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  \hspace{8mm}{\color{gray}\scriptsize SSM gating \quad|\quad arXiv:2609.16540v1}
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\title[On the Importance of Gating]{On the Importance of Gating}
\subtitle{Memorization vs.\ In-Context Learning\\in State Space Models}
\author{%
  \small Paper by William L.\ Tong, Aryo Lotfi, Emmanuel Abbe,\\
  Kostas Vaggelakos, Vishnu Banna, Etai Littwin, Josh Susskind,\\
  Cengiz Pehlevan, and Eran Malach\\[2mm]
  \footnotesize Apple / Harvard University\\[1mm]
  \footnotesize Independent paper-reading presentation}
\date{\small arXiv:2609.16540v1 \quad|\quad 28 September 2026}

\begin{document}

% 1
\begin{frame}
\titlepage
\vspace{-3mm}
\begin{center}
\small\alert{Gating changes both which solution is learned and which history survives.}
\end{center}
\end{frame}

% 2
\begin{frame}{Two questions explain the apparent contradiction}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{block}{Learning a retrieval solution}
A correct solution exists, but strong decay weakens its gradient signal.
\end{block}
\medskip
The model can first learn a stable mapping stored in its weights.
\end{column}
\begin{column}{0.48\textwidth}
\begin{block}{Using it in a longer context}
Retaining every token also retains interference from irrelevant keys.
\end{block}
\medskip
Decay can help when relevant information stays close to the query.
\end{column}
\end{columns}
\bigskip
\alert{Weak gating helps discovery; selective forgetting can help extrapolation.}
\papersource{Paper, Secs.\ 1--2, PDF pp.\ 1--5. The second claim requires locality.}
\end{frame}

% 3
\begin{frame}{Perturbed labels make memorization distinguishable}
\begin{itemize}
\item Fix $P$ keys $x_i\in\R^d$ and base labels $y(x_i)\in\{-1,+1\}$.
\item Resample a flip on each context occurrence:
\[
\widetilde y_i=\eta_i y(x_i),\qquad
\Pr(\eta_i=-1)=\epsilon.
\]
\item Query a key that appears in the context; predict its \alert{current} label.
\end{itemize}
\begin{block}{Illustrative example}
A familiar key usually means $+1$. This prompt assigns it $-1$.\\
A weight lookup returns $+1$; successful context retrieval returns $-1$.
\end{block}
At $\epsilon=0.1$, memorizing base labels gives $90\%$ single-label accuracy.
\papersource{Paper, Sec.\ 2.1, PDF pp.\ 2--3. Example is a teaching illustration.}
\end{frame}

% 4
\begin{frame}{A scalar decay controls how much history reaches the query}
\[
\widetilde x_t=(x_t,\widetilde y_t),\quad
\widetilde x_q=(x_q,0),\quad
h=\widetilde x_q+\sum_{t=1}^{\ell}
e^{-(\ell-t+1)e^\lambda}
(\widetilde x_q^\top\widetilde x_t)\widetilde x_t
\]
\[
f=w^\top h,\qquad
w=(w_x,v)\in\R^{d+1}.
\]
\begin{itemize}
\item Train only the readout $w$ and global log-decay rate $\lambda$.
\item Initialize $w=0$; use logistic loss $\log(1+e^{-\widetilde y f})$.
\item At distance $\Delta$:
\[
\alpha_\Delta=e^{-\Delta e^\lambda}=e^{-\Delta/\tau},
\qquad \tau=e^{-\lambda}.
\]
\end{itemize}
\alert{Larger $\lambda$ means stronger decay, smaller $\alpha_\Delta$, and shorter $\tau$.}
\papersource{Paper, Sec.\ 2.2, PDF p.\ 3. A simplified Mamba-like layer with identity Q/K/V.}
\end{frame}

