% Research discussion: about 25 minutes; frames 23--27 are backup material. % Based on the arxiv-paper-reader Beamer template. Compile with pdfLaTeX. \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{\fig}[3]{\includegraphics[width=#2\linewidth,height=#3\textheight,keepaspectratio]{figures/#1}} \newcommand{\E}{\mathbb E} \newcommand{\mask}{\mathrm m} \title[LoopMDM]{Looped Diffusion Language Models} \subtitle{Selective recurrence in masked diffusion\\Research group discussion} \author{Paper authors: Sanghyun Lee, Chunsan Hong, Seungryong Kim\\ Jonghyun Lee, Jongho Park, Dongmin Park} \date{arXiv:2605.26106v1 (May 25, 2026)\\Reading: October 13, 2026} \begin{document} \begin{frame} \titlepage \vspace{-0.5cm} \begin{center} \alert{Reuse two early layers to improve quality at fixed training FLOPs.} \end{center} \papersource{\href{https://arxiv.org/abs/2605.26106v1}{arxiv.org/abs/2605.26106v1}. Presentation prepared for paper discussion; not an author presentation.} \end{frame} \begin{frame}{A second iteration axis inside each denoising step} \begin{block}{Research question} With fixed parameters and total training FLOPs, should an MDM spend more computation on each example through shared layers? \end{block} \medskip \centering \begin{tabular}{lll} \toprule Resource & What is repeated? & State being updated\\ \midrule Denoising steps $T$ & Entire denoising network & Discrete sequence $x_t$\\ Internal loops $S$ & A shared subset of layers & Hidden states $H_t^{(k)}$\\ \bottomrule \end{tabular} \medskip \begin{itemize} \item Unresolved positions can exchange information before token commitment. \item More work per example means fewer training tokens at fixed FLOPs. \end{itemize} \papersource{Paper, Secs. 1--3, PDF pp. 1--4.} \end{frame} \begin{frame}{Loop only a small block of the denoising network} \begin{columns}[c,onlytextwidth] \begin{column}{0.32\textwidth} \centering\fig{architecture.pdf}{0.94}{0.72} \end{column} \begin{column}{0.65\textwidth} \[ \begin{aligned} H_t^{(0)} &= \operatorname{head}(x_t),\\ H_t^{(k)} &= \operatorname{mid}(H_t^{(k-1)}),\\ p_\theta^S(x_0\mid x_t)&=\operatorname{tail}(H_t^{(S)}). \end{aligned} \] \begin{itemize} \item $H_t^{(k)}\in\mathbb R^{L\times d}$: all token positions. \item The mid-block shares parameters across $S$ applications. \item Head and tail run once; tail includes the vocabulary projection. \end{itemize} \[ D_{\mathrm{eff}}=n_h+Sn_m+n_o \] \end{column} \end{columns} \papersource{Paper, Fig. 1 left panel, PDF p. 2; Eq. (4), PDF p. 3.} \end{frame} \begin{frame}{Random loop counts train one operator at many depths} \[ S\sim\mathcal U\{1,\ldots,S_{\max}\}, \qquad t\sim\mathcal U[0,1] \] \[ \mathcal L = \E_{t,S,x_t}\!\left[ \sum_{i=1}^{L}\mathbb I[x_t^i=\mask]\, \frac{\alpha'_t}{1-\alpha_t} \log\langle p_\theta^{S,i}(x_t,t),x_0^i\rangle \right] \] \begin{itemize} \item Apply the usual masked diffusion loss to the final loop output. \item $\alpha_t$: decreasing token retention probability. \item $x_0^i$: one-hot target; the inner product selects its probability. \item Both $\alpha'_t$ and log probability are nonpositive: this is a weighted negative log-likelihood. \end{itemize} \begin{block}{Observed extrapolation} Train with $S_{\max}=12$, evaluate up to $S=24$ in language modeling. \end{block} \papersource{Paper, Eq. (5), PDF p. 4; Fig. 2, PDF p. 5.