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\title[LangFlow]{LangFlow: Continuous Diffusion Rivals Discrete in Language Modeling}
\subtitle{Paper discussion \;\textbullet\; 25-minute main presentation}
\author{Paper authors: Yuxin Chen, Chumeng Liang, Hangke Sui,\\Ruihan Guo, Chaoran Cheng, Jiaxuan You, and Ge Liu\\[2mm]\small University of Illinois Urbana-Champaign}
\date{arXiv:2604.11748v3 \;\textbullet\; April 15, 2026\\Reading prepared October 14, 2026}

\begin{document}

\begin{frame}
\titlepage
\vspace{-3mm}
\centering\small\href{https://arxiv.org/abs/2604.11748v3}{arxiv.org/abs/2604.11748v3}
\end{frame}

\begin{frame}{Continuous diffusion becomes competitive at 130M}
\begin{block}{Central result}
Token cross-entropy, an information-based noise schedule, and self-conditioning make embedding diffusion competitive with strong discrete DLMs.
\end{block}
\centering
\begin{tabular}{lrrr}
\toprule
Validation PPL $\downarrow$ & AR & MDLM & LangFlow\\
\midrule
LM1B & 22.8 & 31.0 & \alert{30.0}\\
OpenWebText & 17.5 & 23.2 & \alert{24.6}\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
DLM numbers are upper bounds. The claim is competitiveness with diffusion baselines at this scale; AR retains better in-domain PPL.
\src{Sec. 5, Table 2, PDF pp. 9--10.}
\end{frame}

\begin{frame}{A dense state still needs a categorical learning signal}
\begin{columns}[T,onlytextwidth]
\column{0.48\textwidth}
\begin{block}{Representation choice}
\begin{itemize}
\item One-hot / simplex state: vocabulary-sized and sparse.
\item Embedding state: $\z\in\R^{L\times D}$, with $D=768$.
\item A continuous trajectory can support ODE solvers and future distillation.
\end{itemize}
\end{block}
\column{0.48\textwidth}
\begin{block}{Three obstacles}
\begin{enumerate}
\item What objective should learn the continuous flow?
\item How can its likelihood be evaluated?
\item Where should training and sampling spend their steps?
\end{enumerate}
\end{block}
\end{columns}
\medskip
The network still predicts $L\times V$ token probabilities: the smaller state does not remove the vocabulary output layer.
\src{Secs. 1 and 3.1, PDF pp. 1--5.}
\end{frame}

\begin{frame}{Predict tokens, then move through embedding space}
\centering
\includegraphics[width=0.98\linewidth,height=5.5cm,keepaspectratio]{figures/fig-01-pipeline.pdf}
\par\smallskip\raggedright
Train on noisy embeddings with CE. During generation, turn probabilities into an expected embedding and update the entire sequence.
\src{Fig. 1, PDF p. 6. Original author figure.}
\end{frame}

\begin{frame}{The noise coordinate makes denoising difficulty explicit}
\[
\gamma=\log\frac{\sigma_\gamma^2}{\alpha_\gamma^2},\qquad
\z_\gamma=\alpha_\gamma\z+\sigma_\gamma\eps,
\quad\eps\sim\N(0,I)
\]
\[
\sigma_\gamma^2=\sigmoid(\gamma),\qquad
\alpha_\gamma^2=\sigmoid(-\gamma).
\]
\begin{block}{Direction matters}
Large $\gamma$: noisy.\qquad Small $\gamma$: clean.\\[1mm]
Generation travels from $b$ to $a$, with $a<b$.
\end{block}
\begin{itemize}
\item Condition the model on log noise-to-signal ratio, rather than an arbitrary time index.
\item Noise sampling density $\pi(\gamma)$ determines where training effort is spent.
\end{itemize}
\src{Sec. 3.1, Eqs. (7)--(8), PDF p. 4.}
\end{frame}

