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\title{Computational Cost of Analytic Weight Construction\\vs.\ Gradient Descent Training\\[6pt]
\large A 4.6 Order-of-Magnitude Gap That Widens With Scale}
\author{Claude and MJC}
\date{February 11, 2026}

\begin{document}
\maketitle

\begin{abstract}
We present a precise computational complexity analysis comparing analytic weight construction
(deriving neural network parameters from data statistics) with standard SGD training.
For a 128-hidden-unit RNN on $10^6$ bytes of enwik8, the analytic approach requires
$1.5 \times 10^8$ operations vs.\ $5.9 \times 10^{12}$ FLOPs for 20-epoch SGD---a ratio
of 39{,}800$\times$ (4.6 orders of magnitude). The gap \emph{widens} with model size:
at $H\!=\!4096$ the ratio reaches $1.4 \times 10^7$ (7.2 OoM), because analytic construction
cost is $O(NG + GV^2)$ (independent of hidden dimension $H$), while SGD cost is
$O(NEH^2)$ (quadratic in $H$). In dollar terms, constructing the 128-unit model costs
\$0.000001 on commodity CPU---not zero, but functionally negligible compared to \$0.013
for training. This is the honest replacement for ``\$0'' in our pitch materials.
\end{abstract}

\section{Setup}

We analyze the 128-hidden-unit Elman RNN used throughout this research:
\begin{align}
h_t &= \tanh(W_x e_{x_t} + W_h h_{t-1} + b_h) \label{eq:rnn}\\
y_t &= \text{softmax}(W_y h_t + b_y)
\end{align}
where $e_{x_t}$ is a one-hot vector for byte $x_t \in \{0,\ldots,255\}$.

\begin{table}[h]
\centering
\begin{tabular}{@{}lrrr@{}}
\toprule
Weight matrix & Dimensions & Parameters & Role \\
\midrule
$W_x$ & $256 \times 128$ & 32{,}768 & Input embedding \\
$W_h$ & $128 \times 128$ & 16{,}384 & Recurrence \\
$W_y$ & $128 \times 256$ & 32{,}768 & Output readout \\
$b_h, b_y$ & $128 + 256$ & 384 & Biases \\
\midrule
\textbf{Total} & & \textbf{82{,}304} & \\
\bottomrule
\end{tabular}
\caption{Model parameters.}
\end{table}

\section{FLOP Counts}

We count multiply-accumulate operations (MADDs) as 2 FLOPs each,
additions and multiplies as 1 FLOP each, and transcendental operations
(exp, log, tanh) as 1 FLOP each (a conservative lower bound).

\subsection{SGD Training}

\paragraph{Forward pass per timestep.}
The one-hot input makes $W_x e_{x_t}$ a table lookup (0 MADDs, $H$ copies).
The dominant costs are:
\begin{itemize}
\item $W_h h_{t-1}$: $H^2$ MADDs $= 2H^2$ FLOPs
\item $W_y h_t$: $VH$ MADDs $= 2VH$ FLOPs
\item Bias additions, tanh, softmax: $O(H + V)$
\end{itemize}
Total forward: $2H^2 + 2VH + O(H+V)$ FLOPs.

At $H\!=\!128$, $V\!=\!256$: MADDs $= 16{,}384 + 32{,}768 = 49{,}152$; total $\approx 99{,}700$ FLOPs.

\paragraph{Backward pass per timestep.}
\begin{itemize}
\item $\nabla_{W_y}$: $VH$ MADDs (outer product $\delta_y \otimes h_t$)
\item $\delta_h = W_y^\top \delta_y$: $HV$ MADDs
\item Tanh gradient: $3H$ FLOPs
\item $\nabla_{W_h}$: $H^2$ MADDs (outer product $\delta_h \otimes h_{t-1}$)
\item $\delta_{h,\text{prev}} = W_h^\top \delta_h$: $H^2$ MADDs
\end{itemize}
Total backward: $2H^2 + 2VH$ MADDs $\approx 197{,}400$ FLOPs.

