Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
My PhD talk "Application of H-matrices for computing partial inverse"Alexander Litvinenko
This document describes a hierarchical domain decomposition (HDD) method for solving stochastic elliptic boundary value problems with oscillatory or jumping coefficients. HDD constructs mappings between boundary and interface values that allow the solution to be computed locally in each subdomain. These mappings are represented as H-matrices to reduce computational costs. The total storage cost of HDD is O(kn log2nh) and complexity is O(k2nh log3nh), where n is the number of degrees of freedom, k is the H-matrix rank, and h is the mesh size. HDD can also be used to compute solutions when the right-hand side is represented on a coarser grid.
This document provides an introduction and overview of nonparametric predictive regression for modeling nonlinear and time-varying effects. It summarizes the following key points:
1) Nonparametric predictive regression is proposed to model relationships that may change over time using orthogonal series estimation and local smoothing.
2) A two-step estimation procedure is used, with orthogonal series estimation in the first step to reduce bias, followed by local smoothing in the second step.
3) Asymptotic properties of the estimators are derived, showing they are consistent and asymptotically normal under certain conditions, whether the predictors are stationary or nonstationary.
On the-approximate-solution-of-a-nonlinear-singular-integral-equationCemal Ardil
This document summarizes a study on finding approximate solutions to nonlinear singular integral equations. The study proves the existence and uniqueness of solutions to such equations defined on bounded regions of the complex plane. It then presents a method for finding approximate solutions using an iterative fixed-point principle approach. Nonlinear singular integral equations have many applications in fields like elasticity, fluid mechanics, and mathematical physics. The study contributes to improving methods for solving these important types of equations.
This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
My PhD talk "Application of H-matrices for computing partial inverse"Alexander Litvinenko
This document describes a hierarchical domain decomposition (HDD) method for solving stochastic elliptic boundary value problems with oscillatory or jumping coefficients. HDD constructs mappings between boundary and interface values that allow the solution to be computed locally in each subdomain. These mappings are represented as H-matrices to reduce computational costs. The total storage cost of HDD is O(kn log2nh) and complexity is O(k2nh log3nh), where n is the number of degrees of freedom, k is the H-matrix rank, and h is the mesh size. HDD can also be used to compute solutions when the right-hand side is represented on a coarser grid.
This document provides an introduction and overview of nonparametric predictive regression for modeling nonlinear and time-varying effects. It summarizes the following key points:
1) Nonparametric predictive regression is proposed to model relationships that may change over time using orthogonal series estimation and local smoothing.
2) A two-step estimation procedure is used, with orthogonal series estimation in the first step to reduce bias, followed by local smoothing in the second step.
3) Asymptotic properties of the estimators are derived, showing they are consistent and asymptotically normal under certain conditions, whether the predictors are stationary or nonstationary.
On the-approximate-solution-of-a-nonlinear-singular-integral-equationCemal Ardil
This document summarizes a study on finding approximate solutions to nonlinear singular integral equations. The study proves the existence and uniqueness of solutions to such equations defined on bounded regions of the complex plane. It then presents a method for finding approximate solutions using an iterative fixed-point principle approach. Nonlinear singular integral equations have many applications in fields like elasticity, fluid mechanics, and mathematical physics. The study contributes to improving methods for solving these important types of equations.
This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
This document provides an introduction and overview of Sylow's theorem regarding the construction of finite groups with specific numbers of Sylow p-subgroups. It begins with prerequisites and definitions, then presents three theorems:
Theorem 1 proves the existence of a group with qe Sylow p-subgroups for any e in a set E. Corollary 1 extends this to allow constructing groups with qem Sylow p-subgroups for any m. Theorem 2 addresses the special case of 2-subgroups, showing there exists a group with n Sylow 2-subgroups for any odd positive integer n. The document establishes notation and provides proofs of lemmas supporting each theorem. It aims to provide intuition on constructing groups to
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Reinforcement learning: hidden theory, and new super-fast algorithms
Lecture presented at the Center for Systems and Control (CSC@USC) and Ming Hsieh Institute for Electrical Engineering,
February 21, 2018
Stochastic Approximation algorithms are used to approximate solutions to fixed point equations that involve expectations of functions with respect to possibly unknown distributions. The most famous examples today are TD- and Q-learning algorithms. The first half of this lecture will provide an overview of stochastic approximation, with a focus on optimizing the rate of convergence. A new approach to optimize the rate of convergence leads to the new Zap Q-learning algorithm. Analysis suggests that its transient behavior is a close match to a deterministic Newton-Raphson implementation, and numerical experiments confirm super fast convergence.
