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Lecture

Stochastic Optimization and Adaptive Gradient Methods

In course

EE-566: Adaptation and learning

In this course, students learn to design and master algorithms and core concepts related to inference and learning from data and the foundations of adaptation and learning theories with applications.

Description

This lecture covers stochastic optimization and adaptive gradient methods. It discusses the application of these methods in recommender systems and matrix factorization, focusing on user-item rating matrices. The lecture explains the stochastic gradient algorithm and its implementation for predicting missing ratings in the matrix. It also explores the Polyak-Ruppert averaging technique to smooth iterates in the optimization process. Various concepts such as bias terms, feature vectors, and matrix factorization are detailed with examples and theoretical derivations.

Instructor

Ali H. Sayed

Ali H. Sayed is Dean of Engineering at EPFL, Switzerland, where he also leads the Adaptive Systems Laboratory. He has also served as Distinguished Professor and Chairman of Electrical Engineering at UCLA. He is recognized as a Highly Cited Researcher and is a member of the US National Academy of Engineering. He is also a member of the World Academy of Sciences and served as President of the IEEE Signal Processing Society during 2018 and 2019. Dr. Sayed is an author/co-author of over 570 scholarly publications and six books. His research involves several areas including adaptation and learning theories, data and network sciences, statistical inference, and multiagent systems. His work has been recognized with several major awards including the 2022 IEEE Fourier Award, the 2020 Norbert Wiener Society Award and the 2015 Education Award from the IEEE Signal Processing Society, the 2014 Papoulis Award from the European Association for Signal Processing, the 2013 Meritorious Service Award and the 2012 Technical Achievement Award from the IEEE Signal Processing Society, the 2005 Terman Award from the American Society for Engineering Education, the 2005 Distinguished Lecturer from the IEEE Signal Processing Society, the 2003 Kuwait Prize, and the 1996 IEEE Donald G. Fink Prize. His publications have been awarded several Best Paper Awards from the IEEE (2002, 2005, 2012, 2014) and EURASIP (2015). He is a Fellow of IEEE, EURASIP, and the American Association for the Advancement of Science (AAAS); the publisher of the journal Science.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Stochastic gradient descent

Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data).

LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then for all indices and Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

Unitary matrix

In linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

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