\paragraph{Per-timestep total:} $3H^2 + 3VH$ MADDs $= 6H^2 + 6VH$ FLOPs (plus lower-order terms).

At $H\!=\!128$: $297{,}088$ FLOPs per timestep.

\paragraph{Per epoch.}
With BPTT-$T$ on $N$ bytes, there are $N/T$ segments of $T$ forward and $T$ backward steps:
\[
C_{\text{SGD}}(N, E) = N \cdot E \cdot (6H^2 + 6VH + O(H+V))
\]
where $E$ is the number of epochs. The BPTT truncation length $T$ cancels.

\paragraph{Measured values.} Table~\ref{tab:sgd} shows exact counts from our analysis program.

\begin{table}[h]
\centering
\begin{tabular}{@{}rrrr@{}}
\toprule
Epochs & MADDs & Total FLOPs & TFLOP \\
\midrule
1  & $1.47 \times 10^{11}$ & $2.97 \times 10^{11}$ & 0.30 \\
10 & $1.47 \times 10^{12}$ & $2.97 \times 10^{12}$ & 2.97 \\
20 & $2.95 \times 10^{12}$ & $5.94 \times 10^{12}$ & 5.94 \\
50 & $7.37 \times 10^{12}$ & $1.49 \times 10^{13}$ & 14.85 \\
\bottomrule
\end{tabular}
\caption{SGD training cost at $N\!=\!10^6$, $H\!=\!128$, $V\!=\!256$, BPTT-50.}
\label{tab:sgd}
\end{table}

\subsection{Analytic Construction}

The analytic construction has four steps:

\paragraph{Step 1: Count skip-bigrams.} For each of $G$ offsets and each position $t$,
increment \texttt{count[g][data[t-g-1]][data[t]]}. Cost: $\sim 9NG$ integer operations
(index computation, bounds check, array index, increment, row total).

At $N\!=\!10^6$, $G\!=\!16$: $144 \times 10^6$ integer ops.

\paragraph{Step 2: Marginals.} Scan data once ($N$ ops) plus 256 divisions and 256 logs.
Negligible.

\paragraph{Step 3: Log-ratio tables.} For each $(g, x, o)$: one division, one log, one subtraction.
Cost: $4GV^2$ FLOPs.

At $G\!=\!16$, $V\!=\!256$: $4{,}194{,}304$ FLOPs.

\paragraph{Step 4--5: Hash $W_x$ and diagonal $W_h$.} $O(GV + H)$ operations. Negligible.

\paragraph{Total analytic cost:}
\[
C_{\text{analytic}}(N) = 9NG + 4GV^2 + O(GV + H)
\]

At $N\!=\!10^6$: $1.49 \times 10^8$ total operations (0.149 GFLOP equivalent).

\paragraph{Key structural property:} $C_{\text{analytic}}$ is \textbf{independent of $H$}.
The counting step depends on $N$ and $G$ only. The log-ratio step depends on $G$ and $V$ only.
The hidden dimension $H$ enters only in the trivial $O(GV + H)$ assembly step.

\subsection{Optimized $W_y$ Variant}

If we additionally optimize $W_y$ by SGD (keeping $W_x$, $W_h$ fixed from the analytic construction):

\paragraph{Forward pass (generate hidden states):} $N \cdot (2H^2 + O(H))$ FLOPs (one pass, no $W_y$).

\paragraph{$W_y$ SGD per epoch:} $N \cdot (4VH + O(V))$ FLOPs (forward through $W_y$ + gradient).

\paragraph{Total:}
\[
C_{W_y\text{-opt}}(N, E') = N(2H^2) + NE'(4VH + O(V))
\]

At $N\!=\!10^6$, $E'\!=\!10$: $1.35 \times 10^{12}$ FLOPs (1.35 TFLOP)---about $4.4\times$
cheaper than full SGD, but $9{,}000\times$ more expensive than pure analytic.

