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\title[Context Language Models]{Context Language Models}
\subtitle{Learning to manage the live context}
\author{Paper by Rulin Shao et al.\\[4pt]\small Research group reading presentation}
\date{arXiv:2609.37725v1 \enspace | \enspace 10 October 2026}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-5mm}
  \begin{center}
    \alert{Make context an editable file, then learn how to maintain it.}\\[4pt]
    \small\href{https://arxiv.org/abs/2609.37725v1}{arxiv.org/abs/2609.37725v1}\quad\textbar\quad Main discussion: 20--30 minutes
  \end{center}
\end{frame}

\begin{frame}{Different information needs different context operations}
  \begin{tabular}{p{0.29\linewidth}p{0.31\linewidth}p{0.31\linewidth}}
    \toprule
    Information & Required operation & Fixed-summary difficulty \\
    \midrule
    Exact identifiers & Preserve verbatim & Paraphrase loses identity \\[5pt]
    Current board or plan & Update one field & Rewriting adds work/errors \\[5pt]
    Large search results & Offload, evict, retrieve & Storage alone leaves history \\[5pt]
    Multi-agent progress & Maintain a live ledger & Old status accumulates \\
    \bottomrule
  \end{tabular}
  \begin{block}{Research question}
    Can a model choose and implement its own context transformations?
  \end{block}
  \papersource{Paper, Secs. 1--3, PDF pp. 1--4. Examples summarized for this presentation.}
\end{frame}

\begin{frame}{CLM controls the next context, including edits to history}
  \begin{columns}[T]
    \begin{column}{0.47\textwidth}
      \begin{block}{Append-only LM}
        \[c_{t+1}=c_t\oplus f^{\mathrm{LM}}_\theta(c_t)\]
        Output is appended to existing history.
      \end{block}
    \end{column}
    \begin{column}{0.49\textwidth}
      \begin{block}{Context Language Model}
        \[c_{t+1}=f^{\mathrm{CLM}}_\theta(c_t)\]
        The model controls the transformation.
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  \begin{enumerate}
    \item Mirror live context to a file; give its path to the model.
    \item Allow general Bash/programmatic edits.
    \item Synchronize the file with the next model invocation.
  \end{enumerate}
  \smallskip
  $c_t$: context at turn $t$; $\theta$: model parameters; $\oplus$: concatenation.\\
  Existing LMs can implement this interface without additional training.
  \papersource{Paper, Sec. 4.1, Eqs. (1)--(2), PDF p. 4.}
\end{frame}

\begin{frame}{Context editing can express reusable procedures}
  \begin{columns}[T]
    \begin{column}{0.52\textwidth}
      \begin{itemize}
        \item Update task status in place.
        \item Keep conclusions and evidence pointers.
        \item Remove unhelpful search observations.
        \item Define a reusable compaction function.
      \end{itemize}
    \end{column}
    \begin{column}{0.44\textwidth}
      \begin{block}{Observed examples}
        163 ledger edits, with context held at 6--8K tokens.\\[8pt]
        A helper function reused 37 times in a research trajectory.
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  Multiple context files naturally represent multiple agents.\\[4pt]
  An external memory file stores material; a \alert{context file determines the next input}.
  \papersource{Paper, Sec. 4.1 and Fig. 3, PDF pp. 4--5; App. A, PDF p. 14.}
\end{frame}

\begin{frame}{ContextBench isolates retention, updates, and retrieval}
  \begin{tabular}{p{0.29\linewidth}p{0.36\linewidth}p{0.25\linewidth}}
    \toprule
    Task & Required behavior & Metric $\uparrow$ \\
    \midrule
    Needle Retention & Keep specified lines & Verbatim retention \\[4pt]
    Sudoku Sketchpad & Edit a $16\times16$ board & Exact board versions \\[4pt]
    KV Store & Store batches; answer 24 queries & Exact values \\[4pt]
    Log Triage & Offload logs; answer 24 queries & Exact answers \\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item GPT-5.4; 32,768 tokens, including a 2,048-token response reserve.
    \item Input arrives in live context before the agent can act.
    \item Grade from context; files alone receive no credit.
    \item Necessary retained information fits within the budget.
  \end{itemize}
  \papersource{Paper, Sec. 3 and App. D--E, PDF pp. 3--4, 18--22; Table 3.}
\end{frame}