\begin{frame}{Cross-entropy matches the token posterior}
For a one-hot target $p=\mathbf1_{x^{(i)}}$ and prediction $q$,
\[
\E[D_f(p,q)\mid\z_\gamma]
=D_f(\E[p\mid\z_\gamma],q)+C.
\]
\begin{itemize}
\item The best prediction is the posterior probability vector.
\item With $f(p)=\sum_kp_k\log p_k$, Bregman divergence becomes KL; a one-hot target gives cross-entropy.
\end{itemize}
\[
\CE=\E_{\gamma\sim\pi,\x,\eps}
\left[-\frac1L\sum_{i=1}^L\log\hat x_\theta^{(i,x^{(i)})}(\z_\gamma,\gamma)\right].
\]
\alert{A consistency argument for posterior prediction}, for a fixed representation; not an identity between CE and exact model NLL.
\src{Sec. 3.1, Eqs. (9)--(18), PDF pp. 4--5.}
\end{frame}

\begin{frame}{Posterior probabilities determine the continuous velocity}
\[
\zh_\theta^{(i)}=\sum_{k=1}^V\hat x_\theta^{(i,k)}\mathbf e_k
=\mE^\top\xh_\theta^{(i)}.
\]
\begin{block}{The bridge to flow matching}
Correct token posterior $\Rightarrow$ correct expected embedding\\
$\Rightarrow$ correct marginal velocity for an affine Gaussian path.
\end{block}
\[
\mathbf v_\theta(\z_\gamma,\gamma)
=\frac{\dot\sigma_\gamma}{\sigma_\gamma}\z_\gamma
-\frac{\alpha_\gamma}{2}\zh_\theta(\z_\gamma,\gamma).
\]
Dots denote derivatives with respect to $\gamma$. The soft embedding preserves uncertainty during refinement; argmax is deferred to the end.
\src{Eq. (6), p. 3; Eq. (19), p. 5; Eqs. (37)--(38), p. 17.}
\end{frame}

\begin{frame}{Sampling integrates a soft prediction over each interval}
Hold $\zh_k$ fixed within a solver interval. Then
\[
d\!\left(\frac{\z_\gamma}{\sigma_\gamma}\right)
=\zh_\theta\,d\!\left(\frac{\alpha_\gamma}{\sigma_\gamma}\right),
\]
\[
\boxed{\z_{k+1}=\sigma_{k+1}\left[
\frac{\z_k}{\sigma_k}+
\left(\frac{\alpha_{k+1}}{\sigma_{k+1}}-\frac{\alpha_k}{\sigma_k}\right)\zh_k
\right].}
\]
\begin{enumerate}
\item Initialize $\z_0\sim\N(0,\sigma_b^2I)$.
\item Predict token probabilities, embed them, and integrate toward $a$.
\item Run a final prediction and decode each position by argmax.
\end{enumerate}
\src{Appendix A.1--A.2, Algorithm 2 and Eq. (22), PDF pp. 15--16.}
\end{frame}

\begin{frame}{Likelihood needs a decoder and a volume correction}
Use a noisy data endpoint $\z_a\sim\N(\alpha_a\z,\sigma_a^2I)$ and integrate $a\to b$ for evaluation:
\begin{align*}
\log p(\x)\ge\E\Bigg[&\frac{LD}{2}-\frac{\|\z_b\|^2}{2\sigma_b^2}
+\sum_i\log\hat x_\theta^{(i,x^{(i)})}(\z_a,a)\\[-1mm]
&-\int_a^b\frac{\alpha_\gamma}{2}\nabla_{\z_\gamma}\!\cdot\zh_\theta(\z_\gamma,\gamma)\,d\gamma\Bigg].
\end{align*}
\begin{itemize}
\item Gaussian energy + token reconstruction + continuous volume change.
\item Negative ELBO gives an NLL upper bound, hence a PPL upper bound.
\item Evaluation: 128-step Heun-2 and Hutchinson trace estimation.
\end{itemize}
The bound uses a probabilistic token decoder; sample generation uses argmax.
\src{Theorem 3.1, Eq. (20), p. 6; Appendix C.4, p. 19.}
\end{frame}