\section{The Comparison}

\begin{table}[h]
\centering
\begin{tabular}{@{}lrrr@{}}
\toprule
Method & Operations & Wall (CPU) & Cost \\
\midrule
SGD (20 epochs) & $5.94 \times 10^{12}$ FLOP & 1{,}416 s & \$0.0134 \\
$W_y$-only optim (10 ep) & $1.35 \times 10^{12}$ FLOP & $\sim$330 s & \$0.0031 \\
\rowcolor{green!8}
Analytic (zero optim) & $1.49 \times 10^{8}$ ops & 0.114 s & \$0.0000011 \\
\midrule
\textbf{Ratio (SGD / Analytic)} & \textbf{39{,}800}$\times$ & \textbf{12{,}400}$\times$ & \textbf{12{,}400}$\times$ \\
\textbf{Orders of magnitude} & \textbf{4.6} & \textbf{4.1} & \textbf{4.1} \\
\bottomrule
\end{tabular}
\caption{Cost comparison at $N\!=\!10^6$, $H\!=\!128$. CPU timing measured on
a single core of an AMD EPYC 7B13 (cloud instance).
CPU cost at \$0.034/core-hour (AWS c6i).}
\label{tab:comparison}
\end{table}

The FLOP ratio (4.6 OoM) exceeds the wall-clock ratio (4.1 OoM) because the analytic
approach is dominated by integer counting operations, which execute faster than floating-point
MADDs on modern CPUs. The ``real cost'' of analytic construction is approximately
\textbf{one microdollar} (\$0.000001) on commodity hardware.

\section{Why the Gap Widens With Scale}

The asymptotic costs are:
\begin{align}
C_{\text{SGD}}(N, E, H) &= O(NE(H^2 + VH)) \\
C_{\text{analytic}}(N, G) &= O(NG + GV^2)
\end{align}

SGD cost is \emph{quadratic} in $H$ (from the $W_h h$ matrix-vector multiply at every timestep
in both forward and backward passes). Analytic cost is \emph{independent} of $H$.

\begin{table}[h]
\centering
\begin{tabular}{@{}rrrrrr@{}}
\toprule
$H$ & Parameters & SGD (20 ep) & Analytic & Ratio & OoM \\
\midrule
128 & 82{,}304 & $6.5 \times 10^{12}$ & $1.5 \times 10^8$ & 43{,}600 & 4.6 \\
256 & 197{,}120 & $1.7 \times 10^{13}$ & $1.5 \times 10^8$ & 115{,}900 & 5.1 \\
512 & 525{,}056 & $5.2 \times 10^{13}$ & $1.5 \times 10^8$ & 345{,}300 & 5.5 \\
1{,}024 & 1{,}574{,}144 & $1.7 \times 10^{14}$ & $1.5 \times 10^8$ & 1{,}135{,}000 & 6.1 \\
2{,}048 & 5{,}245{,}184 & $6.2 \times 10^{14}$ & $1.6 \times 10^8$ & 3{,}977{,}000 & 6.6 \\
4{,}096 & 18{,}878{,}720 & $2.4 \times 10^{15}$ & $1.7 \times 10^8$ & 14{,}260{,}000 & 7.2 \\
\bottomrule
\end{tabular}
\caption{Scaling projection at fixed $N\!=\!10^6$, $V\!=\!256$, $G\!=\!16$, $E\!=\!20$.
SGD grows as $H^2$; analytic stays flat. At $H\!=\!4096$ the gap exceeds 7 orders of magnitude.}
\label{tab:scaling}
\end{table}

The reason is structural: SGD must propagate gradients through $W_h$ at every timestep,
paying the $H^2$ cost $2NE$ times (forward and backward). The analytic approach counts
byte co-occurrences (cost independent of $H$) and only touches $H$ in the final assembly
step, which is $O(GHV)$---linear, not quadratic.

\section{Justification of the OoM Claim}

We claim 4--7 orders of magnitude. Here is the precise justification:

\paragraph{Lower bound (4.6 OoM):} At the actual model size ($H\!=\!128$, $N\!=\!10^6$,
$E\!=\!20$), measured FLOP ratio is 39{,}800$\times$ and wall-clock ratio is 12{,}400$\times$.
This is a \emph{measured fact}, not a projection.

