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\title[Sigma]{Large Language Continuous\newline Diffusion Models}
\subtitle{Sigma: scalable training, trajectory steering, and few-step generation}
\author[Paper discussion]{Paper by Zhihan Yang, Wei Guo, Jean-Marie Lemercier,\newline Simon Welker, et al.}
\institute{NVIDIA / Cornell University / Georgia Institute of Technology}
\date{Paper discussion \enspace $\cdot$ \enspace 14 October 2026\newline\href{https://arxiv.org/abs/2610.02665v1}{arXiv:2610.02665v1}}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-5mm}
  \begin{center}\small Continuous token trajectories reach competitive 3B/8B reasoning quality.\end{center}
\end{frame}

\begin{frame}{Can continuous diffusion handle conditional reasoning?}
  \begin{block}{The question}
    Scale likelihood-trained diffusion over learned token embeddings to\newline
    3B/8B models, with math and code evaluation against a matched masked dLM.
  \end{block}
  \begin{columns}[T]
    \begin{column}{0.49\textwidth}
      \textbf{Design}\medskip
      \begin{itemize}
        \item 16D continuous state per token.
        \item 32-token blocks; cached clean prefix.
        \item AR warm start and auxiliary AR loss.
      \end{itemize}
    \end{column}
    \begin{column}{0.47\textwidth}
      \textbf{Six-task base-model average}\par\medskip
      \begin{tabular}{lrr}\toprule
        & Sigma & NLD \\\midrule
        3B & 64.34 & 64.74 \\
        8B & 69.67 & 70.22 \\\bottomrule
      \end{tabular}
      \smallskip\par\small Sigma: DDPM-256, diffusion mode.
    \end{column}
  \end{columns}
  \papersource{Paper, Secs. 1, 3; Tables 1--2, PDF pp. 1--5, 8. Scores in percent; quality comparison.}
\end{frame}

\begin{frame}{Generate each block by moving continuous states}
  \begin{columns}[c]
    \begin{column}{0.58\textwidth}
      \includegraphics[width=\linewidth,height=5.1cm,keepaspectratio]{figures/fig01-pipeline.pdf}
    \end{column}
    \begin{column}{0.39\textwidth}
      \begin{enumerate}
        \item Cache the clean prefix.
        \item Draw Gaussian noise for the new block.
        \item Jointly denoise all positions.
        \item Read out tokens and extend the prefix.
      \end{enumerate}
      \medskip
      \alert{Autoregressive across blocks; bidirectional within a noisy block.}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 1 (left), PDF p. 2; Sec. 4.1, PDF p. 6. Original author figure.}
\end{frame}

\begin{frame}{Two streams share a Transformer backbone}
  \begin{columns}[c]
    \begin{column}{0.50\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=4.6cm,keepaspectratio]{figures/fig02-architecture.pdf}
    \end{column}
    \begin{column}{0.47\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=4.6cm,keepaspectratio]{figures/fig02-mask.pdf}
    \end{column}
  \end{columns}
  \medskip
  Clean tokens attend causally. A noisy block sees earlier clean blocks and itself.
  \papersource{Paper, Fig. 2, PDF p. 4. Original architecture and attention-mask panels.}
\end{frame}

\begin{frame}{The likelihood objective also learns the geometry}
  \[
    \E\in\R^{V\times16},\quad \|\E_v\|_2=1,\qquad
    \z_t=\alpha_t\x\E+\sigma_t\epsilon,\quad \alpha_t^2+\sigma_t^2=1
  \]
  \[
    \underbrace{\x_\theta(\z_t,t;\cctx)}_{\text{vocabulary probabilities}}\E
    =\underbrace{\e_\theta(\z_t,t;\cctx)}_{\text{predicted mean embedding}}
  \]
  \begin{block}{A per-block negative ELBO}
    \[
      \mathcal L_{\mathrm{NELBO}}
      =\mathcal L_{\mathrm{prior}}+\mathcal L_{\mathrm{rec}}
      -\tfrac12\mathbb E\!\left[\SNR'(t)\|\e_\theta-\x\E\|_F^2\right]
    \]
  \end{block}
  \begin{itemize}
    \item Prior KL aligns the noisy endpoint; reconstruction CE reads out tokens.
    \item $\SNR'(t)<0$: the denoising MSE has positive weight.
    \item Sum the bound over $p_\theta(\x^{(b)}\mid\x^{(<b)})$ across blocks.
  \end{itemize}
  \papersource{Paper, Eqs. (1)--(4), PDF pp. 2--3. 16D is the latent size; the output still spans all 131,072 tokens.}
\end{frame}

