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\title[Elastic Latent Recurrence]{One Model to Loop Them All}
\subtitle{Elastic Latent Recurrence for Language Models}
\author{Paper: Faisal Ladhak, Shantanu Acharya, Santiago Akle Serano,\\
Simeng Sun, Boris Ginsburg, Krishna C Puvvada}
\date{arXiv:2610.12582v1 \quad 8 October 2026\\
Independent reading discussion \quad 15 October 2026}

\begin{document}

\begin{frame}
\titlepage
\vspace{-4mm}
\begin{center}
\alert{Preserve the useful one-loop endpoint; spend more compute when needed.}
\end{center}
\end{frame}

\begin{frame}{Elastic depth needs a competitive starting point}
\begin{block}{The deployment question}
Can one checkpoint serve a fast tier and a stronger tier while retaining the fast tier's accuracy?
\end{block}
\begin{itemize}
\item Shared layers let the model increase effective depth without storing another set of weights.
\item The \alert{one-loop tax}: a looped model can underperform its non-recurrent counterpart at its cheapest setting.
\item ELR combines a learned update, shallow/deep gradient mixing, and explicit loop conditioning.
\end{itemize}
\papersource{Paper, Secs. 1--2, PDF pp. 1--4. Scope: loop counts within the trained range.}
\end{frame}

\begin{frame}{Only the middle block repeats}
\par
\centering
\includegraphics[width=0.98\linewidth,height=3.4cm,keepaspectratio]{figures/fig01-architecture.pdf}\par
\[
D_{\mathrm{eff}}(k)=L_{\mathrm{in}}+kL_{\mathrm{rec}}+L_{\mathrm{out}}
\]
\raggedright
For the hybrid models, $14/24/14$ gives \textbf{52, 76, 100, 124} executed sublayers at $k=1,2,3,4$.
\papersource{Paper, Fig. 1, PDF p. 2; Eq. (1), p. 3; Appendix A.1, p. 18. Each attention, GDP, or FF sublayer counts once.}
\end{frame}

\begin{frame}{Adaptive delta learns how much to revise each state}
\[
\tilde s_i=s_{i-1}+\hin,\qquad r_i=\R(\tilde s_i),\qquad s_0=\hin
\]
\begin{block}{A gated step toward the candidate state}
\[
s_i=s_{i-1}+\beta_i\odot(r_i-s_{i-1}),\qquad
\beta_i=\sigma(W_\beta\tilde s_i+b_\beta)\odot\operatorname{softplus}(\gamma)
\]
\end{block}
\begin{itemize}
\item A token-dependent, channel-wise step; $W_\beta\in\mathbb R^{d\times d}$ and $b_\beta,\gamma\in\mathbb R^d$.
\item Zero initialization gives $\beta_i=\tfrac12\log 2\approx0.347$.
\item Gate adds $d^2+2d$ parameters, shared across loops. Learned steps need not remain below one.
\end{itemize}
\papersource{Paper, Sec. 3.1, Eq. (2), PDF p. 4; Appendix A, p. 18.}
\end{frame}

\begin{frame}{Mix shallow and deep gradients within one update}
Draw a common $k\sim U\{1,\ldots,K\}$ for each microbatch. A rotating fraction $f$ of data-parallel ranks uses one loop instead.
\[
p_1=\frac1K+\left(1-\frac1K\right)f,
\qquad \bar k=f+(1-f)\frac{K+1}{2}
\]
\begin{block}{Main runs: $K=4$, $f=0.10$}
$(p_1,p_2,p_3,p_4)=(0.325,0.225,0.225,0.225)$; $\bar k=2.35$.
\end{block}
\begin{itemize}
\item Example: when $k=4$, 10 of 100 ranks run one loop and 90 run four.
\item Average their gradients before the optimizer update; backpropagate through every executed loop.
\end{itemize}
\papersource{Paper, Sec. 3.2, Eq. (3), PDF pp. 4--5; Appendix A.3, p. 19. Rank example is explanatory.}
\end{frame}