\begin{frame}{Precise edits remain effective as input pressure rises}
  \centering
  \fig{fig02-contextbench.pdf}{0.92}{0.65}
  \par\smallskip
  \normalsize Pressure = cumulative input / context limit. Higher accuracy is better.\\
  All methods receive instructions adapted to their own mechanisms.
  \papersource{Paper, Fig. 2, PDF p. 4; GPT-5.4, 4 seeds per level, 8 at selected levels.}
\end{frame}

\begin{frame}{An edit can shorten context and still incur recomputation}
  \[
  F_t=C_{\mathrm{token}}(U_t+G_t)
  +C_{\mathrm{attn}}\left[\frac{P_t^2-R_t^2}{2}+G_tP_t+\frac{G_t^2}{2}\right]
  \]
  \begin{itemize}
    \item $P_t$: prompt length; $G_t$: generated length; $R_t$: reusable prefix.
    \item $U_t=P_t-R_t$; trajectory cost is $\sum_t F_t$.
    \item $C_{\mathrm{token}}$: MLP/projection cost per token; $C_{\mathrm{attn}}$: cost per attention pair.
  \end{itemize}
  \medskip
  \begin{tabular}{lccc}
    \toprule
    Reusable prefix $R_t$ & 18K & 10K & 0 \\
    Cost ($10^{14}$ FLOPs) & 1.41 & 5.74 & 10.81 \\
    \bottomrule
  \end{tabular}
  \par\smallskip
  Illustrative Qwen3.6-27B turn: $P_t=20$K, $G_t=500$.\\
  \alert{Optimize the whole trajectory, including cache invalidation.}
  \papersource{Paper, Eq. (3), PDF p. 5; App. C, Eq. (9) and Fig. 14, PDF pp. 17--18.}
\end{frame}

\begin{frame}{A skill document becomes an optimizable context policy}
  \[
  c_{t+1}=f^{\mathrm{CLM}}_\theta(c_t;s),\qquad
  s^*=\arg\max_s\mathbb E_{x\sim\mathcal D}[R(\tau(x;s))]
  \]
  \begin{block}{Textual evolution loop}
    Training rollouts $\longrightarrow$ propose skill rewrites
    $\longrightarrow$ development evaluation $\longrightarrow$ select a skill.
  \end{block}
  \begin{itemize}
    \item $s$: natural-language guidance; $\tau$: resulting task trajectory.
    \item Parameters and execution interface stay fixed.
    \item Assisted evolution: stronger external proposer.
    \item Self-evolution: the agent also proposes the next skill.
  \end{itemize}
  \smallskip
  Freeze selection before the final held-out test evaluation.
  \papersource{Paper, Sec. 4.2, Eqs. (4)--(5), PDF pp. 5--6; App. E, PDF p. 23.}
\end{frame}

\begin{frame}{The efficiency advantage rewards cheaper successes}
  \[
  A_i^{\mathrm{eff}}=
  \begin{cases}
    \operatorname{clip}\!\left(\dfrac{\bar c_g-c_i}{\bar c_g},-1,1\right),
       &i\in\mathcal G_g^+,\\[5pt]
    0,&i\notin\mathcal G_g^+,
  \end{cases}
  \qquad A_i=A_i^{\mathrm{out}}+w_{\mathrm{eff}}A_i^{\mathrm{eff}}.
  \]
  \begin{itemize}
    \item $c_i$: trajectory FLOPs; $\bar c_g$: mean among successes in group $g$.
    \item Fewer than two successes: efficiency advantage is zero.
    \item Stepwise GRPO trains each segment under its original input.
    \item Main CLM recipe: $w_{\mathrm{eff}}=0.25$ on context-management tokens.
  \end{itemize}
  \begin{block}{Teaching example}
    Successful costs 1 and 3 PFLOPs give advantages $+0.5$ and $-0.5$.\\
    A cheap failed trajectory receives no efficiency bonus.
  \end{block}
  \papersource{Paper, Sec. 4.2, Eq. (6), PDF p. 6; App. A/E, PDF pp. 14, 24.}
\end{frame}