% 5
\begin{frame}{Two margins reveal which solution the model uses}
Assume orthonormal keys and fix the matching position.
\[
u=y(x_i)(w_x^\top x_i),\qquad
\delta_m=(1+\alpha)u,\qquad \delta_r=\alpha v.
\]
\begin{block}{Signed classification margin}
\[
z=\widetilde y f=\eta\delta_m+\delta_r
\]
\end{block}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\textbf{Unflipped label:} $\delta_m+\delta_r$.\\
Both mechanisms help.
\end{column}
\begin{column}{0.48\textwidth}
\textbf{Flipped label:} $-\delta_m+\delta_r$.\\
Retrieval must overcome memorization.
\end{column}
\end{columns}
\bigskip
$w_x=0,\ v>0$ already solves this idealized classification task.
\papersource{Paper, Sec.\ 2.3, PDF p.\ 3; App.\ B.1, PDF p.\ 15.}
\end{frame}

% 6
\begin{frame}{Strong decay suppresses the initial retrieval gradient twice}
At zero readout initialization,
\[
\dot{\delta}_m(0)=\frac{(1+\alpha)^2(1-2\epsilon)}{2P},
\qquad
\dot{\delta}_r(0)=\frac{\alpha^2}{2}.
\]
\begin{block}{Proposition 1: initial growth-rate ratio}
\[
\frac{\dot{\delta}_r(0)}{\dot{\delta}_m(0)}
=\frac{P\alpha^2}{(1-2\epsilon)(1+\alpha)^2}.
\]
\end{block}
\begin{itemize}
\item One factor $\alpha$ weakens learning of $v$; another weakens its output.
\item Shared retrieval benefits from all keys; each memory direction sees $1/P$.
\item For small $\alpha$, the crossover scale is
$\alpha^*\approx\sqrt{(1-2\epsilon)/P}$.
\end{itemize}
\papersource{Paper, Proposition 1 / Eq.\ (B.1), PDF p.\ 15. An initial-rate result, not a global phase transition.}
\end{frame}

% 7
\begin{frame}{A memorization plateau can last extremely long}
\begin{columns}[c]
\begin{column}{0.43\textwidth}
\centering\fig{theory-ab.pdf}{5.65cm}
\end{column}
\begin{column}{0.55\textwidth}
\[
\delta_m=\kappa=\log\frac{1-\epsilon}{\epsilon}
\]
On the leading-order plateau dynamics:
\[
T=\Theta\!\left(\frac{1}{\alpha_0\log(1/\alpha_0)}\right).
\]
\medskip
\begin{itemize}
\item Small initial retention makes escape slow.
\item $\alpha_0=e^{-\Delta e^{\lambda_0}}$ creates severe distance sensitivity.
\item The theorem concerns \alert{reduced dynamics}, not full Mamba training.
\end{itemize}
\end{column}
\end{columns}
\papersource{Paper, Fig.\ 1a--b, PDF p.\ 5; Proposition 4 / Eq.\ (B.4), PDF pp.\ 17--18.}
\end{frame}

% 8
\begin{frame}{In long contexts, retained keys also contribute noise}
Now use fresh Gaussian keys and independent random labels.\\
Population gradient flow from zero gives $w_x=0,\ v>0$.
\[
g=
\underbrace{\alpha_\Delta\|x_q\|^2\widetilde y_q}_{\text{signal}}
+
\underbrace{\sum_{t\ne t^*}\alpha_{\ell-t+1}
(x_q^\top x_t)\widetilde y_t}_{\text{interference}}.
\]
With $S_2=\sum_{j=1}^{\ell}\alpha_j^2$, the conditional error is
\[
\Pr(\operatorname{sign}(g)\ne\widetilde y_q\mid x_q)=
\Phi\!\left(-\alpha_\Delta\|x_q\|
\sqrt{\frac{d}{S_2-\alpha_\Delta^2}}\right).
\]
\alert{Weaker decay increases both the retained signal and the noise variance.}
\papersource{Paper, App.\ B.2, Proposition 5 and Lemma 6, PDF pp.\ 18--19. Bayes error is for statistic $g$.}
\end{frame}

% 9
\begin{frame}{No decay is not always the best retrieval regime}
\begin{columns}[c]
\begin{column}{0.57\textwidth}
\centering\fig{theory-c.pdf}{4.7cm}
\end{column}
\begin{column}{0.40\textwidth}
As $\tau\to\infty$ at fixed $\ell$:
\[
p_{\tau,\Delta}(x_q)\to
\Phi\!\left(-\|x_q\|\sqrt{\frac{d}{\ell-1}}\right).
\]
\smallskip
For $d=128,\ell=1024$,\\
accuracy $\approx0.638$.
\medskip