} \end{frame} \begin{frame}{Shared parameters still require repeated computation} \[ F_{\mathrm{loop}}=F_{\mathrm{base}} +(\E[S]-1)n_mF_{\mathrm{layer}}, \qquad \E[S]=\frac{S_{\max}+1}{2} \] \begin{center} \begin{tabular}{lcc} \toprule 12-layer language model & Loops & Layer executions\\ \midrule Non-looped MDM & --- & 12\\ LoopMDM: training average & 6.5 & 23\\ LoopMDM: inference & 12 & 34\\ LoopMDM: inference & 24 & 58\\ \bottomrule \end{tabular} \end{center} \medskip Match total training FLOPs by \alert{reducing LoopMDM training steps}. The models use the same corpora but see different token totals. \papersource{Paper, Eq. (6), PDF p. 4. Layer counts are a teaching calculation, excluding embedding/output costs.} \end{frame} \begin{frame}{Three experiments probe different claims} \begin{center} \begin{tabular}{p{2.1cm}p{5.3cm}p{5.3cm}} \toprule Setting & Model and data & Comparison / evaluation\\ \midrule Language & 170M, 12 layers, $d=768$; LM1B, OWT, FineWeb-Edu & Same-parameter MDM; matched total training FLOPs\\[5pt] Math & 125M, 14 layers, $d=512$; TinyGSM with Python solutions & 14/18/21-layer MDMs; GSM8K, Top-2 and Top-3 unmasking\\[5pt] Sudoku & 1 layer, $d=384$; 100k train / 1,000 test & Fixed left-to-right order; exact-solve evaluation\\ \bottomrule \end{tabular} \end{center} \medskip Main LM configuration: loop zero-based layers 1--2 (the second and third layers), $S_{\max}=12$. Math uses $S_{\max}=8$; Sudoku uses 6. \papersource{Paper, Secs. 4.1--4.3, PDF pp. 4, 7--8; Appendix C, pp. 18--20.} \end{frame} \begin{frame}{Selective looping reaches the target NLL sooner} \centering \fig{fig-02.pdf}{1.0}{0.53} \begin{itemize} \item Solid: LoopMDM at different inference $S$; dashed: MDM. \item $S\ge6$ improves NLL; $S=1$ underperforms; $S=24$ still helps. \item Figure annotations: FineWeb-Edu $3.34\times$, OWT $2.95\times$, LM1B $2.34\times$ less training compute to the target. \end{itemize} \papersource{Paper, Fig. 2, PDF p. 5. Version note: Sec. 4.1, p. 4 reverses the FineWeb-Edu/LM1B ratio assignments. The figure's mapping is shown here.} \end{frame} \begin{frame}{Generation quality improves at fixed denoising steps} \begin{center} \begin{tabular}{cccc} \toprule Steps $T$ & MDM & LoopMDM $S=6$ & LoopMDM $S=12$\\ \midrule 64 & $116.71\pm2.72$ & $105.59\pm3.83$ & $99.06\pm5.91$\\ 256 & $79.43\pm2.60$ & $64.08\pm1.39$ & $56.45\pm0.50$\\ 1024 & $42.56\pm1.09$ & $39.23\pm1.31$ & $38.53\pm1.23$\\ \bottomrule \end{tabular} \end{center} \medskip \begin{itemize} \item Generative perplexity: lower is better; five random seeds. \item Both external denoising steps and internal loops spend inference compute. \item This table fixes $T$, rather than total inference FLOPs. \end{itemize} \papersource{Paper, Table 1, PDF p. 5; selected rows with the original uncertainty values. The scoring model is not specified in the described protocol.