$\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 Buck Shlegeris’s talk at EA Global Bay Area 2023.

How to think about the impact of different types of alignment projects, based on how they help in a realistic near-term scenario.

The scenario

Magma, a safety-conscious AI company, just got a major breakthrough in AGI, and starts thinking about how to make things things go well. They have a trade-off with how much to use it:

  • if they don’t use it at all, a less careful company will create an unaligned AGI and kill everyone;
  • if they use it too much, they create risk themselves.

They could use their AI to help out with:

  • alignment research,
  • enforcing agreements about AI development,
  • making it harder for unaligned systems to cause trouble (e.g. hardening, eating low-hanging fruit, detection).

v0 plan

Magma just go ahead and train the thing to be nice and helpful based on human feedback. Two reasons why things could go wrong:

  • outer alignment failure: they gave incorrect feedback because the model’s outputs were too complicated to understand;
  • inner alignment failure: they gave correct feedback, but when we moved to production, the AI executed a coup.

v1 plan

Magma takes the obvious corresponding precautions, like

  • using other AIs to help evaluate the AI proposals (e.g. debate, IDA);
  • trying to make the AI think it moved to production and seeing if it executes a coup;
  • deployment-time measures, like checks and balances, whistleblowing.

This is not great: coups seems pretty hard, but these techniques are not foolproof (Shlegeris’s P(doom) is 25%). Shlegeris suggests evaluating every alignment proposal in terms of how it incrementally helps makes this plan more hopeful.

Five types of technical research

  1. Ready to go. Techniques that have been demonstrated to work, and are obviously applicable.
    • e.g. critiques, scalable oversight, model psychology
  2. Worst-case techniques. If it works out, give theoretical guarantees that apply no matter what happens.
    • e.g. ARC theory
  3. Gain of function. Experiment with alignment failure in current systems (+training wheels) to understand the threat better.
  4. Low-hanging fruit. Cheap tricks where there’s no great story of why it’ll help but they’re so easy that why not try.
    • e.g. find the “truth vector” in activation space, most of conventional ML research
  5. Big if true. Not clear it’s possible, but extremely great if it works.
    • e.g. interpretability

(It seems to me like 2 and 5 have a lot of overlap.)

As we get closer to AGI:

  • Gain of function and low-hanging fruit will be more realistic.
  • Less time for fancy stuff (e.g. worst-case techniques and big-if-true).
  • Hopefully, fancy stuff will graduate to “ready to go” in time.