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\title[Language Models Need Sleep]{Language Models Need Sleep}
\subtitle{Learning to Self-Modify and Consolidate Memories}
\author{Paper: Ali Behrouz, Farnoosh Hashemi,\\Adel Javanmard, Vahab Mirrokni}
\date{arXiv:2606.03979v2 \; (10 July 2026)\\Paper discussion prepared 29 September 2026}

\begin{document}

\begin{frame}
\titlepage
\vspace{-4mm}
\begin{center}
\small Periodically transfer fast memories into slower parameters,\\then learn to generate useful adaptation data.
\end{center}
\papersource{Independent reading presentation. 22 main slides + 5 backups; intended for 20--30 minutes.}
\end{frame}

\begin{frame}{The problem is retaining knowledge after context ends}
\begin{columns}[T]
\begin{column}{0.47\textwidth}
\begin{block}{Fast adaptation}
New context changes model behavior.\\[2mm]
Removing that context does not automatically preserve its contents in weights.
\end{block}
\end{column}
\begin{column}{0.47\textwidth}
\begin{block}{Persistent adaptation}
Weight updates can store new knowledge.\\[2mm]
Repeated updates can interfere with earlier capabilities.
\end{block}
\end{column}
\end{columns}
\medskip
\textbf{Research question:} How can a model preserve a fast memory before rewriting it, while keeping capacity for future learning?
\papersource{Paper, Secs. 1--2, PDF pp. 1--6. The sleep analogy motivates the design; it is not a biological equivalence claim.}
\end{frame}

\begin{frame}{Sleep separates consolidation from self-improvement}
\centering
\includegraphics[width=\linewidth,height=0.41\textheight,keepaspectratio]{figures/fig02-consolidation.png}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{block}{1. Consolidate}
Expand a slower memory and distill into its new low-rank experts.
\end{block}
\end{column}
\begin{column}{0.48\textwidth}
\begin{block}{2. Dream}
Generate, select and learn from synthetic adaptation data.
\end{block}
\end{column}
\end{columns}
\papersource{Original Fig. 2, PDF p. 7; Sec. 3.4, pp. 9--10. Figure depicts consolidation.}
\end{frame}

\begin{frame}{Different update intervals create a memory hierarchy}
\[
y_t=\mathrm{MLP}^{(f_k)}\!\left(\cdots\mathrm{MLP}^{(f_1)}(x_t)\right),
\qquad C^{(\ell)}=\frac{\max_{\ell'}C^{(\ell')}}{f_\ell}.
\]
\begin{itemize}
\item $f_\ell$: updates per reference time unit; $C^{(\ell)}$: interval between updates.
\item Fast blocks adapt often; slow blocks retain information for longer.
\item Before a scheduled fast-block update, consolidate its memory into the next slower block.
\end{itemize}
\begin{alertblock}{Keep frequency and interval distinct}
The paper sometimes calls 1K/10K update intervals ``frequencies.'' This ambiguity also affects the interpretation of Fig. 4.
\end{alertblock}
\papersource{Sec. 2.2, Definition 1 and Eqs. (1)--(2), PDF p. 5; Sec. 3.2, p. 7; Fig. 7, p. 22.}
\end{frame}

\begin{frame}{New experts provide room for incoming knowledge}
\begin{columns}[T]
\begin{column}{0.51\textwidth}
\includegraphics[width=\linewidth,height=0.52\textheight,keepaspectratio]{figures/fig08-experts.png}
\papersource{Original Fig. 8, PDF p. 22.}
\end{column}
\begin{column}{0.46\textwidth}
\[
A\in\mathbb R^{d\times d_{\rm low}},\qquad
B\in\mathbb R^{d_{\rm low}\times d}
\]
\begin{itemize}
\item Add a low-rank expert in the receiving block.
\item Freeze existing student weights during consolidation.
\item Train only the new parameters.
\item Preallocated, masked experts can implement growth.
\end{itemize}
\end{column}
\end{columns}
\medskip
\textbf{Reading inference:} frozen old weights do not guarantee unchanged old-task outputs; the forward computation still changes.
\papersource{Secs. 3.2--3.3, PDF pp. 7--9. Matrix-pair cost is $2dd_{\rm low}$, excluding routing and optimizer state.}
\end{frame}

