% ============================================================================
\section{Predictive Benchmark: From OLS to Deep Learning}
\label{sec:benchmark}
% ============================================================================

The hedonic OLS of Section~\ref{sec:results} is designed for interpretation: its
coefficients are implicit prices. It is not, however, the most accurate predictor of
price. To map the accuracy frontier---and to gauge how much curvature and interaction
the linear model leaves on the table---we benchmark a ladder of estimators of
increasing flexibility on the \emph{identical} feature set, under a common
5-fold cross-validation protocol. All rows of Table~\ref{tab:models} share the same
folds and random seed, so differences reflect the estimator, not the split.

\subsection{The Estimator Ladder}
\label{subsec:ladder}

\paragraph{(1) Ordinary least squares.} The linear hedonic of
Equation~\eqref{eq:hedonic}, with one-hot city, property-type, and source indicators
and within-property-type median imputation of missing continuous attributes. It is
the interpretable baseline and, mechanically, the least flexible: it imposes constant
marginal effects and additive separability.

\paragraph{(2) Random forest.} A bagged ensemble of regression trees
\citep{mullainathan2017machine}. Trees capture non-linearities and low-order
interactions automatically and handle missing values natively, but averaging
deep decorrelated trees tends to over-smooth sharp price thresholds.

\paragraph{(3) Gradient boosting.} A histogram-based gradient-boosted tree ensemble
(600 boosting iterations, learning rate 0.05, up to 63 leaves, $L_2$ regularization).
Boosting fits trees sequentially to residuals, which lets it model both smooth
curvature (age, area) and abrupt thresholds; like the forest it ingests missing
values directly. It is the strongest single model in our comparison.

\paragraph{(4) Neural networks.} A plain multilayer perceptron and a robust
residual network with entity embeddings, described in
Section~\ref{subsec:nn}. Neural networks excel when high-cardinality categoricals
(here, 195 municipalities) can be represented as dense learned vectors rather than
sparse indicators.

\paragraph{(5) Blend.} A simple 50/50 average of the robust neural network and the
gradient-boosting predictions. Because the two models make partly decorrelated
errors---the network smooths through embeddings while the boosted trees cut sharp
thresholds---their average cancels idiosyncratic noise, a standard stacking result.

\subsection{Formal Definitions}
\label{subsec:formal}

Let $y_i=\ln P_i$ and let $\mathbf{x}_i$ denote the feature vector of
Equation~\eqref{eq:blocks}. Each estimator approximates the conditional mean
$f(\mathbf{x})=\mathbb{E}[y\mid\mathbf{x}]$.

\paragraph{Linear (OLS).} $f(\mathbf{x})=\mathbf{x}^{\!\top}\bm{\theta}$ with the
closed form of Section~\ref{subsec:econometrics}; categoricals enter as one-hot
indicators.

\paragraph{Random forest \citep{breiman2001random}.} An average of $B$ regression
trees, each grown on a bootstrap resample with a random feature subset at every split:
%
\begin{equation}
  \hat f_{\text{RF}}(\mathbf{x})=\frac1B\sum_{b=1}^{B} T_b(\mathbf{x};\Theta_b),
  \label{eq:rf}
\end{equation}
%
where $\Theta_b$ denotes the randomness (resample and split subsets) of tree~$b$.

\paragraph{Gradient boosting \citep{friedman2001greedy}.} A forward stagewise additive
expansion built by fitting each new tree to the negative gradient of the loss:
%
\begin{equation}
  \resizebox{.94\textwidth}{!}{$\displaystyle
  F_M(\mathbf{x}) = F_0 + \nu\sum_{m=1}^{M} f_m(\mathbf{x}),
  \quad
  f_m = \arg\min_{f}\sum_{i=1}^{n}
  \big(\!-g_{mi}-f(\mathbf{x}_i)\big)^2,
  \quad
  g_{mi}=\frac{\partial L\big(y_i,F_{m-1}(\mathbf{x}_i)\big)}{\partial F_{m-1}(\mathbf{x}_i)}
  $}
  \label{eq:gbm}
\end{equation}
%
with learning rate $\nu$ and $L_2$-regularized histogram trees $f_m$; missing values
are routed to a learned default direction at each split.

