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\title[Spectral Unlearning]{Spectral Unlearning}
\subtitle{Transformer Structure-Preserving Updates for Language Models}
\author{Paper: Sung Il Choi, Junhao Cai, Dohun Kim, and Changhee Joo}
\date{Reading seminar \enspace|\enspace arXiv:2611.11902v1\\12 November 2026}
\newcommand{\fig}[2]{\centering\includegraphics[width=\linewidth,height=#2,keepaspectratio]{figures/#1.pdf}\par}
\begin{document}
\begin{frame}\titlepage\end{frame}
\begin{frame}{Update geometry is a second design axis for unlearning}

\begin{block}{Takeaway}
Keep the unlearning objective; change how matrix gradients become updates.
\end{block}
\begin{itemize}
\item Ideal spectral direction: $-UV^\top$ for $G=U\Sigma V^\top$.
\item Practical realizations: Muon, KLShampoo, and SOAP.
\item SOAP often improves balanced forgetting; gains have compute costs and exceptions.
\end{itemize}
\medskip
\small \href{https://arxiv.org/abs/2611.11902v1}{arXiv v1: 9 November 2026}.\\
Accepted at NeurIPS 2026, per \href{https://rain.korea.ac.kr/}{authors' lab} and arXiv.\\
\href{https://github.com/unlearn-ml/spectral-unlearning}{Official code}. Status checked 12 November 2026.

\papersource{Paper, Abstract and Secs. 1, 3--5, PDF pp. 1--10. Acceptance is distinguished from formal publication.}
\end{frame}
\begin{frame}{The same NPO objective follows different optimization paths}

\fig{fig1}{3.8cm}
\begin{itemize}
\item Left: unlearning steps. Right: wall-clock time. Lower objective is preferred.
\item Solid: spectral variants; dashed: entry-wise AdamW and Sophia.
\item Loss convergence motivates the method; forgetting quality needs separate metrics.
\end{itemize}

\papersource{Paper, Fig. 1, PDF p. 2. Llama-3.2-1B on TOFU; NPO unlearning.}
\end{frame}
\begin{frame}{GD and NPO generate different forgetting signals}

\textbf{Gradient Difference (GD)}
\[
\mathcal L_{\rm GD}=\mathbb E_u[\log\pi_w(y\mid x)]
+\alpha\mathbb E_r[-\log\pi_w(y\mid x)]
\]
\textbf{Negative Preference Optimization (NPO): forget term}
\[
\mathcal L_{u,\rm NPO}=\frac2\beta\mathbb E_u
\log\!\left[1+\left(\frac{\pi_w(y\mid x)}{\pi_{w_0}(y\mid x)}\right)^\beta\right]
\]
\begin{itemize}
\item GD's forget term is unbounded below; NPO's is lower-bounded by zero.
\item Experiments add retain NLL to NPO, with $\beta=0.1$ and unit loss weights.
\end{itemize}

\papersource{Paper, Eqs. (1)--(3), PDF p. 3; App. C.3, PDF p. 20. $u/r$: forget/retain data; $w_0$: frozen reference.}
\end{frame}
\begin{frame}{RMU changes representations while anchoring retained examples}

\[
\begin{aligned}
\mathcal L_{\rm RMU}
={}&\mathbb E_u\|h_w^{(\ell)}(x)-cu\|_2^2\\
&+\alpha\mathbb E_r\|h_w^{(\ell)}(x)-h_{w_0}^{(\ell)}(x)\|_2^2.
\end{aligned}
\]
\begin{itemize}
\item $u$: fixed random unit direction; $c$: steering magnitude.
\item Forget representations move to a random target; retained ones stay near the reference.
\item TOFU uses block 7 and $c=2$, with all transformer parameters trainable.
\item The spectral update is applied to this loss without replacing it.
\end{itemize}

\papersource{Paper, Eq. (4), PDF p. 3; App. C.3, PDF p. 20. TOFU setup differs from layer-restricted WMDP RMU.}
\end{frame}
\begin{frame}{A norm decides which change counts as a large step}

\[
\Delta w^*\in\arg\min_{\Delta w}
\left[\langle g,\Delta w\rangle+\frac\lambda2\|\Delta w\|^2\right]
\]
\[
\Delta w^*=-\frac{\|g\|_*}{\lambda}\,t^*(g),\quad
 t^*(g)\in\arg\max_{\|t\|=1}\langle g,t\rangle
\]
\begin{itemize}
\item Direction: maximize first-order loss change per unit step.
\item Magnitude: dual norm divided by local curvature scale $\lambda$.
\item Vector $\ell_\infty$ geometry gives $\operatorname{sign}(g)$.
\item The AdamW/sign connection removes moving averages and decay; it is an idealization.
\end{itemize}

