% Editable research discussion, based on arXiv:2610.19768v1. % Compile with build_slides.py using pdfLaTeX, with shell escape disabled. \documentclass[aspectratio=169,11pt]{beamer} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,graphicx,booktabs} \definecolor{AcademicBlue}{RGB}{26,57,86} \definecolor{AcademicTeal}{RGB}{0,105,112} \setbeamercolor{structure}{fg=AcademicBlue} \setbeamercolor{frametitle}{fg=AcademicBlue,bg=white} \setbeamercolor{title}{fg=AcademicBlue} \setbeamercolor{alerted text}{fg=AcademicTeal} \setbeamercolor{block title}{fg=white,bg=AcademicBlue} \setbeamercolor{block body}{fg=black,bg=AcademicBlue!5} \setbeamertemplate{navigation symbols}{} \setbeamertemplate{itemize items}[circle] \setbeamertemplate{footline}{% \hfill\usebeamerfont{page number in head/foot}% \insertframenumber\,/\,\inserttotalframenumber\hspace{4mm}\vspace{3mm}} \setbeamersize{text margin left=8mm,text margin right=8mm} \setbeamerfont{frametitle}{size=\Large,series=\bfseries} \setbeamerfont{title}{size=\LARGE,series=\bfseries} \newcommand{\papersource}[1]{\par\smallskip{\scriptsize\color{gray}#1\par}} \newcommand{\FO}{\textrm{\textsc{final-only}}} \newcommand{\AP}{\textrm{\textsc{all-prefix}}} \newcommand{\SP}{\textrm{\textsc{sampled-prefix}}} \newcommand{\ME}{Matched exit} \newcommand{\ind}{\mathbf{1}} \title[Compute, retain, reuse]{Learning to Compute Is Not Learning\\to Retain or Reuse in Looped Transformers} \subtitle{A paper discussion: three properties, three tests} \author{Paper authors: Peilong Zhou, Hongtao Zhang, Yulong Huang,\\Cangyuan Li, Mingyu Chen, Ying Wang} \date{arXiv:2610.19768v1 \quad $\vert$ \quad 15 October 2026} \begin{document} % 1 \begin{frame} \titlepage \vspace{-5mm} \begin{center} \alert{A correct answer can arrive before its state is ready to stop updating.} \end{center} \papersource{Independent reading presentation; not authored by the paper's authors. \href{https://arxiv.org/abs/2610.19768v1}{https://arxiv.org/abs/2610.19768v1}} \end{frame} % 2 \begin{frame}{One accuracy curve hides three different questions} \renewcommand{\arraystretch}{1.35} \begin{tabular}{p{2.3cm}p{5.1cm}p{5.2cm}} \toprule Property & Question & Test in this paper\\ \midrule \textbf{Compute} & Was the answer produced by a usable computation? & Read at arrival; redirect exact continuation with a state swap.\\ \textbf{Retain} & Does the answer survive more loops? & Keep the input and model fixed; extend the loop budget.\\ \textbf{Reuse} & Can a solved position stop changing? & Freeze its hidden state; test later positions.\\ \bottomrule \end{tabular} \par\medskip \alert{Correct readout $\not\Rightarrow$ stable readout $\not\Rightarrow$ safe state freezing.} \papersource{Paper, Secs. 1--2 and 5, PDF pp. 1--4, 8. Reuse here follows the freezing test.} \end{frame} % 3 \begin{frame}{Mealy machines make every continuation checkable} \begin{columns}[T,onlytextwidth] \begin{column}{0.49\textwidth} \begin{block}{Input} Initial state $q_0$; transition table; action sequence $a_1,\ldots,a_H$. \end{block} \begin{itemize} \item 4/6/8 states; 2/4 input symbols; 2 output symbols. \item Random symbol names and table-row order. \item Intermediate states and outputs are targets only. \end{itemize} \end{column} \begin{column}{0.47\textwidth} \begin{block}{Exact simulator} \[ q_k=\delta(q_{k-1},a_k) \] \[ y_k=\lambda(q_{k-1},a_k) \] \end{block} Replace the state at step $c$, keep future actions, and simulate the unique counterfactual suffix. \end{column} \end{columns} \medskip A changed prediction is insufficient: the \emph{entire predicted continuation} must match. \papersource{Paper, Sec. 2, Eq. (1), PDF pp. 2--3; Appendix A.1, p. 12.} \end{frame} % 4 \begin{frame}{The same two-layer block evolves every position} \[ X^0=0,\qquad X^r=B_\theta\bigl(X^{r-1}+E(x)\bigr) \] \centering \includegraphics[width=0.97\linewidth,height=0.43\textheight,keepaspectratio]{figures/fig_arch_detail.pdf} \begin{itemize} \item Width 128, 4 heads, MLP width 512; block weights shared across loops. \item No position or loop embeddings; action attention is causal. \item Replace $h_k^r$ after the block, before final LN and both readout heads. \end{itemize} \papersource{Original Fig. A1, PDF p. 13; Eq. (2), p. 3. $X^r,E(x)\in\mathbb{R}^{T\times128}$.