\begin{frame}{AR transfer and self-conditioning make scaling practical}
  \begin{columns}[T]
    \begin{column}{0.60\textwidth}
      \[
        \mathcal L=\mathcal L_{\mathrm{ar}}+w_{\mathrm{diff}}\mathcal L_{\mathrm{BD}}
      \]
      \begin{itemize}
        \item Start from a 1T-token AR checkpoint; randomly initialize 16D embeddings.
        \item Continue for 150K steps, over 300B tokens.
        \item Switch $w_{\mathrm{diff}}:1.0\to0.3$ at 200B tokens.
        \item SC: detached first prediction on 25\% of training samples.
      \end{itemize}
    \end{column}
    \begin{column}{0.36\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=4.6cm,keepaspectratio]{figures/fig03-regularization.pdf}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 3c, Sec. 3.2, App. E; PDF pp. 5, 28. Validation PPL: lower is better.}
\end{frame}

\begin{frame}{DDIM reduces the cost of many-step generation}
  For $s<t$, predict a clean embedding $\hat\e_\theta$ and implied noise:
  \[
    \hat\epsilon_\theta=\frac{\z_t-\alpha_t\hat\e_\theta}{\sigma_t}.
  \]
  \begin{block}{DDIM: deterministic given the initial noise}
    \[\z_s=\alpha_s\hat\e_\theta+\sigma_s\hat\epsilon_\theta.\]
  \end{block}
  DDPM instead samples a Gaussian reverse transition:
  \[
    \z_s=(1-\kappa)\frac{\alpha_s}{\alpha_t}\z_t
      +\kappa\alpha_s\hat\e_\theta
      +\sqrt{\kappa(1-\alpha_s^2)}\,\epsilon,
    \quad \kappa=1-e^{\gamma(s)-\gamma(t)}.
  \]
  \alert{One extra call at $t=0$ performs the vocabulary argmax readout.}
  \papersource{Paper, Sec. 4.1, PDF p. 6; Algorithm 2, PDF p. 27. Main-table step budgets exclude this readout.}
\end{frame}

\begin{frame}{CFG and score temperature steer different offsets}
  Let $\e_c$ use the clean prefix and $\e_\varnothing$ drop the prefix.
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \begin{block}{Classifier-free guidance}
        \[\e^w=\e_\varnothing+w(\e_c-\e_\varnothing)\]
        $w\geq1$: amplify the conditional change.
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Score temperature}
        \[\e^\tau=\frac{\z_t}{\alpha_t}+\frac1\tau\left(\e_c-\frac{\z_t}{\alpha_t}\right)\]
        $0<\tau\leq1$: rescale the score by $1/\tau$.
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  \[
    \e^{w,\tau}=\frac w\tau\e_c+\frac{1-w}{\tau}\e_\varnothing
       +\left(1-\frac1\tau\right)\frac{\z_t}{\alpha_t}
  \]
  \begin{itemize}
    \item Both act in embedding space; ST differs from logit temperature.
    \item The two affine transformations commute.
  \end{itemize}
  \papersource{Paper, Sec. 4.2, PDF pp. 6--7; Proposition 1 / Eq. (28), PDF pp. 45--46.}
\end{frame}