\begin{frame}{Condition shared layers on progress and budget}
\[
\tau_i=i/k,\quad \kappa=k/K,\quad
c_i=\operatorname{MLP}([\phi(\tau_i);\phi(\kappa)])
\]
\[
\begin{aligned}
\tilde x&=(1+\lambda(c_i))\odot\norm(x)+\mu(c_i)\\
\hat x&=\tilde x+A\bigl(g(c_i)\odot B\tilde x\bigr)\\
y&=x+(1+\omega(c_i))\odot f(\hat x)
\end{aligned}
\]
\begin{itemize}
\item Modulate normalized input, add a gated low-rank correction, and scale the residual branch.
\item $A\in\mathbb R^{d\times r}$, $B\in\mathbb R^{r\times d}$; parameters are layer-specific and shared across loops.
\item Input block is unconditioned; output block uses $(1,k/K)$.
\end{itemize}
\papersource{Paper, Sec. 3.3, Eq. (4), PDF p. 5. $\phi$: sinusoidal embedding. $f$: one model sublayer.}
\end{frame}

\begin{frame}{Zero initialization preserves the trained endpoint}
At the switch to mixed depth, add zero-initialized conditioning heads:
\[
s_i=s_{i-1}+(\pi_i+g_{\mathrm{exit}}(c_i))\odot\beta_i\odot(r_i-s_{i-1})
\]
\[
\pi_1=1,\qquad \pi_{i>1}=0.
\]
\begin{enumerate}
\item Layer modulation initially makes no change to the original sublayers.
\item Loop one reproduces the existing ELR checkpoint's update.
\item Later loops have zero exit factors: $s_i=s_{i-1}$.
\end{enumerate}
\begin{block}{Initial function is the same for every budget}
In exact arithmetic, every $k$ starts from the one-loop checkpoint's output. Later loops learn useful corrections during continuation.
\end{block}
\papersource{Paper, Sec. 3.3, PDF p. 5. The starting checkpoint already has the adaptive-delta gate.}
\end{frame}

\begin{frame}{Late mixed-depth training limits added compute}
If mixed depth covers fraction $\rho$ of training tokens,
\[
\frac{C_{\mathrm{ELR}}}{C_{\mathrm{base}}}
\approx(1-\rho)+\rho\frac{D_{\mathrm{eff}}(\bar k)}{D_{\mathrm{eff}}(1)}.
\]
\par
\centering
\begin{tabular}{lccc}
\toprule
Configuration & $\bar k$ & $\rho$ & Estimated cost\\
\midrule
4B / 30B matched runs & 2.35 & 0.25 & $1.16\times$\\
GPT-0.5B late ablation & 2.70 & 0.25 & $1.20\times$\\
Koi-4B pretraining & 2.35 & 0.50 & $1.31\times$\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
Keep optimizer state and the learning-rate schedule. These are \alert{layer-execution estimates}, not measured training wall time.
\papersource{Paper, Eq. (5), PDF p. 6; Sec. 6.5, p. 13; Appendix A.4, pp. 19--20.}
\end{frame}

\begin{frame}{Three experiments answer different questions}
\par
\centering
\begin{tabular}{P{2.7cm}P{4.3cm}P{5.5cm}}
\toprule
Experiment & Controlled comparison & What it establishes\\
\midrule
4B / 30B-A3B & Matched backbone, data and optimizer; 1T / 300B tokens; no conditioning & One-loop parity and gains from depth; about 16\% more estimated training compute\\[2mm]
GPT-0.5B & 300B tokens; validation loss; usually $K=5$, no input re-injection & Effects of updates, sampling, delayed training and conditioning\\[2mm]
Koi-4B & Full recipe plus post-training; external reference models & End-to-end usefulness; no matched non-recurrent training control\\
\bottomrule
\end{tabular}\par
\papersource{Paper, Secs. 4--6, PDF pp. 6--14; Appendix A, pp. 18--20.}
\end{frame}

\begin{frame}{One-loop averages stay close; more loops add accuracy}
\par
\centering
\includegraphics[width=0.96\linewidth,height=4.35cm,keepaspectratio]{figures/fig02-matched-results.pdf}\par
\raggedright
4B: $65.65$ baseline, $65.13\to72.38$ ELR.\quad
30B: $65.04$ baseline, $66.08\to70.83$ ELR.
\smallskip