Error approaches $1/2$ only as $(\ell-1)/d\to\infty$.
\end{column}
\end{columns}
\medskip
The formal appendix uses $d/(\ell-1)$; the main-text formula has a typo.
\papersource{Paper, Fig.\ 1c, PDF p.\ 5; formal Proposition 8, PDF p.\ 20; compare Sec.\ 2.4, PDF p.\ 4.}
\end{frame}

% 10
\begin{frame}{Decay supports local retrieval inside a much longer sequence}
\centering
\includegraphics[width=0.73\linewidth,height=3.4cm,keepaspectratio]{figures/theory-d.pdf}
\[
S_2=\Theta(\tau),\qquad
\mathrm{SNR}\asymp e^{-\Delta/\tau}\sqrt{d/\tau},
\qquad
\Delta_{\max}=\Theta\!\left(\tau\log\frac d\tau\right).
\]
\begin{itemize}
\item Assumptions: $\ell/\tau\to\infty$, $1\le\tau=o(d)$, $\|x_q\|=\Theta(1)$.
\item Arbitrarily many distant distractors can be tolerated if the target stays local.
\end{itemize}
\papersource{Paper, Fig.\ 1d, PDF p.\ 5; Lemma 7 and Proposition 9, PDF pp.\ 19--21. Distances also cannot exceed $\ell$.}
\end{frame}

% 11
\begin{frame}{Three experiments test progressively less idealized retrieval}
\centering
\begin{tabular}{L{2.5cm}L{3.7cm}L{5.6cm}}
\toprule
Task & Shortcut / intervention & What the comparison tests\\
\midrule
Multi-token KV & Fixed values with token perturbations; gate offset & Escape from memorization; early vs.\ late targets\\[2mm]
Logical rules & Fixed clauses, changing numeric names & Retrieve a current name; extrapolate with local relevant clauses\\[2mm]
Tool calling & Pretrained Mamba; decay-rate scale before SFT & Correct calls and functions hallucinated outside the pool\\
\bottomrule
\end{tabular}\par
\bigskip
\alert{Keep target position, pool size, and training distribution separate.}
\papersource{Paper, Secs.\ 3--5, PDF pp.\ 6--10. Each experiment uses different architectures and optimization.}
\end{frame}

% 12
\begin{frame}{Weak gate initialization escapes the KV shortcut faster}
\centering\includegraphics[width=0.82\linewidth,height=3.75cm,keepaspectratio]{figures/kv-ab.pdf}
\begin{itemize}
\item Eight-token values, flip probability $0.1$: memory baseline $0.9^8\approx0.43$.
\item Default initialization does not escape within 100k steps.
\item Five seeds: panel a is the median-escape run; b shows all seeds.
\end{itemize}
\papersource{Paper, Fig.\ 2a--b, PDF p.\ 7; App.\ D.2, PDF pp.\ 23--24. Escape means sequence accuracy $>0.95$.}
\end{frame}

% 13
\begin{frame}{With later targets, stronger gates degrade more slowly}
\begin{columns}[c]
\begin{column}{0.53\textwidth}
\centering\fig{kv-c.pdf}{4.8cm}
\end{column}
\begin{column}{0.44\textwidth}
\begin{itemize}
\item Train with at most 512 KV pairs; test up to 2048.
\item Query the last $10\%$ of keys (best-last).
\item Stronger gates retain higher accuracy as the pool grows.
\item Five-seed means; appendix specifies $95\%$ confidence intervals.
\end{itemize}
\medskip
\alert{Better extrapolation still leaves substantial errors.}
\end{column}
\end{columns}
\papersource{Paper, Fig.\ 2c, PDF p.\ 7; App.\ D.2.3, PDF p.\ 24. This is a separate best-last task.}
\end{frame}

