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% 22 main slides for a 20--30 minute discussion; 5 backup slides.
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\title[Q-Delta]{Q-Delta: Beyond Key--Value\\Associative State Evolution}
\subtitle{Reading discussion: query feedback in recurrent memory}
\author{Paper authors: Sumin Park, Seojin Kim, Noseong Park\\KAIST}
\date{arXiv:2606.08804v1\quad |\quad Reading: October 8, 2026}

\begin{document}

% 1
\begin{frame}
  \titlepage
  \vspace{-3mm}
  \begin{center}
    \alert{Use the query to help decide what the state should remember.}
  \end{center}
  \papersource{A reading presentation, not an author presentation. Paper: ICML 2026; arXiv v1, June 7, 2026.}
\end{frame}

% 2
\begin{frame}{The readout direction can shape future memory}
  \begin{block}{The gap}
    Conventional delta updates correct a key-based prediction.\\
    The query reads the result but does not enter that correction.
  \end{block}
  \[
    \underbrace{S_{t-1}k_t}_{\text{key prediction}}
    \qquad\qquad
    \underbrace{S_{t-1}q_t}_{\text{query prediction}}
  \]
  \begin{itemize}
    \item Both are value-space projections of the same memory.
    \item Q-Delta uses both to form the write residual.
    \item The recurrent state keeps the same dimensions.
  \end{itemize}
  \papersource{Paper, Secs. 1 and 3, PDF pp. 1--5; Appendix B, p. 13.}
\end{frame}

% 3
\begin{frame}{Delta rules correct an existing key--value association}
  \textbf{Additive linear attention}
  \[
    S_t=S_{t-1}+v_tk_t^\top,\qquad o_t=S_tq_t
  \]
  \textbf{DeltaNet: write a prediction residual}
  \[
    S_t=S_{t-1}+\beta_t(v_t-S_{t-1}k_t)k_t^\top
  \]
  \textbf{Gated DeltaNet: decay, then correct}
  \[
    S_t=\alpha_tS_{t-1}
      +\beta_t(v_t-\alpha_tS_{t-1}k_t)k_t^\top
  \]
  \alert{The query is absent from every write residual above.}
  \papersource{Paper, Sec. 2.1, Eqs. (1)--(3), PDF p. 2. Single head: $S\in\R^{d_v\times d_k}$.}
\end{frame}

% 4
\begin{frame}{A query reads another mixture of past values}
  For $S_t=S_{t-1}P_t+\eta_tv_tk_t^\top$ and $S_0=0$:
  \[
    S_{t-1}=\sum_{\tau<t}v_\tau b_{\tau,t-1}^\top,
    \qquad b_{\tau,t}=P_t^\top b_{\tau,t-1}.
  \]
  \begin{block}{Query-conditioned value aggregation}
    \[
      \hat o_t=S_{t-1}q_t
      =\sum_{\tau<t}(b_{\tau,t-1}^\top q_t)v_\tau
    \]
  \end{block}
  \begin{itemize}
    \item State transitions reshape the effective historical keys.
    \item Weights are unnormalized and may be signed.
    \item The identity holds for any probe; $q_t$ is the actual readout direction.
  \end{itemize}
  \papersource{Paper, Sec. 3.1, Eqs. (8)--(15), PDF pp. 3--4; proof in Appendix C.1, pp. 13--14.}
\end{frame}

% 5
\begin{frame}{The two readouts emphasize different value directions}
  \centering
  \includegraphics[width=0.94\linewidth,height=4.25cm,keepaspectratio]{figures/fig-02-complementarity.pdf}
  \begin{itemize}
    \item Mean cosine similarity: \textbf{0.073}; 340M model, layers 5/10/15.
    \item Principal directions of $\hat o$ capture much less energy of $\hat v$.
  \end{itemize}
  \papersource{Paper, Fig. 2, PDF p. 5; Sec. 3.2, pp. 4--5. Author figure; 10,000 sampled timesteps.}
\end{frame}

