$\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}
\renewcommand{\t}{\textrm}
%
% 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}
% .. functions
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\err}{err}
\DeclareMathOperator{\ReLU}{ReLU}
% .. 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{\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{\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{\inter}{\cap}
\newcommand{\UNION}{\bigcup}
\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}}
% .. 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
% .. classical
\newcommand{\Poly}{\mathsf{P}}
\newcommand{\NP}{\mathsf{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}}
% .. keywords
\newcommand{\coclass}{\mathsf{co}}
\newcommand{\Prom}{\mathsf{Promise}}
%
% 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}}$
Personal summary of Thoughts on AI safety by Boaz Barak. See personal takes below.
There are many risks:
- malfunctions (e.g. self-driving cars killing people),
- unfairness (e.g. bias in bail decisions),
- feedback loops between several AIs (e.g. 2010 flash crash),
- societal risks (e.g. job loss, power imbalance),
- AI literacy (e.g. not recognizing deepfakes),
- misuse (e.g. hacking, designing weapons),
- misalignment (e.g. AI takeover).
And different fields of “AI safety”:
- assurance / quality control (clarify and verify specifications),
- ethics,
- security (e.g. malicious users, data suppliers),
- alignment (AI is the adversary).
Two important questions to consider when presented with a safety problem:
- Does safety align with corporate incentives?
- Does safety improve with the capabilities of the model?
Impact of AI on humanity – the baseline expectation
Two “lessons” from history:
- The 2 technological revolutions we’ve seen have both gone well overall.
- There is no reason to expect fairness.
Capabilities
We might expect to go crowd worker -> college student -> PhD -> world expert one OOM of scaling at a time.
Capabilities vs. Safety Round 1: Risk of misuse
When the attacker and the defender are given the same tools, it’s unclear how the balance of power will shift (for example, Barak argues ChatGPT might make cheating harder because it’s easier to check for it).
Capabilities vs. Safety Round 2: Unaligned powerful AI systems
_Also summarizing parts of GPT as an “Intelligence Forklift”.
We can classify worrying AIs into two caricatures:
- Genie: tries reall hard to do what we tell it to do but Goodharts everything.
- Alien: does its own thing, might or might not care about us.
Barak argues that viewing AIs as genies or aliens (indeed, as an individual entity/agent at all) is misguided as long as you believe that the core of the AI’s intelligence is the pre-trained text predictor model. It’s more like a tiny agentic head on top of an intelligent non-agentic language model. The language model is a general imitation of all of humanity that we need to condition into what we want by prompting it. It’s an “intelligence forklift” that humans and AIs will have equal access to.
Additionally (?), Barak believes that intelligence is much more a property of the society than the individual.
Verification
Barak is positive about TCS-like verification tools for AIs. (Some of it sounded to me strikingly similar to what ARC theory is trying to do.)
My personal takes
It seems like the biggest disagreement between this piece by Barak and the AI x-risk community is the assumption that the agency will not be intimately tied with the intelligence.
- This is the case right now: GPT-4 is purely “system 1”, definitely not (explicitly or proficiently) forming, filtering and executing plans to achieve its goals.
- On the other hand, the rationalist community is founded around a strong belief in the power of agency.
It’s not clear whether we’re likely to get a true AGI that is nicely separable into a large “intelligence forklift” (that everyone can use) and a small “agentic head”. I think it’s at least a possibility that being able to reason at the level of intuition rather at the level of English LLM text will be necessary in order to get transformative AI.
If AIs can indeed be separated into a small agentic head and a large intelligence forklift, then it seems like scalable oversight might be easy (since we’re on the same footing as the AIs)… as long as we have enough time to learn to use the forklift as well as the AIs do? It definitely seems like a better situation than one where all the intelligence is bundled up within an intuitive level inside of a fundamentally agentic AI.