\begin{frame}{Uniform time spends most steps on easy token recovery}
\begin{columns}[c,onlytextwidth]
\column{0.55\textwidth}
\includegraphics[width=\linewidth,height=5.0cm,keepaspectratio]{figures/fig-02a-loss-time.pdf}
\column{0.41\textwidth}
\begin{itemize}
\item CE is almost zero for $t>0.2$ in the plotted convention.
\item Finite, separated token endpoints can be identified while noise remains.
\item Concentrate effort around the transition in information.
\end{itemize}
\end{columns}
\src{Sec. 4.1 and Fig. 2 left, PDF pp. 7--8. Original panel.}
\papersource{The caption's time-to-gamma formula has a sign inconsistency; generation here follows Algorithm 2: decreasing gamma.}
\end{frame}

\begin{frame}{Schedule density should follow the change in entropy}
\begin{columns}[c,onlytextwidth]
\column{0.49\textwidth}
\centering\includegraphics[width=\linewidth,height=3.9cm,keepaspectratio]{figures/fig-02b-loss-gamma.pdf}
\column{0.49\textwidth}
\centering\includegraphics[width=\linewidth,height=3.9cm,keepaspectratio]{figures/fig-02c-derivative.pdf}
\end{columns}
\[
\ell_{\rm CE}(\gamma)=H_\gamma+\text{posterior KL}
\ \approx\ H_\gamma,
\qquad \alert{\pi(\gamma)\propto H'_\gamma}.
\]
Stable loss profiles motivate the entropy proxy. The derivative is right-skewed and fits a Gumbel density; this fit is empirical.
\src{Sec. 4.1, Fig. 2 middle/right, PDF pp. 7--8. Original panels.}
\end{frame}

\begin{frame}{A learned Gumbel fit allocates both training and steps}
\[
F(\gamma)=\exp\!\left[-\exp\!\left(-\frac{\gamma-P_\mu}{P_\beta}\right)\right],
\qquad H_\gamma\approx H_{+\infty}F(\gamma).
\]
\begin{columns}[T,onlytextwidth]
\column{0.48\textwidth}
\begin{block}{Training}
Sample $\gamma\sim\mathrm{Gumbel}(P_\mu,P_\beta)$.\\[2mm]
Fit entropy using squared error to detached CE. Detach sampled $\gamma$ from the main model loss.
\end{block}
\column{0.48\textwidth}
\begin{block}{Sampling}
Take descending quantiles:\\[1mm]
$q_k=\operatorname{clip}(1-k/N,\delta,1-\delta)$,\\[2mm]
$\gamma_k=P_\mu-P_\beta\log(-\log q_k)$,\\[1mm]
with $\delta=10^{-5}$.
\end{block}
\end{columns}
\medskip
Equal CDF intervals approximate equal entropy reduction. The paper reports Gen. PPL falling from $\sim1000$ to 154.2 after schedule improvements.
\src{Sec. 4.1, Eq. (21), p. 8; Algorithms 1--2, p. 15.}
\end{frame}

\begin{frame}{Self-conditioning carries an earlier interpretation forward}
\begin{columns}[T,onlytextwidth]
\column{0.48\textwidth}
\begin{block}{Training: enabled with probability 0.25}
\begin{enumerate}
\item Predict once from the current noisy state.
\item Embed and detach that prediction.
\item Predict again using it as an auxiliary input.
\end{enumerate}
Otherwise, use a zero auxiliary input.
\end{block}
\column{0.48\textwidth}
\begin{block}{Sampling: always enabled}
\begin{enumerate}
\item Start with a zero SC input.
\item Predict and integrate.
\item Reuse the preceding predicted embedding at the next step.
\end{enumerate}
No second pass is added to every sampling step.
\end{block}
\end{columns}
\medskip
The auxiliary input enters through a zero-initialized projection before the DiT blocks.
\src{Sec. 4.2, p. 8; Algorithm 1, p. 15; Appendix C.1, p. 18.}
\end{frame}