% 6
\begin{frame}{Q-Delta adds query feedback to the write residual}
  \[
    \hat v_t=S_{t-1}k_t,\qquad \hat o_t=S_{t-1}q_t
  \]
  \begin{block}{Without state decay}
    \[
      S_t=S_{t-1}+\beta_t
      \bigl(v_t-\hat v_t-\alert{\lambda_t\hat o_t}\bigr)k_t^\top
    \]
  \end{block}
  \[
    x_t=k_t+\lambda_tq_t
    \quad\Longrightarrow\quad
    S_t=S_{t-1}(I-\beta_tx_tk_t^\top)+\beta_tv_tk_t^\top
  \]
  \begin{itemize}
    \item Predict along $x_t$; write along $k_t$; output along $q_t$.
    \item This is not a gradient step on $\frac12\|v_t-Sx_t\|^2$:
      that gradient would write along $x_t$.
  \end{itemize}
  \papersource{Paper, Eqs. (17)--(18), PDF p. 5; Sec. 3.3.3, p. 6.}
\end{frame}

% 7
\begin{frame}{The deployed rule first decays the old state}
  \[
    \bar S_{t-1}=\alpha_t S_{t-1},\qquad x_t=k_t+\lambda_tq_t
  \]
  \begin{block}{One recurrent step}
    \[
      S_t=\bar S_{t-1}
      +\beta_t\bigl(v_t-\bar S_{t-1}x_t\bigr)k_t^\top,
      \qquad o_t=S_tq_t
    \]
  \end{block}
  \begin{center}
    \begin{tabular}{lll}
      \toprule
      Gate & Role & Special case \\
      \midrule
      $\alpha_t$ & Retain the previous state & $\alpha_t=1$: no decay \\
      $\beta_t$ & Scale the write residual & $\beta_t=0$: no write \\
      $\lambda_t$ & Scale query feedback & $\lambda_t=0$: Gated DeltaNet \\
      \bottomrule
    \end{tabular}
  \end{center}
  \papersource{Paper, Eqs. (19)--(20), PDF p. 5. The fresh value term is not multiplied by $\alpha_t$.}
\end{frame}

% 8
\begin{frame}{One extra projection controls each head's feedback}
  \begin{columns}[c,onlytextwidth]
    \column{0.55\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=4.9cm,keepaspectratio]{figures/fig-01-architecture.pdf}
    \column{0.42\textwidth}
      \[
        \lambda_t=\sigma(W_\lambda h_t+b)
      \]
      \[
        W_\lambda\in\R^{H\times d_{\mathrm{model}}}
      \]
      \begin{itemize}
        \item Per token, per head.
        \item Scalar bias initialized to $-0.8$.
        \item No extra recurrent state.
        \item Same $q,k,v$ transformations as the underlying block.
      \end{itemize}
  \end{columns}
  \papersource{Paper, Fig. 1, PDF p. 2; Appendix B, p. 13. Author architecture figure.}
\end{frame}

% 9
\begin{frame}{One-step contraction depends on directional alignment}
  \textbf{Lemma 3.1 assumes $\alpha_t=1$.} Define
  \[
    a_t=k_t^\top x_t,\qquad
    \widetilde e_t=v_t-S_{t-1}x_t,\qquad e_t=v_t-S_tx_t.
  \]
  Right-multiplying the update by $x_t$ gives the exact identity:
  \begin{block}{The central algebraic fact}
    \[
      e_t=(1-\beta_ta_t)\widetilde e_t
    \]
  \end{block}
  \begin{itemize}
    \item $0<\beta_ta_t<2$ implies contraction of this mixed error.
    \item It does not establish contraction of the entire state norm.
    \item A uniform rate requires $\sup_t|1-\beta_ta_t|<1$.
  \end{itemize}
  \papersource{Paper, Lemma 3.1, PDF p. 6; Appendix C.2 proof, p. 14.}
\end{frame}

% 10
\begin{frame}{Global tracking also requires bounded residual drift}
  \[
    r_t=\underbrace{v_t-v_{t-1}}_{\text{target drift}}
       -\underbrace{S_{t-1}(x_t-x_{t-1})}_{\text{prediction drift}}
  \]
  Since $\widetilde e_t=e_{t-1}+r_t$, if $\|r_t\|\le r$ and $\rho<1$:
  \[
    \boxed{\|e_t\|\le\rho^t\|e_0\|
      +\frac{\rho(1-\rho^t)}{1-\rho}\,r}
  \]
  \begin{itemize}
    \item Initial error fades; persistent drift sets a nonzero error radius.
    \item Pointwise contraction alone does not supply a uniform margin.
    \item With $\alpha_t<1$, an extra term
      $(1-\alpha_t)S_{t-1}x_t$ enters tracking (backup slide).
  \end{itemize}
  \papersource{Paper, Theorem 3.2, PDF pp. 6--7; proof pp. 14--16. Decay extension: reader derivation.}
\end{frame}