Based on
@article{devmey17a,
Title = {Fastest Convergence for {Q-learning}},
Author = {Devraj, Adithya M. and Meyn, Sean P.},
Journal = {NIPS 2017 and ArXiv e-prints},
Year = 2017}
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
1) This document summarizes a presentation on global sensitivity analysis (GSA) for high-dimensional models. GSA aims to quantify how uncertainties in a model's output can be attributed to uncertainties in its inputs.
2) GSA methods include variance-based (Sobol' indices), regression-based, and derivative-based approaches. Variance-based GSA uses Sobol' indices to apportion a model output's variance to its input parameters.
3) The presentation illustrates GSA on a neurovascular coupling model with 67 variables and 160 uncertain parameters. Dimension reduction via linear regression identified important parameters before building a polynomial chaos surrogate model to compute Sobol' indices.
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...Alexander Litvinenko
We develop hierarchical domain decomposition method to compute a part of the solution, a part of the inverse operator with O(n log n) storage and computing cost.
This document discusses priors for mixture models. It introduces weakly informative priors like symmetric empirical Bayes priors and dependent priors. Improper independent priors are problematic for mixtures. Reparameterization techniques are discussed to define proper Jeffreys priors, including expressing components as local perturbations, using moments, and spherical reparameterization. Specific examples for Gaussian and Poisson mixtures show valid reparameterizations that lead to proper posteriors.
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
Modeling the Dynamics of SGD by Stochastic Differential EquationMark Chang
1) Start with a small learning rate and large batch size to find a flat minimum with good generalization. 2) Gradually increase the learning rate and decrease the batch size to find sharper minima that may improve training accuracy. 3) Monitor both training and validation/test accuracy - similar accuracy suggests good generalization while different accuracy indicates overfitting.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,aijcsa
A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.
The Solovay-Kitaev Theorem guarantees that for any single-qubit gate U and precision ε > 0, it is possible to approximate U to within ε using Θ(logc(1/ε)) gates from a fixed finite universal set of quantum gates. The proof involves first using a "shrinking lemma" to show that any gate in an ε-net can be approximated to within Cε using a constant number of applications of gates from the universal set. This is then iterated to construct an approximation of the desired gate U to arbitrary precision using a number of gates that scales as the logarithm of 1/ε.
1) The first integral evaluates to 4πi using the Cauchy integral formula applied to circles around z=1 and z=2.
2) The second integral evaluates the 4th derivative of e2z at z=-1 using a formula relating derivatives and contour integrals, giving a value of 24.
3) Both integrals are evaluated quickly using results from complex analysis without direct computation.
1. The document contains 30 multiple choice questions about limits, functions, and graphs. Questions cover topics like domains, ranges, limits, periodicity, symmetry, and functional equations.
2. Several questions involve greatest integer functions, fractional parts of numbers, trigonometric functions like sine, cosine, and tangent, and inverse trigonometric functions.
3. Correct answers are requested for questions involving evaluating limits, determining properties of functions, finding function values, and identifying function domains and ranges.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
This document defines and provides examples of curves and parametrized curves. It discusses regular and unit-speed curves. The key points are:
i) A parametrized curve is a continuous function from an interval to Rn. Examples of parametrized curves include ellipses, parabolas, and helices.
ii) A regular curve is one where the derivative of the parametrization is never zero. A unit-speed curve has a derivative of constant length 1.
iii) The arc-length of a curve is defined as the integral of the derivative of the parametrization. Any reparametrization of a regular curve is also regular. A curve has a unit-
2017-07, Research Seminar at Keio University, Metric Perspective of Stochasti...asahiushio1
In this talk, I explain several major stochastic optimizers from the perspective of the metric, that is the definition of the parameter space of the model.
√
(t, d1 , d2 )| ≤ tφ(t) whp
This document summarizes results about the statistics of a random graph model called the Bollobás-Borgs-Chayes-Riordan model. Key points:
1) The expected number of vertices at time t grows as (α + γ)t, where α and γ are parameters of the model.
2) The expected number of vertices with in-degree d follows a power law distribution pdt, where pd decays as d increases.
