$\require{mathtools}
\newcommand{\nc}{\newcommand}
%
%%% GENERIC MATH %%%
%
% Environments
\newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work
\renewcommand{\r}{\mathrm} % BAD!! does cursed things with accents :((
\renewcommand{\t}{\textrm}
\newcommand{\either}[1]{\begin{cases}#1\end{cases}}
%
% Delimiters
% (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing)
% .. one-part
\newcommand{\p}[1]{\mathopen{}\left( #1 \right)}
\renewcommand{\P}[1]{^{\p{#1}}}
\renewcommand{\b}[1]{\mathopen{}\left[ #1 \right]}
\newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}}
\newcommand{\abs}[1]{\mathopen{}\left\lvert #1 \right\rvert}
\newcommand{\floor}[1]{\mathopen{}\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\mathopen{}\left\lceil #1 \right\rceil}
\newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle}
\newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \strut \right\rVert}
\newcommand{\frob}[1]{\norm{#1}_\mathrm{F}}
\newcommand{\mix}[1]{\mathopen{}\left\lfloor #1 \right\rceil}
%% .. two-part
\newcommand{\inco}[2]{#1 \mathop{}\middle|\mathop{} #2}
\newcommand{\co}[2]{ {\left.\inco{#1}{#2}\right.}}
\newcommand{\cond}{\co} % deprecated
\newcommand{\pco}[2]{\p{\inco{#1}{#2}}}
\newcommand{\bco}[2]{\b{\inco{#1}{#2}}}
\newcommand{\setco}[2]{\set{\inco{#1}{#2}}}
\newcommand{\at}[2]{ {\left.#1\strut\right|_{#2}}}
\newcommand{\pat}[2]{\p{\at{#1}{#2}}}
\newcommand{\bat}[2]{\b{\at{#1}{#2}}}
\newcommand{\para}[2]{#1\strut \mathop{}\middle\|\mathop{} #2}
\newcommand{\ppa}[2]{\p{\para{#1}{#2}}}
\newcommand{\pff}[2]{\p{\ff{#1}{#2}}}
\newcommand{\bff}[2]{\b{\ff{#1}{#2}}}
\newcommand{\bffco}[4]{\bff{\cond{#1}{#2}}{\cond{#3}{#4}}}
\newcommand{\sm}[1]{\p{\begin{smallmatrix}#1\end{smallmatrix}}}
%
% Greek
\newcommand{\eps}{\epsilon}
\newcommand{\veps}{\varepsilon}
\newcommand{\vpi}{\varpi}
% the following cause issues with real LaTeX tho :/ maybe consider naming it \fhi instead?
\let\fi\phi % because it looks like an f
\let\phi\varphi % because it looks like a p
\renewcommand{\th}{\theta}
\newcommand{\Th}{\Theta}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
%
% Miscellaneous
\newcommand{\LHS}{\mathrm{LHS}}
\newcommand{\RHS}{\mathrm{RHS}}
\DeclareMathOperator{\cst}{const}
% .. operators
\DeclareMathOperator{\poly}{poly}
\DeclareMathOperator{\polylog}{polylog}
\DeclareMathOperator{\quasipoly}{quasipoly}
\DeclareMathOperator{\negl}{negl}
\DeclareMathOperator*{\argmin}{arg\thinspace min}
\DeclareMathOperator*{\argmax}{arg\thinspace max}
\DeclareMathOperator{\diag}{diag}
% .. functions
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\err}{err}
\DeclareMathOperator{\ReLU}{ReLU}
\DeclareMathOperator{\softmax}{softmax}
% .. analysis
\let\d\undefined
\newcommand{\d}{\operatorname{d}\mathopen{}}
\newcommand{\dd}[1]{\operatorname{d}^{#1}\mathopen{}}
\newcommand{\df}[2]{ {\f{\d #1}{\d #2}}}