\begin{frame}{Preserve the teacher until consolidation is complete}
\begin{enumerate}
\item \textbf{Compute:} obtain prospective fast-block weights from accumulated gradients. Do not overwrite the running model.
\item \textbf{Consolidate:} teacher = pre-update model. Student = prospective fast weights, reset fast experts, and a new slow expert. Optimize the new expert to retain teacher behavior.
\item \textbf{Update:} apply the fast base-weight update, reset its old experts, and activate the trained slow expert.
\end{enumerate}
\medskip
\begin{block}{Why this order matters}
The teacher must still contain the memory that the student is about to lose.
\end{block}
\papersource{Sec. 3.3, compute--consolidate--update protocol, PDF pp. 7--8. The teacher definition has a local layer-index typo.}
\end{frame}

\begin{frame}{Distill on both teacher and student trajectories}
\smallskip
\begin{align*}
\mathcal L_{\rm GKD}={}&(1-\lambda)\E_{(x,y)\sim\mathcal D}
  [\mathcal F(\pT\Vert\pS)(y\mid x)]\\
&+\lambda\E_{x\sim\mathcal D,\,y\sim\pS(\cdot\mid x)}
  [\mathcal F(\pT\Vert\pS)(y\mid x)].
\end{align*}
\begin{itemize}
\item $\mathcal D$: teacher-generated data; $\pT,\pS$: teacher and student.
\item $\lambda$ controls the share of student-generated trajectories.
\item $\mathcal F$ compares output distributions along those trajectories.
\item Stop gradients through sampling in this distillation step; update only expanded parameters.
\end{itemize}
\medskip
\textbf{Intuition:} provide teacher feedback where the student actually goes.
\papersource{Sec. 3.3, unnumbered GKD objective, PDF p. 8. Notation shortened; the paper leaves the divergence general.}
\end{frame}

\begin{frame}{Imitation rewards complement distribution matching}
\textbf{Learning to Imitate:} sample a prefix of a teacher sequence, ask the student to continue, and reward agreement.
\[
r=\gamma r_{\rm sem}+(1-\gamma)r_{\rm abs}
\]
\begin{itemize}
\item $r_{\rm sem}$: binary semantic-equivalence reward from a frozen model.
\item $r_{\rm abs}$: thresholded, normalized Levenshtein similarity.
\end{itemize}
\[
\underbrace{J_{\rm KS}}_{\text{maximize}}=
\E_{x\sim\mathcal D,\,y\sim\pS}
\left[(1-\alpha)r(y)-\alpha\mathcal D(\pT\Vert\pS)(y\mid x)\right].
\]
The paper names this expression $\mathcal L_{\rm KS}$; its signs imply reward maximization, or minimization of its negative.
\papersource{Sec. 3.3, Eqs. (3)--(4) and unnumbered KS objective, PDF p. 9. $\alpha$ is distinct from $\lambda$.}
\end{frame}

\begin{frame}{Dreaming learns which synthetic data improves adaptation}
\begin{enumerate}
\item Generate candidate dreams from context $C$, with an extra random MoE expert during generation.
\item Select promising candidates using gradient information; retain random candidates for diversity.
\item LoRA-finetune an isolated model copy on each selected dream.
\item Reward the dream if downstream score $\tau$ improves; optimize generation using ReST$^{\rm EM}$, following SEAL.
\end{enumerate}
\medskip
\textbf{Reproducibility gap:} a gradient vector is defined, but its conversion to a scalar ranking score is unspecified.
\par\medskip
\textbf{Scope:} this loop requires an evaluation signal; self-generated data alone is insufficient.
\papersource{Sec. 3.4 and Eq. (5), PDF pp. 9--10. The dream utility reward differs from the imitation reward.}
\end{frame}