3) The expected number of edges between a vertex with out-degree d1 and one with in-degree d2 is given by a formula
This document provides an introduction and overview of Sylow's theorem regarding the construction of finite groups with specific numbers of Sylow p-subgroups. It begins with prerequisites and definitions, then presents three theorems:
Theorem 1 proves the existence of a group with qe Sylow p-subgroups for any e in a set E. Corollary 1 extends this to allow constructing groups with qem Sylow p-subgroups for any m. Theorem 2 addresses the special case of 2-subgroups, showing there exists a group with n Sylow 2-subgroups for any odd positive integer n. The document establishes notation and provides proofs of lemmas supporting each theorem. It aims to provide intuition on constructing groups to
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Reinforcement learning: hidden theory, and new super-fast algorithms
Lecture presented at the Center for Systems and Control (CSC@USC) and Ming Hsieh Institute for Electrical Engineering,
February 21, 2018
Stochastic Approximation algorithms are used to approximate solutions to fixed point equations that involve expectations of functions with respect to possibly unknown distributions. The most famous examples today are TD- and Q-learning algorithms. The first half of this lecture will provide an overview of stochastic approximation, with a focus on optimizing the rate of convergence. A new approach to optimize the rate of convergence leads to the new Zap Q-learning algorithm. Analysis suggests that its transient behavior is a close match to a deterministic Newton-Raphson implementation, and numerical experiments confirm super fast convergence.
Based on
@article{devmey17a,
Title = {Fastest Convergence for {Q-learning}},
Author = {Devraj, Adithya M. and Meyn, Sean P.},
Journal = {NIPS 2017 and ArXiv e-prints},
Year = 2017}
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
This document discusses methods for identifying the source node of information spread in networks based on the observed spread over time. It begins by introducing epidemic models like SIS and SI for modeling information spread over networks. It then discusses maximum likelihood methods for identifying the source node on regular tree networks based on the observed subgraph. The accuracy of these methods increases with network size and degree. Extensions to other network structures and SIR models are also proposed. Overall, the document reviews mathematical models and algorithms for source identification in networks from limited observations of information spread.
1) This document summarizes a presentation on global sensitivity analysis (GSA) for high-dimensional models. GSA aims to quantify how uncertainties in a model's output can be attributed to uncertainties in its inputs.
2) GSA methods include variance-based (Sobol' indices), regression-based, and derivative-based approaches. Variance-based GSA uses Sobol' indices to apportion a model output's variance to its input parameters.
3) The presentation illustrates GSA on a neurovascular coupling model with 67 variables and 160 uncertain parameters. Dimension reduction via linear regression identified important parameters before building a polynomial chaos surrogate model to compute Sobol' indices.
Application H-matrices for solving PDEs with multi-scale coefficients, jumpin...Alexander Litvinenko
We develop hierarchical domain decomposition method to compute a part of the solution, a part of the inverse operator with O(n log n) storage and computing cost.
This document discusses priors for mixture models. It introduces weakly informative priors like symmetric empirical Bayes priors and dependent priors. Improper independent priors are problematic for mixtures. Reparameterization techniques are discussed to define proper Jeffreys priors, including expressing components as local perturbations, using moments, and spherical reparameterization. Specific examples for Gaussian and Poisson mixtures show valid reparameterizations that lead to proper posteriors.
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
- Hiroaki Shiokawa's research interests include graph mining, network analysis, and efficient algorithms. He was previously employed at NTT from 2011 to 2015.
- His current research focuses on developing clustering algorithms for large-scale networks and evaluating their performance on real-world network datasets.
- He has published highly cited papers in top data mining and network science conferences such as KDD, CIKM, and WSDM.
Modeling the Dynamics of SGD by Stochastic Differential EquationMark Chang
1) Start with a small learning rate and large batch size to find a flat minimum with good generalization. 2) Gradually increase the learning rate and decrease the batch size to find sharper minima that may improve training accuracy. 3) Monitor both training and validation/test accuracy - similar accuracy suggests good generalization while different accuracy indicates overfitting.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,aijcsa
A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.
The Solovay-Kitaev Theorem guarantees that for any single-qubit gate U and precision ε > 0, it is possible to approximate U to within ε using Θ(logc(1/ε)) gates from a fixed finite universal set of quantum gates. The proof involves first using a "shrinking lemma" to show that any gate in an ε-net can be approximated to within Cε using a constant number of applications of gates from the universal set. This is then iterated to construct an approximation of the desired gate U to arbitrary precision using a number of gates that scales as the logarithm of 1/ε.