\papersource{Paper, Eqs. (5)--(8), PDF pp. 3--4; App. A.1, PDF p. 14.}
\end{frame}
\begin{frame}{The spectral-norm budget gives the polar-factor direction}

\[
G=U\Sigma V^\top,\qquad
\|W\|_2=\sup_{\|x\|_2=1}\|Wx\|_2
\]
\[
\max_{\|T\|_2\le1}\langle G,T\rangle
=\sum_i\sigma_i,\qquad T^*=UV^\top
\]
\[
\boxed{\Delta W^*=-\frac{\operatorname{tr}\Sigma}{\lambda}UV^\top}
\quad\Longrightarrow\quad
W_{t+1}=W_t-\eta UV^\top
\]
\begin{itemize}
\item Keep singular directions and flatten their update amplitudes.
\item Scheduled learning rate $\eta$ replaces the exact local-model magnitude.
\item At rank deficiency, the optimal direction need not be unique.
\end{itemize}

\papersource{Paper, Eqs. (9)--(12), PDF p. 4; App. A.3, PDF p. 15.}
\end{frame}
\begin{frame}{Spectral unlearning reshapes gradients, not the weight spectrum}

\textbf{Teaching example:} a diagonal gradient
\[
G=\begin{bmatrix}10&0\\0&1\end{bmatrix},\qquad
UV^\top=\begin{bmatrix}1&0\\0&1\end{bmatrix}.
\]
\begin{itemize}
\item Euclidean gradient descent: update amplitude ratio $10:1$.
\item Spectral direction: ratio $1:1$ on these singular modes.
\item Weaker modes receive more relative weight.
\item No low-rank truncation or permanent orthogonality constraint is imposed on $W$.
\end{itemize}

\papersource{Teaching illustration of Paper Eq. (12), PDF p. 4; not an experimental result.}
\end{frame}
\begin{frame}{RMS geometry connects normalized features to spectral norms}

\[
\|x\|_{\rm RMS}=\frac{\|x\|_2}{\sqrt d},\qquad
\boxed{\|W\|_{{\rm RMS}\to{\rm RMS}}
=\sqrt{\frac{d_{\rm in}}{d_{\rm out}}}\|W\|_2}
\]
\begin{itemize}
\item RMSNorm in pre-norm blocks motivates measuring feature scale this way.
\item A selected network norm takes the maximum of the layerwise RMS-to-RMS norms.
\item Its local steepest-descent direction is $-U_lV_l^\top$, with dimension-dependent step sizes.
\item This is a geometric derivation; it is not a certified-unlearning guarantee.
\end{itemize}

\papersource{Paper, Lemma 1/Theorem 2, Eqs. (16)--(17), PDF pp. 6--7. Multi-layer derivation in backup.}
\end{frame}
\begin{frame}{Three matrix optimizers realize related, distinct updates}

\begin{tabular}{lp{9.7cm}}
\toprule
Muon & Five Newton--Schulz iterations approximate the polar factor of a momentum matrix.\\[4pt]
KLShampoo & Running row/column second moments; update proportional to $-L_t^{-1/4}G_tR_t^{-1/4}$.\\[4pt]
SOAP & Shampoo-style eigenbases, with Adam-style adaptive updates in the rotated coordinates.\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Momentum, EMA, damping, and basis refreshes distinguish these from exact instantaneous SVD.
\item Attention/MLP matrices use matrix updates; embeddings use AdamW.
\end{itemize}

\papersource{Paper, Secs. 3.2--3.3, PDF pp. 5--6; App. C.4, PDF pp. 20--21.}
\end{frame}
\begin{frame}{One-sided polar identities have specific conditions}

\[
\begin{array}{ll}
G\text{ tall, full column rank}:&G(G^\top G)^{-1/2}=UV^\top,\\[5pt]
G\text{ wide, full row rank}:&(GG^\top)^{-1/2}G=UV^\top.
\end{array}
\]
\begin{itemize}
\item MLP up/gate: precondition on the residual input side; down: on the residual output side.
\item SOAP uses \texttt{max\_precond\_dim}$=d$; larger intermediate axes are skipped.
\item Exact identities use instantaneous inverse square roots and full rank.
\item EMA statistics and SOAP's inner Adam remain practical approximations.
\end{itemize}