% 14
\begin{frame}{Rule names separate context use from a memorized association}
\centering\includegraphics[width=0.95\linewidth,height=2.15cm,keepaspectratio]{figures/horn-task.pdf}
\begin{itemize}
\item Select a named clause that moves toward the goal.
\item Actual task: 3 proposition sets, 128 propositions per set, 2 turns.
\item \textbf{No oracle:} do not inject the correct clause; probes memorization.
\item \textbf{Random oracle name:} inject a correct clause with a new name; probes retrieval.
\end{itemize}
Score the first turn as correct if the selected transition can still reach the goal.
\papersource{Paper, Fig.\ 3a, PDF p.\ 8 (conceptual diagram); Sec.\ 4.2, p.\ 9; actual task in App.\ D.3.3, p.\ 25.}
\end{frame}

% 15
\begin{frame}{Strong gates preserve old names longer during training}
\begin{columns}[c]
\begin{column}{0.68\textwidth}
\centering\fig{horn-b.pdf}{4.1cm}
\end{column}
\begin{column}{0.30\textwidth}
\centering\fig{horn-legend.pdf}{3.7cm}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item Strong decay: high no-oracle accuracy, slow learning of renamed clauses.
\item Weak decay: faster adoption of the name supplied in the context.
\item Train with pools up to 32; rename the first $10\%$ of clauses.
\end{itemize}
\papersource{Paper, Fig.\ 3b and legend, PDF p.\ 8; Sec.\ 4.2, p.\ 9. This is a diagnostic pair, not an exact decomposition.}
\end{frame}

% 16
\begin{frame}{Strong gates extrapolate when useful rules stay local}
\begin{columns}[c]
\begin{column}{0.39\textwidth}
\centering\fig{horn-d-legend.pdf}{5.5cm}
\end{column}
\begin{column}{0.58\textwidth}
\begin{itemize}
\item A separate experiment: all names are resampled.
\item Training pool size $\le8$; evaluate up to 128 clauses.
\item Rank useful clauses closer to the query.
\item Strong gates stay near perfect accuracy; weak gates and this Transformer degrade.
\end{itemize}
\bigskip
\alert{This tests local retrieval amid more distractors, not arbitrary distant recall.}
\end{column}
\end{columns}
\papersource{Paper, Fig.\ 3d and original legend, PDF p.\ 8; Sec.\ 4.2, p.\ 9; App.\ D.3, pp.\ 24--25.}
\end{frame}

% 17
\begin{frame}{BFCL tests a gate intervention after pretraining}
\begin{itemize}
\item Mamba-2 1.5B vs.\ Llama-style Transformer 2B, each pretrained on 300B tokens.
\item SFT on Nemotron Agentic tool calling; train pools up to $k=24$.
\item Modified BFCL \texttt{python simple}: BM25 pool, best-first order; inject the correct tool if missed.
\item Before SFT, scale Mamba's decay rate:
\[
e^\lambda\leftarrow s e^\lambda
\quad\Longleftrightarrow\quad
\lambda\leftarrow\lambda+\log s.
\]
\end{itemize}
\begin{block}{Two distinct metrics}
\textbf{Accuracy:} a fully correct call, including arguments.\\
\textbf{Hallucination:} a syntactically valid call to a function absent from the pool.
\end{block}
\papersource{Paper, Sec.\ 5, PDF pp.\ 9--10; App.\ D.4, p.\ 25. Unequal model sizes; not the full BFCL benchmark.}
\end{frame}

% 18
\begin{frame}{Weaker Mamba gates improve calls and reduce pool violations}
\centering\includegraphics[width=\linewidth,height=3.65cm,keepaspectratio]{figures/fig-04-bfcl.pdf}
\begin{itemize}
\item Low-scale Mamba variants generally score better and hallucinate less.
\item Some weakened gates improve on the pretrained scale $s=1$.
\item All Mamba variants extrapolate better than this Transformer here.
\end{itemize}
\alert{The authors do not find a clear gate-strength effect on length generalization in this range.}
\papersource{Paper, Fig.\ 4 and Sec.\ 5.2, PDF p.\ 10. No confidence intervals reported; small differences are not significance claims.}
\end{frame}

