Depth grows a receptive field, but does not guarantee that the model uses it.

Keep a ruler in input coordinates

Let $R_\ell$ be the receptive-field span of an output after layer $\ell$, and $J_\ell$ the distance in original input coordinates between neighbouring outputs. Initially $R_0=J_0=1$.

A width-$k_\ell$, dilation-$d_\ell$ window reaches $d_\ell(k_\ell-1)$ previous-layer jumps beyond its first position. Each of those jumps spans $J_{\ell-1}$ original positions. Stride then changes the spacing between neighbouring outputs:

This derivation proves I.11.T3 for a sequential stack of local layers. Equation (I.11.6) gives $R=3,5,7$ for three ordinary width-three, stride-one layers. The diagram traces one top output back through three, five and seven positions in successively earlier rows.

Striding increases the jump used by later layers. For width-three layers with strides $1,2,1$, the pairs $(R,J)$ are $(3,1)$, $(5,2)$ and $(9,2)$. Using the new jump too early would incorrectly enlarge the field of the striding layer itself.

Dilation can leave holes inside the span. Branches require the union of their dependency sets. A global average, global attention layer or dense flattened head can make a final prediction depend on positions beyond the field of an individual earlier feature. Name the output whose field you are calculating.

Accessible is not influential

A nonzero derivative cannot arrive through a path that does not exist. The converse fails: a path may exist while its weights are zero, its activation is inactive, or contributions cancel.

Luo and colleagues distinguished the theoretical field from the distribution of effective influence. Their analysis and experiments found concentrated, often approximately Gaussian influence within the wider field. This is not a theorem that every trained CNN has the same Gaussian profile or one universal square-root growth law. Architecture, weights, inputs and the quantity used to measure influence matter.

For a local dense-prediction head with no other route to a distant region, increasing channel count alone does not add that missing route. Increasing depth, changing dilation or adding a global operation changes the dependency graph. Downsampling also changes access, but spends spatial resolution.

Read the architectures as design choices

LeNet combined learned local filters with subsampling for document recognition. Its lesson is a composed spatial hierarchy, not that every later CNN must use exactly its subsampling operator.

AlexNet demonstrated a larger learned visual hierarchy on ImageNet using ReLU activations, GPU training and regularisation including dropout. Its historical architecture is not a prescription for today’s implementation.

VGG stacked small $3$ by $3$ filters to gain depth. Three such stride-one layers have a $7$ by $7$ field. At equal input, hidden and output width $C$ and ignoring biases, they use $27C^2$ weights versus $49C^2$ for one $7$ by $7$ layer. The intermediate nonlinearities make them different functions, not interchangeable factorisations of one linear filter.

ResNet learns a residual branch. When the shapes agree, its pre-activation sum has the form

Equation (I.11.8) explains the identity route, not guaranteed non-vanishing gradients: $F(\vec x)=-\vec x$ makes the sum zero. A following activation supplies another Jacobian factor. If resolution or channel width changes, a projection shortcut must make the shapes compatible. I.11.X09 checks both issues.

EfficientNet studied coordinated scaling of depth, width and resolution. A rough convolutional cost model is proportional to depth times width squared times resolution squared. Its compound-scaling idea allocates a larger budget across all three rather than changing one in isolation. That approximation is a planning tool, not a latency guarantee.

These papers answer different questions. Counting layers alone does not compare their compute, training setup, accuracy or suitability for a new task.