Jinran Wu, You-Gan Wang, Geoffrey J. McLachlanstat.ML cs.LG
We consider semi-supervised classification from a partially classified sample arising from a two-component Weibull mixture. The feature is observed for all data, whereas some class labels are missing. The probability of a missing label is modelled as a function of classification uncertainty, giving a feature-dependent missing-at-random (MAR) mechanism that shares parameters with the Weibull-mixture classifier. The missing-label indicators can therefore provide information about the classifier in addition to the observed features and available class labels. Under a common Weibull shape, a Bayes' rule has at most one positive decision boundary, which is unique when the rule is nonconstant; under unequal shapes, it can have two. We characterise these decision regions, derive the Fisher information for the classifier after adjustment for nuisance parameters in the missingness model, and obtain a decision-boundary expansion of the expected error rate of the plug-in sample rule relative to the Bayes error. The expansion yields classification-specific asymptotic relative efficiency formulas for the one- and two-boundary cases and shows that a positive-definite increase in Fisher information is sufficient, but not necessary, for a smaller first-order expected error rate. Numerical studies and a semi-synthetic analysis based on hard-drive failure data illustrate potential reductions in expected error rate and improvements in decision-boundary estimation from modelling feature-dependent label missingness.
You-Gan Wang, Jinran Wu, Geoffrey J. McLachlanmath.ST stat.ME stat.ML
Missing labels are usually regarded as a source of information loss in classification. We study a semi-supervised setting in which the probability of label missingness depends on the observed features through posterior classification uncertainty. In this setting, the missingness indicator is not only a record of an unobserved label, but also an observable signal generated by a mechanism linked to the classifier. We develop a likelihood-based information theory for such uncertainty-dependent missing labels. Under correct specification, we derive a Fisher-information decomposition that separates a partial-labeling component from a nonnegative mechanism-curvature term. Under joint misspecification of the label model and the missingness mechanism, we obtain the corresponding Godambe--Eicker--Huber--White sensitivity and sandwich-covariance partitions. We also clarify the relevant complete-data benchmark: favorable missingness can increase information relative to ordinary fully labeled or budget-matched non-informative labeling baselines, but cannot exceed the information in the augmented experiment in which labels and mechanism indicators are both observed. For plug-in classifiers, we connect the information decomposition to margin-based excess-risk bounds. In regular two-component mixture settings this yields the parametric \(n^{-1}\) excess-risk rate, with constants determined by the nuisance-adjusted information in discriminant directions. Gaussian-mixture calculations and a medical diagnosis example illustrate how uncertainty-dependent labeling mechanisms can improve estimation and classification under a fixed labeling budget.
Denoising score matching trains diffusion models by regressing onto a conditional score, although generation ultimately requires the marginal score. The two objectives share the same population minimizer, but the conditional target remains random at fixed noisy state and introduces an irreducible excess in the training loss. We isolate this excess and show that, for a general corruption kernel under mild regularity assumptions, it is exactly the trace of the Fisher--Rao metric of the conditional endpoint family, integrated along the diffusion trajectory. This gives an exact conditional-variance decomposition of the denoising objective and identifies the information geometry observed in diffusion latent spaces as an intrinsic component of the training loss. We derive the result from a Schr"odinger bridge variational principle, in which the ideal objective arises as excess path-space relative entropy. For corruption diffusions, the Fisher term is proportional to the rate at which the noisy state loses mutual information about the clean data, separating the loss floor into an information flow determined by the data and a weight determined by the corruption schedule and objective. In the Gaussian case, this yields a closed form for the floor, recovers reparametrization invariance of the continuous-time objective, and relates its high-SNR divergence to the information dimension of the data. Finally, we show that raw losses obtained with different noise ranges or weightings need not rank models consistently because they contain different additive floors, and contrast the second-order geometry seen by training with the third-order conditional statistics entering numerical sampling error.
Generative Flow Networks (GFlowNets) have emerged as a flexible framework for amortised inference over discrete and mixed discrete-continuous objects, requiring only an unnormalised target density specified through a reward. In this work, we formulate forward-policy training in GFlowNets through the information geometry of the induced trajectory sampler. Treating the forward policy as an induced trajectory sampler, we show that its intrinsic first-order geometry is given by the Fisher-Rao metric of the trajectory family, and that the associated natural gradient provides the canonical local update whenever the corresponding Fisher information is computable or accurately approximable. We derive an exact decomposition of the trajectory Fisher into per-step conditional second moments, which clarifies when temporal score interactions vanish and when dense couplings remain under shared parameterisation. This leads to three computational regimes: settings with tractable exact Fisher information, settings where Monte Carlo estimators of the expected Fisher are sufficient, and structure-exploitable settings in which target locality or factorisation yields accurate approximations of the Fisher expectation. In the latter case, graphical-model tools such as exact marginalisation, separator methods, and belief propagation provide principled surrogates for natural-gradient updates. The resulting framework turns target structure into optimisation geometry and yields a tractable route to structure-aware forward-policy training in GFlowNets. We illustrate the framework empirically through examples comparing convergence and exploration behaviour under Riemannian and Euclidean optimisation.