} \end{frame} \begin{frame}{Transfer PPL improves, with task-specific saturation} \begin{columns}[c,onlytextwidth] \begin{column}{0.57\textwidth} \centering \begin{tabular}{lrrr} \toprule Corpus & MDM & $S=6$ & $S=12$\\ \midrule PTB &108.5&100.2&90.2\\ WikiText &35.1&34.2&32.4\\ LM1B &68.1&68.0&67.8\\ Lambada &49.6&44.6&45.8\\ AG News &69.3&59.2&56.6\\ PubMed &43.0&37.2&36.5\\ Arxiv &37.8&33.1&32.7\\ \bottomrule \end{tabular} \end{column} \begin{column}{0.39\textwidth} \begin{itemize} \item All seven corpora improve over MDM at $S=12$. \item Largest change: PTB, $-18.3$ PPL units. \item Lambada is better at 6 loops than at 12. \item At $S=1$, all seven are worse than MDM. \end{itemize} \end{column} \end{columns} \papersource{Paper, Table 2, PDF p. 5. Means over five random seeds; uncertainty omitted here.} \end{frame} \begin{frame}{Average downstream gains are uneven across tasks} \begin{center} \begin{tabular}{lrrr} \toprule Task & MDM (\%) & $S=12$ (\%) & Change (pp)\\ \midrule BoolQ &46.6&57.4&+10.8\\ RACE &28.5&31.7&+3.2\\ HellaSwag &31.9&32.7&+0.8\\ ARC-c &25.5&25.3&$-0.2$\\ OBQA &30.8&28.0&$-2.8$\\ \midrule All nine: average &$40.0\pm0.3$&$41.4\pm0.6$&+1.4\\ \bottomrule \end{tabular} \end{center} \medskip Six tasks improve, one ties, two decline. BoolQ contributes much more than most tasks. \papersource{Paper, Table 3, PDF p. 6; selected task means and original average uncertainty. The text claims all nine match/improve, but ARC-c and OBQA decline in the table.} \end{frame} \begin{frame}{Where and how much to loop are critical} \begin{center} \begin{tabular}{llr} \toprule Factor & Selected configuration & Test NLL $\downarrow$\\ \midrule Non-looped baseline & 12 layers & 3.798\\ Position & Start at 1 or 2 (zero-based) & 3.729\\ Position & Start at 10 & 3.777\\ Looped layers & $n_m=2$ & 3.729\\ Looped layers & $n_m=12$ & 3.933\\ Training max loops & $S_{\max}=12$ & 3.729\\ Training max loops & $S_{\max}=24$ & 3.800\\ \bottomrule \end{tabular} \end{center} More repeated work per example leaves fewer training steps. Two early-middle layers capture most of the benefit. \papersource{Paper, Table 4, PDF p. 6. LM1B, 170M, matched training FLOPs. Training with 24 loops differs from evaluating a 12-loop-trained model at 24.} \end{frame} \begin{frame}{Math gains persist against deeper training baselines} \centering \fig{fig-03.pdf}{0.85}{0.58} \begin{itemize} \item At $S=16$: +8.5 pp (Top-2), +8.1 pp (Top-3) vs.\ 14 layers. \item Matches the 14-layer endpoint with 43\% of its training FLOPs. \item Also exceeds the 21-layer model: reported +2.7 pp at $S=8$. \end{itemize} \papersource{Paper, Sec. 4.2 and Fig. 3, PDF p. 7. TinyGSM-trained 125M model; three random seeds; $S_{\max}=8$.} \end{frame} \begin{frame}{Training and inference have different effective depths} \[ D_{\mathrm{eff}}(S)=14+2(S-1)=12+2S \] \begin{center} \begin{tabular}{lcc} \toprule Model / operating point & Loop count & Layer executions\\ \midrule Deeper non-looped baseline & --- & 21\\ LoopMDM training average & $\E[S]=4.5$ & 21\\ LoopMDM inference & $S=8$ & 28\\ LoopMDM inference & $S=16$ & 44\\ \bottomrule \end{tabular} \end{center} \medskip \begin{itemize} \item Reusing computation improves the training-budget trade-off. \item An equal-inference-FLOPs accuracy frontier remains to be measured. \end{itemize} \papersource{Teaching calculation from the architecture in Sec. 4.2, PDF p. 7. Layer counts approximate compute and exclude embedding/output overhead.