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\title[Self-Improving Branches]{Mixture of Self-Improving Branches for Agent Harness Optimization}
\subtitle{Paper discussion: adaptive search, routing, and evidence}
\author[Dong et al.]{\small Paper authors: Haoyu Dong, Yuhang Zhou, Zihao Lin, Yifan Wu, Bo Peng,\\
Mingyi Wang, Xiangjun Fan, Lizhu Zhang, Zhuokai Zhao}
\date{\href{https://arxiv.org/abs/2609.37834v1}{arXiv:2609.37834v1} \quad 23 October 2026}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Adapt the search process, then route among its products}
  \begin{block}{Core idea}
    Let branches change both the development cases they optimize and the guidance used to propose their next harness; deploy their best heads through an input-level router.
  \end{block}
  \begin{itemize}
    \item Math--Gemini held-out accuracy: \textbf{46.0\% $\rightarrow$ 62.0\%} vs Meta-Harness.
    \item Relative gains: 34.8\% Math--Gemini, 11.6\% Terminal-Bench 2.0, 3.8\% SWE-bench Lite.
    \item Selection and router configuration use development data; test-best analysis is reported separately.
  \end{itemize}
  \papersource{Paper, Abstract and Table 1, PDF pp. 1, 6; Sec. 4.6, PDF p. 9.}
\end{frame}

\begin{frame}{A fixed development set can hide complementary harnesses}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Harness}
        Executable code around the action model: retrieval, prompts, tool interfaces, verification, and control flow.
      \end{block}
      \vspace{2mm}
      Meta-Harness iteratively proposes code and scores it on a fixed development set.
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Failure mode motivating branches}
        A candidate that solves cases missed by the leading harness can rank lower overall, so later proposals may never build on it.
      \end{block}
      \vspace{2mm}
      The paper also changes proposal instructions as each branch learns from its own history.
    \end{column}
  \end{columns}
  \papersource{Paper, Sec. 1, PDF pp. 1--2.}
\end{frame}

\begin{frame}{Each branch maintains an objective and a proposal policy}
  \centering
  \begin{tabular}{c@{\hspace{5mm}}c@{\hspace{5mm}}c}
    \fbox{Branch-local history $\mathcal D_b^t$} &
    \fbox{Development subset $\mathcal X_b^t$} &
    \fbox{Guidance $S_b^t$} \\
    $\downarrow$ & $\downarrow$ & $\downarrow$ \\
    \multicolumn{3}{c}{\fbox{Shared proposer $P$ generates a new harness $H_b^t$}} \\
    \multicolumn{3}{c}{$\downarrow$} \\
    \multicolumn{3}{c}{\fbox{Evaluate with action model $M$; update history and branch head}} \\
  \end{tabular}
  \vspace{5mm}
  \begin{itemize}
    \item Two branches start from the same seeds, full development set, and initial guidance.
    \item One candidate per branch per iteration; branches run in parallel.
  \end{itemize}
  \papersource{Paper, Sec. 3 and Algorithm 1, PDF pp. 4--5. Diagram is an explanatory redrawing.}
\end{frame}

\begin{frame}{Changing the development subset changes the preferred parent}
  \[
    J_b^t(H)=\frac{1}{|\mathcal X_b^t|}
    \sum_{x\in\mathcal X_b^t}\mathbb E[r(f_M(H,x),x)]
  \]
  \begin{itemize}
    \item $H$: candidate harness; $M$: action model; $f_M(H,x)$: execution trajectory; $r$: task reward.
    \item The branch head $\widehat H_b^t$ is the candidate with highest observed mean reward on its current subset.
    \item A subset update can change the ranking and head even when candidate code stays the same.
  \end{itemize}
  \papersource{Paper, Sec. 3, Eq. (1), PDF p. 4.}
\end{frame}