The second loop supplies 79\% / 87\% of the one-to-four-loop gain. Individual tasks can regress: 4B HumanEval is 66.5 at $k=2$, 64.0 at $k=4$.
\papersource{Paper, Fig. 2, PDF p. 7; Tables 8--9, pp. 21--22. Ten-task averages; changes are percentage points.}
\end{frame}

\begin{frame}{Input-dependent step sizes give the best ablation curve}
\begin{columns}[c]
\begin{column}{0.51\textwidth}
\includegraphics[width=\linewidth,height=4.9cm,keepaspectratio]{figures/ablation_state_update_rule.pdf}\par
\end{column}
\begin{column}{0.46\textwidth}
\begin{itemize}
\item Replace: $s_i=r_i$.
\item Residual: $s_i=s_{i-1}+r_i$.
\item Delta: a learned, input-independent channel-wise step.
\item Adaptive delta is best at every tested depth; improves loss by 0.006--0.007 over delta.
\end{itemize}
\end{column}
\end{columns}
\papersource{Paper, Fig. 4a / Sec. 6.1, PDF p. 11. GPT-0.5B, 300B tokens, no input re-injection; lower loss is better.}
\end{frame}

\begin{frame}{Depth marginals alone do not determine the result}
Same marginal distribution: $p_1=0.32$, $p_{2,3,4,5}=0.17$.
\medskip
\par
\centering
\begin{tabular}{lcccc}
\toprule
Sampling & Depths/update & Step time & Loss at 1 & Loss at 5\\
\midrule
Synchronized, weighted & 1 & $1.00\times$ & 1.6871 & 1.6316\\
15\% rank override & $\le2$ & $1.00\times$ & 1.6830 & 1.6273\\
Independent ranks & $\le5$ & $1.36\times$ & 1.6774 & 1.6194\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item Non-recurrent baseline loss: \textbf{1.6852}.
\item Mixing more depths in an update improves the observed loss.
\item Independent ranks wait for the deepest rank; overrides retain the synchronized step time.
\end{itemize}
\papersource{Paper, Table 3 / Sec. 6.3, PDF p. 12. GPT-0.5B; lower loss is better.}
\end{frame}

\begin{frame}{Conditioning strengthens a late-trained loop range}
\begin{columns}[c]
\begin{column}{0.52\textwidth}
\includegraphics[width=\linewidth,height=4.9cm,keepaspectratio]{figures/ablation_conditioning.pdf}\par
\end{column}
\begin{column}{0.45\textwidth}
\begin{itemize}
\item Add conditioning after 75\% of tokens.
\item One-loop loss: $1.6748\to1.6739$.
\item Five-loop loss falls by about 0.022.
\item Five-loop result: 1.6147 vs. 1.6273 for unconditioned training from scratch.
\end{itemize}
\end{column}
\end{columns}
\smallskip
Conditioning adds 31.6M parameters (+6.34\%). The comparison changes both the schedule and the model.
\papersource{Paper, Fig. 6 / Sec. 6.6, PDF pp. 13--14. No conditioned-from-scratch control is reported.}
\end{frame}

\begin{frame}{Route replay mainly accelerates training}
Reuse first-loop expert assignments \emph{and weights}; experts still process each loop's updated representation.
\medskip
\par
\centering
\begin{tabular}{lcc}
\toprule
Relative to dynamic routing & Inference-only replay & Replay training\\
\midrule
Four-loop validation loss & +0.0055 & +0.0038\\
Mixed-depth update time & --- & $-11.7\%$\\
Fixed four-loop update time & --- & $-19.6\%$\\
8K inference forward time & $-1.2\%$ & $-1.2\%$\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
Router calls fall from 53 to 23 at $k=4$. Expert work and all-to-all communication remain. Four-loop average score: 70.83 dynamic vs. 70.90 replay.
\papersource{Paper, Table 1 / Sec. 4.3, PDF p. 8; Appendix E, pp. 25--26. The 11.7\% saving applies to the mixed-depth phase.}
\end{frame}