\newcommand{\ds}[2]{ {\sl{\d #1}{\d #2}}}
\newcommand{\ddf}[3]{ {\f{\dd{#1} #2}{\p{\d #3}^{#1}}}}
\newcommand{\dds}[3]{ {\sl{\dd{#1} #2}{\p{\d #3}^{#1}}}}
\renewcommand{\part}{\partial}
\newcommand{\partf}[2]{\f{\part #1}{\part #2}}
\newcommand{\parts}[2]{\sl{\part #1}{\part #2}}
\newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}}
% .. sets
\newcommand{\es}{\emptyset}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Rge}{\R_{\ge 0}}
\newcommand{\Rgt}{\R_{> 0}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\zo}{\set{0,1}}
\newcommand{\pmo}{\set{\pm 1}}
\newcommand{\zpmo}{\set{0,\pm 1}}
% .... set operations
\newcommand{\sse}{\subseteq}
\newcommand{\out}{\not\in}
\newcommand{\minus}{\setminus}
\newcommand{\inc}[1]{\union \set{#1}} % "including"
\newcommand{\exc}[1]{\setminus \set{#1}} % "except"
% .. over and under
\renewcommand{\ss}[1]{_{\substack{#1}}}
\newcommand{\OB}{\overbrace}
\newcommand{\ob}[2]{\OB{#1}^\t{#2}}
\newcommand{\UB}{\underbrace}
\newcommand{\ub}[2]{\UB{#1}_\t{#2}}
\newcommand{\ol}{\overline}
\newcommand{\tld}{\widetilde} % deprecated
\renewcommand{\~}{\widetilde}
\newcommand{\HAT}{\widehat} % deprecated
\renewcommand{\^}{\widehat}
\newcommand{\rt}[1]{ {\sqrt{#1}}}
\newcommand{\for}[2]{_{#1=1}^{#2}}
\newcommand{\sfor}{\sum\for}
\newcommand{\pfor}{\prod\for}
% .... two-part
\newcommand{\f}{\frac}
\renewcommand{\sl}[2]{#1 /\mathopen{}#2}
\newcommand{\ff}[2]{\mathchoice{\begin{smallmatrix}\displaystyle\vphantom{\p{#1}}#1\\[-0.05em]\hline\\[-0.05em]\hline\displaystyle\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}{\begin{smallmatrix}\vphantom{\p{#1}}#1\\[-0.1em]\hline\\[-0.1em]\hline\vphantom{\p{#2}}#2\end{smallmatrix}}}
% .. arrows
\newcommand{\from}{\leftarrow}
\DeclareMathOperator*{\<}{\!\;\longleftarrow\;\!}
\let\>\undefined
\DeclareMathOperator*{\>}{\!\;\longrightarrow\;\!}
\let\-\undefined
\DeclareMathOperator*{\-}{\!\;\longleftrightarrow\;\!}
\newcommand{\so}{\implies}
% .. operators and relations
\renewcommand{\*}{\cdot}
\newcommand{\x}{\times}
\newcommand{\ox}{\otimes}
\newcommand{\OX}[1]{^{\ox #1}}
\newcommand{\sr}{\stackrel}
\newcommand{\ce}{\coloneqq}
\newcommand{\ec}{\eqqcolon}
\newcommand{\ap}{\approx}
\newcommand{\ls}{\lesssim}
\newcommand{\gs}{\gtrsim}
% .. punctuation and spacing
\renewcommand{\.}[1]{#1\dots#1}
\newcommand{\ts}{\thinspace}
\newcommand{\q}{\quad}
\newcommand{\qq}{\qquad}
%
% Levels of closeness
\newcommand{\scirc}[1]{\sr{\circ}{#1}}
\newcommand{\sdot}[1]{\sr{.}{#1}}
\newcommand{\slog}[1]{\sr{\log}{#1}}
\newcommand{\createClosenessLevels}[7]{
\newcommand{#2}{\mathrel{(#1)}}
\newcommand{#3}{\mathrel{#1}}
\newcommand{#4}{\mathrel{#1\!\!#1}}
\newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}}
\newcommand{#6}{\mathrel{(\sdot{#1})}}
\newcommand{#7}{\mathrel{(\slog{#1})}}
}
\let\lt\undefined
\let\gt\undefined
% .. vanilla versions (is it within a constant?)