\begin{frame}{Frontier coverage assigns each development case}
  \[
    \mathcal F_b^t=\operatorname{Top}_q(\{H\}_b^t),\qquad
    c_b^t(x)=\sum_{H\in\mathcal F_b^t}\mathbf 1[H\text{ solves }x]
  \]
  \centering
  \begin{tabular}{ll}
    \toprule
    Coverage condition for case $x$ & Subset update \\
    \midrule
    Every frontier harness solves $x$ & Remove from all branches \\
    One branch leads by at least $\Delta$ votes & Keep only in that branch \\
    Otherwise & Preserve current membership \\
    \bottomrule
  \end{tabular}
  \vspace{3mm}
  \begin{itemize}
    \item Default $q=5$, $\Delta=2$: votes $4{:}1$ assign ownership; $5{:}5$ remove; $3{:}2$ leave unchanged.
  \end{itemize}
  \papersource{Paper, Sec. 3, Eq. (2), PDF p. 4. Vote examples expand the stated rule.}
\end{frame}

\begin{frame}{Math--Gemini's development objective changes substantially}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.59\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.65\textheight,keepaspectratio]{figures/fig3-math-gemini-crop.pdf}
    \end{column}
    \begin{column}{0.39\textwidth}
      \textbf{At iteration 20, of 250 cases:}
      \begin{itemize}
        \item 25 shared.
        \item 14 Branch 1 only.
        \item 6 Branch 2 only.
        \item 205 removed from both.
      \end{itemize}
      \vspace{1mm}
      Final head selection uses each branch's retained subset.
    \end{column}
  \end{columns}
  \papersource{Original paper Fig. 3, cropped Math--Gemini panel, PDF p. 14. Bar widths use a log scale.}
\end{frame}

\begin{frame}{Proposal guidance also learns from branch-local history}
  \begin{itemize}
    \item Every $C_s=5$ iterations, the same proposer $P$ revises each branch's \texttt{SKILL.md} guidance.
    \item The update prompt compares candidates within the same development stage and distinguishes repeated evidence from isolated outcomes.
    \item At iteration 15 on Math--Gemini, Branch 1 emphasizes concrete information from retrieved solutions; Branch 2 emphasizes free-form derivation and two-sided closure.
  \end{itemize}
  \begin{block}{Different levers}
    Development updates change which harness scores well; guidance updates change what the proposer is encouraged to try next.
  \end{block}
  \papersource{Paper, Sec. 3, PDF pp. 4--5; Sec. 4.3, PDF p. 7; Appendix C, PDF pp. 14--16.}
\end{frame}

\begin{frame}{A head replacement redirects subsequent proposals}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.24\textwidth}
      \begin{block}{Iteration 13}
        Branch 2 drops four cases: 44 $\rightarrow$ 40.
      \end{block}
    \end{column}
    \begin{column}{0.24\textwidth}
      \begin{block}{New head}
        Free-form derivation replaces a geometry-aware dual solver.
      \end{block}
    \end{column}
    \begin{column}{0.24\textwidth}
      \begin{block}{14--15}
        Targeted retrieval and two-sided closure extend that head.
      \end{block}
    \end{column}
    \begin{column}{0.24\textwidth}
      \begin{block}{17--19}
        Rehearsal, agreement challenge, and gap-directed construction follow.
      \end{block}
    \end{column}
  \end{columns}
  \vspace{4mm}
  Recorded dependencies support a search-redirection mechanism; they do not isolate the cause of the eventual test gain.
  \papersource{Paper, Sec. 4.3, PDF p. 7; Appendix B.2, PDF pp. 13--14.}
\end{frame}

\begin{frame}{The router chooses a branch head before execution}
  \begin{enumerate}
    \item Select one final head per branch using its own final development subset.
    \item Label development cases solved by exactly one head; provide expert code, these examples, and both heads' \emph{development-example} outputs.
    \item Optimize the router instruction with GEPA on those development labels.
    \item For a new input, select one head \emph{before} running it: no test answer or expert output on that input is available.
  \end{enumerate}
  \papersource{Paper, Sec. 3, PDF p. 5; Appendix D.1--D.2, PDF pp. 17--19.}
\end{frame}

