The Effective Depth Paradox: Topology and Trainability in Deep CNNs

This paper presents a controlled comparative study of convolutional neural network (CNN) topologies and image classification performance across the architectural families VGG, ResNet, and GoogLeNet, evaluated on CIFAR-10 using a unified experimental protocol. We revisit and formalize the distinction between nominal depth (Dnom), the physical count of weight bearing layers, and effective depth (Deff ), an operational metric that quantifies the expected length of forward information paths. The effective-depth construction builds directly on the ensemble interpretation of residual networks introduced by Veit et al. 2016. Our contribution is to (i) extend this pathaveraging logic into closed-form, pre-training proxies that span sequential, residual, and multi-branch topologies alike, (ii) validate a computationally cheap structural proxy against a gradient-weighted variant computed from observed backpropagation signal, and (iii) test the resulting "Effective Depth Paradox" — the empirical pattern whereby nominal depth alone is a poor predictor of trainability and accuracy — against results from eight representative models (VGG-11/13/16/19, ResNet-18/34/50, GoogLeNet). In our protocol, plain VGG-style stacks show early saturation in accuracy as Dnom grows, while ResNet and GoogLeNet continue to benefit from additional depth by keeping Deff low relative to Dnom. A pooled correlation analysis shows that both Dnom and Deff are strongly associated with precision (r = 0.94 and r = 0.93, respectively); given the small family-clustered sample, this analysis alone cannot cleanly separate the two metrics, and we treat the accompanying evidence of gradient-norm and gradient-weighted validation as complementary mechanistic support rather than as decisive statistical proof. All claims in this paper are scoped to CIFAR-10-scale training of the three architecture families studied; we explicitly do not claim validation at ImageNet scale or generalization to modern architectures such as EfficientNet, ConvNeXt, or Vision Transformers, which we identify as necessary future work.