Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.00157

Interpreting FCDNNs via RG on Exponential Family

Fuzhou Gong·Zigeng Xia
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider establishing the interpretability theory of deep learning through constructing a corresponding relationship between the renormalization group (RG) method in statistical physics and the training process of deep neural networks (DNNs). We have proved the constructed relationship using the one-dimensional Ising model as the input data. In this paper we generalize our results to the case of continuous input data, which is a necessary preparation for applying the corresponding framework to real-world data. To be representative, we consider a class of data distribution in the exponential family. We prove that when the parameters of fully connected (FC) DNNs achieve their optimal value after training, the characteristic parameters of the feature layer output of DNNs are equal to the fixed points of the characteristic parameters of input data under RG method for continuous fields. This conclusion shows that the training process of DNNs is equivalent to RG calculation on this kind of data and therefore the network can extract main features from the input data just like RG. Also, the equivalence further validates the correspondence framework we have established, providing an explanation for the outstanding performance of DNNs on real-world data.

Explore related subjects

stat.MLcs.AIcs.LGmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fuzhou Gong, Zigeng Xia. 2026-05-29. Interpreting FCDNNs via RG on Exponential Family. https://arxiv.org/abs/2606.00157

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Attack-Resistant Uniform Fairness for Linear and Smooth Contextual Bandits

Modern digital platforms use contextual bandits to allocate valuable exposure and opportunities among competing participants. Fair treatment is therefore an important concern, yet reward maximization alone does not ensure that preferential allocation reflects participants' merits. We develop algorithms for linear and smooth contextual bandits under uniform merit-based fairness, requiring the reward ordering to justify preferential allocation across all contexts and rounds, and study how these guarantees are affected by adversarial reward corruption. Our algorithms achieve \((1-\widetilde O(1/T))\)-fairness, with regret that is minimax optimal among fair policies for linear rewards and nearly minimax optimal for smooth rewards. In the linear setting, matching lower bounds identify the price of fairness exactly: minimax regret increases from \(\log T\) to \(\log^2 T\). For smooth rewards, the cost of fairness is at most polylogarithmic. We further establish a separation between regret and fairness robustness: an \(\widetilde O(1)\) corruption budget can cause substantial fairness violations without worsening the regret order. We therefore develop robust algorithms that adapt sampling, estimation, and fairness certification to corruption, which preserve uniform fairness and achieve minimax-optimal and nearly optimal regrets for linear and smooth rewards, respectively. Numerical and semi-synthetic experiments illustrate these findings.

stat.ML↗
arXiv2026

The Cost of Privacy: Rates of Convergence for Parameter Estimation with Differential Privacy

We study the minimax cost of $(\varepsilon,δ)$-differential privacy for mean estimation and Gaussian linear regression in low and high dimensions. For low-dimensional mean estimation, a resampling reduction to fingerprinting yields the privacy contribution $d^2\log(1/δ)/(n^2\varepsilon^2)$ in the stated polynomial-$δ$ regime. For low-dimensional regression, a tracing argument gives the contribution $d^2/(n^2\varepsilon^2)$ under an explicit approximate-DP remainder condition. For sparse mean estimation and sparse regression, a constant-weight packing and a private Fano lemma produce an effective privacy entropy of order $\min\{s\log(ed/s),[\log((e^\varepsilon-1)/δ)]_+\}$ for $δ>0$, up to universal constants and a fixed threshold; for pure DP it is $s\log(ed/s)$. Thus, when $δ$ is polynomially smaller than $\varepsilon$, the pure-DP dependence is retained up to polylogarithmic factors whenever the effective dimension is polylogarithmic in $n$, including regimes with $\varepsilon=o(1)$. Coordinatewise-clipping estimators for means and split-sample noisy-gradient estimators for regression attain the lower bounds up to explicit logarithmic factors. Simulations and data examples illustrate related implementations.

stat.ML↗
arXiv2026

Robust Mixture Models for Algorithmic Fairness Under Latent Heterogeneity

Machine learning models optimized for average performance can perform poorly on vulnerable subpopulations. Existing approaches often rely on groups specified in advance, yet fairness-relevant subgroup structure may be latent, intersectional, and driven by complex interactions among continuous and discrete attributes. We introduce \textbf{ROME} (\textbf{\underline{RO}}bust \textbf{\underline{M}}ixture \textbf{\underline{E}}nsemble), a framework that learns latent group structure while optimizing worst-group predictive performance. ROME connects latent-variable modeling with distributionally robust optimization (DRO) through two complementary approaches: an Expectation-Maximization formulation with robust aggregation for linear models and a neural Mixture-of-Experts formulation for nonlinear settings. Across simulations and three real-world regression datasets, ROME improves worst-group performance while maintaining competitive overall accuracy, including in comparisons with established group-aware and group-label-free robust learning methods. ROME provides a flexible approach to robust prediction when fairness-relevant attributes are available for subgroup discovery but their direct use in group-specific outcome models is restricted.

stat.ML↗