$\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}}} % % 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} % 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}}$

Summary of (part of) https://www.alignmentforum.org/posts/CoZhXrhpQxpy9xw9y/where-i-agree-and-disagree-with-eliezer.

1. We will be able to test many things out before the first critical try.
2. We’ll get superhuman (and extremely useful) systems before we get nanotech.
3. AI systems that hide their true capabilities will get weeded out by SGD.
4. Self-improvement is just human-led R&D; AI will be slow at improving themselves at first.
5. Intermediate AIs can reduce risks too (e.g. by consuming “free energy”).
6. Pivotal acts are super dangerous and toxic; policy is more promising and reasonable.
7. Intermediate AIs will be equally useful for AI capabilities and alignment.
8. Eliezer is too sloppy about generalization, convergent incentives, consequentialism.
9. Intelligence increase might be gradual.
10. Takeoff speeds will not be that fast, threshold not that hard.
11. MIRI failing doesn’t mean the problem is impossible.
12. Eliezer doesn’t understand how scientific progress happens.
13. Eliezer is strawmanning alignment strategies.
14. Eliezer is strawmanning interpretability.
15. Big tasks are made of smaller tasks with good feedback loops.
16. We’ll be able to study deceptive alignment in advance.
17. The list isn’t engaging with our most serious alignment hopes.
18. Natural selection is a bad analogy because we deliberately shape ML training.
19. We could probably verify plans for pivotal acts: it’s easier to verify than to find.
20. AIs will be able to imitate human thoughts just fine.
21. If any of our hopes work out, we win.
22. AIs might be very human-like, especially in early superhuman stages.
23. Coordination between AIs won’t be that easy in early stages because they’ll be messy.
24. Eliezer misunderstands the checks and balances proposal.
25. AIs might find pivotal acts before they’re superhuman at persuasion.
26. We might succeed without a clear plan.
27. Eliezer isn’t the only person meaningfully contributing to AI safety research.

Overall themes

• Intermediate systems will exist for some time and help us diminish the risk in many ways.
• Eliezer is being too confident.
• Eliezer underestimates the potential effectiveness of alignment research.