\begin{frame}{Packed guidance has a small marginal latency cost}
  \begin{columns}[c]
    \begin{column}{0.47\textwidth}
      \includegraphics[width=\linewidth,height=3.8cm,keepaspectratio]{figures/fig04-mask.pdf}
    \end{column}
    \begin{column}{0.49\textwidth}
      \includegraphics[width=\linewidth,height=3.8cm,keepaspectratio]{figures/fig04-steering.pdf}
    \end{column}
  \end{columns}
  \medskip
  \begin{itemize}
    \item Conditional and unconditional queries share one packed forward pass.
    \item SC reuses the previous \emph{unsteered} conditional mean; unconditional SC is zero.
    \item H100, 3B, batch 1, block 32: 19.08 $\to$ 19.89 ms per NFE (+4.2\%).
  \end{itemize}
  \papersource{Paper, Fig. 4, PDF p. 7; Table 20, PDF p. 45. Timing covers the first generated block.}
\end{frame}

\begin{frame}{Read the evaluation as several distinct protocols}
  \begin{tabular}{P{2.2cm}P{5.1cm}P{5.4cm}}\toprule
    Evaluation & What is scored? & Relevant conditions \\\midrule
    Base models & 2 math + 4 coding tasks; pass@1 & Few-shot prompts; 256/512 output tokens \\[3pt]
    Instruct & 10 tasks: math, code, knowledge, instructions & Non-thinking SFT; task-specific output caps \\[3pt]
    Multiple choice & Candidate ranking by negative ELBO & Monte Carlo likelihood surrogate \\[3pt]
    Pass@$k$ & At least one correct candidate among $k$ & Paired uncertainty across problems \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item NLD is the main aligned masked-diffusion control.
    \item Sigma main-table errors: mean $\pm2$ SE over 10 sampling lanes.
    \item Quality tables do not match end-to-end inference latency.
  \end{itemize}
  \papersource{Paper, Apps. F, G.1, G.8, H.2; PDF pp. 31--34, 42--44, 55.}
\end{frame}

\begin{frame}{Base quality is close to NLD, with task-level tradeoffs}
  \centering
  \begin{tabular}{lrrrr}\toprule
    Diffusion mode & GSM8K & HumanEval & MBPP & Six-task avg. \\\midrule
    NLD-3B & 83.40 & 56.10 & 57.60 & 64.74 \\
    Sigma-3B, DDPM-256 & 84.50 & 59.45 & 52.32 & 64.34 \\
    Sigma-3B, DDIM-32 & 81.91 & 56.59 & 47.26 & 60.45 \\\midrule
    NLD-8B & 87.26 & 62.80 & 67.00 & 70.22 \\
    Sigma-8B, DDPM-256 & 89.07 & 61.40 & 64.78 & 69.67 \\
    Sigma-8B, DDIM-32 & 87.78 & 58.54 & 59.54 & 66.72 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item At 3B, stronger HumanEval accompanies weaker MBPP.
    \item At 8B, strong math scores coexist with a slightly lower overall mean.
    \item Reducing diffusion steps from 256 to 32 loses quality, especially in coding.
  \end{itemize}
  \papersource{Paper, Tables 1--2, PDF p. 8. Percent, higher is better. Average also includes Minerva, HumanEval+, MBPP+.}
\end{frame}

\begin{frame}{MBPP shows a quality--coverage tradeoff}
  \centering
  \includegraphics[width=0.93\linewidth,height=4.25cm,keepaspectratio]{figures/fig16-mbpp.pdf}
  \medskip
  \begin{itemize}
    \item Fixed configuration: pass@256 is 91.8\% vs. 87.6\%; paired CI for the gain: [+1.7,+6.7] pp.
    \item Exact-string repetition: 10.4\% vs. 53.1\% in the selected configurations.
  \end{itemize}
  \papersource{Paper, Fig. 16 / App. G.8.1, PDF pp. 42, 51. Sigma: DDPM-256; NLD: 32 transfer steps.}
\end{frame}