} \end{frame} % 5 \begin{frame}{Position coverage changes what must survive} \[ \mathcal L_S=\frac{1}{\sum_i|S_i|}\sum_i\sum_{k\in S_i}\ell_{i,k}^{R}, \qquad R=H \] \renewcommand{\arraystretch}{1.25} \begin{tabular}{lll} \toprule Objective & Supervised positions $S_i$ & Pairs per example\\ \midrule \FO & $\{H\}$ & 1\\ \AP & $\{1,\ldots,H\}$ & $H$\\ \SP & $\{K_i\}$, $K_i\sim U\{1,\ldots,H\}$ & 1\\ \bottomrule \end{tabular} \medskip \begin{itemize} \item All losses are applied \alert{only at the exit}, not at every loop. \item Shorter-length training constrains $h_k^k$ by causal prefix invariance. \item Broader coverage also asks position $k$ to remain correct at $H>k$. \end{itemize} \papersource{Paper, Eq. (3) and Sec. 2, PDF pp. 3--4. $\ell$ sums state/output cross-entropies.} \end{frame} % 6 \begin{frame}{The readout time is part of the metric} \begin{block}{Endpoint joint accuracy} \[ A(H,r)=\frac1N\sum_i \ind[\hat q_{i,H}^{r}=q_{i,H}\ \land\ \hat y_{i,H}^{r}=y_{i,H}] \] \end{block} \begin{itemize} \item $T(H,r)$: \textbf{every} position has both labels correct at the same loop $r$. \item Frontier readout: position $k$ is read at its own loop $s(k)=k$. \item Main behavior: 512 traces/model/length = 16 machines $\times$ 32 traces. \item Average within each model, then equally across model seeds. \end{itemize} \medskip Common training comparison: 3 seeds, 132k updates, curriculum to $H=32$. \papersource{Paper, Sec. 2 and Appendix A.2--A.3, PDF pp. 3--4, 12--14.} \end{frame} % 7 \begin{frame}{One random supervised position preserves answers} $H=32$, common evaluation stream; three-seed means (\%). \medskip \centering \renewcommand{\arraystretch}{1.25} \begin{tabular}{lrrrrr} \toprule Objective & $A_{32}$ & $A_{64}$ & $A_{96}$ & $T_{32}$ & $T_{64}$\\ \midrule \FO & 100.00 & 22.33 & 22.07 & 0.00 & 0.00\\ \AP & 100.00 & 94.27 & 82.16 & 100.00 & 74.28\\ \SP & 100.00 & \alert{99.80} & 96.61 & 100.00 & \alert{96.03}\\ \ME & 100.00 & 97.85 & 86.26 & 74.09 & 29.56\\ \bottomrule \end{tabular} \raggedright \medskip \begin{itemize} \item \SP\ keeps one supervised position per example, as in \FO. \item Strong endpoint retention can coexist with weak whole-trajectory retention. \item At loop 96, \SP\ trajectory exactness is only 74.09\%. \end{itemize} \papersource{Paper, Table A2, PDF p. 16; Fig. 2, p. 5. Matched exit = matched random exit.} \end{frame} % 8 \begin{frame}{Variable exits help beyond their training support} \[ \underbrace{R=H+4}_{\text{Slack-4}},\qquad \underbrace{R=H+U\{0,\ldots,8\}}_{\text{Jitter-8}},\qquad \mathbb E[R\mid H]=H+4 \] \centering \renewcommand{\arraystretch}{1.3} \begin{tabular}{lrrrr} \toprule Endpoint accuracy (\%) & $r=36$ & $r=48$ & $r=64$ & $r=96$\\ \midrule Slack-4 & 93.98 & 38.15 & 30.01 & 26.79\\ Jitter-8 & 87.27 & 87.17 & \alert{85.55} & 79.79\\ \bottomrule \end{tabular} \raggedright \medskip \begin{itemize} \item $H=32$; six paired seeds; final-position supervision in both conditions. \item Loop-64 gain: 55.54 percentage points; lower mean at loop 36. \item Cumulative block counts differ by 0.062\%; total FLOPs are not matched. \end{itemize} \papersource{Paper, Tables A7--A9, PDF pp. 18--19. Historical cohort, separate from slide 7.} \end{frame} % 9 \begin{frame}{The matched gradient control changes the conclusion} Both conditions optimize $\frac12L_m+\frac12L_{n+k}$, with $m=H+8$ and $H\le n+k\le m$. \medskip \begin{columns}[T,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Progressive} Full $m$-loop branch.\\ Random branch: detach after $n$ loops; backpropagate through $k$ loops. \end{block} Two seeds peak below 60\%. The successful seed falls from 100\% at loops 40--48 to 26.56\% at loop 96. \end{column} \begin{column}{0.48\textwidth} \begin{block}{Matched random exit} Same examples, exits and loss weights.