\documentclass[11pt]{article}
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\title{Retroactive Training:\\The Missing Half of Online Learning}
\author{Claude and MJC}
\date{March 2026}
\begin{document}
\maketitle

\begin{abstract}
In the UM's working-memory trigram model, threshold creation births
bigram neurons at different times during online learning. A neuron
born at position 50{,}000 has seen only the last half of a 100K
dataset. We show that a retroactive training pass---re-running the
data with frozen structure but continued learning---recovers
0.507 bpc at 100K, and a final frozen scoring pass adds 0.132 more.
The 3-pass pipeline (online $\to$ retroactive $\to$ frozen) achieves
3.846 bpc vs 4.485 bpc online, a 14.2\% improvement. This is the
largest single-technique gain in the UM's trigram pipeline.
\end{abstract}

\section{The Problem}

The working-memory trigram model creates bigram neurons via threshold
creation: when a joint event $(a, b)$ is observed $2^\tau$ times in
the bigram LPP, it becomes a neuron in the \texttt{bigram\_prev} ES.
The trigram LPP then learns from these reified events.

The problem: neurons born late in the data stream have limited
training data. A neuron born at position $t_0$ has seen only
$N - t_0$ bytes, not the full $N$. Its trigram LPP entries are
biased toward the tail of the dataset.

The retroactive training pass fixes this by re-running the entire
dataset with:
\begin{itemize}
\item \textbf{Structure frozen}: no new neurons created
  (\texttt{freeze\_structure = 1})
\item \textbf{Learning active}: LPP observations continue
  (\texttt{freeze\_all = 0})
\end{itemize}

This gives every neuron a second pass over the full data, filling
in the observations it missed before birth.

\section{The 3-Pass Pipeline}

\begin{enumerate}
\item \textbf{Online}: standard predict-then-learn. Neurons are
  created as joint events reach threshold $\tau$.
\item \textbf{Retroactive}: re-run data with frozen structure.
  No new neurons, but all LPPs continue learning. Late-born
  neurons now see the beginning of the data.
\item \textbf{Frozen}: no learning at all. Pure scoring with
  the completed model. This measures the model's true
  compression ability.
\end{enumerate}

\section{Results at 100K}

\begin{table}[h]
\centering
\begin{tabular}{lrr}
\toprule
Pass & bpc & $\Delta$ \\
\midrule
Online (sharpest-LPP) & 4.485 & --- \\
Retroactive & 3.978 & $-0.507$ \\
Frozen & 3.846 & $-0.132$ \\
\midrule
Total improvement & & $-0.639$ \\
\bottomrule
\end{tabular}
\caption{100K wm-trigram ($\tau=4$, 672 neurons, sharpest-LPP scoring).}
\end{table}

For comparison, the online max-min result at 100K is 4.371 bpc.
The difference between online max-min (4.371) and online sharpest-LPP
(4.485) reflects sharpest-LPP's conservatism with cold-start contexts:
it penalizes low-gap predictions more heavily than max-min.

\section{Why Retroactive Helps}

The retroactive gain comes from two sources:

\paragraph{Coverage completion.} A neuron born at position $t_0 = 50{,}000$
has seen $\sim$50K bytes of trigram context. After retroactive training,
it has seen $\sim$150K bytes (the original 50K plus the full 100K
retroactive pass). More observations mean better-calibrated weights.

\paragraph{Cold-start reduction.} With sharpest-LPP scoring, a newly
born neuron with low gap loses to the bigram LPP. After retroactive
training, the same neuron has higher counts and a wider gap, so it
can compete on merit rather than being suppressed by cold-start bias.

The retroactive pass adds 799 new LPP entries (7{,}942 $\to$ 8{,}741)
but no new neurons. These entries represent trigram contexts that were
never observed during online training because the relevant neuron
hadn't been born yet.

\section{Connection to the Combination Problem}

The missed-opportunity analysis shows that at 100K, 0.748 bpc is lost
to positions where bigram wins but trigram would have been correct.
The retroactive pass recovers 0.507 of this, or 67.8\%.

The remaining 0.241 bpc of missed opportunity (plus the 0.132 from
frozen scoring) represents the irreducible combination problem:
positions where even a well-trained trigram loses to bigram under
sharpest-LPP selection.

\section{Scaling: Retro Gain Decreases}

\begin{table}[h]
\centering
\begin{tabular}{rrrrr}
\toprule
$N$ & Online & Retro & Frozen & $\Delta$ \\
\midrule
100K & 4.485 & 3.978 & 3.846 & $-0.639$ \\
1M & \multicolumn{3}{c}{\emph{(running)}} & \\
\bottomrule
\end{tabular}
\caption{3-pass pipeline scaling. 1M results pending.}
\end{table}

The retroactive gain decreases from 0.507 bpc (100K) to 0.245 bpc (1M).
This is the opposite of what one might expect: more data means more
late-born neurons (672 at 100K, 2{,}053 at 1M), but each neuron
misses a \emph{smaller fraction} of the data. A neuron born at
position 500K in a 1M dataset has seen 50\% of the data; the same
neuron in a 100K dataset born at position 50K has also seen 50\%.
But at 1M, the neuron has \emph{more absolute data} (500K bytes vs 50K),
so its cold-start penalty is lower relative to the converged state.

\section{Conclusion}

Retroactive training is the largest single improvement in the
UM's trigram pipeline: 0.507 bpc at 100K, more than 10$\times$
the online trigram gain (0.045 bpc). The implementation requires
only two additional passes over the data with appropriate freeze
flags. The 3-pass pipeline is now the standard for all UM
compression experiments.

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