\paragraph{Neural network (robust).} High-cardinality categoricals are mapped to
learned embeddings \citep{guo2016entity}, concatenated with the batch-normalized
numeric block and projected to a $d$-dimensional trunk:
%
\begin{equation}
  \mathbf{h}_0 = \mathbf{W}_0
  \Big[\, \textstyle\bigoplus_{c\in\{\text{city},\text{type},\text{src}\}}
  \mathbf{E}_c(\text{cat}_{ci})
  \;\Vert\; \mathrm{BN}(\mathbf{x}^{\text{num}}_i)\,\Big].
  \label{eq:nn_in}
\end{equation}
%
The trunk passes through $L=3$ pre-activation residual blocks
\citep{he2016deep,gorishniy2021revisiting},
%
\begin{equation}
  \mathbf{h}_{\ell}
  = \mathbf{h}_{\ell-1}
  + \mathrm{Drop}\!\Big(\mathbf{W}_2^{(\ell)}\,
    \mathrm{Drop}\!\big(\phi(\mathbf{W}_1^{(\ell)}\,
    \mathrm{BN}(\mathbf{h}_{\ell-1}))\big)\Big),
  \qquad \phi=\text{GELU},
  \label{eq:resblock}
\end{equation}
%
and a normalization head yields $\hat y_i = \mathbf{w}^{\!\top}\phi(\mathrm{BN}(\mathbf{h}_L))+b$.
Parameters minimize the Huber (smooth-$L_1$) loss, robust to price outliers,
%
\begin{equation}
  L_\delta(r)=
  \begin{cases}
    \tfrac12 r^2, & |r|\le \delta,\\[2pt]
    \delta\big(|r|-\tfrac12\delta\big), & |r|>\delta,
  \end{cases}
  \qquad r_i=\hat y_i-y_i,\;\delta=1,
  \label{eq:huber}
\end{equation}
%
optimized with AdamW (decoupled weight decay) \citep{loshchilov2019decoupled} under a
cosine-annealed learning rate $\eta_t=\tfrac12\eta_0\big(1+\cos(\pi t/T)\big)$,
gradient-norm clipping, and early stopping. Three seeds are averaged per fold.

\paragraph{Blend.} A convex combination of the two strongest models in log space,
%
\begin{equation}
  \widehat{\ln P}^{\,\text{blend}}
  = w\,\widehat{\ln P}^{\,\text{GBM}}
  + (1-w)\,\widehat{\ln P}^{\,\text{NN}},
  \qquad w=0.5.
  \label{eq:blend}
\end{equation}

\paragraph{Evaluation.} With $\hat y_{-k(i)}$ the prediction for $i$ from the model
trained on the folds excluding $i$'s fold, we report out-of-sample fit and level
accuracy,
%
\begin{equation}
  R^2_{\text{OOS}} = 1-\frac{\sum_i\big(y_i-\hat y_{-k(i)}\big)^2}
  {\sum_i\big(y_i-\bar y\big)^2},
  \qquad
  \text{MdAPE}=\operatorname*{median}_i\Big|e^{\,\hat y_{-k(i)}-y_i}-1\Big|.
  \label{eq:cvmetrics}
\end{equation}

\subsection{Results}
\label{subsec:bench_results}

\input{tables/models}

Table~\ref{tab:models} reports the ladder in two panels. In Panel~A (structural,
amenity, and text features) out-of-sample $R^2$ rises monotonically with flexibility:
from 0.628 for OLS, to 0.677 for the random forest, 0.711 for the plain network, and a
near-tie at the top between the robust network (0.741) and gradient boosting (0.743);
their blend attains $R^2_{\text{OOS}}=0.760$. Panel~B adds the twelve façade vision
attributes. Vision lifts \emph{every} estimator---OLS gains 5.8 points
(0.628 $\to$ 0.686), the random forest 2.7 (0.677 $\to$ 0.704), and gradient boosting
2.7 (0.743 $\to$ 0.770), the single largest increment from any one feature layer at
the frontier---and the \textbf{text\,+\,vision blend attains
$R^2_{\text{OOS}}=0.779$ with a median absolute error of 14.0\%}, the best model in
the study.