\papersource{Paper, App. B, Table 4 and Eqs. (46)--(48), PDF pp. 17--19. Not every projection directly receives RMSNorm output.}
\end{frame}
\begin{frame}{The experiments compare objectives and optimizers separately}

\begin{tabular}{ll}
\toprule
TOFU & Llama-3.2-1B/3B; forget 1\%, 5\%, 10\%\\
RWKU & Llama-3-8B-Instruct; real-person knowledge\\
WMDP & Zephyr-7B-$\beta$; Bio/Cyber, with MMLU retention\\
Main combinations & 3 losses $\times$ 5 optimizers = 15 methods\\
Training & 10 epochs; first epoch warmup; LR sweep per cell\\
Seeds & 3 trajectories; initial checkpoint and data split fixed\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Original: full-data fine-tuned checkpoint. Retrain: retain-only fine-tuning reference.
\item The oracle concerns this fine-tuning task, not deletion from all pretraining data.
\end{itemize}

\papersource{Paper, Sec. 5.1 and App. C, PDF pp. 7, 19--22. Extra baselines use their canonical training settings.}
\end{frame}
\begin{frame}{Balanced evaluation needs forgetting, privacy, and utility}

\[
\mathrm{Agg}=\mathrm{HM}(\mathrm{Mem},\mathrm{Priv},\mathrm{Util}),\qquad
\mathrm{HM}(x_1,\ldots,x_n)=\frac n{\sum_i1/\max(x_i,10^{-12})}
\]
\begin{itemize}
\item \textbf{Mem $\uparrow$:} HM of inverted extraction, token agreement, forget probability, and a symmetric truth ratio.
\item \textbf{Priv $\uparrow$:} similarity of four MIA AUCs to the retain-only reference.
\item \textbf{Util $\uparrow$:} ten retention/knowledge/fluency sub-metrics.
\item Higher Agg can reflect a better balance while one component declines.
\end{itemize}

\papersource{Paper, App. D, Eqs. (49)--(56), PDF pp. 22--23. Privacy is an empirical score, not a formal guarantee.}
\end{frame}
\begin{frame}{SOAP improves the main TOFU aggregate comparisons}

\centering
\begin{tabular}{lrrr}
\toprule
Setting & GD & NPO & RMU\\
\midrule
1B / 10\% & .5704 $\to$ .6043 & .6876 $\to$ .6926 & .6888 $\to$ .6982\\
1B / 5\% & .5653 $\to$ .6075 & .6599 $\to$ .6711 & .6801 $\to$ .6862\\
3B / 10\% & .5839 $\to$ .6091 & .7243 $\to$ .7262 & .7038 $\to$ .7165\\
3B / 5\% & .5737 $\to$ .6455 & .7062 $\to$ .7211 & .7317 $\to$ .7360\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Each cell: best entry-wise mean $\to$ same-loss SOAP mean. Agg $\uparrow$.
\item SOAP wins these 12 comparisons; margins vary substantially.
\item Other spectral variants can lose: 3B/10\% GD-Muon .5016 vs. Sophia .5839.
\end{itemize}

\papersource{Paper, Table 1, PDF p. 7; precise values from Table 12, PDF p. 36. No significance claim from small mean gaps.}
\end{frame}
\begin{frame}{A similar aggregate score does not mean the oracle is matched}

\centering
\begin{tabular}{lrrrr}
\toprule
3B / 5\%, RMU & Agg & Mem & Priv & Util\\
\midrule
AdamW & .7151 & .8440 & .6856 & .6444\\
Sophia & .7317 & .6920 & .9010 & .6474\\
SOAP & \alert{.7360} & .7160 & .8468 & .6673\\
Retrain & .7402 & .6362 & .9999 & .6753\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Both SOAP and Retrain round to .74 in the main table.
\item Precise aggregate gap: .0042; privacy behavior still differs visibly.
\item SOAP improves balance vs. AdamW, even though its Mem score falls.
\end{itemize}