\begin{frame}{SCR reuses surviving suffix states after an edit}
  \centering
  \fig{fig04-scr.png}{0.98}{0.38}
  \par\medskip
  \begin{itemize}
    \item Diff old and new prompts; relocate up to $K=6$ surviving spans.
    \item Re-rotate cached keys to their new positions; prefill new text.
    \item Old states still encode the old prefix: \alert{reuse is approximate}.
    \item Session-private cache slots; fall back if allocation fails.
  \end{itemize}
  \papersource{Paper, Sec. 4.3, Fig. 4, PDF p. 6; App. B, PDF pp. 14--16.}
\end{frame}

\begin{frame}{Evaluation spans minutes, hours, and multi-agent work}
  \begin{tabular}{p{0.26\linewidth}p{0.32\linewidth}p{0.33\linewidth}}
    \toprule
    Setting & Main model & Outcome / resource measure \\
    \midrule
    Coding + research & Qwen3.6-27B & Accuracy / PFLOPs per question \\[5pt]
    Mathematical search & Claude 4.6 Sonnet & Best score / 100 attempts or 5 h \\[5pt]
    EdgeBench-10 & Qwen3.6-27B; Sonnet & Best-of-three score / 12 h \\[5pt]
    Software World & GPT-5.6-Sol, 6 agents & Downstream speedup / API spend \\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Zero-shot baselines: Summary, MEM1, Self-Compact, ACM, RLM, Base.
    \item Fig. 5's 32K/100-turn caption differs from detailed budgets in App. E; backup slide lists both.
  \end{itemize}
  \papersource{Paper, Sec. 5.1, PDF pp. 7--8; App. E, PDF pp. 19, 22--24.}
\end{frame}

\begin{frame}{CLM improves the accuracy--compute frontier}
  \centering
  \fig{fig05-pareto.pdf}{0.82}{0.53}
  \par\smallskip
  \begin{tabular}{lcc}
    \toprule
    Task & Quality vs. Summary & Compute vs. Summary \\
    \midrule
    BrowseComp-Plus & 59.4\%; +11.4\% relative & $-21.5\%$ \\
    TerminalBench 2.1 & Matched accuracy & $-29.5\%$ \\
    TBLite & 73.7\% vs. 67.0\% & About $-9\%$ \\
    \bottomrule
  \end{tabular}
  \papersource{Paper, Fig. 5 and Sec. 5.1.1, PDF p. 7; TB compute reduction also in introduction, p. 3.}
\end{frame}

\begin{frame}{General context control competes with specialized search}
  \begin{center}
  \setlength{\tabcolsep}{7pt}
  \begin{tabular}{lrrrr}
    \toprule
    Method & Circle $\uparrow$ & Heilbronn $\uparrow$ & Ratio $\uparrow$ & Overlap $\downarrow$ \\
    \midrule
    OpenEvolve & 2.541 & 0.03127 & 0.07690 & 0.38123 \\
    OE-Agent & 2.525 & 0.03053 & 0.07724 & 0.38167 \\
    CLM & 2.618 & \textbf{0.03653} & \textbf{0.07758} & \textbf{0.38094} \\
    CLM + subagents & \textbf{2.636} & 0.03617 & \textbf{0.07758} & 0.38109 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \medskip
  \begin{itemize}
    \item Same Claude 4.6 Sonnet model and evaluators; 32K context.
    \item One run per method; stop after 100 scored attempts or 5 hours.
    \item CLM vs. OpenEvolve: +16.8\% Heilbronn; +3.0\% circle packing.
    \item Best-of-run solutions; statistical repeatability is not measured here.
  \end{itemize}
  \papersource{Paper, Table 1, PDF p. 7; Fig. 18, PDF p. 25; App. E, PDF p. 22.}
\end{frame}