1) The first integral evaluates to 4πi using the Cauchy integral formula applied to circles around z=1 and z=2.
2) The second integral evaluates the 4th derivative of e2z at z=-1 using a formula relating derivatives and contour integrals, giving a value of 24.
3) Both integrals are evaluated quickly using results from complex analysis without direct computation.
1. The document contains 30 multiple choice questions about limits, functions, and graphs. Questions cover topics like domains, ranges, limits, periodicity, symmetry, and functional equations.
2. Several questions involve greatest integer functions, fractional parts of numbers, trigonometric functions like sine, cosine, and tangent, and inverse trigonometric functions.
3. Correct answers are requested for questions involving evaluating limits, determining properties of functions, finding function values, and identifying function domains and ranges.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
This document defines and provides examples of curves and parametrized curves. It discusses regular and unit-speed curves. The key points are:
i) A parametrized curve is a continuous function from an interval to Rn. Examples of parametrized curves include ellipses, parabolas, and helices.
ii) A regular curve is one where the derivative of the parametrization is never zero. A unit-speed curve has a derivative of constant length 1.
iii) The arc-length of a curve is defined as the integral of the derivative of the parametrization. Any reparametrization of a regular curve is also regular. A curve has a unit-
2017-07, Research Seminar at Keio University, Metric Perspective of Stochasti...asahiushio1
In this talk, I explain several major stochastic optimizers from the perspective of the metric, that is the definition of the parameter space of the model.
√
(t, d1 , d2 )| ≤ tφ(t) whp
This document summarizes results about the statistics of a random graph model called the Bollobás-Borgs-Chayes-Riordan model. Key points:
1) The expected number of vertices at time t grows as (α + γ)t, where α and γ are parameters of the model.
2) The expected number of vertices with in-degree d follows a power law distribution pdt, where pd decays as d increases.
3) The expected number of edges between a vertex with out-degree d1 and one with in-degree d2 is given by a formula
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
Recurrence Relation for Achromatic Number of Line Graph of GraphIRJET Journal
This document presents a mathematical analysis of graph coloring and derives a recurrence relation for the achromatic number A(n) of the line graph of the complete graph K_n.
The key points are:
1) It defines relevant graph coloring concepts like chromatic number, achromatic number, and line graphs.
2) It proves two lemmas about upper bounds on A(n) in terms of functions g(n,t) and h(n,t) related to the number of edges.
3) Using the lemmas, it derives that A(n+2) is greater than or equal to A(n)+2 for n greater than 4, providing a recurrence
Murphy: Machine learning A probabilistic perspective: Ch.9Daisuke Yoneoka
This document summarizes key concepts about the exponential family and generalized linear models (GLMs). It defines the exponential family and provides examples like the Bernoulli, multinomial, and Gaussian distributions. The exponential family has important properties like finite sufficient statistics, existence of conjugate priors, and convexity. Maximum likelihood estimation for the exponential family involves matching sample moments to population moments. Conjugate priors allow tractable Bayesian inference for the exponential family. The document outlines maximum entropy derivation of the exponential family and how GLMs can generate classifiers.
A common random fixed point theorem for rational inequality in hilbert spaceAlexander Decker
This document presents a common random fixed point theorem for four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. It begins with introducing basic concepts such as separable Hilbert spaces, random operators, and common random fixed points. It then defines a condition (A) that the four mappings must satisfy. The main result is Theorem 2.1, which proves the existence of a unique common random fixed point for the four operators under condition (A) and a rational inequality condition. The proof constructs a sequence of measurable functions and shows it converges to the common random fixed point. This establishes the common random fixed point theorem for these operators.
This document presents a Bayesian joint model for longitudinal and time-to-event outcomes that allows for subpopulation heterogeneity using latent variables. It begins with motivational examples in HIV and prostate cancer research. It then reviews existing joint modeling approaches before introducing a new latent process model that models the longitudinal and survival outcomes conditional on a shared latent process. The document describes prior distributions, a Gibbs sampling algorithm, and simulation studies to evaluate the model under both Gaussian and non-Gaussian longitudinal distributions.
Fixed point theorems for random variables in complete metric spacesAlexander Decker
This document presents two fixed point theorems for random variables in complete metric spaces. Theorem 1 proves that if a self-mapping E on a complete metric space satisfies certain rational inequalities involving distances between random variables, then E has a fixed point. Theorem 2 proves a similar result for a self-mapping E satisfying alternative rational inequalities, assuming E is onto. Both theorems use properties of complete metric spaces and rational inequalities to show the existence of fixed points for random variables under the given conditions.