} \end{frame} \begin{frame}{Sudoku improves even with a fixed generation order} \centering \fig{fig-04.pdf}{1.0}{0.49} \begin{itemize} \item Fixed left-to-right unmasking removes the advantage of solving easy cells first. \item One-layer baseline: 10.9\% exact solves; looped model at $S=1$: 10.6\%. \item Shown puzzle: 24 errors $\to$ 7 $\to$ 0 at $S=1,2,3$. \end{itemize} \papersource{Paper, Fig. 4 and Sec. 4.3, PDF p. 8. The illustration is one puzzle; aggregate exact-solve rates for larger $S$ are not provided.} \end{frame} \begin{frame}{Loops increase interactions between unresolved positions} \centering \fig{fig-05.pdf}{0.98}{0.55} \begin{itemize} \item Left: mask-to-mask attention at $t=0.5$ grows and then saturates. \item Right: $\mathrm{NLL}_t(1)-\mathrm{NLL}_t(S)$ peaks at intermediate noise. \item Interpretation: useful context and unresolved positions coexist there. \end{itemize} \papersource{Paper, Fig. 5, PDF p. 9. OWT, length 1024, $S_{\max}=12$. Attention trends support the workspace interpretation; they are not a causal intervention.} \end{frame} \begin{frame}{Stop looping when hidden-state changes become small} \[ r_k=\frac{\|H^{(k)}-H^{(k-1)}\|}{\|H^{(k)}\|}; \qquad \text{stop if }r_k<\epsilon\text{ or the budget is reached.} \] \begin{center} \begin{tabular}{lccc} \toprule Metric & Fixed $S=12$ & Adaptive & Mean loops\\ \midrule Zero-shot PPL $\downarrow$ &51.7&52.9&5.1\\ Generative PPL $\downarrow$ &38.53&39.11&5.2\\ Downstream accuracy $\uparrow$ &41.4\%&41.3\%&4.9\\ \bottomrule \end{tabular} \end{center} \medskip With $\epsilon=0.1$, about five loops retain most of the gains. This monitors state stability rather than answer correctness. \papersource{Paper, Sec. 4.3 and Table 5, PDF p. 9. Norm type and batch aggregation are not specified in the text.} \end{frame} \begin{frame}{Adaptive loops concentrate on intermediate timesteps} \centering \fig{fig-06.pdf}{0.95}{0.50} \[ \frac{\text{adaptive layer executions}}{\text{fixed-12 layer executions}} \approx \frac{10+2(5.2)}{10+2(12)}=0.60 \] About 40\% fewer layer executions in this teaching estimate. The loop-count ratio alone excludes head and tail computation. \papersource{Paper, Fig. 6, PDF p. 21; OWT, $\epsilon=0.1$, 100 sequences. Time runs from 1 to 0. Cost estimate uses Table 5's generative mean, not measured latency.} \end{frame} \begin{frame}{Simple uniform loop sampling beats the tested variants} \begin{columns}[c,onlytextwidth] \begin{column}{0.50\textwidth} \begin{itemize} \item Injecting Gaussian noise into mask embeddings worsens NLL relative to MDM. \item Log-normal Poisson loop sampling gives only a modest benefit. \item Uniform sampling performs best in this comparison. \end{itemize} \medskip The tested input injection is specifically a mask-noise variant. \end{column} \begin{column}{0.46\textwidth} \centering\fig{fig-07.pdf}{1.0}{0.57} \end{column} \end{columns} \papersource{Paper, Appendix D.2 and Fig. 7, PDF pp. 20--21.