\begin{frame}{High-pass@$k$ gains depend on the task and control}
  \begin{tabular}{P{2.5cm}P{4.2cm}P{6.0cm}}\toprule
    Task & High-budget result & Interpretation \\\midrule
    Geometry & About 95.4\% vs. 94.2\% at $k=512$ & CI spans zero. At matched pass@1, Sigma ties best-tuned NLD. \\[5pt]
    GSM8K & 1314 vs. 1305 of 1319 solved at least once & Small hard-tail gain; pass@256 difference CI excludes zero. \\[5pt]
    AIME 2025 & 63.3\% vs. 53.3\% at $k=256$ & Only 30 problems. Best-pair CI spans zero. \\\bottomrule
  \end{tabular}
  \medskip
  \begin{block}{What pass@$k$ tells us}
    Additional samples can expand correct-solution coverage.\newline
    Coverage depends on sampler tuning and the per-sample compute budget.
  \end{block}
  \papersource{Paper, Tables 17--19, Figs. 17--19; PDF pp. 43--44, 51--52. 3B base / 3B instruct as appropriate.}
\end{frame}

\begin{frame}{Low-step sampling favors DDIM and a log-SNR grid}
  \begin{columns}[c]
    \begin{column}{0.67\textwidth}
      \includegraphics[width=\linewidth,height=4.8cm,keepaspectratio]{figures/fig13-samplers.pdf}
    \end{column}
    \begin{column}{0.30\textwidth}
      \textbf{Uniform-$\gamma$ grid}\medskip
      \[
        \gamma_i=\gamma_{\max}+\frac iN(\gamma_{\min}-\gamma_{\max})
      \]
      \smallskip
      Training learns a variance-efficient schedule.\par\medskip
      Inference chooses a finite integration grid.\par\medskip
      \alert{These are different optimization problems.}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 13; Sec. 5.1 / App. G.6; PDF pp. 8--9, 38--39, 49. CFG/ST tuned per configuration.}
\end{frame}

\begin{frame}{SUSReg reduces clustering of the token embeddings}
  \begin{columns}[c]
    \begin{column}{0.50\textwidth}
      \includegraphics[width=\linewidth,height=4.6cm,keepaspectratio]{figures/fig08-geometry.pdf}
    \end{column}
    \begin{column}{0.47\textwidth}
      \begin{itemize}
        \item Unregularized embeddings use about 4 effective principal dimensions.
        \item SUSReg matches random 1D projections to a uniform sphere distribution.
        \item Less clustering may reduce token ambiguity under coarse updates.
        \item Validation PPL changes little; low-step generation improves.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 8 / Eq. (16), Sec. 5.2 / App. E.1; PDF pp. 9, 28--29. Random sphere projection; density=1 is uniform.}
\end{frame}

\begin{frame}{Embedding regularization helps most below 32 steps}
  \centering
  \begin{tabular}{lrrr}\toprule
    DDIM steps per block & Sigma & +SUSReg & Gain (pp) \\\midrule
    256 & 63.88 & 64.34 & +0.46 \\
    32  & 60.41 & 61.85 & +1.44 \\
    16  & 55.17 & 58.50 & +3.33 \\
    8   & 37.06 & 42.76 & +5.70 \\
    4   & 7.74  & 11.80 & +4.06 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Six-task mean accuracy (\%), 3B, block size 32, self-conditioning on.
    \item Guidance settings are selected for each model and step budget.
    \item Four-step decoding still needs trajectory distillation.
  \end{itemize}
  \papersource{Paper, Table 8, PDF p. 33. Table 1 reports 60.45 for the ordinary DDIM-32 row; Table 8 reports 60.41.}
\end{frame}