\begin{frame}{Self-conditioning changes the continuous model's ranking}
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lrr}
\toprule
LM1B setting & Gen. PPL $\downarrow$ & PPL $\downarrow$\\
\midrule
MDLM & 103.9 & 31.0\\
MDLM + SC & 94.9 & 32.7\\
\midrule
LangFlow & 154.2 & 49.0\\
LangFlow + SC & \alert{81.5} & \alert{30.0}\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
SC improves both metrics for LangFlow. MDLM's sample score improves while its validation PPL worsens.
\begin{block}{Keep the reported results separate}
This ablation reports 81.5 Gen. PPL; the main comparison reports 92.2. The paper does not explain the difference in protocol.
\end{block}
\src{Table 1, PDF p. 8; contrast Table 2, p. 9, and Table 5, p. 20.}
\end{frame}

\begin{frame}{Two datasets test different sequence lengths}
\centering
\begin{tabular}{lcc}
\toprule
 & LM1B & OpenWebText\\
\midrule
Context length & 128 & 1024\\
Tokenizer & BERT uncased & GPT-2 Large\\
Training steps / batch & 1M / 512 & 1M / 512\\
Main sampling steps & 128 & 1024\\
Generated samples & 1024 & 1024\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
\begin{itemize}
\item Approx. 130M parameters; hidden size 768, 12 heads, RoPE.
\item Validation PPL: likelihood on real data; DLM values are bounds.
\item Gen. PPL: generated samples scored by GPT-2 Large.
\end{itemize}
\src{Sec. 5, PDF p. 9. The text specifies 12 layers; Table 4 says 16.}
\papersource{Plaid uses its own 108M architecture and 16-dimensional embeddings. FLM uses 1024 LM1B sampling steps.}
\end{frame}

\begin{frame}{Validation PPL closes the gap with discrete diffusion}
\centering
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lrr}
\toprule
Model & LM1B PPL $\downarrow$ & OWT PPL $\downarrow$\\
\midrule
AR Transformer & 22.8 & 17.5\\
MDLM & 31.0 & 23.2\\
SEDD Absorb & 32.0 & 24.1\\
Plaid & 32.4 & ---\\
UDLM & 33.8 & 27.4\\
Duo & 33.6 & 25.2\\
\midrule
\textbf{LangFlow} & \alert{\textbf{30.0}} & \alert{\textbf{24.6}}\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
Best reported DLM bound on LM1B; third-best on OWT. Different DLM bounds need not have the same tightness.
\src{Table 2, PDF p. 9. Selected rows; all DLM PPLs are upper bounds.}
\end{frame}

\begin{frame}{OWT generation scores approach the AR reference}
\centering
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lrr}
\toprule
Model & LM1B Gen. PPL $\downarrow$ & OWT Gen. PPL $\downarrow$\\
\midrule
AR Transformer & 66.7 & 35.9\\
MDLM & 103.9 & 104.9\\
Plaid & 77.3 & ---\\
Duo & 97.6 & 77.6\\
FLM & 96.9* & 62.2\\
\midrule
\textbf{LangFlow} & \alert{\textbf{92.2}} & \alert{\textbf{36.5}}\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
Best DLM sample score on OWT; second on LM1B. This measures a reference model's judgment of generated text, not the generator's likelihood.
\src{Table 2, PDF p. 9. GPT-2 Large scores 1024 generated sequences.}
\papersource{* FLM: 1024 LM1B sampling steps; other listed diffusion models: 128. OWT: 1024 steps.}
\end{frame}

\begin{frame}{Zero-shot gains over AR occur on four of seven corpora}
\centering
\begin{tabular}{lrrr}
\toprule
Target corpus & AR & MDLM & LangFlow\\
\midrule
PTB & 82.05 & 95.26 & \alert{81.20}\\
Wikitext & 25.75 & 32.83 & 32.28\\
LM1B & 51.25 & 67.01 & 68.21\\
Lambada & 51.28 & 47.52 & \alert{46.93}\\
AG News & 52.09 & 61.15 & 69.41\\
Pubmed & 49.01 & 41.89 & \alert{46.74}\\
Arxiv & 41.73 & 37.37 & \alert{38.47}\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
All are \textbf{corpus perplexities} after OWT training; Lambada is not a last-word accuracy result. LangFlow also beats MDLM on PTB, Wikitext, and Lambada.
\src{Table 3, PDF p. 10. DLM perplexities are upper bounds.}
\end{frame}