\begin{frame}{Extra depth has a measurable latency price}
\par
\centering
\includegraphics[width=0.95\linewidth,height=4.2cm,keepaspectratio]{figures/fig07-inference-cost.pdf}\par
\raggedright
At $k=4$: $2.33\times$ TTFT and $2.06\times$ per-token decode latency.\par
At $k=3$: 96\% of the 4B accuracy gain at $1.87\times$ TTFT.
\papersource{Paper, Fig. 7, PDF p. 25. H100 80GB, BF16, batch 1, 1,024 input + 128 fixed output tokens; vLLM 0.20.2, eager prefill, CUDA-graph decode. Bars measure timing uncertainty.}
\end{frame}

\begin{frame}{Koi carries mixed depth through post-training}
\par
\centering
\begin{tabular}{P{2.5cm}P{4.8cm}P{4.9cm}}
\toprule
Stage & Budget & Loop schedule\\
\midrule
Pretraining & 7T tokens at 8K context & First half $k=1$; second half mixed + conditioning\\[1mm]
Context extension & 50B tokens at 256K & Uniform $k=1\ldots4$, 10\% override\\[1mm]
SFT & Text-only data, four epochs & Same mixed-depth schedule\\[1mm]
RLVR & 200 async GRPO steps & $4:1:1:1$ prompt allocation across depths\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
Conditioning adds 155.6M parameters (3.94\% of the backbone). Koi has \alert{no matched non-recurrent training control}.
\papersource{Paper, Sec. 5.1, PDF p. 9; Appendix A.4 / Table 6, pp. 19--20.}
\end{frame}

\begin{frame}{Koi improves scores while producing fewer tokens}
\par
\centering
\includegraphics[width=0.96\linewidth,height=4.25cm,keepaspectratio]{figures/fig03-koi-scaling.pdf}\par
\raggedright
Unrounded index: \textbf{17.52, 25.57, 27.93, 29.23}.\par
Authors' harness estimates, not official AA results. CritPt is unrun and contributes zero; reference models are evaluated in the same harness.
\papersource{Paper, Fig. 3, PDF p. 9; Tables 10--11, pp. 22--23. Token ratios are geometric means across eight benchmarks.}
\end{frame}

\begin{frame}{Shorter outputs partly offset per-token compute}
\[
\bar r_k=\left(\prod_{b=1}^{8}\frac{T_b(k)}{T_b(1)}\right)^{1/8}
\]
$T_b(k)$ is mean completion tokens per rollout, summed over all requests and turns, including reasoning and capped requests.
\begin{block}{Approximate decode-cost arithmetic}
\[
k=2:\quad1.36\times0.7821\approx1.06
\qquad
k=4:\quad2.06\times0.6732\approx1.39
\]
\end{block}
This combines fixed-workload GDP-Hybrid-4B token latency with Koi's benchmark length ratios. It is an estimate of decode work, not measured end-to-end task latency.
\papersource{Paper, Sec. 5.2, PDF p. 10; Eq. (6) / Appendix C.1, p. 23; Appendix D, pp. 24--25.}
\end{frame}

\begin{frame}{One KV cache changes which history each loop reads}
\begin{block}{During decoding, token $t$ at loop $i$ attends to}
Earlier tokens' \textbf{final-loop} keys and values, plus token $t$'s own loop-$i$ keys and values. The current slot is overwritten each loop.
\end{block}
Three of six attention layers are recurrent:
\[
\frac{M_{\mathrm{per-loop}}(k)}{M_1}=\frac{3+3k}{6},
\qquad \frac{M_{\mathrm{last-loop}}(k)}{M_1}=1.
\]
\begin{itemize}
\item At $k=4$, 128K context, BF16: \textbf{7.5 GiB $\to$ 3.0 GiB} attention KV per request.
\item GDP keeps per-loop state, below 25 MiB per request at $k=4$.
\item No retraining; the cache policy changes the attention computation.
\end{itemize}
\papersource{Paper, Secs. 3.4 / 5.3, PDF pp. 6, 10--11. Memory ratio derived from the reported layer layout.}
\end{frame}