\begin{frame}{PDD shares one backbone call across fine-grid updates}
  \begin{block}{A segment of the teacher's integration grid}
    One student Transformer call $\longrightarrow$ $\Delta$ parallel prediction heads\newline
    $\longrightarrow$ $\Delta$ inexpensive DDIM updates $\longrightarrow$ next segment
  \end{block}
  \[
    \Delta=G/M,\qquad
    \hat\e_j^{S,\mathrm{raw}}
      =\exp(2\tanh\mathbf s_j)\odot\hat\e^{S,\mathrm{anchor}}+\mathbf r_j
  \]
  \begin{itemize}
    \item $G$: fine intervals; $M$: expensive student evaluations per block.
    \item Zero-initialized residual heads start from the pretrained prediction.
    \item Distill CFG into one conditional student call; apply ST analytically.
    \item Disable SC in teacher and student: local heads are predicted in parallel.
  \end{itemize}
  \papersource{Paper, App. I.1--I.2, Eqs. (32)--(37), PDF pp. 57--58. Top flow is an explanatory schematic.}
\end{frame}

\begin{frame}{PDD recovers quality at four denoising calls}
  \centering
  \begin{tabular}{lrrrr}\toprule
    3B + SUSReg & GSM8K & HumanEval & MBPP & Six-task avg. \\\midrule
    DDIM-16, SC & 80.85 & 53.60 & 45.22 & 58.50 \\
    DDIM-8, SC  & 70.88 & 34.45 & 27.98 & 42.76 \\
    PDD-8, no SC & 72.48 & 43.90 & 29.00 & 46.79 \\\midrule
    DDIM-4, SC & 26.04 & 11.59 & 4.32 & 11.80 \\
    PDD-4, no SC & 67.60 & 31.10 & 16.20 & 35.45 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item The largest recovery occurs at four diffusion evaluations.
    \item Code remains sensitive to aggressive step reduction.
    \item PDD-4 uses distilled $(w,\tau)=(4,0.4)$; PDD-8 uses $(3,0.3)$.
  \end{itemize}
  \papersource{Paper, Table 8, PDF p. 33; App. I, PDF pp. 57--60. Percent; each row uses its reported sampling recipe.}
\end{frame}

\begin{frame}{Higher-order integration can damage endpoint geometry}
  \begin{columns}[c]
    \begin{column}{0.62\textwidth}
      \includegraphics[width=\linewidth,height=4.35cm,keepaspectratio]{figures/fig15-endpoints.pdf}
    \end{column}
    \begin{column}{0.35\textwidth}
      \textbf{GSM8K, 16 calls}\par\medskip
      \begin{tabular}{lr}\toprule
        DDIM & 78.08 \\
        DPM++(2M) & 33.18 \\
        $\rho$AB1-DEIS & 48.96 \\\bottomrule
      \end{tabular}
      \medskip\par
      Coarse extrapolation can place terminal states far from the vocabulary sphere.
    \end{column}
  \end{columns}
  \medskip
  Reconstruction is trained near the clean endpoint; off-manifold states are harder to decode.
  \papersource{Paper, Table 16 / Fig. 15, PDF pp. 40--41, 50. Uniform-$\gamma$, SC on; guidance selected for DDIM.}
\end{frame}

\begin{frame}{SFT preserves average parity, with different strengths}
  \begin{columns}[T]
    \begin{column}{0.46\textwidth}
      \textbf{Ten-task mean (\%)}\par\medskip
      \begin{tabular}{lrr}\toprule
        Diffusion mode & 3B & 8B \\\midrule
        NLD & 52.89 & 59.92 \\
        Sigma DDPM-256 & 52.15 & 59.27 \\
        Sigma DDIM-32 & 49.86 & 57.01 \\\bottomrule
      \end{tabular}
      \medskip\par
      Assistant-only supervision, joint diffusion + AR objective; geometry frozen by default.
    \end{column}
    \begin{column}{0.50\textwidth}
      \textbf{8B DDPM-256 vs. NLD}\par\medskip
      \begin{tabular}{lrr}\toprule
        Task & Sigma & NLD \\\midrule
        GPQA & 44.70 & 33.33 \\
        IFEval & 79.14 & 73.02 \\
        MATH-500 & 78.88 & 84.60 \\
        AIME 2025 & 16.33 & 23.33 \\
        HumanEval & 79.76 & 83.54 \\\bottomrule
      \end{tabular}
    \end{column}
  \end{columns}
  \papersource{Paper, Tables 3--4, PDF p. 10; App. H, PDF pp. 54--56. Non-thinking protocol; IFEval uses instruction-level loose accuracy.}
\end{frame}