\begin{frame}{Long repository work benefits from compact working state}
  \begin{center}
  \begin{tabular}{llrr}
    \toprule
    Model & Method (32K context) & Score $\uparrow$ & PFLOPs/run $\downarrow$ \\
    \midrule
    Qwen3.6-27B & Summary & 42.3 & 437 \\
     & CLM & \textbf{44.6} & \textbf{179} \\
     & CLM + subagents & 44.2 & 181 \\
    \midrule
    Claude 4.6 Sonnet & Summary & 42.3 & --- \\
     & CLM & \textbf{51.0} & --- \\
     & CLM + subagents & 50.4 & --- \\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item EdgeBench-10: 12 hours; best score over 3 seeds per task.
    \item Qwen CLM: +2.3 score points with 59\% less compute.
    \item Subagents add little at 32K; the 128K result differs (backup).
  \end{itemize}
  \papersource{Paper, Sec. 5.1.2 and Fig. 6a, PDF p. 8; compute is mean per run, including subagents.}
\end{frame}

\begin{frame}{Software World tests transfer to unseen downstream work}
  \begin{columns}[T]
    \begin{column}{0.43\textwidth}
      \centering
      \fig{software-world.pdf}{0.78}{0.62}
      \par\smallskip
      \small\textcolor{CLMBlue}{CLM swarm}\quad\textcolor{SummaryPink}{Summary swarm}
    \end{column}
    \begin{column}{0.53\textwidth}
      \begin{itemize}
        \item Six agents optimize interdependent repositories.
        \item GPT-5.6-Sol; 272K context; over 24 hours.
        \item Four unseen packages; 17 evaluation tasks.
        \item Geometric mean of speedup in executed instructions.
      \end{itemize}
      \begin{block}{Interpret the 65\% claim carefully}
        The paper reports 65\% more \emph{speedup gain} above 1.0 at matched spend.\\[3pt]
        Rounded final ratios: 1.044 vs. 1.026.
      \end{block}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 6b and Sec. 5.1.2, PDF p. 8; App. E, PDF p. 23.}
\end{frame}

\begin{frame}{A sentence can change when and how context is edited}
  \centering
  \fig{fig07-steering.pdf}{0.91}{0.58}
  \par\medskip
  \begin{itemize}
    \item Threshold requests 16K/24K/32K $\rightarrow$ first edits at 16.0K/23.6K/30.9K.
    \item Compaction within two turns of a boundary: 0.40 $\rightarrow$ 0.88.
    \item Full backup before an edit: 0.00 $\rightarrow$ 0.68; partial backup adds 0.09.
  \end{itemize}
  \papersource{Paper, Fig. 7, PDF p. 9; Claude 4.6 Sonnet. These are behavior measures, not answer accuracy.}
\end{frame}

\begin{frame}{Evolved skills improve held-out context management}
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \centering
      \fig{fig08-evolution-kv.pdf}{0.98}{0.60}
      \papersource{Fig. 8: KV Store evolution trajectories.}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Assisted evolution: held-out test}
        KV Store: \alert{38.3\% $\rightarrow$ 74.2\%}\\[3pt]
        $+35.9$ percentage points.
      \end{block}
      \begin{itemize}
        \item Qwen3.6-27B agent; Claude Fable 5.1 proposer.
        \item Development result is different: 22.3\% $\rightarrow$ 83.8\%.
        \item Opus 5 self-evolves from a much stronger initial policy.
      \end{itemize}
    \end{column}
  \end{columns}
  \smallskip
  Qwen cost excludes some unsuccessful/over-budget runs; it is not total search cost.
  \papersource{Paper, Figs. 8/20, PDF pp. 9/26; held-out result, App. F, p. 25; cost protocol, App. E, p. 23.}
\end{frame}