\begin{frame}{The experiments test several distinct capabilities}
\centering
\begin{tabular}{p{0.30\linewidth}p{0.39\linewidth}p{0.20\linewidth}}
\toprule
Question & Benchmark & Reported metric\\
\midrule
Retain old classes? & CLINC / Banking / DBpedia & Accuracy $\uparrow$\\
Use long contexts? & MK-NIAH / LongHealth & Accuracy $\uparrow$\\
 & QASPER & Perplexity $\downarrow$\\
Learn sequential skills? & Manchu / Kalamang & ChRF $\uparrow$\\
Scale context length? & BABILong & Accuracy $\uparrow$\\
Improve reasoning? & AIME / HMMT & average@16 $\uparrow$\\
Store new facts? & SQuAD, no passage at test & Accuracy $\uparrow$\\
Adapt from few examples? & Filtered ARC subset & Attempt success $\uparrow$\\
\bottomrule
\end{tabular}
\papersource{Sec. 4, PDF pp. 10--13; Appendix B, pp. 25--26. Early experiments emphasize consolidation; SQuAD and ARC use full Sleep.}
\end{frame}

\begin{frame}{Sleep improves class-incremental accuracy over Hope}
\begin{columns}[T]
\begin{column}{0.33\textwidth}
\centering CLINC\\
\includegraphics[width=\linewidth,height=0.44\textheight,keepaspectratio]{figures/fig03-clinc.png}
\end{column}
\begin{column}{0.33\textwidth}
\centering Banking\\
\includegraphics[width=\linewidth,height=0.44\textheight,keepaspectratio]{figures/fig03-banking.png}
\end{column}
\begin{column}{0.33\textwidth}
\centering DBpedia\\
\includegraphics[width=\linewidth,height=0.44\textheight,keepaspectratio]{figures/fig03-dbpedia.png}
\end{column}
\end{columns}
\begin{itemize}
\item Both 3B and 8B backbones show higher Sleep bars than Hope.
\item ICL, EWC, InCA and Hope represent different retention mechanisms.
\item End-point bars do not reveal a full forgetting trajectory; no error bars or exact per-bar numbers are supplied.
\end{itemize}
\papersource{Original Fig. 3, PDF p. 10; Sec. 4.1, p. 11; dataset definitions, Appendix B.1, p. 25.}
\end{frame}

\begin{frame}{More levels generally help long-context behavior}
\begin{columns}[T]
\begin{column}{0.49\textwidth}
\centering LongHealth: accuracy $\uparrow$\\
\includegraphics[width=\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig04-longhealth.png}
\end{column}
\begin{column}{0.49\textwidth}
\centering QASPER: perplexity $\downarrow$\\
\includegraphics[width=\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig04-qasper.png}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item Stronger performance with more memory levels is the overall trend.
\item It is not strictly monotonic: QASPER's 8K curve worsens from one to two levels. Some LongHealth settings trail Cartridges.
\end{itemize}
\papersource{Original Fig. 4 (middle/right), PDF p. 10. Frequency labels are reproduced as printed; their units are ambiguous.}
\end{frame}

\begin{frame}{Sleep retains more after learning two new languages}
\begin{columns}[T]
\begin{column}{0.55\textwidth}
\includegraphics[width=\linewidth,height=0.64\textheight,keepaspectratio]{figures/fig05-translation.png}
\end{column}
\begin{column}{0.42\textwidth}
\begin{itemize}
\item Both axes are ChRF; upper-right is better.
\item Caption: red = single language; blue = sequential learning.
\item Sequential ICL falls far below Sleep variants.
\item Blue Sleep-2/3 points exceed their red points; the plot is stronger than ``nearly recovering'' alone.
\end{itemize}
\end{column}
\end{columns}
\papersource{Original Fig. 5, PDF p. 11; discussion, pp. 11--12. Two-language evidence, not an indefinite task stream.}
\end{frame}

\begin{frame}{BABILong remains strong at 10M, with qualifications}
\begin{columns}[T]
\begin{column}{0.59\textwidth}
\includegraphics[width=\linewidth,height=0.65\textheight,keepaspectratio]{figures/fig06-babilong.png}
\end{column}
\begin{column}{0.38\textwidth}
\begin{itemize}
\item Sleep is about \textbf{91\%} at 10M (visual estimate).
\item Small models receive benchmark-specific finetuning.
\item The caption's red/blue explanation does not match the lines.
\item Several baselines discussed in the appendix are absent from this figure.
\end{itemize}
\end{column}
\end{columns}
\papersource{Original Fig. 6, PDF p. 11; Sec. 4.1, p. 12; Appendix B.2, p. 26. Length bins are not plotted on a linear numeric scale.}
\end{frame}