\\ Backpropagate through all $n+k$ loops. \end{block} All three seeds reach 100\% at loop 32; mean accuracy is 97.85\% at loop 64. \end{column} \end{columns} \medskip \alert{Separate failure to learn from forgetting an answer already learned.} \papersource{Paper, Sec. 3 and Appendix B.1, PDF pp. 5--6, 15--16. Forward work matched; backward work differs. FSM adaptation of progressive training.} \end{frame} % 10 \begin{frame}{Length extrapolation has a different ranking} All models trained only through $H=32$; evaluate $H=64$. \medskip \centering \renewcommand{\arraystretch}{1.22} \begin{tabular}{lrr} \toprule Objective & $A(64,64)$ (\%) & $A(64,192)$ (\%)\\ \midrule \FO & \alert{97.79} & 17.12\\ \AP & 76.69 & \alert{72.72}\\ \SP & 85.87 & 60.61\\ \ME & 86.07 & 53.71\\ Progressive & 33.92 & 17.19\\ \bottomrule \end{tabular} \raggedright \medskip \begin{block}{Two axes of evaluation} Compute a longer input correctly; keep a solved input correct under more loops. Improving one does not guarantee improving the other. \end{block} \papersource{Paper, Table A4, PDF p. 17. Common stream, three-seed means.} \end{frame} % 11 \begin{frame}{Swap a state, then demand an exact counterfactual} \begin{enumerate} \item Match source and base machines, symbol names and table presentation. \item Change prefix actions through $c$; require $q_c^S\ne q_c^B$. \item Keep actions after $c$ identical and replace one complete vector: \[ h_c^r\leftarrow h_{c,S}^r. \] \item Continue the model and compare \textbf{every} later state/output to simulation from $q_c^S$. \end{enumerate} \begin{block}{Eligibility prevents a trivial success} The factual and counterfactual suffixes must differ somewhere. A model that ignores the swap cannot pass suffix exactness. \end{block} \papersource{Paper, Sec. 4, Eq. (5), PDF p. 6; Appendix A.3--A.4, p. 14.} \end{frame} % 12 \begin{frame}{One vector redirects 24 steps of exact execution} \centering \includegraphics[width=0.99\linewidth,height=0.51\textheight,keepaspectratio]{figures/fig_transport.pdf} \medskip \alert{237/237 eligible suffixes in each of 10 models.}\par \raggedright \smallskip Swap at $(k,r)=(8,8)$; read each suffix position $k$ at loop $k$.\\ The bright diagonal fades above the frontier: readouts are transient. \papersource{Original Fig. 3, PDF p. 7; Appendix A.4, p. 14. $H=32$. Grids average 256 pairs/model; exactness uses 237 eligible pairs/model.} \end{frame} % 13 \begin{frame}{Caching an answer and freezing a state differ} \begin{columns}[T,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Readout cache} Save the clean prediction at $s(k)=k$ outside the model. \end{block} \begin{itemize} \item All hidden states continue updating. \item Tests whether answers were available. \item No downstream computation is altered. \end{itemize} \end{column} \begin{column}{0.48\textwidth} \begin{block}{State freeze} Capture the arm's own $h_k^{k+d}$; write it back after every later block. \end{block} \begin{itemize} \item Other positions remain live. \item Frozen vectors remain visible to attention. \item Tests whether downstream work can use a fixed state. \end{itemize} \end{column} \end{columns} \medskip No source run or symbolic answer is inserted. Table tokens stay live. \papersource{Paper, Sec. 5 and Appendix D.1/D.3, PDF pp. 8, 25--28. Experimental freezing executes all blocks and does not save FLOPs.} \end{frame} % 14 \begin{frame}{Correct cached answers coexist with failed freezing} \centering \includegraphics[width=0.99\linewidth,height=0.57\textheight,keepaspectratio]{figures/fig_freeze.pdf} \raggedright \begin{itemize} \item Original 12 models: cache retains all 32 answers; immediate freezing leaves at most 2\% of trajectories fully correct. \item Failure appears in both supervision baselines and larger models. \end{itemize} \papersource{Original Fig. 4, PDF p. 7; Appendix D, pp. 25--28. Panels b/c: original cohort; d: new controls. Larger block: width 512, four layers, about 32 times the block parameters.