This document contains 12 math problems involving algebra concepts like solving equations, logarithms, trigonometry, and geometry. The problems cover topics such as solving systems of equations, evaluating logarithmic and trigonometric expressions, finding slopes of lines, and more. Detailed step-by-step workings are shown for each problem.
This document summarizes the derivation of Planck's law for blackbody radiation. It begins by defining the spectral radiance as a function of wavelength and temperature. Taking the derivative and setting it equal to zero allows solving for the wavelength of maximum power emission. Integrating the spectral radiance over all wavelengths provides an expression for the total emissive power in terms of the Stefan-Boltzmann constant. The document then derives expressions for the spectral and total hemispherical emissivity of a surface in terms of the absorptivity. It shows that the emissivity equals the absorptivity for thermal equilibrium. Approximations are made for the Planck function in the limits of long and short wavelengths compared to temperature.
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This document contains a chapter about mathematical descriptions of continuous-time signals. It includes examples of signal functions, operations like shifting and scaling on signals, derivatives and integrals of signals, properties of even and odd signals, and exercises with answers related to these topics. The exercises involve graphing signals, finding signal values at times, manipulating signals using operations, and identifying signal properties.
A common unique random fixed point theorem in hilbert space using integral ty...Alexander Decker
This document presents a common unique random fixed point theorem for two continuous random operators defined on a non-empty closed subset of a Hilbert space.
The theorem proves that if two continuous random operators S and T satisfy a certain integral type condition (Condition A), then S and T have a unique common random fixed point.
The proof constructs a sequence of measurable functions {ng} and shows that it converges to the common unique random fixed point of S and T. It utilizes a rational inequality and the parallelogram law to show {ng} is a Cauchy sequence that converges, and its limit is the random fixed point.
A common random fixed point theorem for rational ineqality in hilbert space ...Alexander Decker
This document presents a common random fixed point theorem for four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. It begins with introducing relevant definitions including measurable functions, random operators, and random fixed points. It then states the main theorem (Theorem 2.1) which shows that if four random operators satisfy a certain condition (Condition A), then they have a unique common random fixed point. The proof of Theorem 2.1 is also presented, showing that a sequence constructed from the random operators converges to the common random fixed point.
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ATT00002
ATT00003
ATT00004
ATT00005
CARD.DTA
Card_1995_geo_var_schooling.pdf
Exam2_2014.pdf
ADVANCED ECONOMETRICS
Midterm 2 (Take Home)
Due: Dec.25, 2014
Answer all questions. You should not discuss solutions with your peers but me. Good luck!
Prof. Dr. H. Taştan ,
First Name:...................................................
Last Name:................................................
No:...................................................
1 (20) In class we have shown that when the number of instrumental variables is larger than the number
of endogenous variables the generalized IV estimator (or 2SLS) can be written as
β̂IV =
(
X>PzX
)−1
X>Pzy
where Pz = Z(Z
>Z)−1Z>. In this formulation X is n×k and Z is n× l, l > k.
(a) Show that β̂IV can be obtained as a solution to the following minimization problem
min
β
Q(β) = (y−Xβ)>Pz (y−Xβ)
(b) Show that when k = l the generalized IV estimator reduces to the simple IV estimator:
β̂IV =
(
Z>X
)−1
Z>y
2 (20) Consider the following simple consumption model as a function of permanent income
ci = β1 + β2y
∗
i + ui, ui ∼ iid (0,σ
2
u)
where ci is the logarithm of consumption by household i, and y
∗
i is the permanent income of household
i which is not observed. Instead we observe current income, yi
yi = y
∗
i + vi, vi ∼ iid (0,σ
2
v)
where vi is assumed to be uncorrelated with y
∗
i and ηi. We run the following regression
ci = β1 + β2yi + ηi
(a) Show that yi is negatively correlated with ηi. You can assume β2 > 0.
(b) Evaluate the plim of the OLS estimator β̂2:
β̂2 =
∑n
i=1(yi − ȳ)ci∑n
i=1(yi − ȳ)2
In particular, show that this plim is less than the true β2.
1
3 (30) Use card.dta to answer the following questions. Also read Card (1993), “Using Geographic
Variation in College Proximity to Estimate the Return to Schooling”, NBER Working Paper.
