Efficient and convergent natural gradient based optimization algorithms for machine learning
[eng] Many times machine learning is casted as an optimization problem. This is the case when an objective function assesses the success of an agent in a certain task and hence, learning is accomplished by optimizing that function. Furthermore, gradient descent is an optimization algorithm that has...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/202192 |
| Acceso en línea: | https://hdl.handle.net/2445/202192 http://hdl.handle.net/10803/689000 |
| Access Level: | acceso abierto |
| Palabra clave: | Aprenentatge automàtic Optimització matemàtica Programació (Matemàtica) Algorismes computacionals Convergència (Matemàtica) Machine learning Mathematical optimization Mathematical programming Computer algorithms Convergence |
| Sumario: | [eng] Many times machine learning is casted as an optimization problem. This is the case when an objective function assesses the success of an agent in a certain task and hence, learning is accomplished by optimizing that function. Furthermore, gradient descent is an optimization algorithm that has proven to be a powerful tool, becoming the cornerstone to solving most machine learning challenges. Among its strengths, there are the low computational complexity and the convergence guarantee property to the optimum of the function, after certain regularities on the function. Nevertheless, large dimension scenarios show sudden drops in convergence rates which inhibit further improvements in an acceptable amount of time. For this reason, the field has contemplated the natural gradient to tackle this issue. The natural gradient is defined on a Riemannian manifold (M, g). A Riemannian manifold is a manifold M equipped with a metric g. The natural gradient vector of a function f at a point p in (M, g) is a vector in the tangent space at p that points to the direction in which f locally increases its value faster taking into account the metric attached to the manifold. It turns out that the manifold of probability distributions of the same family, usually considered in machine learning, has a natural metric associated, namely the fisher information metric. While natural gradient based algorithms show a better convergence speed in some limited examples, they often fail in providing good estimates or they even diverge. Moreover, they demand more calculations than the ones performed by gradient descent algorithms, increasing the computational complexity order. This thesis explores the natural gradient descent algorithm for the function optimization task. Our research aims at designing a natural gradient based algorithm to solve a function optimization problem, whose computational complexity is comparable to those gradient based and such that it benefits from higher rates of convergence compared to standard gradient based methods. To reach our objectives, the hypothesis formulated in this thesis is that the convergence property guarantee stabilizes natural gradient algorithms and it gives access to fast rates of convergence. Furthermore, the natural gradient can be computed fast for particular manifolds named dually flat manifolds, and hence, fast natural gradient optimization methods become available. The beginning of our research is mainly focused on the convergence property for natural gradient methods. We develop some strategies to define natural gradient methods whose convergence can be proven. The main assumptions require (M, g) to be a Riemannian manifold and f to be a differentiable function on M. Moreover, it turns out that the multinomial logistic regression problem, a widely considered machine learning problem, can be adapted and solved by taking a dually flat manifolds as the model. Hence, this problem is our most promising target in which the objective of the thesis can be completely accomplished. |
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