\begin{frame}{Count the readout before interpreting speed}
  \[
    \text{Forward calls per committed token}\ \approx\ \frac{N+1}{S}
  \]
  $N$: diffusion updates per window; $S$: tokens committed from each 32-token window.
  \par\medskip
  \begin{tabular}{lrr}\toprule
    Default full-block decoding & Calls/block & Calls/token \\\midrule
    DDPM-256 & 257 & 8.03 \\
    DDIM-32  & 33  & 1.03 \\
    DDIM-4   & 5   & 0.156 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item One call processes a whole block and may contain two CFG branches.
    \item Small CFG overhead does not establish end-to-end speed over AR.
    \item The paper identifies high-step decoding speed and low-step code quality as limitations.
  \end{itemize}
  \papersource{Paper, Algorithm 2 / Apps. B, G.5, G.10; PDF pp. 22, 27, 38--40, 44--45. Ratios computed from the reported formula.}
\end{frame}

\begin{frame}{Three connected contributions}
  \begin{enumerate}
    \item \textbf{A scalable continuous language-model recipe}\medskip\newline
      AR transfer, learned 16D embeddings, and a blockwise likelihood objective.
      \medskip
    \item \textbf{A useful inference control space}\medskip\newline
      CFG and ST trade single-sample quality for broader correct-solution coverage.
      \medskip
    \item \textbf{A route toward few-step generation}\medskip\newline
      Embedding geometry and trajectory distillation jointly improve low-NFE behavior.
  \end{enumerate}
  \medskip
  \alert{The next bottleneck: robust latent-to-token decoding at low compute.}
  \papersource{Synthesis of Secs. 3--7 and App. B, PDF pp. 3--10, 22.}
\end{frame}

\begin{frame}{Reference and discussion material}
  \textbf{Zhihan Yang, Wei Guo, Jean-Marie Lemercier, Simon Welker, et al.}\medskip

  \emph{Large Language Continuous Diffusion Models}.\medskip

  arXiv:2610.02665v1, submitted 2 October 2026.\medskip

  \href{https://arxiv.org/abs/2610.02665v1}{https://arxiv.org/abs/2610.02665v1}
  \bigskip
  \begin{block}{Companion material}
    Chinese reading notes, editable Beamer source, original figure assets,\newline
    and the versioned source paper are archived together in the paper library.
  \end{block}
  \papersource{All experimental results and original figures in this deck come from this paper. Backup slides follow.}
\end{frame}

\appendix

\begin{frame}{Backup: unbiased training with one loss branch per block}
  \[
    \widehat{\mathcal L}
     =\mathcal L_{\mathrm{prior}}
      +\frac{R}{p_{\mathrm{rec}}}\mathcal L_{\mathrm{rec}}
      +\frac{1-R}{1-p_{\mathrm{rec}}}\mathcal L_{\mathrm{diff}},
      \quad R\sim\operatorname{Bern}(0.1)
  \]
  \begin{itemize}
    \item Select reconstruction at $t=0$ on 10\% of blocks; otherwise sample a diffusion time.
    \item Inverse-probability weighting preserves the expected NELBO.
    \item All blocks fit into one forward with independent times and branch choices.
  \end{itemize}
  \begin{block}{Two noise-schedule objectives}
    Endpoints $\gamma_0,\gamma_1$: optimize the likelihood bound.\medskip\newline
    Interior shape $\tilde\gamma$: reduce Monte Carlo variance through the second moment.
  \end{block}
  \papersource{Paper, Eq. (5), Algorithm 1 / App. D; PDF pp. 3, 26--27.}
\end{frame}