\begin{frame}{RL closes the small model's accuracy gap at lower cost}
  \begin{center}
  \begin{tabular}{lcc}
    \toprule
    Qwen3.5-9B & Accuracy before $\rightarrow$ after & PFLOPs/Q before $\rightarrow$ after \\
    \midrule
    Summary & 34.7\% $\rightarrow$ 42.1\% & 4.01 $\rightarrow$ 2.19 \\
    CLM & 28.8\% $\rightarrow$ \textbf{42.5\%} & 1.52 $\rightarrow$ \textbf{1.34} \\
    \bottomrule
  \end{tabular}
  \end{center}
  \medskip
  \begin{itemize}
    \item CLM gains 13.7 points (47.6\% relative); own compute falls about 12\%.
    \item After training: +0.4 points vs. Summary, with 38.8\% less compute.
    \item Train on OpenResearcher; select on its validation set; test on BCP.
  \end{itemize}
  \smallskip
  Appendix E describes the main Summary baseline as task-reward-only; CLM adds efficiency and tool/format terms.
  \papersource{Paper, Table 2, PDF p. 9; App. E, PDF pp. 23--24. Checkpoints selected by held-out validation.}
\end{frame}

\begin{frame}{The reward ablation exposes instability in Summary}
  \centering
  \fig{fig21-rl-curves.pdf}{0.99}{0.54}
  \par\medskip
  \begin{itemize}
    \item Pink solid curve: Summary + FLOPs reward drops to about 1\% at step 40.
    \item CLM's reward curves retain much higher accuracy at lower compute.
    \item This qualifies the p. 9 statement of no clear accuracy loss for either harness.
  \end{itemize}
  \papersource{Paper, Fig. 21, PDF p. 26; discrepancy with Sec. 5.2, PDF p. 9. Interpretation from the plotted curves.}
\end{frame}

\begin{frame}{SCR cuts measured compute by 35\% at matched accuracy}
  \centering
  \fig{fig09-scr-results.pdf}{0.99}{0.48}
  \par\medskip
  \begin{itemize}
    \item Qwen3.6-27B, 830 BCP questions: 60.2\% accuracy for both servers.
    \item 10.98 $\rightarrow$ 7.14 PFLOPs/question; these are separate SGLang runs.
    \item Of 7.8\% extra token reuse, 5.3 points follow reasoning stripping; 2.5 follow other edits.
    \item Token fractions in the right panel are not FLOPs fractions or latency.
  \end{itemize}
  \papersource{Paper, Fig. 9, PDF p. 9; stripping decomposition, Fig. 12, PDF p. 16.}
\end{frame}

\begin{frame}{Three conditions shape how these results should be used}
  \begin{block}{Model capability and task organization}
    Smaller models edit less effectively. Larger context can change whether subagents help.
  \end{block}
  \begin{block}{State semantics and context integrity}
    SCR retains old-prefix information. Writable context can also preserve injected or self-generated instructions.
  \end{block}
  \begin{block}{Evaluation protocol}
    Best-of-run math, best-of-three EdgeBench, and matched-spend software gains measure different outcomes. Some budget descriptions differ across sections.
  \end{block}
  \papersource{Paper, Sec. 6, PDF p. 10; App. B/E/F, PDF pp. 15, 22--25.}
\end{frame}

\begin{frame}{The main contribution is a learnable context interface}
  \begin{enumerate}
    \item \textbf{Expressiveness:} live context becomes a model-editable artifact.
    \item \textbf{Learning:} skills and RL optimize how that artifact evolves.
    \item \textbf{Serving:} cache reuse can make the resulting edits cheaper.
  \end{enumerate}
  \medskip
  \begin{block}{Useful next questions}
    When should an agent pay a one-time cache penalty to save future work?\\[5pt]
    How quickly does reused state forget a corrected or deleted fact?\\[5pt]
    Can length awareness and reliable editing be learned together?
  \end{block}
  \papersource{Synthesis of the paper. Future questions are presentation analysis. Paper: arXiv:2609.37725v1.}
\end{frame}

