Наумов Алексей Александрович Неасимптотический анализ случайных объектов в пространствах высокой размерности и приложения к задачам машинного обучения

Краткое описание

Наумов Алексей Александрович Неасимптотический анализ случайных объектов в пространствах высокой размерности и приложения к задачам машинного обучения
ОГЛАВЛЕНИЕ ДИССЕРТАЦИИ
доктор наук Наумов Алексей Александрович
Contents

1 Introduction

2 Notations

3 Large-ball probabilities and applications to bootstrap and Bayesian inference

3.1 Gaussian comparison and anti-concentration inequalities

3.1.1 Main results

3.2 Application examples

3.2.1 Bootstrap validity for the Maximum Likelihood Estimation (MLE)

3.2.2 Prior impact in linear Gaussian modeling

3.2.3 Nonparametric Bayes approach

3.2.4 Central Limit Theorem in finite- and infinite-dimensional spaces

3.2.5 Bootstrap confidence sets for spectral projectors of sample covariance

4 On the Stability of Random Matrix Product: Application to Linear Stochastic Approximation and TD Learning

4.1 LSA driven by general state space Markov chain

4.1.1 Main Results

4.1.2 Application to Linear Stochastic Approximation

4.1.3 Temporal Difference Learning Algorithms

4.2 Tight High Probability Bounds for Linear Stochastic Approximation

with Fixed Stepsize

4.2.1 Moment and High-probability Bounds for Products of Random Matrices

5 Variance reduction in MCMC algorithms

5.1 Empirical Spectral Variance Minimization

5.1.1 Method

5.1.2 Theoretical analysis

5.2 Applications

5.2.1 Langevin dynamics

5.2.2 Extension to the Stochastic Gradient Langevin Dynamics

5.3 Experiments

5.3.1 Toy example

5.3.2 Gaussian mixture model

5.3.3 Bayesian logistic regression

5.3.4 Bayesian Probabilistic Matrix Factorization

6 Conclusion

7 Acknowledgements

References

A Paper "Large ball probabilities, Gaussian comparison and anticoncentration"

B Paper "Nonasymptotic Estimates for the Closeness of Gaussian

Measures on Balls"

C Paper "Bootstrap confidence sets for spectral projectors of sample covariance"

D Paper "Confidence Sets for Spectral Projectors of Covariance Matrices"

E Paper "Finite Time Analysis of Linear Two-timescale Stochastic Approximation with Markovian Noise"

F Paper "On the Stability of Random Matrix Product with Markovian Noise: Application to Linear Stochastic Approximation and TD Learning"

G Paper "Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize"

H Paper "Variance reduction for Markov chains with application to MCMC"

I Paper "Variance Reduction for Dependent Sequences with Applications to Stochastic Gradient MCMC"

J Paper "Distribution of linear statistics of singular values of the product of random matrices"