% 11
\begin{frame}{Measured dynamics fall within the contraction regime}
  \centering
  \includegraphics[width=0.94\linewidth,height=4.25cm,keepaspectratio]{figures/fig-03-stability.pdf}
  \begin{itemize}
    \item Mean $\beta_ta_t$: \textbf{0.043}; mean residual drift norm: \textbf{0.538}.
    \item Prediction drift is smaller than target drift in these samples.
  \end{itemize}
  \papersource{Paper, Fig. 3, PDF p. 6; discussion p. 7. Sample means do not specify worst-case $\rho$ or $r$.}
\end{frame}

% 12
\begin{frame}{Chunking turns token dependencies into triangular solves}
  Stack $X,K$ by rows. The interaction is
  \[
    (XK^\top)_{ri}=x_r^\top k_i,
    \qquad x_r=k_r+\lambda_rq_r.
  \]
  With $B=\diag(\beta)$ and causal decay ratios $H_{ri}$:
  \[
    L=\slower(BXK^\top),\qquad
    L_\gamma=\slower\bigl(B(H\odot XK^\top)\bigr)
  \]
  \[
    W=(I+L)^{-1}BX,\qquad
    \widetilde U=(I+L_\gamma)^{-1}BV
  \]
  \begin{itemize}
    \item Chunk-local matrix operations; recurrent state passed between chunks.
    \item Same state size; linear scaling in length for fixed chunk/head sizes.
    \item Appendix D.2 prints $KX^\top$; the recurrence requires $XK^\top$.
  \end{itemize}
  \papersource{Paper, Eqs. (23)--(27), PDF pp. 5--6; Appendix D, pp. 16--18. $\slower$: strict lower triangle.}
\end{frame}

% 13
\begin{frame}{Comparisons match data and optimization settings}
  \begin{center}
    \begin{tabular}{lrr}
      \toprule
      Model size & FineWeb-Edu tokens & Peak learning rate \\
      \midrule
      340M & 15B & $10^{-3}$ \\
      1.3B & 30B & $4\times10^{-4}$ \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{itemize}
    \item FLA framework; 4 RTX Pro 6000 Blackwell GPUs; bfloat16.
    \item AdamW, cosine schedule, gradient clipping; baselines retrained.
    \item Baselines: RetNet, Mamba, Mamba2, DeltaNet, Gated DeltaNet.
    \item Language: both scales. Synthetic retrieval: 1.3B at 1K--4K.
    \item Real-world recall: 340M, inputs capped at 2K, cloze prompts.
  \end{itemize}
  \papersource{Paper, Secs. 4.1--4.2, PDF pp. 7--8; Appendix A, p. 13.}
\end{frame}

% 14
\begin{frame}{Training throughput stays close to Gated DeltaNet}
  \centering
  \includegraphics[width=0.94\linewidth,height=4.3cm,keepaspectratio]{figures/fig-04-training-throughput.pdf}
  \begin{itemize}
    \item 340M; single GPU; 50 training steps per throughput setting.
    \item Fixed 32,768 tokens/step; lengths 2K--16K. DeltaNet is faster.
  \end{itemize}
  \papersource{Paper, Fig. 4, PDF p. 7. Left: loss over 28,600 steps. Right: training throughput, not decoding.}
\end{frame}

% 15
\begin{frame}{Average zero-shot accuracy improves at both scales}
  \begin{center}
    \begin{tabular}{lrr}
      \toprule
      Model & 340M / 15B tokens & 1.3B / 30B tokens \\
      \midrule
      RetNet & 44.55 & 50.31 \\
      Mamba & 46.64 & 52.39 \\
      Mamba2 & 46.27 & 52.46 \\
      DeltaNet & 43.82 & 52.53 \\
      Gated DeltaNet & 46.01 & 52.77 \\
      \textbf{Q-Delta} & \textbf{47.24} & \textbf{53.47} \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{itemize}
    \item Over Gated DeltaNet: \textbf{+1.23 pp} at 340M, \textbf{+0.70 pp} at 1.3B.
    \item At 340M, the gain over the best other average is +0.60 pp.
  \end{itemize}
  \papersource{Paper, Table 2, PDF p. 8. Unweighted mean of 8 accuracy tasks; higher is better.}
\end{frame}