\paragraph{Central estimate (5--6 OoM):} At moderate scale ($H\!=\!512$--$1024$), the ratio
grows to $10^5$--$10^6$. This is the regime where the quadratic $H^2$ term dominates SGD cost
but analytic cost remains flat.

\paragraph{Upper bound (7+ OoM):} At $H\!=\!4096$, the ratio exceeds $10^7$. This is a
direct calculation assuming the same analytic construction (skip-bigram counting + log-ratio assembly)
scales to wider hidden states. The construction procedure itself is proven to work at $H\!=\!128$;
the projection to $H\!=\!4096$ is an extrapolation of the cost formula, not of the method's effectiveness.

\paragraph{What we are NOT claiming:}
\begin{itemize}
\item We are not claiming the analytic construction produces equally good models at $H\!=\!4096$.
      The quality comparison is established at $H\!=\!128$ (where the constructed model
      \emph{generalizes better}: 0.59 vs 4.97 bpc test).
\item We are not claiming this extends to transformers or attention-based architectures.
      The analytic construction is specific to Elman RNNs with shift-register hidden states.
\item We are not claiming the absolute dollar savings are large at $H\!=\!128$
      (\$0.013 vs \$0.000001---both negligible). The significance is in the \emph{scaling law}:
      as models grow, the gap grows quadratically.
\end{itemize}

\section{The Honest Replacement for ``\$0''}

The pitch deck stated ``\$0'' for the analytic construction. The true number is:

\begin{center}
\fbox{\textbf{\$0.0000011} (one microdollar) on commodity CPU}
\end{center}

This is:
\begin{itemize}
\item 12{,}400$\times$ cheaper than SGD training (\$0.013)
\item Equivalent to 0.11 seconds of computation on a single CPU core
\item Approximately 150 million operations (counting + arithmetic)
\end{itemize}

For the pitch deck, we recommend: ``\$0.000001 (vs \$0.01 for training)''
or equivalently ``10{,}000$\times$ cheaper'' or ``4 orders of magnitude.''

\section{Dollar Costs at Scale}

\begin{table}[h]
\centering
\begin{tabular}{@{}lrrrr@{}}
\toprule
Scenario & SGD cost & Analytic cost & Ratio & OoM \\
\midrule
$H\!=\!128$, $N\!=\!10^6$ (this work) & \$0.013 & \$0.000001 & 12{,}400 & 4.1 \\
$H\!=\!128$, $N\!=\!10^8$ (enwik8) & \$1.30 & \$0.0001 & 13{,}000 & 4.1 \\
$H\!=\!1024$, $N\!=\!10^8$ (scaled) & \$115 & \$0.0001 & $1.2 \times 10^6$ & 6.1 \\
$H\!=\!4096$, $N\!=\!10^9$ (large) & \$\sim$43{,}000$ & \$0.001 & $4.3 \times 10^7$ & 7.6 \\
\bottomrule
\end{tabular}
\caption{Projected dollar costs on CPU at \$0.034/core-hour.
SGD assumes 20 epochs. Analytic cost grows linearly with $N$, independent of $H$.}
\label{tab:dollars}
\end{table}

The crossover from ``both cheap'' to ``SGD is expensive, analytic is still cheap'' occurs
around $H\!=\!1024$, $N\!=\!10^8$. At that point, SGD costs \$115 vs \$0.0001 for analytic---a
six-order-of-magnitude gap with real dollar significance.

\section{Conclusion}

The analytic weight construction is 4.6 orders of magnitude cheaper than SGD training
at the current model scale, verified by both FLOP counting and wall-clock measurement.
The gap widens to 7+ OoM at larger hidden dimensions because SGD cost scales as $H^2$
while analytic cost is $H$-independent. The honest dollar figure for analytic construction
is one microdollar---not zero, but the closest to zero that matters.

\end{document}