\begin{frame}{Reasoning gains over OPSD are 1.0--2.6 points at 8B}
\centering
Qwen3-8B; average@16 $\uparrow$
\medskip

\begin{tabular}{lrrr}
\toprule
Method & AIME-24 & AIME-25 & HMMT-25\\
\midrule
Base (Instruct) & 73.8 & 68.1 & 42.4\\
SFT & 75.5 & 66.4 & 43.7\\
GRPO & 76.4 & 68.1 & 44.9\\
OPSD & 76.6 & 67.4 & 45.1\\
\textbf{Sleep} & \textbf{79.2} & \textbf{69.0} & \textbf{46.1}\\
\midrule
Sleep $-$ OPSD & +2.6 & +1.6 & +1.0\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item average@16 is not pass@16: do not interpret it as at least one success.
\item The paper gives no uncertainty estimates for these differences.
\end{itemize}
\papersource{Table 2, PDF p. 12. Values transcribed; differences calculated for this presentation. 1.7B results in backup.}
\end{frame}

\begin{frame}{Four-level Sleep improves no-context factual recall}
\centering
SQuAD mean accuracy (\%); original passage absent at evaluation
\medskip

\begin{tabular}{lrr}
\toprule
Method & One passage & CPT: 200 passages\\
\midrule
Base & 31.9 & 31.9\\
Finetuning, no Dreaming & 33.4 & 32.0\\
SEAL & 46.7 & 43.2\\
Sleep: two-level Transformer & 48.1 & 44.3\\
Sleep: four levels & \textbf{48.9} & \textbf{46.2}\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Four levels vs. SEAL: \textbf{+2.2 / +3.0 percentage points}.
\item CPT uses 974 questions; five dreams per passage form one training corpus.
\item This is one 200-passage run, not 200 separately evaluated learning cycles.
\end{itemize}
\papersource{Table 3, PDF p. 12; protocol, pp. 12--13. Backbone settings are referred to SEAL rather than fully restated.}
\end{frame}

\begin{frame}{ARC's 80\% is an adaptation-attempt success rate}
\begin{columns}[T]
\begin{column}{0.37\textwidth}
\centering
\begin{tabular}{lr}
\toprule
Method & Success (\%)\\
\midrule
ICL & 0\\
TTT & 10\\
SEAL & 72.5\\
Sleep & \textbf{80}\\
\bottomrule
\end{tabular}
\end{column}
\begin{column}{0.59\textwidth}
\begin{itemize}
\item Llama-3.2-1B; filtered tasks: 11 train, 8 held out.
\item Training: generate 60 dreams, reject 45 per task.
\item Test: five dreams per task, each applied independently.
\item Report the fraction of dream adaptations yielding a correct answer.
\end{itemize}
\end{column}
\end{columns}
\medskip
\begin{alertblock}{The denominator changes the interpretation}
The protocol implies 40 attempts: 80\% = 32/40, vs. 29/40 for SEAL. These counts are inferred, not individually reported.
\end{alertblock}
\papersource{Table 4, PDF p. 12; Sec. 4.1, p. 13; Appendix B.3, p. 26. Not a full ARC benchmark task pass rate.}
\end{frame}

\begin{frame}{Expansion explains part of the reasoning gain}
\centering
Qwen3-8B; average@16 $\uparrow$
\medskip

\begin{tabular}{lrrr}
\toprule
Method & AIME-24 & AIME-25 & HMMT-25\\
\midrule
Sleep & \textbf{79.2} & 69.0 & \textbf{46.1}\\
Without imitation learning & 76.8 & 67.9 & 45.0\\
Without semantic reward & 78.9 & \textbf{69.2} & 44.5\\
Without expansion & 78.2 & 67.9 & 44.9\\
OPSD & 76.6 & 67.4 & 45.1\\
OPSD + expansion & 77.9 & 68.2 & 45.9\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Sleep vs. expanded OPSD: only +1.3 / +0.8 / +0.2 points.
\item Removing semantic reward slightly improves AIME-25; component gains are not universal across metrics.
\end{itemize}
\papersource{Table 1, PDF p. 11; Sec. 4.2, p. 13 (its ``Figure 1'' reference should point to this table).}
\end{frame}