} \end{frame} \begin{frame}{More loops can move a correct prediction away} \begin{block}{A failure trajectory from OpenWebText} Ground-truth token: \texttt{with} \end{block} \begin{center} \begin{tabular}{lrrrr} \toprule Loop count $S$ & 1 & 6 & 8 & 24\\ \midrule Prediction & \texttt{with} & \texttt{who} & \texttt{whether} & \texttt{whether}\\ Target NLL & 0.75 & 2.38 & 5.31 & 6.69\\ \bottomrule \end{tabular} \end{center} \medskip \begin{itemize} \item Other examples improve copying, semantic choice and syntax. \item Aggregate gains coexist with local prediction drift. \item NLL here scores the true token, not confidence in the wrong token. \end{itemize} \papersource{Paper, Fig. 8(A--E), PDF p. 22. Selected values from panel E.} \end{frame} \begin{frame}{The theory isolates workspace as an extra resource} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Construction} For fixed $k$, add $n^k$ masked slots, one for each candidate $k$-tuple in an $n$-vertex graph. \end{block} Each slot checks its edges; an answer slot aggregates the results. Constant internal depth can evaluate candidates in parallel. \end{column} \begin{column}{0.48\textwidth} \begin{block}{Conditional separation} Assume the unpadded architecture has an $\mathrm{AC}^0$ simulation of size $O(N^C)$, with $N=\Theta(n^2)$. \end{block} For $k>8C$, the invoked Clique circuit lower bound rules out that unpadded model. \end{column} \end{columns} \medskip \alert{Extra slots increase computation.} Equal constant loop budgets do not establish equal FLOPs. \papersource{Paper, Example 1 and Appendix B, PDF pp. 8, 16--18. The circuit simulation is an explicit assumption.} \end{frame} \begin{frame}{The result is a training-budget trade-off worth testing} \begin{enumerate} \item \textbf{Architecture:} a small shared block can be more useful than broadly adding repeated depth. \item \textbf{Mechanism:} unresolved positions offer space for iterative communication before discrete commitment. \item \textbf{Deployment:} choose $S$, denoising steps and stopping rules jointly; high-loop quality has a compute cost. \end{enumerate} \begin{block}{Discussion} At fixed inference FLOPs, how should compute be split between loops and denoising steps? Does the result persist at billion-parameter scale? \end{block} \papersource{Paper, Sec. 5 and Appendix E, PDF pp. 9--10, 22--23. Main experiments: 125M--170M; manually selected loop block and stopping rule.} \end{frame} \begin{frame}{Paper reference and discussion guide} \textbf{Sanghyun Lee, Chunsan Hong, Seungryong Kim, Jonghyun Lee, Jongho Park, Dongmin Park.}\\ \medskip \emph{Looped Diffusion Language Models}.\\ arXiv:2605.26106v1, submitted May 25, 2026. \[ \href{https://arxiv.org/abs/2605.26106v1}{\texttt{https://arxiv.org/abs/2605.26106v1}} \] \medskip \begin{itemize} \item Method: Sec. 3; quality and efficiency: Secs. 4.1--4.2. \item Mechanism and adaptive inference: Sec. 4.3. \item Proof assumptions and extra results: Appendices B and D. \end{itemize} \papersource{All experimental values and reused figures in this deck come from this version. Arithmetic estimates and interpretation are identified on their slides.} \end{frame} \appendix \begin{frame}{Backup: masking and the sign of the NELBO} \[ q(x_t^i\mid x_0^i)= \alpha_t\delta_{x_0^i}(x_t^i)+(1-\alpha_t)\delta_{\mask}(x_t^i) \] \[ w(t)=-\frac{\alpha'_t}{1-\alpha_t}\ge0 \] \[ \mathcal L = \E_{t,S,x_t}\left[ \sum_{i:x_t^i=\mask}w(t) \bigl(-\log p_\theta^{S,i}(x_0^i\mid x_t,t)\bigr) \right] \] \begin{itemize} \item $t=0$: nearly clean; $t=1$: nearly fully masked. \item Only masked positions contribute to the denoising loss. \item Predictions share contextual hidden states even when reverse sampling factorizes across positions. \end{itemize} \papersource{Paper, Eqs. (1), (3), (5), PDF pp. 3--4. Last equation is an algebraic rewriting for explanation.