\newcommand{\ez}{\scirc=}
\newcommand{\eq}{\simeq}
\newcommand{\eqq}{\mathrel{\eq\!\!\eq}}
\newcommand{\eqqq}{\mathrel{\eq\!\!\eq\!\!\eq}}
\newcommand{\lez}{\scirc\le}
\renewcommand{\lq}{\preceq}
\newcommand{\lqq}{\mathrel{\lq\!\!\lq}}
\newcommand{\lqqq}{\mathrel{\lq\!\!\lq\!\!\lq}}
\newcommand{\gez}{\scirc\ge}
\newcommand{\gq}{\succeq}
\newcommand{\gqq}{\mathrel{\gq\!\!\gq}}
\newcommand{\gqqq}{\mathrel{\gq\!\!\gq\!\!\gq}}
\newcommand{\lz}{\scirc<}
\newcommand{\lt}{\prec}
\newcommand{\ltt}{\mathrel{\lt\!\!\lt}}
\newcommand{\lttt}{\mathrel{\lt\!\!\lt\!\!\lt}}
\newcommand{\gz}{\scirc>}
\newcommand{\gt}{\succ}
\newcommand{\gtt}{\mathrel{\gt\!\!\gt}}
\newcommand{\gttt}{\mathrel{\gt\!\!\gt\!\!\gt}}
% .. dotted versions (is it equal in the limit?)
\newcommand{\ed}{\sdot=}
\newcommand{\eqd}{\sdot\eq}
\newcommand{\eqqd}{\sdot\eqq}
\newcommand{\eqqqd}{\sdot\eqqq}
\newcommand{\led}{\sdot\le}
\newcommand{\lqd}{\sdot\lq}
\newcommand{\lqqd}{\sdot\lqq}
\newcommand{\lqqqd}{\sdot\lqqq}
\newcommand{\ged}{\sdot\ge}
\newcommand{\gqd}{\sdot\gq}
\newcommand{\gqqd}{\sdot\gqq}
\newcommand{\gqqqd}{\sdot\gqqq}
\newcommand{\ld}{\sdot<}
\newcommand{\ltd}{\sdot\lt}
\newcommand{\lttd}{\sdot\ltt}
\newcommand{\ltttd}{\sdot\lttt}
\newcommand{\gd}{\sdot>}
\newcommand{\gtd}{\sdot\gt}
\newcommand{\gttd}{\sdot\gtt}
\newcommand{\gtttd}{\sdot\gttt}
% .. log versions (is it equal up to log?)
\newcommand{\elog}{\slog=}
\newcommand{\eqlog}{\slog\eq}
\newcommand{\eqqlog}{\slog\eqq}
\newcommand{\eqqqlog}{\slog\eqqq}
\newcommand{\lelog}{\slog\le}
\newcommand{\lqlog}{\slog\lq}
\newcommand{\lqqlog}{\slog\lqq}
\newcommand{\lqqqlog}{\slog\lqqq}
\newcommand{\gelog}{\slog\ge}
\newcommand{\gqlog}{\slog\gq}
\newcommand{\gqqlog}{\slog\gqq}
\newcommand{\gqqqlog}{\slog\gqqq}
\newcommand{\llog}{\slog<}
\newcommand{\ltlog}{\slog\lt}
\newcommand{\lttlog}{\slog\ltt}
\newcommand{\ltttlog}{\slog\lttt}
\newcommand{\glog}{\slog>}
\newcommand{\gtlog}{\slog\gt}
\newcommand{\gttlog}{\slog\gtt}
\newcommand{\gtttlog}{\slog\gttt}
%
%
%%% SPECIALIZED MATH %%%
%
% Logic and bit operations
\newcommand{\fa}{\forall}
\newcommand{\ex}{\exists}
\renewcommand{\and}{\wedge}
\newcommand{\AND}{\bigwedge}
\renewcommand{\or}{\vee}
\newcommand{\OR}{\bigvee}
\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\bigoplus}
\newcommand{\union}{\cup}
\newcommand{\dunion}{\sqcup}
\newcommand{\inter}{\cap}
\newcommand{\UNION}{\bigcup}
\newcommand{\DUNION}{\bigsqcup}
\newcommand{\INTER}{\bigcap}
\newcommand{\comp}{\overline}
\newcommand{\true}{\r{true}}
\newcommand{\false}{\r{false}}
\newcommand{\tf}{\set{\true,\false}}
\DeclareMathOperator{\One}{\mathbb{1}}
\DeclareMathOperator{\1}{\mathbb{1}} % use \mathbbm instead if using real LaTeX
\DeclareMathOperator{\LSB}{LSB}
%
% Linear algebra
\newcommand{\spn}{\mathrm{span}} % do NOT use \span because it causes misery with amsmath
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\Img}{Im}
\newcommand{\transp}{\mathsf{T}}
\newcommand{\T}{^\transp}