} \end{frame} % 15 \begin{frame}{A short delay restores safe holding in these models} Endpoint joint accuracy at \alert{actual capture loop $32+d$} (\%). \medskip \centering \renewcommand{\arraystretch}{1.35} \begin{tabular}{lrrrrr} \toprule Objective & Clean $r=32$ & $d=0$ & $d=1$ & $d=2$ & $d=4$\\ \midrule \SP & 100.00 & 4.10 & 80.21 & \alert{99.80} & 100.00\\ \ME & 99.93 & 6.25 & 20.18 & 87.24 & \alert{99.80}\\ \bottomrule \end{tabular} \raggedright \medskip \begin{itemize} \item Clean is 100\% at delayed readouts $r=33,34,36$. \item After capture, frozen endpoint scores stay unchanged through loop 64. \item The delay needed differs across models; one extra loop is insufficient. \end{itemize} \begin{block}{Retention did not supply this guarantee} \SP\ retains 99.80\% at loop 64 in the behavior study, yet immediate state freezing collapses to 4.10\% here. \end{block} \papersource{Paper, Table A20, PDF p. 27; Table A2, p. 16. Freeze follow-up uses a fresh stream and three seeds per condition.} \end{frame} % 16 \begin{frame}{The first error follows a missed upstream update} \begin{block}{Immediate freezing at $s(k)=k$} Positions 1 and 2 stay correct. Position 3 fails first: before loop 3, position 1 is missing its loop-2 update, after its own answer was already correct. \end{block} \begin{itemize} \item \FO\ position-3 accuracy drops to about 68\%; errors compound later. \item Freeze only positions $1{:}8$: the endpoint remains live, but accuracy is only 7.49\% for \SP\ and 7.42\% for matched exit. \item Later captures occur on an already intervened trajectory; they need not themselves be correct. \end{itemize} \medskip Possible role: continuing updates carry order, timing or compatible representations. The experiment does not isolate the internal mechanism. \papersource{Paper, Sec. 5, PDF p. 8; Tables A20/A25, pp. 27/30. Prefix results use the new freeze cohort.} \end{frame} % 17 \begin{frame}{A retained source also becomes hard to transplant} \AP, position $c=16$: collect and insert the source at loop $16+a$. Read suffix position $k$ at loop $k+a$. \medskip \centering \renewcommand{\arraystretch}{1.3} \begin{tabular}{lrrrr} \toprule Suffix exactness (\%) & $a=0$ & $a=1$ & $a=2$ & $a=4$\\ \midrule Clean: factual target & 100.0 & 100.0 & 100.0 & 100.0\\ Swap: counterfactual target & 93.9 & 17.7 & 10.2 & 0.0\\ \bottomrule \end{tabular} \raggedright \medskip \begin{itemize} \item Post-swap update counts are fixed, but source and receiver phases both change. \item Even a fresh loop-16 source inserted at loop 20 gives 0\% one-step transfer. \item Next test: keep the late background fixed and reset one downstream vector. \end{itemize} \papersource{Paper, Tables A12--A13, PDF pp. 21--22. Six seeds; clean uses 256 pairs, swap 242 eligible pairs. Different targets/denominators: not a paired effect subtraction.} \end{frame} % 18 \begin{frame}{Fresh source plus receiver reset restores one step} \[ h_{16}^{20}\leftarrow h_{16,S}^{16},\qquad h_{17}^{20}\leftarrow h_{17,B}^{16}. \] \centering \includegraphics[width=0.98\linewidth,height=0.43\textheight,keepaspectratio]{figures/fig_receiver.pdf} \raggedright \begin{itemize} \item Fresh + reset recovers early transfer; aged + reset adds little. \item Both source ages decode correctly; clean retains its factual answer. \end{itemize} \papersource{Original Fig. A4 / Eq. (7), PDF p. 23; Table A15, p. 24. All-prefix: six seeds, 242 eligible pairs. Replace at loop 20; read position 17 at loop 21. Scores measure counterfactual match.