Behraj Khan, Behroz Mirza, Syed Ahmad Chan Bukhari +1cs.LG
Covariate shift across training-data partitions biases model selection and parameter estimation in cross-validation, lifelong learning, and federated learning. We propose \textit{Partition-Induced Covariate-shift Correction} (\texttt{PIcsC}), a Fisher information-based regularization framework that mitigates distribution mismatch between data partitions and a reference distribution. \texttt{PIcsC} approximates partition divergence using the Fisher Information Matrix (FIM) and incorporates the resulting statistic as a regularizer during optimization. The same formulation applies to both centrally partitioned datasets (batches or cross-validation folds) and inherently distributed data (federated clients or decentralized nodes), requiring only partition-local gradient statistics rather than raw data. We further introduce a conditional adaptation mechanism that combines FIM shift with KL divergence to detect significant distribution shifts and activates regularization only when necessary. Experiments on more than 40 datasets demonstrate consistent improvements under both natural and synthetic covariate shift. On fragmented batch and fold settings, \texttt{PIcsC} reduces fragmentation-induced performance degradation by more than 20\% and 25\%, respectively. On seven federated learning benchmarks, it consistently outperforms FedAvg, FedProx, and SCAFFOLD by 3 -5 percentage points without requiring client-specific personalization. These results demonstrate that Fisher information provides an effective and unified mechanism for mitigating partition-induced covariate shift across both centralized and distributed learning.
Let $p(x)$ be the joint density of variables $X$, and let $ψ(x)=\nabla_x\log p(x)$ be its score field. Geometry constructed from $p$ and $ψ$ alone cannot identify causal direction: structural models with the same observational distribution have the same score geometry. I develop an interventional analogue. A hard intervention $\operatorname{do}(X_k=ξ)$ does not merely reweight the joint law; it restricts the distribution to the submanifold ${x_k=ξ}$. Its score should therefore be defined on the remaining $d-1$ free coordinates. I define causal influence $X_k\rightsquigarrow X_j$ as variation of the interventional marginal distribution of $X_j$ with $ξ$, and show that the corresponding derivative of the marginal interventional score gives a local sufficient condition for influence. Projecting the observational score onto admissible intervention directions does not generally recover causal response: two models may share the same observational score and admissible set yet respond differently. I therefore introduce an interventional response field supplied by structural information. A causal metric is defined as the Fisher information metric on a family of interventions with a common target, avoiding ill-posed comparisons across targets. The framework yields a geometric dictionary for randomized trials, instrumental variables, and conditional-independence designs, clarifying what each does and does not identify. A bivariate Gaussian example gives two models with the same observational score but different interventional score derivatives. The framework organizes relations among designs, interventions, and score fields, but adds no identification beyond the underlying assumptions. In Pearl's Ladder of Causation, observational score geometry belongs to association, intervention-indexed score fields to intervention, and unit-level counterfactual geometry is left for future work.
We study Gaussian-width complexity on statistical manifolds through a pair of functionals: the primal Fisher width $w_G(T) = w(G^{1/2}T)$, induced by the Fisher metric, and the inverse-Fisher width $w_{G^{-1}}(T) = w(G^{-1/2}T)$, induced by the inverse Fisher metric. The two widths play complementary statistical roles. On the learning side, the Fisher width measures the size of local parameter fluctuations in the geometry induced by the Fisher information. For Fisher-regular losses, we prove that the scale \(w_G(H_r)/\sqrt n\) is attained on sufficiently small Fisher balls. On the recovery side, the inverse-Fisher width captures the effect of anisotropic Gaussian measurements whose covariance is determined by the inverse Fisher information. For sparse recovery, the resulting geometry depends not only on sparsity but also on the position of the active coordinates in the Fisher spectrum. We obtain a two-sided estimate for the corresponding statistical dimension, together with support-sensitive recovery estimates and a natural ordering of supports with different curvature profiles. Finally, we establish a sharp relation between the primal and inverse-Fisher widths. On any common compact coordinate set $T$, they satisfy \[ w_G(T)w_{G^{-1}}(T)\geq w(T)^2. \] Thus, Fisher anisotropy may transfer complexity from one geometry to the other, but cannot reduce both widths relative to the Euclidean scale.
Md Sakir Ahmed, Kumaresh Sarmah, Hemen Duttacs.LG cs.CG
A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with $η/B$ matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.
We introduce the Generalized Neural Distributional Regression (GNDR) framework, which seamlessly embeds deep neural networks into the parameter space of classical probability distributions. To reconcile the inherent non-identifiability of deep architectures with maximum likelihood theory, we propose a two-step semi-parametric estimation procedure. By isolating the terminal prediction heads and treating the upstream network as a fixed, non-linear basis expansion, GNDR enables the extraction of analytical Fisher Information matrices. This facilitates rigorous uncertainty quantification, generating observation-specific confidence bands and tolerance intervals via the multivariate Delta method. We demonstrate the framework's versatility and superior distributional calibration across diverse data modalities, including overdispersed clinical counts, right-censored transcriptomic survival profiles under a mixture cure framework, and zero-truncated age distributions derived directly from unstructured facial images. The methodology is natively implemented in the open-source Python package \textit{thetaflow}.