\begin{frame}{Fewer steps preserve a lead, but quality still degrades}
\centering
\begin{tabular}{rrrrr}
\toprule
OWT budget & \multicolumn{3}{c}{Gen. PPL $\downarrow$} & LF entropy\\
\cmidrule(lr){2-4}
Reported NFE & LangFlow & Duo & MDLM & \\
\midrule
1024 & 36.53 & 77.69 & 104.85 & 5.25\\
512 & 41.52 & 78.14 & 104.43 & 5.31\\
256 & 49.24 & 78.62 & 112.70 & 5.37\\
128 & 60.09 & 80.02 & 120.77 & 5.43\\
64 & 80.34 & 85.62 & 143.88 & 5.51\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
One LangFlow checkpoint, no distillation or special few-step training. The gap to Duo narrows at 64 steps; no end-to-end speedup is measured here.
\src{Appendix D.2, Table 6, PDF p. 21. Baselines quoted from Duo. Data entropy: 5.44.}
\papersource{Paper budget labels retained. Algorithm 2 also includes a final decoding forward pass.}
\end{frame}

\begin{frame}{Lower sample entropy leaves a quality question open}
\begin{block}{What entropy measures here}
Token-frequency entropy within each sequence, averaged across samples. It mixes vocabulary diversity, topic concentration, and repetition.
\end{block}
\begin{itemize}
\item OWT at 1024 steps: LangFlow 5.25; data reference 5.44.
\item Authors attribute part of the gap to content words recurring across sentences.
\item A selected LangFlow example repeats \textit{health} 11 times; this is not an aggregate repetition statistic.
\item The shown texts still contain malformed words and semantic errors; the sample score alone does not establish coherence.
\end{itemize}
\src{Appendix C.4, p. 19; Tables 6--7, p. 21; qualitative OWT samples, pp. 26--30.}
\end{frame}

\begin{frame}{The contribution is a coherent recipe for continuous DLMs}
\begin{enumerate}
\item \textbf{A categorical objective can learn a continuous flow.}\\
Posterior matching connects CE to an embedding denoiser.
\medskip
\item \textbf{Noise geometry is modality-dependent.}\\
An entropy-change proxy guides both training and sampling.
\medskip
\item \textbf{Self-conditioning materially changes the comparison.}\\
Its treatment in training and evaluation matters.
\end{enumerate}
\begin{block}{What remains to establish}
Scaling beyond 130M, coherent long generation, and actual few-step latency gains. ODE distillation is a proposed next step, not a result of this paper.
\end{block}
\src{Secs. 3--5 and 7, PDF pp. 4--11; Appendix D.2, pp. 20--21.}
\end{frame}

\begin{frame}{Reference and discussion prompts}
\textbf{Yuxin Chen, Chumeng Liang, Hangke Sui, Ruihan Guo,\\
Chaoran Cheng, Jiaxuan You, and Ge Liu.}\\[2mm]
\emph{LangFlow: Continuous Diffusion Rivals Discrete in Language Modeling.} arXiv:2604.11748v3, April 15, 2026.\\[2mm]
\href{https://arxiv.org/abs/2604.11748v3}{Paper: arxiv.org/abs/2604.11748v3}\\
\href{https://github.com/nealchen2003/LangFlow}{Author project: github.com/nealchen2003/LangFlow}
\medskip
\begin{itemize}
\item Does an entropy-change schedule retain its shape at larger scale?
\item Can distillation preserve the quality--diversity balance?
\item How should the SC history be represented in likelihood evaluation?
\end{itemize}
\papersource{All experimental numbers in this presentation come from this paper. Backup slides follow.}
\end{frame}