% .. named tensors
\newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut
\newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}}
\newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}}
\newcommand{\ndot}[1]{\nbin{#1}{\odot}}
\newcommand{\ncat}[1]{\nbin{#1}{\oplus}}
\newcommand{\nsum}[1]{\sum\limits_{\substack{#1}}}
\newcommand{\nfun}[2]{\mathop{\underset{\substack{#1}}{\namedtensorstrut\mathrm{#2}}}}
\newcommand{\ndef}[2]{\newcommand{#1}{\name{#2}}}
\newcommand{\nt}[1]{^{\transp(#1)}}
%
% Probability
\newcommand{\tri}{\triangle}
\newcommand{\Normal}{\mathcal{N}}
\newcommand{\Exp}{\mathcal{Exp}}
% .. operators
\DeclareMathOperator{\supp}{supp}
\let\Pr\undefined
\DeclareMathOperator*{\Pr}{Pr}
\DeclareMathOperator*{\G}{\mathbb{G}}
\DeclareMathOperator*{\Odds}{Od}
\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
\DeclareMathOperator*{\Cov}{Cov}
\DeclareMathOperator*{\K}{K}
\DeclareMathOperator*{\corr}{corr}
\DeclareMathOperator*{\median}{median}
\DeclareMathOperator*{\maj}{maj}
% ... information theory
\let\H\undefined
\DeclareMathOperator*{\H}{H}
\DeclareMathOperator*{\I}{I}
\DeclareMathOperator*{\D}{D}
\DeclareMathOperator*{\KL}{KL}
% .. other divergences
\newcommand{\dTV}{d_{\mathrm{TV}}}
\newcommand{\dHel}{d_{\mathrm{Hel}}}
\newcommand{\dJS}{d_{\mathrm{JS}}}
%
%%% SPECIALIZED COMPUTER SCIENCE %%%
%
% Complexity classes
% .. keywords
\newcommand{\coclass}{\mathsf{co}}
\newcommand{\Prom}{\mathsf{Promise}}
% .. classical
\newcommand{\PTIME}{\mathsf{P}}
\newcommand{\NP}{\mathsf{NP}}
\newcommand{\coNP}{\coclass\NP}
\newcommand{\PH}{\mathsf{PH}}
\newcommand{\PSPACE}{\mathsf{PSPACE}}
\renewcommand{\L}{\mathsf{L}}
% .. probabilistic
\newcommand{\formost}{\mathsf{Я}}
\newcommand{\RP}{\mathsf{RP}}
\newcommand{\BPP}{\mathsf{BPP}}
\newcommand{\MA}{\mathsf{MA}}
\newcommand{\AM}{\mathsf{AM}}
\newcommand{\IP}{\mathsf{IP}}
\newcommand{\RL}{\mathsf{RL}}
% .. circuits
\newcommand{\NC}{\mathsf{NC}}
\newcommand{\AC}{\mathsf{AC}}
\newcommand{\ACC}{\mathsf{ACC}}
\newcommand{\ThrC}{\mathsf{TC}}
\newcommand{\Ppoly}{\mathsf{P}/\poly}
\newcommand{\Lpoly}{\mathsf{L}/\poly}
% .. resources
\newcommand{\TIME}{\mathsf{TIME}}
\newcommand{\SPACE}{\mathsf{SPACE}}
\newcommand{\TISP}{\mathsf{TISP}}
\newcommand{\SIZE}{\mathsf{SIZE}}
% .. custom
\newcommand{\NCP}{\mathsf{NCP}}
%
% Boolean analysis
\newcommand{\harpoon}{\!\upharpoonright\!}
\newcommand{\rr}[2]{#1\harpoon_{#2}}
\newcommand{\Fou}[1]{\widehat{#1}}
\DeclareMathOperator{\Ind}{\mathrm{Ind}}
\DeclareMathOperator{\Inf}{\mathrm{Inf}}
\newcommand{\Der}[1]{\operatorname{D}_{#1}\mathopen{}}
% \newcommand{\Exp}[1]{\operatorname{E}_{#1}\mathopen{}}
\DeclareMathOperator{\Stab}{\mathrm{Stab}}
\DeclareMathOperator{\Tau}{T}
\DeclareMathOperator{\sens}{\mathrm{s}}
\DeclareMathOperator{\bsens}{\mathrm{bs}}
\DeclareMathOperator{\fbsens}{\mathrm{fbs}}
\DeclareMathOperator{\Cert}{\mathrm{C}}
\DeclareMathOperator{\DT}{\mathrm{DT}}
\DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical
\DeclareMathOperator{\ECDT}{\mathrm{ECDT}}
\DeclareMathOperator{\CDTv}{\mathrm{CDT_{vars}}}
\DeclareMathOperator{\ECDTv}{\mathrm{ECDT_{vars}}}
\DeclareMathOperator{\CDTt}{\mathrm{CDT_{terms}}}
\DeclareMathOperator{\ECDTt}{\mathrm{ECDT_{terms}}}
\DeclareMathOperator{\CDTw}{\mathrm{CDT_{weighted}}}
\DeclareMathOperator{\ECDTw}{\mathrm{ECDT_{weighted}}}
\DeclareMathOperator{\AvgDT}{\mathrm{AvgDT}}
\DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree
\DeclareMathOperator{\DTsize}{\mathrm{DT_{size}}}
\DeclareMathOperator{\W}{\mathbf{W}}
% .. functions (small caps sadly doesn't work)
\DeclareMathOperator{\Par}{\mathrm{Par}}
\DeclareMathOperator{\Maj}{\mathrm{Maj}}
\DeclareMathOperator{\HW}{\mathrm{HW}}
\DeclareMathOperator{\Thr}{\mathrm{Thr}}
\DeclareMathOperator{\Tribes}{\mathrm{Tribes}}
\DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}}
\DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}}
\DeclareMathOperator{\SAT}{\mathrm{SAT}}
\DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}}
%
% Dynamic optimality
\newcommand{\OPT}{\mathsf{OPT}}
\newcommand{\Alt}{\mathsf{Alt}}
\newcommand{\Funnel}{\mathsf{Funnel}}
%
% Alignment
\DeclareMathOperator{\Amp}{\mathrm{Amp}}
%
%%% TYPESETTING %%%
%
% In "text"
\newcommand{\heart}{\heartsuit}
\newcommand{\nth}{^\t{th}}
\newcommand{\degree}{^\circ}
\newcommand{\qu}[1]{\text{``}#1\text{''}}
% remove these last two if using real LaTeX
\newcommand{\qed}{\blacksquare}
\newcommand{\qedhere}{\tag*{$\blacksquare$}}
%
% Fonts
% .. bold
\newcommand{\BA}{\boldsymbol{A}}
\newcommand{\BB}{\boldsymbol{B}}
\newcommand{\BC}{\boldsymbol{C}}
\newcommand{\BD}{\boldsymbol{D}}
\newcommand{\BE}{\boldsymbol{E}}
\newcommand{\BF}{\boldsymbol{F}}
\newcommand{\BG}{\boldsymbol{G}}
\newcommand{\BH}{\boldsymbol{H}}
\newcommand{\BI}{\boldsymbol{I}}
\newcommand{\BJ}{\boldsymbol{J}}
\newcommand{\BK}{\boldsymbol{K}}
\newcommand{\BL}{\boldsymbol{L}}
\newcommand{\BM}{\boldsymbol{M}}
\newcommand{\BN}{\boldsymbol{N}}
\newcommand{\BO}{\boldsymbol{O}}
\newcommand{\BP}{\boldsymbol{P}}
\newcommand{\BQ}{\boldsymbol{Q}}
\newcommand{\BR}{\boldsymbol{R}}
\newcommand{\BS}{\boldsymbol{S}}
\newcommand{\BT}{\boldsymbol{T}}
\newcommand{\BU}{\boldsymbol{U}}
\newcommand{\BV}{\boldsymbol{V}}
\newcommand{\BW}{\boldsymbol{W}}
\newcommand{\BX}{\boldsymbol{X}}
\newcommand{\BY}{\boldsymbol{Y}}
\newcommand{\BZ}{\boldsymbol{Z}}
\newcommand{\Ba}{\boldsymbol{a}}
\newcommand{\Bb}{\boldsymbol{b}}
\newcommand{\Bc}{\boldsymbol{c}}
\newcommand{\Bd}{\boldsymbol{d}}
\newcommand{\Be}{\boldsymbol{e}}
\newcommand{\Bf}{\boldsymbol{f}}
\newcommand{\Bg}{\boldsymbol{g}}
\newcommand{\Bh}{\boldsymbol{h}}
\newcommand{\Bi}{\boldsymbol{i}}
\newcommand{\Bj}{\boldsymbol{j}}
\newcommand{\Bk}{\boldsymbol{k}}
\newcommand{\Bl}{\boldsymbol{l}}
\newcommand{\Bm}{\boldsymbol{m}}
\newcommand{\Bn}{\boldsymbol{n}}
\newcommand{\Bo}{\boldsymbol{o}}
\newcommand{\Bp}{\boldsymbol{p}}
\newcommand{\Bq}{\boldsymbol{q}}