\papersource{Paper, Tables 12/16, PDF pp. 36, 38. All scores higher is better; comparisons use reported means.}
\end{frame}
\begin{frame}{The 1\% split breaks a uniform optimizer ranking}

\begin{tabular}{lrr}
\toprule
Setting & SOAP Agg & Comparator Agg\\
\midrule
1B / NPO / 1\% & .6445 $\pm$ .0128 & Sophia: .6486 $\pm$ .0042\\
3B / RMU / 1\% & .7139 $\pm$ .0033 & AdamW: .7169 $\pm$ .0006\\
3B / GD / 1\% & .6217 $\pm$ .0190 & Muon: .6392 $\pm$ .0040\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Across all 18 loss/model/split cells, SOAP beats the best entry-wise mean in 16.
\item The small split gives few updates: about 13 steps at 1B, 25 at 3B.
\item A strong overall optimizer is not the winner in every configuration.
\end{itemize}

\papersource{Paper, Table 12, PDF p. 36; App. C.2, PDF p. 19. The 16/18 count is calculated from table means.}
\end{frame}
\begin{frame}{Matrix statistics capture gradient correlations missed by diagonal ones}

\fig{fig2}{3.7cm}
\begin{itemize}
\item Output, MLP up, MLP down projections; cosine similarity to measured covariance.
\item SOAP tracks the optimal Kronecker reference most closely here.
\item This measures covariance alignment, not exact update alignment with $UV^\top$.
\end{itemize}

\papersource{Paper, Fig. 2, PDF p. 5; depth/loss extensions in Figs. 7--8, PDF pp. 24--25.}
\end{frame}
\begin{frame}{A broader useful learning-rate range still has a failure threshold}

\fig{fig4}{3.6cm}
\begin{itemize}
\item GD, 3B, forget 5\%; left: Agg, right: Mem/Priv/Util decomposition.
\item SOAP's peak is higher; its effective LR region is wider.
\item At extreme LR, truth-ratio and retention failures can collapse Agg.
\end{itemize}

\papersource{Paper, Fig. 4, PDF p. 8; normalized sensitivity/JSD and sub-metrics in Figs. 9--10, PDF pp. 26--27.}
\end{frame}
\begin{frame}{RMU updates become more selective across blocks}

\fig{fig5}{3.6cm}
\begin{itemize}
\item Left: mean update magnitude by block; right: hidden-state drift.
\item SOAP places more update mass early/mid-stack, especially at L1.
\item Forget drift increases more than retain drift, supporting selective change.
\end{itemize}

\papersource{Paper, Fig. 5, PDF p. 9. The reference line is called forget gradient in the figure and gradient disagreement in the text.}
\end{frame}
\begin{frame}{Better optimization carries substantial time and memory costs}

\centering
\begin{tabular}{lrrr}
\toprule
3B / TOFU 10\% & Time (min) & Peak (GiB) & Agg\\
\midrule
GD-AdamW & 8.79 & 14.64 & .3813\\
GD-SOAP & 18.08 & 36.76 & .6091\\
NPO-AdamW & 15.69 & 20.55 & .7243\\
NPO-SOAP & 24.32 & 42.75 & .7262\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item GD-SOAP: $2.06\times$ total time, $2.51\times$ peak memory at equal epochs.
\item NPO: higher cost buys a much smaller score gain in this setting.
\item Matched-time trajectories can favor SOAP; per-step computation is not cheaper.
\end{itemize}

\papersource{Paper, Table 14, PDF p. 37; Agg from Table 12, PDF p. 36. Ratios calculated from reported means.}
\end{frame}
\begin{frame}{RWKU gains can come from retaining more neighboring knowledge}

\centering
\begin{tabular}{llrrrr}
\toprule
Loss & Update & Forget $\downarrow$ & Neighbor $\uparrow$ & MMLU $\uparrow$ & Gain $\uparrow$\\
\midrule
GD & AdamW & .1127 & .5657 & .6392 & .1117\\
GD & SOAP & .1460 & .6539 & .6376 & .1295\\
NPO & AdamW & .3250 & .7520 & .6292 & .0997\\
NPO & SOAP & .2034 & .7334 & .6281 & .1336\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Llama-3-8B-Instruct; real-person knowledge removal.
\item GD-SOAP has higher joint Gain, although remaining target knowledge rises.
\item Gain averages signed changes from Original across all three metrics.
\end{itemize}

\papersource{Paper, Table 3 and Sec. 5.5, PDF pp. 9--10.}
\end{frame}
\begin{frame}{WMDP improves the forgetting--retention balance}