% 16
\begin{frame}{Perplexity gains depend on scale and comparison}
  \begin{center}
    \begin{tabular}{llrr}
      \toprule
      Scale & Model & LAMBADA ppl. $\downarrow$ & WikiText ppl. $\downarrow$ \\
      \midrule
      340M & Mamba2 & \textbf{30.35} & \textbf{26.60} \\
            & Gated DeltaNet & 36.27 & 27.82 \\
            & Q-Delta & 32.67 & 26.89 \\
      \midrule
      1.3B & Mamba2 & 17.40 & 19.47 \\
            & Gated DeltaNet & 15.32 & 19.61 \\
            & Q-Delta & \textbf{15.19} & \textbf{19.21} \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{itemize}
    \item Q-Delta improves both perplexities over Gated DeltaNet.
    \item Mamba2 leads at 340M; Q-Delta leads all listed baselines at 1.3B.
  \end{itemize}
  \papersource{Paper, Table 2, PDF p. 8. Selected rows; the accuracy average excludes perplexity.}
\end{frame}

% 17
\begin{frame}{The largest synthetic retrieval gains occur at 4K}
  \begin{center}
    \begin{tabular}{lrrrr}
      \toprule
      & \multicolumn{3}{c}{4K accuracy (\%)} & Mean over \\
      Model & Pass-key & Number & UUID & all 9 settings \\
      \midrule
      Mamba2 & 99.0 & 57.0 & 11.6 & 76.58 \\
      DeltaNet & 100.0 & 49.6 & 25.4 & 81.29 \\
      Gated DeltaNet & 100.0 & 76.6 & 21.4 & 83.51 \\
      \textbf{Q-Delta} & \textbf{100.0} & \textbf{94.2} & \textbf{48.0} & \textbf{90.02} \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{block}{Relative to Gated DeltaNet, in percentage points}
    Number at 4K: \textbf{+17.6}\quad UUID at 4K: \textbf{+26.6}\\
    Mean over 3 tasks $\times$ 3 lengths: \textbf{+6.51}
  \end{block}
  \papersource{Paper, Table 3, PDF p. 8. 1.3B models; RULER S-NIAH at 1K, 2K, and 4K.}
\end{frame}

% 18
\begin{frame}{UUID retrieval still degrades sharply with length}
  \begin{center}
    \begin{tabular}{lrrr}
      \toprule
      Context & DeltaNet & Gated DeltaNet & Q-Delta \\
      \midrule
      1K & 87.4 & 83.8 & \textbf{94.6} \\
      2K & \textbf{75.8} & 70.0 & 74.0 \\
      4K & 25.4 & 21.4 & \textbf{48.0} \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{itemize}
    \item At 2K, DeltaNet retains the best UUID result.
    \item At 4K, Q-Delta improves substantially but succeeds only 48\% of the time.
    \item Quality is evaluated up to 4K; the 16K experiment measures throughput.
  \end{itemize}
  \papersource{Paper, Table 3, PDF p. 8, accuracy in percent; throughput settings in Fig. 4, p. 7.}
\end{frame}

% 19
\begin{frame}{Real-world recall shows gains and regressions}
  \begin{center}
    \begin{tabular}{lrrr}
      \toprule
      Task & Gated DeltaNet & Q-Delta & Difference \\
      \midrule
      SWDE & 29.1 & 31.6 & +2.5 \\
      SQuAD & 56.0 & 52.3 & $-3.7$ \\
      FDA & 18.9 & 22.0 & +3.1 \\
      TriviaQA & 48.9 & 48.6 & $-0.3$ \\
      NQ & 18.2 & 18.4 & +0.2 \\
      DROP & 34.2 & 33.2 & $-1.0$ \\
      \bottomrule
    \end{tabular}
  \end{center}
  \textbf{Reader-computed unweighted means:} 34.22 vs. 34.35.\\
  The average advantage is only \alert{0.13 score points}.
  \papersource{Paper, Table 4, PDF p. 8. 340M, 2K cap, cloze prompts. Original table scores; mean not printed there.}
\end{frame}