\begin{frame}{Backup: why guidance and temperature commute}
  Set $\mu_t=\z_t/\alpha_t$. The Gaussian denoising identity gives
  \[
    \mathbf s=\frac{\alpha_t}{\sigma_t^2}(\e-\mu_t).
  \]
  Define $T(\e)=\mu_t+\tau^{-1}(\e-\mu_t)$ and
  $G(\e_c,\e_\varnothing)=w\e_c+(1-w)\e_\varnothing$. Then
  \[
    T\big(G(\e_c,\e_\varnothing)\big)
      =wT(\e_c)+(1-w)T(\e_\varnothing).
  \]
  The resulting score is
  \[
    \frac1\tau\big(w\mathbf s_c+(1-w)\mathbf s_\varnothing\big)
    =\nabla_{\z_t}\log\!\left[
       p_t(\z_t\mid\cctx)^{w/\tau}
       p_t(\z_t\mid\varnothing)^{(1-w)/\tau}\right].
  \]
  This is a per-noise-level score identity; finite-step sampling remains approximate.
  \papersource{Paper, Proposition 1, Eqs. (28)--(29), PDF pp. 45--46.}
\end{frame}

\begin{frame}{Backup: SUSReg compares projected distributions}
  \[
    \phi_{\mathbf a_m}(u)=\frac1V\sum_{v=1}^{V}
        e^{iu\langle\mathbf a_m,\e_v\rangle},
    \qquad \mathbf a_m\sim\operatorname{Unif}(\mathbb S^{15})
  \]
  \[
    s(\E)=\frac VM\sum_{m=1}^{M}\int_{-u_{\max}}^{u_{\max}}
      |\phi_{\mathbf a_m}(u)-\phi_{d_e}(u)|^2e^{-u^2/2}\,du
  \]
  \begin{itemize}
    \item $\phi_{d_e}$: characteristic function of a uniform-sphere projection.
    \item 512 random directions; 17 quadrature points over $[-5,5]$.
    \item Add $\lambda_{\mathrm{susreg}}s(\E)$ per token, $\lambda_{\mathrm{susreg}}=0.005$.
    \item Here $u$ denotes Fourier frequency, distinct from diffusion time.
  \end{itemize}
  \papersource{Paper, App. E.1, Eq. (16), PDF pp. 28--29. Integration variable renamed for clarity.}
\end{frame}

\begin{frame}{Backup: PDD trains on locally visited student states}
  \[
    \mathcal L_{\mathrm{PDD}}
     =\frac1P\sum_{p=1}^{P}\mathbb E
        \|\hat\e^{S,\mathrm{raw}}_{j_p}-\hat\e^{T,\mathrm{raw}}_{j_p}\|^2
  \]
  \[
    \mathcal L_{\mathrm{distill}}=5\mathcal L_{\mathrm{PDD}}
       +0.1\mathcal L_{\mathrm{teacher-rec}}
       +0.3\mathcal L_{\mathrm{teacher-ar}}
  \]
  \begin{itemize}
    \item Segment starts: forward marginal. Later nodes: student rollout.
    \item Frozen teacher predicts CFG targets before ST at those visited states.
    \item Forward KL terms preserve reconstruction and AR distributions.
    \item LoRA rank 128; 32 residual MLP heads; 10K steps; 512-token prefix.
  \end{itemize}
  \smallskip
  \small Implementation ambiguity: App. I.5 reports $P=8$ nodes per segment,\newline
  but a 32-head grid at eight evaluations gives only four local nodes.
  \papersource{Paper, App. I.3--I.5, Eqs. (38)--(43), PDF pp. 58--60.}
\end{frame}