\centering
\begin{tabular}{llrrrr}
\toprule
Loss & Update & Bio $\downarrow$ & Cyber $\downarrow$ & MMLU $\uparrow$ & Gain $\uparrow$\\
\midrule
GD & AdamW & .2916 & .2574 & .4527 & .1271\\
GD & SOAP & .2789 & .2473 & .5084 & .1533\\
RMU & AdamW & .3079 & .2723 & .5387 & .1454\\
RMU & SOAP & .3028 & .2581 & .5476 & .1548\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Zephyr-7B-$\beta$; four-way multiple-choice Bio/Cyber tests.
\item Chance-level reference is .25; it does not test every form of hazardous capability.
\item SOAP improves joint Gain for GD, NPO, and RMU.
\end{itemize}

\papersource{Paper, Table 3, PDF p. 10; three-seed standard deviations in Table 13, PDF p. 36.}
\end{frame}
\begin{frame}{Relearning recovers less content after spectral updates}

\begin{columns}[c]
\begin{column}{0.55\textwidth}\fig{extraction_strength}{4.5cm}\end{column}
\begin{column}{0.42\textwidth}
\begin{itemize}
\item Forget 10\% of all data; relearn on 10\% of that set (1\% of all data).
\item Solid SOAP curves end below dashed AdamW curves for each loss.
\item NPO stays closest to the retain-only oracle.
\item Recovery still occurs; this is empirical resistance under a limited attack.
\end{itemize}
\end{column}
\end{columns}

\papersource{Paper, Fig. 6 and Sec. 5.6, PDF p. 10. Extraction strength is lower-is-better.}
\end{frame}
\begin{frame}{The contribution is useful geometry, with concrete boundaries}

\begin{enumerate}
\item \textbf{Mechanism:} flatten gradient singular amplitudes while retaining matrix directions.
\item \textbf{Evidence:} improved multi-axis results, useful LR ranges, and reduced relearning recovery.
\item \textbf{Conditions:} practical optimizers approximate ideal directions; small splits have counterexamples.
\item \textbf{Trade-off:} extra time/memory, with objective-dependent gains.
\end{enumerate}
\medskip
\alert{Open questions:} distinguish general optimization gains from unlearning-specific gains, and compare alternative layerwise norms.

\papersource{Paper, Sec. 6 and App. H, PDF pp. 10, 43; numerical boundaries from Appendix G.}
\end{frame}
\begin{frame}{References and official implementation}

\begin{itemize}
\item Choi, Cai, Kim, and Joo.\\
\emph{Spectral Unlearning: Transformer Structure-Preserving Updates for Language Models}.\\
\href{https://arxiv.org/abs/2611.11902v1}{arXiv:2611.11902v1}, 9 November 2026.
\item Accepted at NeurIPS 2026, per \href{https://rain.korea.ac.kr/}{authors' laboratory} and arXiv comments.
\item \href{https://github.com/unlearn-ml/spectral-unlearning}{Official repository}: implementations, configurations, and experiment scripts.
\end{itemize}
\medskip
Figures are reused from the authors' arXiv v1 source. Calculated counts and ratios are labeled. This is a reading seminar, not an author presentation.

\papersource{Publication/code checked 12 November 2026. Formal proceedings publication was not independently established.}
\end{frame}
\appendix
\begin{frame}{Backup: the layerwise maximum norm couples step magnitudes}

\[
\|\Delta W\|_{\rm net}=\max_l\|\Delta W_l\|_{{\rm RMS}\to{\rm RMS}},\quad
a_l=\sqrt{d_{\rm out,l}/d_{\rm in,l}}
\]
\[
s_l=a_l\operatorname{tr}\Sigma_l,\qquad
c^*=\frac1\lambda\sum_ks_k,\qquad
\Delta W_l^*=-c^*a_lU_lV_l^\top
\]
\begin{itemize}
\item Fix the maximum step budget: every nonzero-gradient layer can use it without increasing the penalty.
\item All such layers take the same RMS-to-RMS magnitude $c^*$.
\item Square matrices have $a_l=1$; rectangular matrices retain dimensional factors.
\item The empirical advantage of this max aggregation over layerwise sums is open.
\end{itemize}

\papersource{Paper, Theorem 2, Eq. (17), PDF pp. 6--7; proof in App. A.4, PDF pp. 16--17; App. H, PDF p. 43.}
\end{frame}
\begin{frame}{Backup: the memorization score penalizes overconfident wrong answers}

\[
\mathrm{Mem}=\mathrm{HM}(1-\mathrm{ES},1-\mathrm{EM},1-\mathrm{F.Prob},\mathrm{F.TR})
\]
\[
R_z=\frac{p_z^{\rm pert}}{p_z^{\rm para}+10^{-10}},\qquad
\mathrm{F.TR}=\mathbb E_z\min(R_z,(R_z+10^{-10})^{-1})
\]
\begin{itemize}
\item ES: contiguous suffix extraction; EM: token-level greedy agreement.
\item F.Prob: length-normalized likelihood on original forget prompts.
\item F.TR peaks when paraphrased correct and perturbed wrong answers have comparable likelihood.
\item Paraphrase probability is reported separately, but excluded from this Mem aggregate.
\end{itemize}