} \end{frame} % 19 \begin{frame}{A maze model keeps solutions but cannot freeze early} Released recurrent recall network; one checkpoint, 32 mazes per size. \medskip \centering \renewcommand{\arraystretch}{1.15} \begin{tabular}{lrr} \toprule Solved at iteration 120 & $9\times9$ & $33\times33$\\ \midrule Clean / answer cache & 32 / 32 & 19 / 32\\ Freeze at arrival & 0 / 32 & 0 / 32\\ Arrival $+1$ & 3 / 32 & 0 / 32\\ Arrival $+2$ & 28 / 32 & 2 / 32\\ Arrival $+4$ & 32 / 32 & 18 / 32\\ \bottomrule \end{tabular} \raggedright \medskip \begin{itemize} \item All small mazes stay solved from iteration 25 to 120. \item Arrival = last clean prediction change per pixel: an offline diagnostic. \item Spatial output heads couple pixels; holding a pixel need not pin its output. \end{itemize} \papersource{Paper, Appendix D.4 and Table A26, PDF pp. 28, 30. Trained on 9x9 mazes for 30 iterations; larger size is outside training.} \end{frame} % 20 \begin{frame}{Halting must account for unfinished downstream work} \begin{block}{What the experiments establish} At a position's answer arrival, its state can still require updates for other positions to compute correctly. \end{block} \begin{enumerate} \item Evaluate computation, retention and state holding separately. \item State freezing needs a downstream criterion, beyond local readout correctness. \item Cache correctness demonstrates prior computation; it does not measure saved compute. \end{enumerate} \medskip \alert{Discussion idea:} learn when an answer is ready and when its supporting state is safe to hold as distinct decisions. \papersource{Paper, Secs. 5 and 7, PDF pp. 8--10. The last proposal is a reading interpretation, not a tested method in this paper.} \end{frame} % 21 \begin{frame}{The causal result is functional, not a circuit model} \begin{itemize} \item \textbf{Task and scale:} one synthetic execution family and principal architecture; a limited external maze check. \item \textbf{Training evidence:} new controls have three seeds; sampled supervision still needs intermediate labels. \item \textbf{Intervention scope:} full vectors can carry state, history and timing. A readable vector need not be a unique symbolic code. \item \textbf{Time scope:} finite retention windows, model-dependent freeze delays, and one-step receiver recovery. \end{itemize} \medskip \begin{block}{Discussion question} What is a state still doing after its own answer has stopped changing? \end{block} \papersource{Paper, Sec. 7 and Appendix C.4, PDF pp. 9--10, 25.} \end{frame} % 22 \begin{frame}{Source and evidence guide} Peilong Zhou, Hongtao Zhang, Yulong Huang, Cangyuan Li, Mingyu Chen, and Ying Wang. \medskip \textit{Learning to Compute Is Not Learning to Retain or Reuse in Looped Transformers.} arXiv:2610.19768v1, submitted 14 October 2026. \medskip \href{https://arxiv.org/abs/2610.19768v1}{\textcolor{AcademicTeal}{arxiv.org/abs/2610.19768v1}} \bigskip \begin{tabular}{ll} \toprule Result & Main source locations\\ \midrule Retention controls & Fig. 2; Tables A2--A4, A8\\ Exact counterfactual computation & Fig. 3; Appendix A.4\\ State freezing and delayed capture & Fig. 4; Tables A19--A26\\ Local receiver compatibility & Fig. A4; Tables A15--A17\\ Natural first-failure recovery & Table A27\\ \bottomrule \end{tabular} \papersource{All figures are reproduced from the authors' v1 TeX source. Tables select original reported values. Backup slides follow.} \end{frame} \appendix % 23 \begin{frame}{Backup: sampled supervision estimates the dense loss} At a fixed batch, parameters and shared horizon $H$: \[ J_i\sim U\{1,\ldots,H\},\qquad L_{\rm sampled}=\frac1B\sum_{i=1}^{B}\ell_{i,J_i}. \] \[ \mathbb E[L_{\rm sampled}] =\frac1B\sum_{i=1}^{B}\frac1H\sum_{k=1}^{H}\ell_{i,k} =L_{\rm all\text{-}prefix}. \] \begin{itemize} \item Position coverage is broadened without increasing selected pairs per example. \item Intermediate labels are still required. \item Equal expected losses do not imply identical optimization trajectories. \end{itemize} \papersource{Paper, Appendix B.1, PDF p. 15; Eq. (3), p. 3.} \end{frame} % 24 \begin{frame}{Backup: the random-exit law is not uniform} \[ m=H+8,\quad n\sim U\{0,\ldots,m-1\},\quad k\mid n\sim U\{1,\ldots,m-n\}. \] Reject and redraw the whole pair if $n+k