Three lessons follow. First, on the full feature set roughly nine points of
out-of-sample $R^2$ separate the interpretable linear model (0.686) from the
predictive frontier (0.779)---the price of interpretability, and a measure of the
curvature and interactions the hedonic form omits. Notably, the vision layer buys the
linear model more accuracy (+5.8 points) than switching from OLS to a random forest
does (+4.9), so better measurement rivals better estimation. Second, on this tabular
problem gradient boosting and a carefully regularized neural network are
statistically comparable, consistent with the broader evidence that tree ensembles
remain strong baselines on tabular data \citep{grinsztajn2022why}; the gains come
from \emph{combining} them. Third, the façade-vision layer is not redundant with the
text and structural fields: it moves the frontier more than any other block we add,
confirming that image-derived quality is genuinely orthogonal information.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=.72\textwidth]{figures/fig_ablation.pdf}
  \caption{Out-of-sample $R^2$ as feature layers are added (nested, left to right).
  Each layer improves both the linear and boosted models; the façade-vision layer
  delivers the largest single jump---about $+5.8$ points for OLS and $+2.7$ for
  gradient boosting.}
  \label{fig:ablation}
\end{figure}

\subsection{Feature Ablation and Importance}
\label{subsec:ablation}

Table~\ref{tab:ablation} and Figure~\ref{fig:ablation} decompose the accuracy gains
layer by layer. Reading down the table, room detail and listing signals add the first
large increment, the LLM text layer a smaller one, and the vision layer the single
largest jump for every estimator---raising OLS from 0.628 to 0.686 and gradient
boosting from 0.743 to 0.770 out of sample. The pattern is monotone: no layer degrades
fit, and the two AI layers together account for a substantial share of the distance
between the structural baseline and the frontier.

\input{tables/ablation}

Figure~\ref{fig:importance} aggregates the gradient-boosting permutation importances
into feature blocks. Structural attributes and location fixed effects dominate, as
expected, but the \textbf{vision block alone accounts for 13\% of total importance}---
more than the rule-based amenities and the LLM text block combined---reinforcing that
the façade photograph is a first-order, not marginal, source of price information.
Figure~\ref{fig:calibration} shows the blend is well calibrated across the price range,
with an approximately symmetric, tight error distribution centered near zero.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=.72\textwidth]{figures/fig_importance.pdf}
  \caption{Share of gradient-boosting permutation importance by feature block
  (text\,+\,vision model). The vision block accounts for 13\% of total importance,
  exceeding the rule-based amenity and LLM-text blocks.}
  \label{fig:importance}
\end{figure}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{figures/fig_calibration.pdf}
  \caption{Out-of-sample calibration of the text\,+\,vision blend. Left: predicted
  versus actual price (log--log; dashed line is the 45$^\circ$ identity). Right:
  distribution of percentage prediction errors, with median absolute error of 14.0\%.}
  \label{fig:calibration}
\end{figure}

\subsection{A Robust Tabular Neural Network}
\label{subsec:nn}

Figure~\ref{fig:nn} details the robust network. High-cardinality categoricals enter
as learned \textbf{entity embeddings}: each municipality, property type, and source is
mapped to a dense vector (dimension $\approx 1.6\,C^{0.56}$ in cardinality $C$, so
about 29 dimensions for the 195 cities). The embeddings are concatenated with the
batch-normalized numeric block and projected to a 256-dimensional trunk, which passes
through three \textbf{residual blocks}
(BatchNorm $\to$ GELU $\to$ Linear$_{512}$ $\to$ Dropout $\to$ Linear$_{256}$, with a
skip connection), followed by a normalization head that outputs $\ln(\text{Price})$.

Robustness comes from the training recipe rather than raw capacity:
a \textbf{Huber (smooth-$L_1$) loss} that damps the influence of price outliers;
\textbf{AdamW} with decoupled weight decay and \textbf{cosine-annealed} learning rate;
\textbf{gradient-norm clipping} at 5; \textbf{dropout} and batch normalization
throughout; \textbf{learned missingness indicators} appended to the numeric block so
the network can exploit the non-random pattern of missing fields; and
\textbf{early stopping} on a held-out 10\% of each training fold. Finally, three
networks with different random seeds are trained per fold and \textbf{averaged}, which
reduces prediction variance. The result (0.741) is competitive with gradient boosting
and, once blended with it, produces the best model in the study.

\begin{figure}[htbp]
  \centering
  \input{figures/nn_diagram}
  \caption{Architecture of the robust tabular neural network. High-cardinality
  categoricals enter as learned embeddings; numeric features are batch-normalized and
  augmented with missingness indicators. The concatenated representation is projected
  to a 256-dimensional trunk, refined by three residual blocks, and mapped to
  $\ln(\text{Price})$. Three seeds are ensembled per fold; the final model blends this
  network 50/50 with gradient boosting.}
  \label{fig:nn}
\end{figure}