\newcommand{\Br}{\boldsymbol{r}}
\newcommand{\Bs}{\boldsymbol{s}}
\newcommand{\Bt}{\boldsymbol{t}}
\newcommand{\Bu}{\boldsymbol{u}}
\newcommand{\Bv}{\boldsymbol{v}}
\newcommand{\Bw}{\boldsymbol{w}}
\newcommand{\Bx}{\boldsymbol{x}}
\newcommand{\By}{\boldsymbol{y}}
\newcommand{\Bz}{\boldsymbol{z}}
\newcommand{\Balpha}{\boldsymbol{\alpha}}
\newcommand{\Bbeta}{\boldsymbol{\beta}}
\newcommand{\Bgamma}{\boldsymbol{\gamma}}
\newcommand{\Bdelta}{\boldsymbol{\delta}}
\newcommand{\Beps}{\boldsymbol{\eps}}
\newcommand{\Bveps}{\boldsymbol{\veps}}
\newcommand{\Bzeta}{\boldsymbol{\zeta}}
\newcommand{\Beta}{\boldsymbol{\eta}}
\newcommand{\Btheta}{\boldsymbol{\theta}}
\newcommand{\Bth}{\boldsymbol{\th}}
\newcommand{\Biota}{\boldsymbol{\iota}}
\newcommand{\Bkappa}{\boldsymbol{\kappa}}
\newcommand{\Blambda}{\boldsymbol{\lambda}}
\newcommand{\Bmu}{\boldsymbol{\mu}}
\newcommand{\Bnu}{\boldsymbol{\nu}}
\newcommand{\Bxi}{\boldsymbol{\xi}}
\newcommand{\Bpi}{\boldsymbol{\pi}}
\newcommand{\Bvpi}{\boldsymbol{\vpi}}
\newcommand{\Brho}{\boldsymbol{\rho}}
\newcommand{\Bsigma}{\boldsymbol{\sigma}}
\newcommand{\Btau}{\boldsymbol{\tau}}
\newcommand{\Bupsilon}{\boldsymbol{\upsilon}}
\newcommand{\Bphi}{\boldsymbol{\phi}}
\newcommand{\Bfi}{\boldsymbol{\fi}}
\newcommand{\Bchi}{\boldsymbol{\chi}}
\newcommand{\Bpsi}{\boldsymbol{\psi}}
\newcommand{\Bom}{\boldsymbol{\om}}
% .. calligraphic
\newcommand{\CA}{\mathcal{A}}
\newcommand{\CB}{\mathcal{B}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CD}{\mathcal{D}}
\newcommand{\CE}{\mathcal{E}}
\newcommand{\CF}{\mathcal{F}}
\newcommand{\CG}{\mathcal{G}}
\newcommand{\CH}{\mathcal{H}}
\newcommand{\CI}{\mathcal{I}}
\newcommand{\CJ}{\mathcal{J}}
\newcommand{\CK}{\mathcal{K}}
\newcommand{\CL}{\mathcal{L}}
\newcommand{\CM}{\mathcal{M}}
\newcommand{\CN}{\mathcal{N}}
\newcommand{\CO}{\mathcal{O}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CQ}{\mathcal{Q}}
\newcommand{\CR}{\mathcal{R}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\CT}{\mathcal{T}}
\newcommand{\CU}{\mathcal{U}}
\newcommand{\CV}{\mathcal{V}}
\newcommand{\CW}{\mathcal{W}}
\newcommand{\CX}{\mathcal{X}}
\newcommand{\CY}{\mathcal{Y}}
\newcommand{\CZ}{\mathcal{Z}}
% .. typewriter
\newcommand{\TA}{\mathtt{A}}
\newcommand{\TB}{\mathtt{B}}
\newcommand{\TC}{\mathtt{C}}
\newcommand{\TD}{\mathtt{D}}
\newcommand{\TE}{\mathtt{E}}
\newcommand{\TF}{\mathtt{F}}
\newcommand{\TG}{\mathtt{G}}
\renewcommand{\TH}{\mathtt{H}}
\newcommand{\TI}{\mathtt{I}}
\newcommand{\TJ}{\mathtt{J}}
\newcommand{\TK}{\mathtt{K}}
\newcommand{\TL}{\mathtt{L}}
\newcommand{\TM}{\mathtt{M}}
\newcommand{\TN}{\mathtt{N}}
\newcommand{\TO}{\mathtt{O}}
\newcommand{\TP}{\mathtt{P}}
\newcommand{\TQ}{\mathtt{Q}}
\newcommand{\TR}{\mathtt{R}}
\newcommand{\TS}{\mathtt{S}}
\newcommand{\TT}{\mathtt{T}}
\newcommand{\TU}{\mathtt{U}}
\newcommand{\TV}{\mathtt{V}}
\newcommand{\TW}{\mathtt{W}}
\newcommand{\TX}{\mathtt{X}}
\newcommand{\TY}{\mathtt{Y}}
\newcommand{\TZ}{\mathtt{Z}}$