\begin{frame}{Most depth gains survive KV sharing, with losses}
\par
\centering
\begin{tabular}{lrrrr}
\toprule
Benchmark & One loop & Four: per & Four: last & $\Delta$ pp\\
\midrule
Terminal-Bench 2.1 & 12.1 & 28.0 & 27.4 & $-0.6$\\
Humanity's Last Exam & 5.9 & 11.9 & 10.5 & $-1.4$\\
AA-LCR & 51.3 & 68.3 & 65.3 & $-3.0$\\
LiveCodeBench v6 & 50.5 & 71.9 & 70.2 & $-1.6$\\
MATH-500 & 88.8 & 97.2 & 97.0 & $-0.2$\\
AA-Omniscience non-halluc. & 86.1 & 85.9 & 88.2 & $+2.3$\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
Differences use unrounded scores. Across depths, seven of 27 paired comparisons reach uncorrected $|z|\ge2$; uncertainty is within a single run.
\papersource{Paper, Table 2, PDF p. 10; Appendix C.2, pp. 23--24. Scores are percentages; deltas are percentage points.}
\end{frame}

\begin{frame}{GDPval exposes a consequential cache-policy failure}
\par
\centering
\begin{tabular}{lrrr}
\toprule
Failures among 220 tasks & $k=2$ & $k=3$ & $k=4$\\
\midrule
Per-loop KV & 1 & 6 & 12\\
Last-loop KV & 13 & 29 & 52\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item Most additional failures are timeouts; agent turns rise 17--19\%.
\item Timed-out tasks receive no Elo judgment, so the scored task sets differ.
\item The paper reports \alert{no AA-Index composite for last-loop KV}.
\end{itemize}
\begin{block}{Practical implication}
Memory savings need task-completion and wall-time evaluation alongside benchmark scores.
\end{block}
\papersource{Paper, Appendix C.2, PDF p. 24.}
\end{frame}

\begin{frame}{ELR makes a useful compute knob within one model}
\begin{enumerate}
\item \textbf{Protect the endpoint.} Adaptive updates and within-update depth mixing preserve near-baseline one-loop averages.
\item \textbf{Learn extra depth incrementally.} Zero-initialized conditioning enables a function-preserving transition; the late recipe improves deeper loss.
\item \textbf{Account for each cost separately.} More loops increase compute; route replay saves training work; KV sharing saves memory with task-dependent costs.
\end{enumerate}
\medskip
Open directions: later-loop training exposure, conditioned-from-scratch controls, adaptive stopping, and post-trained recurrent MoE.
\papersource{Paper, Sec. 8, PDF p. 15, and limitations from Secs. 4--6 / Appendix C.2. Main discussion ends here; backup slides follow.}
\end{frame}

\appendix

\begin{frame}{Backup: fixed depth does not yield elastic inference}
\begin{columns}[c]
\begin{column}{0.54\textwidth}
\includegraphics[width=\linewidth,height=5.15cm,keepaspectratio]{figures/ablation_fixed_mixed.pdf}\par
\end{column}
\begin{column}{0.42\textwidth}
Rows: separate training runs.\par
Columns: evaluation depth.\medskip

