text{L}}%
\global\long\def\dge{\succeq}%
\global\long\def\dle{\preceq}%
\global\long\def\dsqe{\succ}%
\global\long\def\dsle{\prec}%
\global\long\def\C{\mathcal{C}}%
\global\long\def\P{\mathcal{P}}%
\global\long\def\df{\mathrm{d}}
\global\long\def\tim{{t}}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{assumption}{Assumption}

\maketitle

\begin{abstract}
Many modern machine learning applications, such as multi-task learning, require finding optimal model parameters to trade-off multiple objective functions that may conflict with each other.
The notion of the Pareto set allows us to focus on the set of (often infinite number of)
models that cannot be strictly improved. But it does not provide an actionable procedure for picking one or a few special models to return to practical users. In this paper, we consider \emph{optimization in Pareto set (OPT-in-Pareto)}, the problem of finding Pareto models that optimize an extra reference criterion function within the Pareto set. This function can either encode a specific preference from the users, or represent a generic diversity measure for obtaining a set of diversified Pareto models that are representative of the whole Pareto set.
Unfortunately, despite being a highly useful framework, efficient algorithms for OPT-in-Pareto have been largely missing, especially for large-scale, non-convex, and non-linear objectives in deep learning. A naive approach is to apply Riemannian manifold gradient descent on the Pareto set, which yields a high computational cost due to the need for eigen-calculation of Hessian matrices. We propose a first-order algorithm that approximately solves OPT-in-Pareto using only gradient information, with both high practical efficiency and theoretically guaranteed convergence property. Empirically, we demonstrate that our method works efficiently for a variety of challenging multi-task-related problems.
\end{abstract}


\input{tex/introduction}
\input{tex/background_qiang}
\input{tex/problem}
\input{tex/algorithm}
\input{tex/related}
\input{tex/application}
\input{tex/conclusion}

% \begin{acknowledgements} % will be removed in pdf for initial submission,
%                          % so you can already fill it to test with the
%                          % ‘accepted’ class option
%     Briefly acknowledge people and organizations here.

%     \emph{All} acknowledgements go in this section.
% \end{acknowledgements}

\bibliography{uai2022-template}

%\appendix
%\onecolumn
%\title{Pareto Navigation Gradient Descent: a First-Order Algorithm for Optimization in Pareto Set}
%\maketitle
%\input{tex/continous_proof} 
%\clearpage
%\input{tex/experiment_appendix}

\end{document}