\appendix

\begin{frame}{Backup: hybrid SCR carries a pre-edit recurrent state}
  \centering
  \fig{fig10-hybrid-scr.png}{0.99}{0.48}
  \par\medskip
  \begin{itemize}
    \item Qwen3.6-27B: 16 full-attention and 48 linear-attention layers.
    \item Full attention relocates KV; linear attention reuses a state snapshot.
    \item Edited history is not replayed through the linear recurrence.
    \item New text can still influence later layers through full-attention outputs.
  \end{itemize}
  \papersource{Paper, App. B and Fig. 10, PDF p. 15.}
\end{frame}

\begin{frame}{Backup: prefix-reuse FLOPs are an explicit approximation}
  \centering
  \fig{fig14-flops.pdf}{0.91}{0.55}
  \par\medskip
  \begin{itemize}
    \item Includes MLP/projections and full-attention pairwise operations.
    \item Omits embedding/output layers, Gated DeltaNet state updates, normalization, activations, and softmax.
    \item FLOPs, wall-clock time, and API spend answer different questions.
  \end{itemize}
  \papersource{Paper, App. C, Eqs. (7)--(9), Fig. 14, PDF pp. 17--18. Lengths are illustrative.}
\end{frame}

\begin{frame}{Backup: 128K context changes the subagent tradeoff}
  \begin{center}
  \begin{tabular}{lrr}
    \toprule
    Qwen3.6-27B, EdgeBench-10, 128K & Score $\uparrow$ & PFLOPs/run $\downarrow$ \\
    \midrule
    Summary & 47.8 & 222 \\
    CLM & 47.3 & \textbf{142} \\
    CLM + subagents & \textbf{50.2} & 219 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \medskip
  \begin{itemize}
    \item Single-agent CLM is cheaper but slightly lower scoring at 128K.
    \item Subagents improve score here, unlike the 32K comparison.
    \item Model scale also matters: on TB2.1, 9B edits 1.4 times/task vs. 2.6 for 27B.
    \item Median peak contexts: 30.2K vs. 17.6K under a nominal 32K limit.
  \end{itemize}
  \papersource{Paper, App. F, Figs. 17/19, PDF pp. 24--25. EdgeBench scores use best-of-three.}
\end{frame}

\begin{frame}{Backup: detailed budgets qualify the shared-caption setup}
  \begin{center}
  \begin{tabular}{lcc}
    \toprule
    Source / task & Context budget & Turn limit \\
    \midrule
    Fig. 5 caption, all three tasks & 32K & 100 \\
    App. E, TB2.1 / TBLite & 32,000 & 64 \\
    App. E, BrowseComp-Plus & 23,560 & 100 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \medskip
  \begin{itemize}
    \item BCP excludes CLM editing turns; overflow can roll back up to 6 times.
    \item EdgeBench: text says up to 5 subagents; appendix says up to 6.
    \item Table 2 and zero-shot 9B BCP use different evaluation configurations.
    \item A reproduction should resolve these details before matching results.
  \end{itemize}
  \papersource{Paper, Fig. 5, PDF p. 7; Sec. 5.1.2, p. 8; App. E, pp. 22--24.}
\end{frame}

\begin{frame}{Reference and discussion materials}
  \textbf{Context Language Models}\\[7pt]
  Rulin Shao, Shannon Zejiang Shen, Junjie Oscar Yin, Yuetai Li,\\
  Minheng Wang, Hamish Ivison, Radha Poovendran, Nathan Lambert,\\
  Teng Xiao, Mike Lewis, Wen-tau Yih, Luke Zettlemoyer, Pang Wei Koh.\\[9pt]
  arXiv:2609.37725v1, submitted 29 September 2026.\\[8pt]
  \href{https://arxiv.org/abs/2609.37725v1}{Paper: arxiv.org/abs/2609.37725v1}\\[5pt]
  \href{https://github.com/facebookresearch/context-language-models}{Author code link: facebookresearch/context-language-models}\\[10pt]
  Figures are reproduced from the versioned paper and its author source.\\
  Equations and selected tables are typeset for this reading presentation.\\
  Companion file: \texttt{reading-notes.zh.md}.
\end{frame}

\end{document}