(a) Run the OLS regression of log(wage) on educ, exper, exper2, black, smsa, south, smsa66, reg662
to reg669. Comment on the coefficient estimate of educ.
(b) Estimate the same model by 2SLS using nearc4 as an instrument for educ. Compare the OLS
and IV coefficient estimates on educ. (Note that we partly did this in class). Carry out the
Hausman test.
(c) Use both nearc2 and nearc4 as instruments for educ. Run the reduced form model for educ.
Compare 2SLS estimates to the results obtained in the previous section. Carry out the OID
test.
(d) Discuss the plausibility of Card (1993)’s econometric methodology and empirical findings. Do
you agree with his conclusions?
4 (30) A continuous time model for short term interest rates may be written as a stochastic differential
equation
dr = (α + βr)dt + σrγ�
√
dt
where r is the short term interest rate, � is standard normal random variable, dt is a short time
interval and α,β,γ,σ are parameters. Discrete time approximation is given as
rt+1 − rt = α + βrt + �t+1
with
E(�t+1) = 0, E(�
2
t+1) = σ
2 ...
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and
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Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
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Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
Updates provided to the D-STOP Business Advisory Council at the 2017 Symposium and Board Meeting: https://ctr.utexas.edu/2018/04/12/d-stop-2017-symposium-archive/
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The Business Advisory Council meeting covered the following topics in 3 sentences or less:
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Statistical Inference Using Stochastic Gradient Descent
1. Statistical inference using
stochastic gradient descent
Constantine Caramanis1
Liu Liu1 Anastasios (Tasos) Kyrillidis2 Tianyang Li1
1The University of Texas at Austin
2IBM T.J. Watson Research Center, Yorktown Heights → Rice University
2. Statistical inference is important
Quantifying uncertainty
Signal? Noise?
Skill? Luck?
Frequentist inference
confidence interval
hypothesis testing
3. Statistical inference is important
Quantifying uncertainty
Signal? Noise?
Skill? Luck?
Frequentist inference
confidence interval
hypothesis testing
Confidence intervals can be used to detect adversarial
attacks.
4. Outline of This Work
(a) Large Scale Problems: Point Estimates computed via SGD
(b) Confidence Intervals computed by Boostrap: too expensive.
(c) This talk: we can compute using SGD.
(d) Application to adversarial attacks: implicitly learning the
manifold.
5. SGD in ERM – mini batch SGD
To solve empirical risk minimization (ERM)
f (θ) =
1
n
n
i=1
fi (θ),
where fi (θ) = θ(Zi ).
At each step:
Draw S i.i.d. uniformly random indices It from [n] (with
replacement)
Compute stochastic gradient gs(θt) = 1
S i∈It
fi (θt)
θt+1 = θt − ηgs(θt)
6. Asymptotic normality – classical results
M-estimator – statistics
When number of samples n → ∞,
√
n(θ − θ∗
) N(0, H∗−1
G∗
H∗−1
),
where G∗ = EZ [ θ θ∗ (Z) θ θ∗ (Z) ] and H∗ = EZ [ 2
θ θ∗ (Z)].
Stochastic approximation – optimization
When number of steps t → ∞,
√
t
1
t
t
i=1
θt − θ N(0, H−1
GH−1
),
where G = E[gs(θ)gs(θ) |= θ] and H = 2f (θ).
7. Asymptotic normality – classical results
M-estimator – statistics
When number of samples n → ∞,
√
n(θ − θ∗
) N(0, H∗−1
G∗
H∗−1
),
where G∗ = EZ [ θ θ∗ (Z) θ θ∗ (Z) ] and H∗ = EZ [ 2
θ θ∗ (Z)].
Stochastic approximation – optimization
When number of steps t → ∞,
√
t
1
t
t
i=1
θt − θ N(0, H−1
GH−1
),
where G = E[gs(θ)gs(θ) |= θ] and H = 2f (θ).
SGD not only useful for optimization,
but also useful for statistical inference!
8. Statistical inference using mini batch SGD
burn in
θ−b, θ−b+1, · · · θ−1, θ0,
¯θ
(i)
t =1
t
t
j=1 θ
(i)
j
θ
(1)
1 , θ
(1)
2 , · · · , θ
(1)
t
discarded
θ
(1)
t+1, θ
(1)
t+2, · · · , θ
(1)
t+d
θ
(2)
1 , θ
(2)
2 , · · · , θ
(2)
t θ
(2)
t+1, θ
(2)
t+2, · · · , θ
(2)
t+d
...