} \end{frame} \begin{frame}{Backup: all nine downstream tasks} \begin{center} \begin{tabular}{lrrrr} \toprule Accuracy (\%) & MDM & $S=1$ & $S=6$ & $S=12$\\ \midrule ARC-c &25.5&23.6&25.7&25.3\\ ARC-e &47.7&45.2&47.7&47.7\\ BoolQ &46.6&53.8&53.8&57.4\\ HellaSwag &31.9&31.1&32.5&32.7\\ OBQA &30.8&28.2&28.0&28.0\\ PIQA &60.6&58.5&58.9&61.1\\ RACE &28.5&27.2&31.3&31.7\\ SIQA &36.8&35.9&37.4&37.3\\ WinoGrande &51.3&50.9&51.4&51.5\\ \midrule Average &40.0&39.4&40.7&41.4\\ \bottomrule \end{tabular} \end{center} \papersource{Paper, Table 3, PDF p. 6. Means; original uncertainty omitted. Five evaluation seeds do not by themselves establish five independent pretraining runs.} \end{frame} \begin{frame}{Backup: the three architecture sweeps} \begin{columns}[t,onlytextwidth] \begin{column}{0.31\textwidth} \centering\textbf{Start position}\\[5pt] \begin{tabular}{rr} \toprule Position & NLL\\ \midrule 0&3.744\\1&3.729\\2&3.729\\ 5&3.734\\8&3.747\\10&3.777\\ \bottomrule \end{tabular} \end{column} \begin{column}{0.31\textwidth} \centering\textbf{Block size}\\[5pt] \begin{tabular}{rr} \toprule $n_m$ & NLL\\ \midrule 1&3.743\\2&3.729\\3&3.734\\ 6&3.784\\7&3.857\\12&3.933\\ \bottomrule \end{tabular} \end{column} \begin{column}{0.31\textwidth} \centering\textbf{Training loop cap}\\[5pt] \begin{tabular}{rr} \toprule $S_{\max}$ & NLL\\ \midrule 4&3.741\\8&3.733\\12&3.729\\ 16&3.740\\20&3.763\\24&3.800\\ \bottomrule \end{tabular} \end{column} \end{columns} \medskip Baseline NLL: 3.798. The best reported configuration uses a small block and a moderate training cap. \papersource{Paper, Table 4, PDF p. 6. Selected rows; matched training FLOPs. Position is zero-based.} \end{frame} \begin{frame}{Backup: threshold ablation has a different baseline} \begin{center} \begin{tabular}{lrrrr} \toprule Setting & Zero-shot PPL & $\bar S$ & Gen.\ PPL & $\bar S$\\ \midrule Fixed $S=12$ &55.58&12.00&62.44&12.00\\ $\epsilon=0.02$ &55.57&11.73&62.66&10.73\\ $\epsilon=0.10$ &56.68&6.20&64.39&5.45\\ $\epsilon=0.20$ &59.45&3.74&66.03&3.41\\ \bottomrule \end{tabular} \end{center} \medskip \begin{itemize} \item Larger thresholds generally trade quality for fewer iterations. \item Table 6's fixed baselines and loop means differ from Table 5. The paper does not fully reconcile the protocols. \item Its Compute percentages track $\bar S/12$ numerically, excluding the non-looped network portion. \end{itemize} \papersource{Paper, Table 6, PDF p. 20; compare Table 5, PDF p. 9. These tables are kept separate rather than merged into one trade-off curve.} \end{frame} \begin{frame}{Backup: the conditional Clique separation} \[ c_{\mathbf v}= \left(\bigwedge_{r8C \quad\Longrightarrow\quad O(N^C)=O(n^{2C})=o(n^{k/4}). \] This contradicts the invoked $\omega(n^{k/4})$ Clique circuit lower bound. Extra workspace is the distinguishing computational resource. \papersource{Paper, Appendix B.2--B.3, PDF pp. 16--18; the paper cites Rossman (2008) for the circuit lower bound.} \end{frame} \end{document}