\begin{frame}{Four held-out settings test deployment performance}
  \centering
  \begin{tabular}{lccr}
    \toprule
    Setting & Action model & Dev / test & Metric \\
    \midrule
    Olympiad math & Gemini 3 Flash & 250 / 200 & Accuracy \\
    Olympiad math & Sonnet 4.5 & 250 / 200 & Accuracy \\
    Terminal-Bench 2.0 & Sonnet 4.5 & 30 / 58 & Pass rate \\
    SWE-bench Lite & Sonnet 4.5 & 50 / 250 & Resolution \\
    \bottomrule
  \end{tabular}
  \vspace{5mm}
  \begin{itemize}
    \item $B=2$ branches, $N=20$ iterations, one candidate per branch per iteration.
    \item Meta-Harness uses fixed development cases and proposal guidance; both systems select from development data.
  \end{itemize}
  \papersource{Paper, Sec. 4.1, PDF p. 6.}
\end{frame}

\begin{frame}{Held-out gains are largest for Math--Gemini}
  \centering
  \begin{tabular}{lrrrr}
    \toprule
    Setting & Meta-Harness & Routed system & Absolute gain & Relative gain \\
    \midrule
    Math--Gemini & 46.0\% & \textbf{62.0\%} & \textbf{16.0 pp} & \textbf{34.8\%} \\
    Math--Sonnet & 29.0\% & 30.5\% & 1.5 pp & 5.2\% \\
    Terminal-Bench 2.0 & 44.8\% & 50.0\% & 5.2 pp & 11.6\% \\
    SWE-bench Lite & 63.6\% & 66.0\% & 2.4 pp & 3.8\% \\
    \bottomrule
  \end{tabular}
  \vspace{5mm}
  \begin{block}{Read the percentage correctly}
    $46.0\%\rightarrow62.0\%$ is a 16.0 percentage-point increase, or a 34.8\% relative increase.
  \end{block}
  \papersource{Adapted from paper Table 1, PDF p. 6.}
\end{frame}

\begin{frame}{The heads solve complementary held-out cases}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.37\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.68\textheight,keepaspectratio]{figures/fig2-math-gemini-venn.pdf}
    \end{column}
    \begin{column}{0.60\textwidth}
      \begin{block}{Math--Gemini, 200 test cases}
        Head 1 solves 56.0\%; Head 2 solves 58.0\%. Their union reaches \textbf{67.0\%}, a 9.0-point opportunity above the stronger head.
      \end{block}
      \vspace{2mm}
      \textbf{Other union rates:} Math--Sonnet 36.0\%, Terminal-Bench 55.2\%, SWE-bench Lite 70.4\%.
    \end{column}
  \end{columns}
  \papersource{Original paper Fig. 2(b), cropped Math--Gemini panel, PDF p. 7. Union is an oracle upper bound.}
\end{frame}

\begin{frame}{The router uses some, but not all, complementarity}
  \centering
  \begin{tabular}{lrrrr}
    \toprule
    Setting & Head 1 & Head 2 & Router & Union \\
    \midrule
    Math--Gemini & 56.0\% & 58.0\% & \textbf{62.0\%} & 67.0\% \\
    Math--Sonnet & 26.0\% & \textbf{31.0\%} & 30.5\% & 36.0\% \\
    Terminal-Bench 2.0 & 44.8\% & 48.3\% & \textbf{50.0\%} & 55.2\% \\
    SWE-bench Lite & 65.6\% & \textbf{66.4\%} & 66.0\% & 70.4\% \\
    \bottomrule
  \end{tabular}
  \vspace{3mm}
  \begin{itemize}
    \item Router exceeds the stronger head by 4.0 pp on Math--Gemini and 1.7 pp on Terminal-Bench.
    \item It falls short by one test case in each of Math--Sonnet and SWE-bench Lite.
  \end{itemize}
  \papersource{Paper, Table 2, PDF p. 8; union coverage in Fig. 2(b), PDF p. 7.}
\end{frame}