θ
(R)
1 , θ
(R)
2 , · · · , θ
(R)
t θ
(R)
t+1, θ
(R)
t+2, · · · , θ
(R)
t+d
At each step:
Draw S i.i.d. uniformly random
indices It from [n] (with replacement)
Compute stochastic gradient
gs(θt) = 1
S i∈It
fi (θt)
θt+1 = θt − ηgs(θt)
Use an ensemble of i = 1, 2, . . . , R estima-
tors for statistical inference:
θ(i)
= θ +
√
S
√
t
√
n
(¯θ
(i)
t − θ).
9. Advantages of SGD inference
empirically not more expensive, uses
many fewer operations than
bootstrap
can be used when training neural
networks with SGD
easy to plug into existing SGD code
Other statistical inference
methods
directly computing inverse
Fisher information matrix
resampling:
bootstrap, subsampling
10. Advantages of SGD inference
empirically not more expensive, uses
many fewer operations than
bootstrap
can be used when training neural
networks with SGD
easy to plug into existing SGD code
Other statistical inference
methods
directly computing inverse
Fisher information matrix
resampling:
bootstrap, subsampling
Too computationally expensive,
not suited for “big data”!
11. Intuition – Ornstein-Uhlenbeck process approximation
In SGD, denote ∆t = θt − θ, and we have
∆t+1 = ∆t − ηgs(θ + ∆t).
∆t can be approximated by the Ornstein-Uhlenbeck process
d∆(T) = −H∆ dT +
√
ηG
1
2 dB(T),
where B(T) is a standard Brownian motion.
12. Intuition – Ornstein-Uhlenbeck process approximation
Denote ¯θt = 1
t
t
i=1 θt.
√
t(¯θt − θ) can be approximated as
√
t(¯θt − θ) = 1√
t
t
i=1
(θi − θ)
= 1
η
√
t
t
i=1
(θi − θ)η ≈ 1
η
√
t
tη
0
∆(T) dT,
(1)
where we use the approximation that η ≈ dT. By rearranging terms and multiplying both sides by H−1,
we can rewrite the stochastic differential equation as ∆(T) dT = −H−1 d∆(T) +
√
ηH−1G
1
2 dB(T).
Thus, we have
tη
0
∆(T) dT = −H−1
(∆(tη) − ∆(0)) +
√
ηH−1
G
1
2 B(tη). (2)
After plugging (2) into (1) we have
√
t ¯θt − θ ≈ − 1
η
√
t
H−1
(∆(tη) − ∆(0)) + 1√
tη
H−1
G
1
2 B(tη).
When ∆(0) = 0, the variance Var −1/η
√
t · H−1 (∆(tη) − ∆(0)) = O (1/tη). Since 1/√
tη ·
H−1G
1
2 B(tη) ∼ N(0, H−1GH−1), when η → 0 and ηt → ∞, we conclude that
√
t(¯θt − θ) ∼ N(0, H−1
GH−1
).
13. Theoretical guarantee
Theorem
For a differentiable convex function f (θ) = 1
n
n
i=1 fi (θ), with gradient f (θ), let θ ∈ Rp be
its minimizer, and denote its Hessian at θ by H := 2f (θ) . Assume that ∀θ ∈ Rp, f satisfies:
(F1) Weak strong convexity: (θ − θ) f (θ) ≥ α θ − θ 2
2, for constant α > 0,
(F2) Lipschitz gradient continuity: f (θ) 2 ≤ L θ − θ 2, for constant L > 0,
(F3) Bounded Taylor remainder: f (θ) − H(θ − θ) 2 ≤ E θ − θ 2
2, for constant E > 0,
(F4) Bounded Hessian spectrum at θ: 0 < λL ≤ λi (H) ≤ λU < ∞, ∀i.
Furthermore, let gs(θ) be a stochastic gradient of f , satisfying:
(G1) E [gs(θ) | θ] = f (θ),
(G2) E gs(θ) 2
2 | θ ≤ A θ − θ 2
2 + B,
(G3) E gs(θ) 4
2 | θ ≤ C θ − θ 4
2 + D,
(G4) E gs(θ)gs(θ) | θ − G 2
≤ A1 θ − θ 2 + A2 θ − θ 2
2 + A3 θ − θ 3
2 + A4 θ − θ 4
2,
for positive, data dependent constants A, B, C, D, Ai , for i = 1, . . . , 4. Assume that
θ1 − θ 2
2 = O(η); then for sufficiently small step size η > 0, the average SGD sequence
θt = 1
t
n
i=1 θi satisfies:
tE[(¯θt − θ)(¯θt − θ) ] − H−1
GH−1
2
√
η + 1
tη + tη2,
where G = E[gs(θ)gs(θ) | θ].
