aiqbench analysis
This arXiv preprint argues that PAC-Bayes generalization bounds for neural networks can avoid scaling with parameter count by placing the prior over data distributions instead of weights. The reported certificates (8.8%–9.4% on full MNIST across models up to 4.2M parameters) are notable because classical weight-space bounds become vacuous, but the evidence comes from a single benchmark and an unreviewed preprint. Practitioners should check how the bounds depend on the Dirichlet Process hyperparameters and whether the assumptions hold on harder datasets.
Dimension-Free PAC-Bayes Generalization for Deep Neural Networks via Dirichlet Processes
arXiv:2610.21340v1 Announce Type: cross Abstract: Classical PAC-Bayesian generalization theory places prior distributions over Euclidean parameter space $\Theta \subseteq \mathbb{R}^D$, causing relative entropy to scale with network dimension $D$. For modern overparameterized neural networks where $D \gg n$, this parameter-counting dependency yields vacuous generalization bounds ($> 1{,}000\%$). In this work, we resolve this barrier by shifting prior specification from internal parameters to the non-parametric space of data distributions $\mathcal{M}_1(\mathcal{Z})$ via a \textbf{Dirichlet Process (DP)}, $\mathcal{DP}(\alpha, F_0)$, inducing parameter distributions through push-forward measures under the learning algorithm. By the Data Processing Inequality, parameter-space relative entropy is strictly bounded by the data-space DP divergence, which we evaluate in exact closed form using Digamma functions. The resulting PAC-Bayesian certificate is strictly invariant to network parameter count $D$, width, and depth. On intrinsic $d^*$-dimensional submanifolds, optimal partition scaling achieves the minimax non-parametric rate $\mathcal{O}(n^{-1/(d^*+2)} \sqrt{\ln n})$. Across vision architectures scaling up to 4.2 million parameters ($D = 7{,}850$ to $D = 4{,}203{,}522$) on the full MNIST benchmark, while classical weight-space bounds scale past $940\%$ (becoming vacuous), our data-space DP-PAC-Bayes certificate remains stable, non-vacuous, and certified at \textbf{$8.8\%$--$9.4\%$}.
Abstract:Classical PAC-Bayesian generalization theory places prior distributions over Euclidean parameter space $\Theta \subseteq \mathbb{R}^D$, causing relative entropy to scale with network dimension $D$. For modern overparameterized neural networks where $D \gg n$, this parameter-counting dependency yields vacuous generalization bounds ($> 1{,}000\%$). In this work, we resolve this barrier by shifting prior specification from internal parameters to the non-parametric space of data distributions $\mathcal{M}_1(\mathcal{Z})$ via a \textbf{Dirichlet Process (DP)}, $\mathcal{DP}(\alpha, F_0)$, inducing parameter distributions through push-forward measures under the learning algorithm. By the Data Processing Inequality, parameter-space relative entropy is strictly bounded by the data-space DP divergence, which we evaluate in exact closed form using Digamma functions. The resulting PAC-Bayesian certificate is strictly invariant to network parameter count $D$, width, and depth. On intrinsic $d^*$-dimensional submanifolds, optimal partition scaling achieves the minimax non-parametric rate $\mathcal{O}(n^{-1/(d^*+2)} \sqrt{\ln n})$. Across vision architectures scaling up to 4.2 million parameters ($D = 7{,}850$ to $D = 4{,}203{,}522$) on the full MNIST benchmark, while classical weight-space bounds scale past $940\%$ (becoming vacuous), our data-space DP-PAC-Bayes certificate remains stable, non-vacuous, and certified at \textbf{$8.8\%$--$9.4\%$}.
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2610.21340 [stat.ML]
(or arXiv:2610.21340v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.21340 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 277 words
