This work derives a finite-dimensional dual formulation of PrO inference that separates sampling fluctuation, approximation under a divergence budget, regularization, and numerical optimization error and uses an exactly solvable categorical example to show that predictive-risk convergence can imply convergence to a unique predictive distribution even though the parameter distributions have no weak limit on the original parameter space.
Abstract
Predictively oriented (PrO) inference quantifies uncertainty by selecting a distribution over model parameters to optimize a scoring rule applied to the induced predictive distribution, together with a divergence penalty from a reference distribution. By applying the scoring rule after averaging model densities, PrO inference targets predictive performance, accounting for model misspecification. We focus on the logarithmic score with general $\phi$-divergence regularization. Our contributions are twofold. First, we derive a finite-dimensional dual formulation of PrO inference. For $n$ observations, the dual problem has $n+1$ variables. We establish zero-duality-gap criteria and optimality conditions that relate the primal and dual solutions. When primal and dual solutions exist, these conditions yield a semi-analytical representation of the PrO posterior and certificates for assessing the accuracy of numerical solutions. For Kullback--Leibler regularization, the posterior has an exponential form. Second, we derive a finite-sample excess predictive-risk bound for approximate PrO posteriors that separates sampling fluctuation, approximation under a divergence budget, regularization, and numerical optimization error. The result applies even when the benchmark predictive risk is not attained by any probability distribution over the model parameters having finite divergence from the reference distribution. We use an exactly solvable categorical example to show that predictive-risk convergence can imply convergence to a unique predictive distribution even though the parameter distributions have no weak limit on the original parameter space. The example also shows that different $\phi$-divergences can require different regularization schedules. We conclude with a misspecified Gaussian location-mixture example that illustrates the dual computation, primal recovery, and numerical accuracy checks.
This work proposes a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems and provides explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation.
Rufeng Xiao, Rujun Jiang, Xudong Li et al.· 0 citations
Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by \(40.69\%\) and the mean PAC--Bayes certificate by \(21.40\%\) in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.
A more flexible framework in which a predictive model determines the nominal distribution and a separate model estimates a data-dependent radius is developed, which treats calibration as a practical mechanism for reliable decision making rather than a universal guarantee of improved optimization performance.
Marginalized importance weighting evaluates a target policy by reweighting offline state-action samples with its discounted occupancy ratio, characterized by an adjoint Bellman equation. Existing minimax, primal-dual, and fitted fixed-point estimators can leave residual occupancy-balance violations because of function-class approximation, regularization, or incomplete optimization. These violations are difficult to diagnose and reduce because the objectives generally lack a direct supervised validation loss for hyperparameter tuning, model selection, and early stopping. We introduce isotonic Bellman calibration, a one-dimensional, model-agnostic post-processing method that reduces these violations while preserving the ranking information in any initial occupancy-ratio estimate. The method corrects the estimate's scale and shape by applying fitted occupancy-ratio evaluation (FORE) over a one-dimensional class of nondecreasing transformations. We characterize Bellman calibration as a conditional fixed-point property equivalent to occupancy-balance against every test function of the calibrated ratio. More generally, we derive a calibration-refinement bound showing that any fitted ratio with small calibration error performs nearly as well as the best post-processing based on its fitted values. For isotonic Bellman calibration, we establish finite-sample calibration guarantees and a KL oracle inequality relative to the best monotone transformation of the initial estimate. Consequently, isotonic Bellman calibration achieves small calibration error and KL risk within statistical error of the best monotone correction, with guarantees for downstream target-occupancy functionals, including policy-value estimation.
It is shown that the classical Bayesian bootstrap closes this gap in U-calibration, which asks one online probability fore-caster to have low regret for every bounded proper loss, including losses unknown when the forecasts are made.
We consider the optimization of the Optimized Certainty Equivalent (OCE) risk, with applications including portfolio optimization in finance, and uncertainty quantification, classification, and regression in machine learning. Our contributions cover popular special cases of OCE, such as entropic risk, mean-variance risk, and smooth variants of Conditional Value-at-Risk. Our treatment sets out the conditions that facilitate the extension of OCE to unbounded r.v.s.. We provide a useful characterization of OCE that links OCE to utility-based shortfall risk (UBSR). Our characterization enables us to form an OCE estimator from the classic sample-average approximation (SAA) of UBSR. We derive mean-squared error (MSE) bounds for our proposed OCE estimator. For OCE optimization, we first derive an expression for the OCE gradient using the characterization linking OCE to UBSR. This expression serves as the basis for a gradient estimator for the OCE. We derive non-asymptotic bounds on the MSE for the proposed OCE gradient estimator. We incorporate the aforementioned gradient estimator into a stochastic gradient (SG) algorithm to optimize OCE and quantify its convergence rate using non-asymptotic bounds that we derive. Finally, we present three experiments that use our OCE optimization algorithm to solve portfolio optimization and uncertainty quantification problems.
Sumedh Gupte, A. PrashanthL., Sanjay P. Bhat· 1 citation