14. Theoretical guarantee
Theorem
For a differentiable convex function f (θ) = 1
n
n
i=1 fi (θ), with gradient f (θ), let θ ∈ Rp be
its minimizer, and denote its Hessian at θ by H := 2f (θ) . Assume that ∀θ ∈ Rp, f satisfies:
(F1) Weak strong convexity: (θ − θ) f (θ) ≥ α θ − θ 2
2, for constant α > 0,
(F2) Lipschitz gradient continuity: f (θ) 2 ≤ L θ − θ 2, for constant L > 0,
(F3) Bounded Taylor remainder: f (θ) − H(θ − θ) 2 ≤ E θ − θ 2
2, for constant E > 0,
(F4) Bounded Hessian spectrum at θ: 0 < λL ≤ λi (H) ≤ λU < ∞, ∀i.
Furthermore, let gs(θ) be a stochastic gradient of f , satisfying:
(G1) E [gs(θ) | θ] = f (θ),
(G2) E gs(θ) 2
2 | θ ≤ A θ − θ 2
2 + B,
(G3) E gs(θ) 4
2 | θ ≤ C θ − θ 4
2 + D,
(G4) E gs(θ)gs(θ) | θ − G 2
≤ A1 θ − θ 2 + A2 θ − θ 2
2 + A3 θ − θ 3
2 + A4 θ − θ 4
2,
for positive, data dependent constants A, B, C, D, Ai , for i = 1, . . . , 4. Assume that
θ1 − θ 2
2 = O(η); then for sufficiently small step size η > 0, the average SGD sequence
θt = 1
t
n
i=1 θi satisfies:
tE[(¯θt − θ)(¯θt − θ) ] − H−1
GH−1
2
√
η + 1
tη + tη2,
where G = E[gs(θ)gs(θ) | θ].
Proof idea: H−1 = η i≥0(I − ηH)i
16. 95% confidence interval coverage simulation
η t = 100 t = 500 t = 2500
0.1 (0.957, 4.41) (0.955, 4.51) (0.960, 4.53)
0.02 (0.869, 3.30) (0.923, 3.77) (0.918, 3.87)
0.004 (0.634, 2.01) (0.862, 3.20) (0.916, 3.70)
(a) Bootstrap (0.941, 4.14), normal approximation (0.928, 3.87)
η t = 100 t = 500 t = 2500
0.1 (0.949, 4.74) (0.962, 4.91) (0.963, 4.94)
0.02 (0.845, 3.37) (0.916, 4.01) (0.927, 4.17)
0.004 (0.616, 2.00) (0.832, 3.30) (0.897, 3.93)
(b) Bootstrap (0.938, 4.47), normal approximation (0.925, 4.18)
Table 1: Linear regression: dimension = 10, 100 samples. (a) diagonal
covariance (b) non-diagonal covariance
η t = 100 t = 500 t = 2500
0.1 (0.872, 0.204) (0.937, 0.249) (0.939, 0.258)
0.02 (0.610, 0.112) (0.871, 0.196) (0.926, 0.237)
0.004 (0.312, 0.051) (0.596, 0.111) (0.86, 0.194)
(a) Bootstrap (0.932, 0.253), normal approximation (0.957, 0.264)
η t = 100 t = 500 t = 2500
0.1 (0.859, 0.206) (0.931, 0.255) (0.947, 0.266)
0.02 (0.600, 0.112) (0.847, 0.197) (0.931, 0.244)
0.004 (0.302, 0.051) (0.583, 0.111) (0.851, 0.195)
(b) Bootstrap (0.932, 0.245), normal approximation (0.954, 0.256)
Table 2: Logistic regression: dimension = 10, 1000 samples. (a) diagonal
covariance (b) non-diagonal covariance
Better when
each replicate’s average uses a longer consecutive sequence
larger step size
(coverage probability, confidence interval width)
17. Adversarial Attacks
Neural network classifiers with very high accuracy on test sets are
extremely susceptible to nearly imperceptible adversarial attacks.