\begin{frame}{Backup: SFT geometry freezing is a simple default}
  \begin{tabular}{lrr}\toprule
    3B instruct, ten-task mean & Frozen & Unfrozen \\\midrule
    DDPM-256 & 52.15 & 51.65 \\
    DDIM-32 & 49.86 & 50.84 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Freeze embeddings, noise endpoints, and the schedule network.
    \item Train the Transformer, conditioning projections, and both output heads.
    \item Unfrozen geometry uses a $1000\times$ learning-rate multiplier.
    \item Ordering reverses across samplers; freezing is a simpler recipe.
  \end{itemize}
  \medskip
  SFT: 8K steps, batch 256, LR $2.5\times10^{-6}$; left-pad prompts during training so answers start on the 32-token grid.
  \papersource{Paper, App. H, Tables 21--22, PDF pp. 54--56.}
\end{frame}

\begin{frame}{Backup: guidance ablations support embedding-space CFG}
  \textbf{Best score within each reported sweep}\par\medskip
  \begin{tabular}{lrr}\toprule
    Setting & Embedding CFG & Logit CFG \\\midrule
    GSM8K, DDIM-64, $\tau=0.3$ & 83.19 & 83.49 \\
    HumanEval, DDIM-64, $\tau=0.3$ & 56.22 & 53.78 \\
    GSM8K, DDPM-256, $\tau=1$ & 82.43 & 76.76 \\
    HumanEval, DDPM-256, $\tau=1$ & 46.95 & 29.70 \\\bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Logit CFG peaks early and degrades at stronger guidance.
    \item Retaining context but dropping SC is a weaker bad-prediction choice.
    \item Plain logit temperature helps, but adds no gain over tuned CFG+ST here.
  \end{itemize}
  \papersource{Paper, Tables 9--13 / App. G.3--G.4, PDF pp. 34--37. Percent, higher is better.}
\end{frame}

\begin{frame}{Backup: partial commitment trades context for step count}
  Denoise 32 positions; commit only the first $S$; discard and re-noise the tail.
  \[
    \text{Nominal NFE per 32 tokens}=\frac{32N}{S},\qquad
    \text{actual calls}=\frac{32(N+1)}{S}.
  \]
  \begin{tabular}{lrrrr}\toprule
    3B sampling & $N$ & $S$ & Actual calls & Six-task avg. \\\midrule
    DDPM, nominal 256 & 256 & 32 & 257 & 64.34 \\
    DDIM, nominal 256 & 64 & 8 & 260 & 67.00 \\\midrule
    DDIM, nominal 32 & 32 & 32 & 33 & 60.45 \\
    DDIM, nominal 32 & 16 & 16 & 34 & 61.82 \\
    DDIM, nominal 32 & 8 & 8 & 36 & 57.76 \\\bottomrule
  \end{tabular}
  \par\medskip
  Smaller stride helps until it forces too few denoising steps per window.
  \papersource{Paper, Table 14 / App. G.5, PDF pp. 38--39. Actual calls calculated using the paper's counting rule.}
\end{frame}

\begin{frame}{Backup: initialization and likelihood scoring}
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \textbf{MDLM versus AR warm start}\par\medskip
      \begin{tabular}{lrr}\toprule
        3B base avg. & AR init & MDLM init \\\midrule
        DDPM-256 & 64.34 & 64.67 \\
        DDIM-32 & 60.45 & 61.64 \\\bottomrule
      \end{tabular}
      \medskip\par
      Same 300B-token continuation.\medskip\par
      MBPP improves; HumanEval weakens.
    \end{column}
    \begin{column}{0.48\textwidth}
      \textbf{Likelihood-based multiple choice}\par\medskip
      \begin{tabular}{lrr}\toprule
        Diffusion avg. & Sigma & NLD \\\midrule
        3B & 68.40 & 68.03 \\
        8B & 72.28 & 73.02 \\\bottomrule
      \end{tabular}
      \medskip\par
      Sigma ranks with negative ELBO:\medskip\par
      128 stratified time samples\newline plus 8 reconstruction samples.
    \end{column}
  \end{columns}
  \papersource{Paper, Tables 5--7 / Apps. E.2, F, PDF pp. 29--32. These MMLU scores use candidate ranking, not instruct generation.}
\end{frame}

\end{document}