\begin{frame}{The two branches develop different math mechanisms}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Branch 1: answer verification}
        \begin{itemize}
          \item Dual-path adjudication.
          \item Concrete answer checks.
          \item Proof-obligation routing.
        \end{itemize}
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Branch 2: solution construction}
        \begin{itemize}
          \item Free-form derivation.
          \item Two-sided closure.
          \item Challenge an agreed answer.
        \end{itemize}
      \end{block}
    \end{column}
  \end{columns}
  \vspace{3mm}
  Fig. 2(a) plots \emph{retrospective test-best} progress; it was not the criterion used to propose or deploy a harness.
  \papersource{Paper, Sec. 4.3 and Fig. 2(a), PDF p. 7; Appendix E, PDF pp. 19--20.}
\end{frame}

\begin{frame}{Both adaptive components outperform either one alone}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.43\textwidth}
      \begin{block}{Update interval}
        \centering
        \begin{tabular}{lr}
          \toprule
          Dev update every & Accuracy \\
          \midrule
          5 iterations & 53.0\% \\
          3 iterations & 56.0\% \\
          1 iteration & \textbf{62.0\%} \\
          \bottomrule
        \end{tabular}
      \end{block}
    \end{column}
    \begin{column}{0.54\textwidth}
      \begin{block}{Component combination}
        \centering
        \begin{tabular}{lr}
          \toprule
          Enabled components & Accuracy \\
          \midrule
          Neither & 51.0\% \\
          Guidance only & 50.0\% \\
          Dev pruning only & 54.0\% \\
          Both & \textbf{62.0\%} \\
          \bottomrule
        \end{tabular}
      \end{block}
    \end{column}
  \end{columns}
  \papersource{Paper, Table 3 and Sec. 4.5, PDF pp. 8--9. Math--Gemini; default router.}
\end{frame}

\begin{frame}{Router prompt optimization has setting-dependent effects}
  \centering
  \begin{tabular}{lrrrr}
    \toprule
    Full router context & Math--Gemini & Math--Sonnet & Terminal-Bench & SWE-bench \\
    \midrule
    Without GEPA & 59.5\% & 29.0\% & 46.6\% & \textbf{66.4\%} \\
    With GEPA & \textbf{62.0\%} & \textbf{30.5\%} & \textbf{50.0\%} & 66.0\% \\
    \bottomrule
  \end{tabular}
  \vspace{5mm}
  \begin{itemize}
    \item Full context: expert source code, head-exclusive development examples, and their outputs.
    \item Adding example context alone is not consistently beneficial across settings.
  \end{itemize}
  \papersource{Paper, Appendix D.2, Table 6, PDF pp. 18--19.}
\end{frame}

\begin{frame}{Retrospective test-best results answer a different question}
  \centering
  \small
  \begin{tabular}{lrrrr}
    \toprule
    Method & Math--Gemini & Math--Sonnet & TB2 & SWE Lite \\
    \midrule
    Meta-Harness & 49.0\% & 32.5\% & 46.6\% & 63.6\% \\
    Branching search & \textbf{58.0\%} & 32.5\% & \textbf{50.0\%} & \textbf{66.4\%} \\
    \bottomrule
  \end{tabular}
  \vspace{4mm}
  \begin{itemize}
    \item Each pool contains ten candidates chosen by development performance; the maximum shown here uses test outcomes \emph{retrospectively}.
    \item The deployable Math--Gemini router scores 62.0\%, above the 58.0\% best individual harness in this pool.
  \end{itemize}
  \papersource{Paper, Sec. 4.6 and Table 4, PDF p. 9.}
\end{frame}