% 20
\begin{frame}{Fixed feedback captures most of the accuracy gain}
  \begin{center}
    \begin{tabular}{lrrr}
      \toprule
      340M configuration & Wiki ppl. $\downarrow$ & Lamb. ppl. $\downarrow$ & Avg. acc. $\uparrow$ \\
      \midrule
      Learnable $\lambda_t$ & 26.89 & \textbf{32.67} & \textbf{47.24} \\
      Fixed $\lambda=0.2$ & 26.96 & 35.39 & 46.99 \\
      Fixed $\lambda=0.5$ & 26.86 & 33.31 & 47.20 \\
      Fixed $\lambda=0.8$ & 26.61 & 33.58 & 46.42 \\
      No decay: $\alpha_t=1$ & \textbf{26.52} & 32.97 & 45.86 \\
      Full feedback: $\lambda_t=1$ & 26.55 & 35.21 & 46.36 \\
      \bottomrule
    \end{tabular}
  \end{center}
  \begin{itemize}
    \item Learnable vs. fixed 0.5: only \textbf{0.04 pp} in mean accuracy.
    \item No-decay Q-Delta (45.86) still exceeds DeltaNet (43.82).
  \end{itemize}
  \papersource{Paper, Table 5, PDF p. 9; DeltaNet from Table 2, p. 8. ``No gating'' means $\lambda=1$, not $0$.}
\end{frame}

% 21
\begin{frame}{Query feedback is a useful, compact memory modification}
  \begin{enumerate}
    \item \textbf{Design:} separate the mixed prediction direction, the write
      address, and the output query.
      \medskip
    \item \textbf{Evidence:} better average language scores and stronger 4K
      synthetic retrieval, at training throughput close to Gated DeltaNet.
      \medskip
    \item \textbf{Trade-offs:} real-world recall is mixed; adaptive feedback
      barely improves over a well-chosen fixed coefficient.
      \medskip
    \item \textbf{Theory:} conditional mixed-error tracking; the decay gate
      needs an additional drift argument.
  \end{enumerate}
  \papersource{Synthesis of Sec. 3.3, Tables 2--5, and Fig. 4; PDF pp. 5--9.}
\end{frame}

% 22
\begin{frame}{Paper and directions for discussion}
  \textbf{Sumin Park, Seojin Kim, Noseong Park.}\\
  \emph{Q-Delta: Beyond Key--Value Associative State Evolution.}\\
  ICML 2026; arXiv:2606.08804v1, June 7, 2026.
  \medskip

  \href{https://arxiv.org/abs/2606.08804v1}{arxiv.org/abs/2606.08804v1}\\
  Author-listed code: \href{https://github.com/psmiz/Q-Delta}{github.com/psmiz/Q-Delta}
  \medskip
  \begin{itemize}
    \item Extend tracking theory to the actual decay-gated recurrence.
    \item Evaluate retrieval beyond 4K and measure recurrent decoding cost.
    \item Identify tasks where token-dependent $\lambda_t$ matters.
  \end{itemize}
  \papersource{All empirical figures and results in this deck come from v1. Supplemental algebra is labeled as reader derivation.}
\end{frame}

\appendix

% 23
\begin{frame}{Backup: a small example separates prediction and write}
  \textbf{Reader-created example; $\alpha=1$:}
  \[
    S_{t-1}=[1\;2],\quad k=[1\;0]^\top,\quad q=[0\;1]^\top,
    \quad v=3,\quad\lambda=\beta=\tfrac12.
  \]
  \[
    x=[1\;\tfrac12]^\top,\qquad
    \widetilde e=3-S_{t-1}x=1
  \]
  \[
    S_t=[1\;2]+\tfrac12\cdot1\cdot[1\;0]=[1.5\;2]
  \]
  \begin{itemize}
    \item Mixed error becomes $3-S_tx=0.5$: contraction factor $1-\beta k^\top x=0.5$.
    \item The immediate output remains $S_tq=2$ because $k^\top q=0$.
    \item The memory has changed and can affect later queries.
  \end{itemize}
  \papersource{Teaching example from Eq. (17), PDF p. 5, and Lemma 3.1, p. 6; not a paper experiment.}
\end{frame}