\begin{frame}{Dreaming has the largest SQuAD removal effect}
\centering
Mean no-context SQuAD accuracy (\%)
\medskip

\begin{tabular}{lrr}
\toprule
Configuration & One passage & CPT: 200 passages\\
\midrule
Four-level Sleep & \textbf{48.9} & \textbf{46.2}\\
Remove gradient selection & 47.1 & 45.2\\
Remove random expert & 48.0 & 44.7\\
Remove Dreaming & 35.7 & 36.2\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Relative to the four-level row: removing Dreaming costs \textbf{13.2 / 10.0 percentage points}.
\item Selection and random experts show smaller gains.
\item Removal rows do not restate the backbone; the four-level reference follows the table's ablation context.
\end{itemize}
\papersource{Table 3, PDF p. 12; Sec. 4.2, p. 13. No repeated-seed or matched-compute uncertainty analysis is given.}
\end{frame}

\begin{frame}{Efficiency depends on what is held constant}
\begin{columns}[T]
\begin{column}{0.47\textwidth}
\begin{block}{Same number of steps}
The authors report SFT as about \textbf{4$\times$ more efficient} than Sleep.
\end{block}
\end{column}
\begin{column}{0.47\textwidth}
\begin{block}{Matched target performance}
Reported SFT wall-clock multipliers: \textbf{4.3$\times$, 3.6$\times$, 4.8$\times$} on the three math benchmarks.
\end{block}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item These statements can coexist: a costly step may make more progress.
\item Absolute times, hardware and exact target scores are not supplied here.
\item Constant active parameter count does not imply constant total storage or training cost.
\end{itemize}
\papersource{Appendix B.5, PDF p. 26; active-parameter claim, Appendix B, p. 25. These are author-reported efficiency ratios.}
\end{frame}

\begin{frame}{The evidence supports gains, not indefinite retention}
\begin{itemize}
\item \textbf{Capacity:} experts grow or are unlocked; the slowest memory still has a finite budget.
\item \textbf{Forgetting:} frozen weights and self-distillation do not prove invariance of earlier task behavior.
\item \textbf{Feedback:} Dreaming needs a usable downstream evaluator, and imitation uses a reward model.
\item \textbf{Reproduction:} gradient scalarization, schedules, routing and key objective hyperparameters are incompletely specified.
\item \textbf{Evaluation:} filtered ARC tasks, BABILong finetuning and absent uncertainty estimates limit generalization.
\end{itemize}
\papersource{Reading analysis based on Secs. 3--4 and Appendix B. No general zero-forgetting theorem is presented.}
\end{frame}

\begin{frame}{A useful design principle: consolidate before rewriting}
\begin{enumerate}
\item Preserve a teacher containing the current memory; train a student that anticipates the next update.
\item Use slower, newly available parameters to carry that memory forward.
\item Separate retaining knowledge from learning which synthetic data improves future adaptation.
\end{enumerate}
\medskip
\textbf{Best supported conclusion:} the combined design improves several reported benchmarks under their specific protocols.
\par\medskip
\textbf{Next test (reading proposal):} match total capacity and compute, then measure old-task retention across many Sleep cycles.
\papersource{Ali Behrouz, Farnoosh Hashemi, Adel Javanmard and Vahab Mirrokni. Language Models Need Sleep: Learning to Self-Modify and Consolidate Memories. \href{https://arxiv.org/abs/2606.03979v2}{arXiv:2606.03979v2}. Original figures credited to the authors, CC BY 4.0.}
\end{frame}