\appendix

\begin{frame}{Backup: the likelihood bound follows three steps}
\textbf{1. Introduce a continuous latent endpoint.}
\[
\log p(\x)\ge\E_{q(\z_a\mid\x)}
[\log p_a(\z_a)+\log\hat p(\x\mid\z_a)-\log q(\z_a\mid\x)].
\]
\textbf{2. Track the density along the ODE.}
\[
\log p_a(\z_a)=\log p_b(\z_b)+\int_a^b\nabla\!\cdot\mathbf v_\theta\,d\gamma.
\]
\textbf{3. Cancel the Gaussian scale terms.}
\[
\nabla\!\cdot\mathbf v_\theta=LD\frac{\dot\sigma_\gamma}{\sigma_\gamma}
-\frac{\alpha_\gamma}{2}\nabla\!\cdot\zh_\theta.
\]
The $\log\sigma$ terms cancel against encoder entropy, leaving $LD/2$. A regular differentiable flow is required for the change-of-variables step.
\src{Appendix B.1, Eqs. (30)--(43), PDF pp. 16--17.}
\end{frame}

\begin{frame}{Backup: the architecture adds three small mechanisms}
\begin{enumerate}
\item \textbf{Input adaptation for SC:}
\[
\z_\gamma\leftarrow\z_\gamma+W_{\rm in}\z_\gamma+W_{\rm SC}\z_{\rm SC}.
\]
Both projections are zero-initialized.
\item \textbf{Embedding normalization:} each embedding has norm $\sqrt D$.
\item \textbf{Analytic logit bias for candidate token $k$:}
\[
r\log p(\z_\gamma^{(i)}\mid x^{(i)}=k)
=r\frac{\alpha_\gamma}{\sigma_\gamma^2}\mathbf e_k^\top\z_\gamma^{(i)}+C.
\]
Ramp $r$ from 0 to 1 during the first 5000 training steps.
\end{enumerate}
\src{Appendix C.1, Eq. (44), PDF pp. 17--18.}
\end{frame}

\begin{frame}{Backup: embedding geometry motivates avoiding MSE}
\begin{columns}[c,onlytextwidth]
\column{0.57\textwidth}
\includegraphics[width=\linewidth,height=4.5cm,keepaspectratio]{figures/fig-03-embeddings.pdf}
\column{0.39\textwidth}
\begin{itemize}
\item Nearest-neighbor distance on normalized embeddings.
\item Plaid's vectors are much closer together.
\item MSE can reduce error by contracting the representation.
\end{itemize}
\end{columns}
\medskip
This is a cross-model observation: Plaid uses 16-dimensional embeddings; LangFlow uses 768. The comparison does not isolate loss choice.
\src{Appendix C.3, Fig. 3 and Eqs. (45)--(46), PDF pp. 18--19. Original figure.}
\end{frame}

\begin{frame}{Backup: SC changes how probability shifts under noise}
\centering
\includegraphics[width=0.90\linewidth,height=4.4cm,keepaspectratio]{figures/fig-04-self-conditioning.png}
\par\medskip\raggedright
Higher $\gamma$ moves from the correct token toward related words, then frequent function words. Dashed: pass 1. Solid: pass 2 with SC. This is one illustrative token position.
\src{Appendix D.1, Fig. 4, PDF p. 20. Original figure.}
\end{frame}

\begin{frame}{Backup: LM1B budget and reproduction details}
\begin{columns}[T,onlytextwidth]
\column{0.42\textwidth}
\centering
\begin{tabular}{rrr}
\toprule
NFE & Gen. PPL & Entropy\\
\midrule
128 & 92.24 & 4.31\\
64 & 104.83 & 4.32\\
32 & 127.32 & 4.33\\
16 & 179.60 & 4.35\\
\bottomrule
\end{tabular}
\par\medskip\raggedright
Data entropy: 4.32.\\
No distillation.
\column{0.54\textwidth}
\begin{itemize}
\item Table 1 gives 81.5 Gen. PPL; Tables 2/5 give 92.2/92.24.
\item Sec. 5 and C.1 specify 12 layers; Table 4 specifies 16.
\item SC uses history in Algorithm 2; the likelihood proof does not spell out its treatment in the state and divergence.
\end{itemize}
\end{columns}
\src{Table 5, p. 20; Table 1, p. 8; Table 2, p. 9; Table 4, p. 18; Appendix B.1, pp. 16--17.}
\end{frame}

\end{document}