Estimating Mixture Distributions via Stochastic Mirror Descent

• URL: http://arxiv.org/abs/2605.24929v1
• Date: Sun, 24 May 2026 08:19:42 GMT
• Title: Estimating Mixture Distributions via Stochastic Mirror Descent
• Abstract summary: We revisit the problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss.<n>We propose a family of estimators derived from the mirror descent (SMD) algorithm.<n>We demonstrate that under mild conditions, the proposed estimators achieve near-optimal convergence rates.
• Score: 15.58350224981605
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $ M $-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $ f_i $; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $ \varphi $-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $ \ell_2 $-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

Related papers

• Robust Variational Bayes by Min-Max Median Aggregation [13.102667562202386]
  We propose a robust variational Bayes framework to handle contamination and outliers in dataset.<n>Our approach partitions the data into $m$ disjoint subsets and formulates a joint optimization problem based on robust aggregation principles.<n>Our findings indicate that the two-stage approach yields a smaller approximation error compared to directly aggregating the $m$-powered local posteriors.
  arXiv  Detail & Related papers  (2025-12-14T13:02:00Z)
• On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization [57.179679246370114]
  A potential limitation of existing methods is the bias inherent in most perturbation estimators unless a stepsize is proposed.<n>We propose a novel family of unbiased gradient scaling estimators that eliminate bias while maintaining favorable construction.
  arXiv  Detail & Related papers  (2025-10-22T18:25:43Z)
• Learning Parametric Distributions from Samples and Preferences [19.879505582147807]
  We show that preference-based M-estimators achieve a better variance than sample-only M-estimators.<n>We propose an estimator achieving an estimation error scaling of $mathcalO (1/n)$ -- a significant improvement over the $Theta (1/sqrtn)$ rate attainable with samples alone.
  arXiv  Detail & Related papers  (2025-05-29T15:33:43Z)
• Ensemble Multi-Quantiles: Adaptively Flexible Distribution Prediction for Uncertainty Quantification [4.728311759896569]
  We propose a novel, succinct, and effective approach for distribution prediction to quantify uncertainty in machine learning. It incorporates adaptively flexible distribution prediction of $mathbbP(mathbfy|mathbfX=x)$ in regression tasks. On extensive regression tasks from UCI datasets, we show that EMQ achieves state-of-the-art performance.
  arXiv  Detail & Related papers  (2022-11-26T11:45:32Z)
• Optimization of Annealed Importance Sampling Hyperparameters [77.34726150561087]
  Annealed Importance Sampling (AIS) is a popular algorithm used to estimates the intractable marginal likelihood of deep generative models. We present a parameteric AIS process with flexible intermediary distributions and optimize the bridging distributions to use fewer number of steps for sampling. We assess the performance of our optimized AIS for marginal likelihood estimation of deep generative models and compare it to other estimators.
  arXiv  Detail & Related papers  (2022-09-27T07:58:25Z)
• A Unified Framework for Multi-distribution Density Ratio Estimation [101.67420298343512]
  Binary density ratio estimation (DRE) provides the foundation for many state-of-the-art machine learning algorithms. We develop a general framework from the perspective of Bregman minimization divergence. We show that our framework leads to methods that strictly generalize their counterparts in binary DRE.
  arXiv  Detail & Related papers  (2021-12-07T01:23:20Z)
• Learning to Estimate Without Bias [57.82628598276623]
  Gauss theorem states that the weighted least squares estimator is a linear minimum variance unbiased estimation (MVUE) in linear models. In this paper, we take a first step towards extending this result to non linear settings via deep learning with bias constraints. A second motivation to BCE is in applications where multiple estimates of the same unknown are averaged for improved performance.
  arXiv  Detail & Related papers  (2021-10-24T10:23:51Z)
• Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence [77.06203118175335]
  Variational Inference (VI) is a popular alternative to exact sampling in Bayesian inference. Importance sampling (IS) is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. We propose a novel combination of optimization and sampling techniques for approximate Bayesian inference.
  arXiv  Detail & Related papers  (2021-06-30T11:00:24Z)
• Unified Convergence Analysis for Adaptive Optimization with Moving Average Estimator [75.05106948314956]
  We show that an increasing large momentum parameter for the first-order moment is sufficient for adaptive scaling.<n>We also give insights for increasing the momentum in a stagewise manner in accordance with stagewise decreasing step size.
  arXiv  Detail & Related papers  (2021-04-30T08:50:24Z)
• Optimal Algorithms for Stochastic Multi-Armed Bandits with Heavy Tailed Rewards [24.983866845065926]
  We consider multi-armed bandits with heavy-tailed rewards, whose $p$-th moment is bounded by a constant $nu_p$ for $1pleq2$. We propose a novel robust estimator which does not require $nu_p$ as prior information. We show that an error probability of the proposed estimator decays exponentially fast.
  arXiv  Detail & Related papers  (2020-10-24T10:44:02Z)
• Distributionally Robust Bayesian Quadrature Optimization [60.383252534861136]
  We study BQO under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d. samples. A standard BQO approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set. We propose a novel posterior sampling based algorithm, namely distributionally robust BQO (DRBQO) for this purpose.
  arXiv  Detail & Related papers  (2020-01-19T12:00:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.