% 24
\begin{frame}{Backup: a consistent chunk output formula}
  Row stacks: $Q,K,W\in\R^{C\times d_k}$,
  $\widetilde U\in\R^{C\times d_v}$,
  $S_{\rm in}\in\R^{d_v\times d_k}$.
  \[
    \gamma_i=\prod_{j\le i}\alpha_j,\quad D_\gamma=\diag(\gamma),\quad
    H_{ij}=1_{\{j\le i\}}\,\gamma_i/\gamma_j
  \]
  \[
    Z=\widetilde U-D_\gamma W S_{\rm in}^{\top},\qquad
    K_{\rm out}=\diag(\gamma_C/\gamma_i)K
  \]
  \[
    S_{\rm out}=\gamma_C S_{\rm in}+Z^\top K_{\rm out}
  \]
  \[
    O=D_\gamma Q S_{\rm in}^{\top}+(H\odot QK^\top)Z
  \]
  \textbf{In v1 Eq. (26):} the entrance-state product needs $S^\top$
  under row stacking, and intra-chunk output needs the decay ratios.
  \papersource{Reader derivation from Eq. (25), PDF p. 5. Compare Eq. (26), p. 6; WY proof, pp. 16--17.}
\end{frame}

% 25
\begin{frame}{Backup: decay changes the cross-time tracking residual}
  For the actual gated recurrence:
  \[
    e_t=(1-\beta_ta_t)\bigl(v_t-\alpha_tS_{t-1}x_t\bigr)
  \]
  Using the theorem's original $e_{t-1}$ and $r_t$:
  \[
    v_t-\alpha_tS_{t-1}x_t
    =e_{t-1}+r_t+\underbrace{(1-\alpha_t)S_{t-1}x_t}_{\text{additional drift}}.
  \]
  \begin{itemize}
    \item A gated tracking bound must control this term too.
    \item For unit-norm $q,k$, $a_t\in[1-\lambda_t,1+\lambda_t]$.
    \item This gives pointwise contraction for $0<\lambda_t<1$ and
      $0<\beta_t\le1$, but not a uniform margin away from 0 and 2.
  \end{itemize}
  \papersource{Reader derivation from Eq. (19), PDF p. 5; normalization in Fig. 1, p. 2; theorem pp. 6, 14--16.}
\end{frame}

% 26
\begin{frame}{Backup: zero-shot accuracy has task-level trade-offs}
  \begin{center}
    \begin{tabular}{lrrrr}
      \toprule
      & \multicolumn{2}{c}{340M} & \multicolumn{2}{c}{1.3B} \\
      Task & GDN & Q-Delta & GDN & Q-Delta \\
      \midrule
      ARC-Easy & 60.02 & 59.51 & 68.60 & 68.27 \\
      ARC-Challenge & 25.94 & 28.50 & 33.28 & 36.60 \\
      HellaSwag & 40.25 & 41.61 & 52.60 & 53.46 \\
      LAMBADA & 31.52 & 33.90 & 43.80 & 43.28 \\
      PIQA & 67.30 & 67.63 & 70.84 & 71.44 \\
      WinoGrande & 51.54 & 52.88 & 54.78 & 54.93 \\
      BoolQ & 57.13 & 59.48 & 59.42 & 61.41 \\
      OpenBookQA & 34.40 & 34.40 & 38.80 & 38.40 \\
      \bottomrule
    \end{tabular}
  \end{center}
  \papersource{Paper, Table 2, PDF p. 8. Accuracy in percent; GDN = Gated DeltaNet.}
\end{frame}

% 27
\begin{frame}{Backup: a surrogate objective explains the update}
  A mixed squared error has gradient
  \[
    \ell(S)=\tfrac12\|v_t-Sx_t\|^2,
    \qquad -\nabla_S\ell=(v_t-Sx_t)x_t^\top.
  \]
  Q-Delta instead writes along $k_t$. Fix the previous state and set
  $\delta_t=v_t-\alpha_tS_{t-1}x_t$. Then
  \[
    J_t(S)=\|S-\alpha_tS_{t-1}\|_F^2
      -2\beta_t\langle Sk_t,\delta_t\rangle
  \]
  has the minimizer
  \[
    \arg\min_S J_t(S)=\alpha_tS_{t-1}+\beta_t\delta_tk_t^\top.
  \]
  \alert{A per-step surrogate is compatible with the update; direct
  gradient descent on the mixed squared loss is a different rule.}
  \papersource{Paper, Table 1, PDF p. 3; Sec. 3.3.3, p. 6. Algebra of the surrogate expanded for teaching.}
\end{frame}
\end{document}