\begin{frame}{Search gains come with more total token use}
  \centering
  \begin{tabular}{lrrr}
    \toprule
    Sonnet 4.5 action-model setting & Meta-Harness & Branching & Ratio \\
    \midrule
    Math & 79.92 M & 172.90 M & \textbf{2.16$\times$} \\
    Terminal-Bench 2.0 & 2,366.80 M & 4,001.39 M & \textbf{1.69$\times$} \\
    SWE-bench Lite & 1,907.17 M & 2,951.48 M & \textbf{1.55$\times$} \\
    \bottomrule
  \end{tabular}
  \vspace{4mm}
  \begin{itemize}
    \item Totals combine task-solving and harness-authoring tokens; cached input is counted once.
    \item The math cost row is for Sonnet 4.5, not the largest-gain Gemini 3 Flash setting.
  \end{itemize}
  \papersource{Paper, Appendix A, Table 5, PDF pp. 12--13. Ratios calculated from reported totals.}
\end{frame}

\begin{frame}{What the evidence supports, and what remains open}
  \begin{enumerate}
    \item Dynamic development subsets and branch-local guidance produce distinct harness histories and stronger development-selected systems in four tested settings.
    \item Routing uses development data only and sometimes exceeds the stronger individual head; it still leaves 4.4--5.5 pp of union coverage unrealized.
    \item Full search token use is higher, and independent repeated-search variability is not reported; performance per unit search budget remains open.
    \item Exact coverage-vote handling for newly proposed harnesses on cases outside their current subset is not specified in the algorithm description.
  \end{enumerate}
  \papersource{Paper, Secs. 3--5, PDF pp. 4--10; Appendix A, PDF pp. 12--13. Last item is a reproducibility observation from Eqs. (1)--(2) and Algorithm 1.}
\end{frame}

\begin{frame}{Take-home messages}
  \begin{block}{Search mechanism}
    Relative frontier coverage changes each branch's scoring cases; local guidance turns branch experience into future proposal priorities.
  \end{block}
  \begin{block}{Deployment mechanism}
    A development-trained router combines two final heads without test labels or new-input expert outputs.
  \end{block}
  \begin{block}{Measured outcome}
    The strongest gain is Math--Gemini: 46.0\% $\rightarrow$ 62.0\% held-out accuracy, with higher search token use in the costed Sonnet settings.
  \end{block}
  \papersource{Paper, Tables 1--2, PDF pp. 6, 8; Table 5, PDF p. 13.}
\end{frame}

\appendix

\begin{frame}{Backup: full search schedule and ownership cases}
  \begin{itemize}
    \item $B=2$ branches; $N=20$ iterations; top-$q$ frontier with $q=5$; ownership margin $\Delta=2$.
    \item Development pruning interval $C_p=1$ after both branches have at least five candidates.
    \item Guidance update interval $C_s=5$; when both updates are due, prune first, then revise guidance.
    \item If votes are $5{:}5$, remove a case from both; if $4{:}1$, assign to the leading branch; if $3{:}2$, preserve existing assignment.
  \end{itemize}
  \papersource{Paper, Secs. 3 and 4.1, PDF pp. 4--6.}
\end{frame}

\begin{frame}{Backup: example mechanisms in discovered harnesses}
  \begin{itemize}
    \item \textbf{Math--Gemini:} construct an attainable answer, derive necessary conditions, then test gaps or challenge agreement.
    \item \textbf{Math--Sonnet:} clean retrieved solutions, append reference answers, and conditionally critique disagreements.
    \item \textbf{Terminal-Bench:} request verification on the first completion signal, then allow a later completion.
    \item \textbf{SWE-bench Lite:} find issue identifiers and relevant source/test files before editing.
  \end{itemize}
  \papersource{Paper, Appendix E, PDF pp. 19--20. These are descriptions of test-best harnesses, not a claim that each component alone caused the final gain.}
\end{frame}

\end{document}