Each fixed-depth model works best at its own training depth.\medskip

Mixed depth gives a usable curve in one checkpoint, with some loss relative to each specialist.
\end{column}
\end{columns}
\papersource{Paper, Fig. 4b / Sec. 6.2, PDF pp. 11--12. GPT-0.5B validation loss; logarithmic color scale.}
\end{frame}

\begin{frame}{Backup: training schedules trade quality for cost}
\par
\centering
\includegraphics[width=0.95\linewidth,height=4.4cm,keepaspectratio]{figures/fig05-training-schedule.pdf}\par
\raggedright
Each x-axis point is a separately trained model. Late training reduces overhead; larger override fractions favor the shallow endpoint over deeper loops.
\papersource{Paper, Fig. 5 / Secs. 6.4--6.5, PDF p. 13. GPT-0.5B; estimated compute is specific to $K=5$, partition $8/14/8$.}
\end{frame}

\begin{frame}{Backup: aggregate route stability hides token drift}
\par
\centering
\begin{tabular}{lr}
\toprule
First versus fourth loop, dynamic model & Value\\
\midrule
Aggregate expert-frequency cosine & 0.978\\
First-loop experts retained per token & 53.17\%\\
Routed-output cosine: replay / shifted control & 0.767 / 0.227\\
Full-MoE cosine: replay / shifted control & 0.918 / 0.718\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
256 held-out 2K-token windows, averaged over ten recurrent MoE layers. Output comparisons hold the fourth-loop input fixed; shifted routes preserve usage counts but break token correspondence.
\papersource{Paper, Appendix E / Table 13, PDF pp. 25--26. Retention 95\% document-bootstrap interval: [52.37, 53.89]\%.}
\end{frame}

\begin{frame}{Backup: entry and exit conditioning complete the rule}
\[
z_i=\begin{cases}
\tilde s_i,&i=1,\\
(1+\lambda_e(c_i))\odot\norm(\tilde s_i)+\mu_e(c_i)
+\eta(c_i)\odot\hin,&i\ge2.
\end{cases}
\]
\[
\begin{aligned}
r_i&=\R(z_i;c_i),\\
\beta_i&=\sigma(W_\beta\tilde s_i+b_\beta)\odot\operatorname{softplus}(\gamma),\\
s_i&=s_{i-1}+(\pi_i+g_{\mathrm{exit}}(c_i))\odot\beta_i\odot(r_i-s_{i-1}).
\end{aligned}
\]
\begin{itemize}
\item First loop receives $z_1=2\hin$ in the hybrid models.
\item The adaptive gate uses the unmodulated $\tilde s_i$.
\item Zero heads make later exit factors vanish; $\pi_1=1$, $\pi_{i>1}=0$.
\end{itemize}
\papersource{Paper, Sec. 3.3, PDF p. 5. GPT-0.5B default ablations omit input re-injection.}
\end{frame}

\begin{frame}{Backup: input re-injection is not essential at 0.5B}
\par
\centering
\begin{tabular}{lrr}
\toprule
Recurrent input & Loss at $k=1$ & Loss at $k=5$\\
\midrule
Non-recurrent baseline & 1.6852 & ---\\
$s_{i-1}$ & 1.6830 & 1.6273\\
$s_{i-1}+\hin$ & 1.6858 & 1.6325\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item Re-injection slightly worsens this configuration's loss.
\item The paper reports unchanged ordering after repeating sampling, fixed/mixed, override and delayed-training ablations.
\item Transfer of the state-update ranking is expected by the authors, not demonstrated in this appendix.
\end{itemize}
\papersource{Paper, Appendix F.1 / Table 14, PDF p. 26.}
\end{frame}

\begin{frame}{Reference and reading materials}
\textbf{Faisal Ladhak, Shantanu Acharya, Santiago Akle Serano, Simeng Sun, Boris Ginsburg, and Krishna C Puvvada.}
\medskip

\emph{One Model to Loop Them All: Elastic Latent Recurrence for Language Models.}
\medskip

arXiv:2610.12582v1, 8 October 2026.\par
\href{https://arxiv.org/abs/2610.12582v1}{https://arxiv.org/abs/2610.12582v1}
\medskip

All empirical figures are from this version of the paper. Selected tables are transcribed from the cited source tables. Explanatory examples and memory/cost arithmetic are identified where used.
\medskip

Companion: Chinese reading notes, editable Beamer source, and local paper PDF in the reading archive.
\end{frame}

\end{document}