% 19
\begin{frame}{With imperfect tool retrieval, a larger pool has two effects}
\centering{\small\textbf{Left:} near-random ($\beta=0.05$) \qquad \textbf{Right:} noisy ordered ($\beta=10$)\par}
\includegraphics[width=\linewidth,height=3.5cm,keepaspectratio]{figures/weak-tools.pdf}
\begin{itemize}
\item Remove the guarantee that the correct tool is in the pool.
\item Larger pools improve recall but can reduce conditional call accuracy.
\item In the ordered setting, moderate gating remains useful as the pool grows.
\end{itemize}
\papersource{Paper, App.\ C / Fig.\ 5 bottom, PDF p.\ 22. Left: near-random retrieval, $\beta=0.05$; right: noisy ordered retrieval, $\beta=10$.}
\end{frame}

% 20
\begin{frame}{The strongest evidence is mechanistic and task dependent}
\begin{enumerate}
\item \textbf{Theory:} orthogonal or Gaussian keys, simplified readout, and population gradient flow; escape time uses reduced dynamics.
\medskip
\item \textbf{Locality:} long sequences can remain manageable when targets stay close; arbitrary-position retrieval is a different requirement.
\medskip
\item \textbf{Experiments:} controlled synthetic tasks and one modified BFCL split; model sizes are not always matched.
\medskip
\item \textbf{Reporting:} five seeds for KV; fewer statistical details for the other studies and incomplete SFT hyperparameters.
\end{enumerate}
\papersource{Paper, Sec.\ 6, PDF pp.\ 10--11; Apps.\ B--D, pp.\ 15--25. Scope assessment by this reading.}
\end{frame}

% 21
\begin{frame}{Gate design must address learning and interference together}
\begin{block}{What this paper establishes}
A retrieval solution can exist while optimization strongly favors a memorization shortcut.
\end{block}
\begin{block}{Why removing decay is insufficient}
Long contexts introduce interference; reducing it can require forgetting, with a cost to distant signals.
\end{block}
\begin{block}{A useful next experiment}
Jointly vary target distance, distractor count, gate schedule, and tool-pool ordering under the same training budget.
\end{block}
\papersource{Paper, Secs.\ 2 and 6, PDF pp.\ 2--5 and 10--11. The proposed experiment is reader analysis, not a paper result.}
\end{frame}

% 22
\begin{frame}{Source and reading map}
\textbf{William L.\ Tong et al.}\\
\emph{On the Importance of Gating: Memorization vs.\ In-Context Learning in State Space Models.}\\
arXiv:2609.16540v1, submitted 15 September 2026.
\medskip

\href{https://arxiv.org/abs/2609.16540v1}{arxiv.org/abs/2609.16540v1}
\bigskip
\begin{itemize}
\item Mechanism and formal proofs: Sec.\ 2 and App.\ B.
\item Experiments and protocols: Secs.\ 3--5 and App.\ D.
\item Imperfect external retrieval: App.\ C.
\item Companion files: Chinese reading notes, original paper, and figure provenance.
\end{itemize}
\papersource{All empirical plots are author figures, with source page and panel identified. No new training experiments were run.}
\end{frame}

\appendix

% 23
\begin{frame}{Backup: the escape-time proof uses the early plateau}
On $\delta_m=\kappa$, the leading-order equations are
\[
\dot v=\alpha\phi(\alpha v),\qquad
\dot\alpha=(\alpha\log\alpha)^2v\phi(\alpha v),
\]
where $\phi$ is bounded above and below by positive constants before escape.
\[
b=\log(1/\alpha),\quad
\frac{db}{dv}=-b^2v,\quad
\frac{1}{b(v)}=\frac{1}{b_0}+\frac{v^2}{2}.
\]
Since $v_T=\Theta(1)$ at $\alpha(v_T)v_T=\kappa$,
\[
T=\Theta\!\left[
\int_0^{v_T}\exp\!\left(\frac{b_0}{1+b_0v^2/2}\right)\dd v
\right]
=\Theta(e^{b_0}/b_0).
\]
\alert{The proof does not supply a uniform approximation theorem for full-model training.}
\papersource{Paper, Lemmas 2--3 and Proposition 4, PDF pp.\ 16--18.}
\end{frame}