\papersource{Paper, App. D.1, Eqs. (51)--(53), PDF pp. 22--23. EM here is not whole-answer exact match.}
\end{frame}
\begin{frame}{Backup: privacy targets the retain oracle, not minimum AUC}

\[
s_k=1-|\mathrm{AUC}_k-\mathrm{AUC}^{\rm retain}_k|,\qquad
\mathrm{Priv}=\mathrm{HM}(s_{\rm LOSS},s_{\rm ZLib},s_{\rm MinK},s_{\rm MinK++})
\]
\begin{itemize}
\item LOSS, compression-adjusted loss, bottom-40\% token scores, and standardized bottom-token scores.
\item AUC below .5 does not automatically mean better privacy under this protocol.
\item Util gives equal weight to ten sub-metrics: three measures on each of retain/real-author/world-facts, plus fluency.
\item A non-gibberish classifier may reward fluent repetition; examine outputs as well as scores.
\end{itemize}

\papersource{Paper, App. D.2--D.3, Eqs. (54)--(56), PDF p. 23; high-LR failure discussed in Sec. 5.3, PDF p. 9.}
\end{frame}
\begin{frame}{Backup: matched-time curves reveal objective-dependent behavior}

\fig{wallclock-3b10}{3.7cm}
\begin{itemize}
\item Fig. 13 panels (g)--(i): 3B, forget 10\%; GD, NPO, RMU.
\item Baselines run past SOAP's endpoint; NPO can approach a similar plateau.
\item The full figure covers 5\%/10\%, not 1\%; NPO at 3B/5\% even shows a slightly higher late AdamW score.
\end{itemize}

\papersource{Paper, Fig. 13, PDF p. 29. Shared legend and selected original panels composed without changing data.}
\end{frame}
\begin{frame}{Backup: flattening also benefits retain-only fine-tuning}

\begin{columns}[c]
\begin{column}{0.48\textwidth}\fig{spectral_exponent_sft_paper_style}{3.8cm}\end{column}
\begin{column}{0.49\textwidth}
\[
\Delta W_p=-\eta\frac{U\Sigma^pV^\top}{\operatorname{RMS}(U\Sigma^pV^\top)}
\]
\begin{itemize}
\item $p=1$: momentum direction; $p=0$: flattened spectrum.
\item Lower $p$ improves mean utility for GD, NPO, and retain SFT.
\item General optimization gains may contribute to the unlearning improvement.
\end{itemize}
\end{column}
\end{columns}

\papersource{Paper, App. E.7, Fig. 14, PDF p. 30. Shading: seed variation after averaging over learning rates.}
\end{frame}
\begin{frame}{Backup: parameter trajectories support the optimization account}

\fig{npo-trajectory-ab}{3.6cm}
\begin{itemize}
\item NPO, 1B, forget 10\%: joint objective space and forget objective over steps.
\item SOAP reaches low forget-objective values with competitive retain loss.
\item Forget loss is the NPO objective. Fig. 12 adds PCA paths; these visuals do not isolate direction from all optimizer factors.
\end{itemize}

\papersource{Paper, App. E.5, Fig. 11 panels (a)--(b), PDF p. 28; Fig. 12 provides PCA parameter paths.}
\end{frame}
\begin{frame}{Backup: qualitative examples expose failure modes hidden by scores}

\begin{tabular}{lp{9.8cm}}
\toprule
Forget side & GD can reproduce the target; NPO can produce fluent unrelated hallucinations; RMU can produce degenerate sequences.\\[5pt]
Retain side & NPO/RMU with SOAP often preserve details better in the selected examples.\\[5pt]
Persistent failure & GD variants still answer some general-knowledge questions incorrectly, including the SOAP variant.\\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Changing an answer is not enough to establish useful unlearning.
\item Five selected examples illustrate mechanisms; they do not estimate failure frequencies.
\end{itemize}

\papersource{Paper, App. F, Tables 7--11, PDF pp. 31--34. Generations are truncated in the source tables.}
\end{frame}
\end{document}