\appendix

\begin{frame}{Backup: gains at 1.7B are small on AIME-25}
\centering
Qwen3-1.7B; average@16 $\uparrow$
\medskip

\begin{tabular}{lrrr}
\toprule
Method & AIME-24 & AIME-25 & HMMT-25\\
\midrule
Base (Instruct) & 49.8 & 34.5 & 25.7\\
SFT & 47.3 & 36.1 & 22.9\\
GRPO & 51.0 & 38.6 & 26.1\\
OPSD & 51.6 & 40.0 & 28.1\\
Sleep & \textbf{53.2} & \textbf{40.2} & \textbf{29.3}\\
\midrule
Sleep $-$ OPSD & +1.6 & +0.2 & +1.2\\
\bottomrule
\end{tabular}
\par\medskip
The 0.2-point AIME-25 difference has no accompanying uncertainty estimate. It should not be called statistically significant.
\papersource{Table 2, PDF p. 12. Differences calculated for this presentation.}
\end{frame}

\begin{frame}{Backup: the literal CMS update needs index care}
\[
\boldsymbol\theta^{(f_\ell)}_{i+1}
=\boldsymbol\theta^{(f_\ell)}_i-\boldsymbol e_{i,\ell}
\]
\[
\boldsymbol e_{i,\ell}=\begin{cases}
\displaystyle\sum_{t=i-C^{(\ell)}}^i\eta_t^{(\ell)}
f(\boldsymbol\theta_t^{(f_\ell)};x_t),
&i\equiv0\pmod{C^{(\ell)}},\\
0,&\text{otherwise}.
\end{cases}
\]
\begin{itemize}
\item $f(\cdot)$ is an optimizer error component, such as a gradient; it is not the frequency $f_\ell$.
\item The printed sum has $C^{(\ell)}+1$ terms if both endpoints are included.
\item The intended accumulation interval needs clarification before a literal implementation.
\end{itemize}
\papersource{Sec. 2.2, Eq. (2), PDF p. 5. Reproduced as printed, with an explicit indexing caveat.}
\end{frame}

\begin{frame}{Backup: imitation uses two distinct similarity signals}
\[
r_{\rm abs}(\hat d,d)=\begin{cases}
1-\dfrac{z(\hat d,d)}{\max\{|\hat d|,|d|\}},&z(\hat d,d)\leq z_0,\\[2mm]
0,&\text{otherwise}.
\end{cases}
\]
\begin{itemize}
\item $z$: Levenshtein distance; $z_0$: similarity threshold.
\item $r_{\rm sem}$ is 1 for semantic equivalence, 0 otherwise.
\item $\gamma$ mixes these rewards; $\alpha$ mixes imitation and distillation.
\item The paper does not fully report their task-specific settings or the semantic reward model's identity.
\end{itemize}
\papersource{Sec. 3.3, Eqs. (3)--(4), PDF p. 9. This reward is distinct from Dreaming's post-adaptation improvement reward.}
\end{frame}

\begin{frame}{Backup: MK-NIAH improves with more memory levels}
\begin{columns}[T]
\begin{column}{0.60\textwidth}
\includegraphics[width=\linewidth,height=0.67\textheight,keepaspectratio]{figures/fig04-niah.png}
\end{column}
\begin{column}{0.37\textwidth}
\begin{itemize}
\item Accuracy measures multi-key retrieval.
\item Four-level, ``512'' curve reaches approximately 100\%.
\item ICL is around 88.5\%; values here are visual estimates.
\item The printed frequency labels lack a clear unit interpretation.
\end{itemize}
\end{column}
\end{columns}
\papersource{Original Fig. 4 (left), PDF p. 10; Sec. 4.1, p. 11; Appendix B.1, p. 25.}
\end{frame}

\begin{frame}{Backup: reported training settings and source scope}
\centering
\begin{tabular}{lrrr}
\toprule
Setting & GRPO & SFT & Sleep\\
\midrule
Learning rate & $5\cdot10^{-6}$ & $5\cdot10^{-6}$ & $5\cdot10^{-6}$\\
Effective batch & 32 & 32 & 32\\
Training steps & 500 & 100 & 100\\
LoRA rank & 64 & 64 & 64\\
LoRA alpha & 128 & 128 & 128\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Table 5 does not provide a separate OPSD column or a complete per-task configuration map.
\item Read material: version-matched PDF, main TeX, all included body and appendix files, and original figures.
\item This presentation analyzes reported evidence; no training results were independently reproduced.
\end{itemize}
\papersource{Table 5, Appendix B.4, PDF p. 26. Bibliographic and source details are retained with the Chinese reading notes.}
\end{frame}

\end{document}