% 24
\begin{frame}{Backup: experimental configurations must stay separate}
\small
\begin{tabular}{L{2.25cm}L{5.3cm}L{5.45cm}}
\toprule
Setting & Architecture / data & Optimizer / measurement\\
\midrule
Theory & Identity projections; Fig.\ 1a: $d=512$, $P=32$, $\ell=8$, $\Delta=4$ & SGD, batch 128, lr 0.1; Fig.\ 1b uses lr 0.3\\[2mm]
KV retrieval & Mamba-2: 24 layers, width 768; 8-token keys and values; 10k pairs & AdamW, peak lr $3\times10^{-4}$; five seeds; escape at accuracy $>0.95$\\[2mm]
Logical rules & Mamba-2: 4 layers, 16 heads, head/state size 256 & Muon, peak lr $10^{-3}$, batch 128; first-turn accuracy\\[2mm]
Tool calling & Mamba-2: 32 layers, width 2048; 1.5B vs.\ 2B Transformer & AdamW for pretraining and SFT; SFT details incompletely reported\\
\bottomrule
\end{tabular}
\papersource{Paper, App.\ D, PDF pp.\ 23--25. See companion notes for remaining settings and baseline architectures.}
\end{frame}

% 25
\begin{frame}{Backup: trained gates preserve the broad initialization ordering}
\begin{columns}[c]
\begin{column}{0.55\textwidth}
\centering
\begin{tabular}{rrr}
\toprule
Initial $\tau$ & After 200k & $|\Delta\alpha|$\\
\midrule
$2.1\times10^4$ & $1.2\times10^4$ & $4\times10^{-5}$\\
297 & 315 & 0.001\\
17 & 14 & 0.024\\
2.8 & 3.0 & 0.083\\
1.4 & 1.4 & 0.067\\
0.8 & 0.7 & 0.053\\
0.2 & 0.2 & 0.007\\
\bottomrule
\end{tabular}
\end{column}
\begin{column}{0.42\textwidth}
\begin{itemize}
\item Logical-rules length experiment.
\item $\tau$ is the median effective length across heads, in tokens.
\item Ordering survives a sweep of roughly five orders of magnitude.
\item This supports a regime proxy, not an exactly static gate.
\end{itemize}
\end{column}
\end{columns}
\papersource{Paper, App.\ B.3 / Table 1, PDF p.\ 21. Values transcribed exactly; no recomputation from aggregate medians.}
\end{frame}

% 26
\begin{frame}{Backup: clause recall can rise while model accuracy saturates}
\centering\includegraphics[width=\linewidth,height=3.65cm,keepaspectratio]{figures/weak-horn.pdf}
\begin{itemize}
\item Left: random candidate ranking ($\beta=0$); right: noisy ordered ranking ($\beta=1$).
\item Gray dashed curve: probability that a relevant clause enters the context.
\item Recall gains alone do not guarantee end-to-end gains.
\end{itemize}
\papersource{Paper, App.\ C / Fig.\ 5 top, PDF p.\ 22. The $\beta$ scale differs from the BFCL experiment.}
\end{frame}

% 27
\begin{frame}{Backup: distinguish exact results from approximations}
\begin{enumerate}
\item \textbf{Initial crossover.} With $s_0=\sqrt{(1-2\epsilon)/P}$, equality of initial rates gives $\alpha=s_0/(1-s_0)$ when feasible. The paper uses the small-$\alpha$ approximation $\alpha^*\approx s_0$.
\medskip
\item \textbf{No-decay limit.} Use $d/(\ell-1)$ from formal Proposition 8, not $d/\ell-1$ printed in the main text.
\medskip
\item \textbf{Offset scaling.} At fixed distance and base rate,
\[
\alpha_\Delta(\delta)=
\exp[-\Delta e^{\lambda_{\rm base}}e^\delta].
\]
The linear fit of escape time against $e^\delta$ ($R^2=0.816$) is empirical; $\alpha\propto e^{-\delta}$ does not follow.
\end{enumerate}
\papersource{Reader algebra checked against Eq.\ (B.1), PDF p.\ 15; Sec.\ 2.4, p.\ 4; Proposition 8, p.\ 20; Sec.\ 3.2, p.\ 6.}
\end{frame}

\end{document}