Assume that two features $i$ and $j$ never co-occur. Then there are two main ways that they can share the same neuron with zero loss:
- $i$ and $j$ activate the neuron with opposite signs;[^1]
- $i$ and $j$ activate the neuron with the same sign, but some other feature can help disambiguate them.
Toy model
Model
The model has three layers (input, hidden, output):
- $n$ input coordinates;
- $m$ hidden coordinates, with $l_1$ regularization on “activations”;
- $n$ output coordinates, with biases and ReLU.
The encoding (input -> hidden) and decoding (hidden -> output) weights can be tied together for simplicity, and this is what we’ll assume for the rest of the write-up, even though I think the story is slightly more natural without tying (because then each output can individually choose to use the disambiguation neuron or not).
Task
The input vectors $x \in \zo^n$ are uniform under the condition
\[
\set{|x_{1, \ldots, n/2}|, |x_{n/2+1, \ldots n}|} = \set{0, n/4}.
\]
That is, we choose either the first half or the second half of the coordinates at random, and turn on exactly half of the coordinates within that half.
Output and loss
- Autoencoder task: the output is equal to the input.
- The loss is the squared norm of the difference between input and output.
Possible solution
Here is a description of possible trained model that gets zero loss despite having some pairs of features be represented on the same hidden neuron and with the same sign.
- One “disambiguation neuron” which has small positive weights on the first half and small negative weights on the second half (specifically, $\sqrt{4/n}$ and $-\sqrt{4/n}$).
- This contributes $+1$ to the output of each feature that’s in the chosen half, and $-1$ to the output of each feature in the other half.
- Other than that, each input feature is represented in exactly one hidden neuron, under the constraints:
- no two features from the same half share the same hidden neuron;
- each hidden neuron represents at most two features (from different halves);
- all weights are $\pm 1$.
- All biases are $-1$.
Schematic view of the trained model:
Schematic view of the weight matrix:
This works because:
- The output of all features in the chosen half will be
- $+1+1-1 = 1$ if the feature is on,
- $+1-1 = 0$ if the feature is off.
- The (pre-ReLU) output of all features in the other half will be
- $-1+1-1=-1$ if the hidden neuron that the feature is represented on gets turned on,
- $-1-1 = -2$ otherwise.
Disclaimers:
- I am not claiming that this covers all possible zero-loss solutions (it clearly doesn’t: for example you could make the weights on the disambiguation neuron bigger and the bias correspondingly bigger).
- I am even less sure that SGD would find such a solution.
- I am not even sure that this solution is stable once taking the pressure from the $l_1$ regularization into account.
[^1]: note that this only really makes sense if the neuron does not have a non-linearity