Fast Incremental SVDD Learning Algorithm with the Gaussian Kernel

Support vector data description (SVDD) is a machine learning technique that is used for single-class classification and outlier detection. The idea of SVDD is to find a set of support vectors that defines a boundary around data. When dealing with online or large data, existing batch SVDD methods have to be rerun in each iteration. We propose an incremental learning algorithm for SVDD that uses the Gaussian kernel. This algorithm builds on the observation that all support vectors on the boundary have the same distance to the center of sphere in a higher-dimensional feature space as mapped by the Gaussian kernel function. Each iteration involves only the existing support vectors and the new data point. Moreover, the algorithm is based solely on matrix manipulations; the support vectors and their corresponding Lagrange multiplier $\alpha_i$'s are automatically selected and determined in each iteration. It can be seen that the complexity of our algorithm in each iteration is only $O(k^2)$, where $k$ is the number of support vectors. Experimental results on some real data sets indicate that FISVDD demonstrates significant gains in efficiency with almost no loss in either outlier detection accuracy or objective function value.Comment: 18 pages, 1 table, 4 figure

Suboptimal Solution Path Algorithm for Support Vector Machine

We consider a suboptimal solution path algorithm for the Support Vector Machine. The solution path algorithm is an effective tool for solving a sequence of a parametrized optimization problems in machine learning. The path of the solutions provided by this algorithm are very accurate and they satisfy the optimality conditions more strictly than other SVM optimization algorithms. In many machine learning application, however, this strict optimality is often unnecessary, and it adversely affects the computational efficiency. Our algorithm can generate the path of suboptimal solutions within an arbitrary user-specified tolerance level. It allows us to control the trade-off between the accuracy of the solution and the computational cost. Moreover, We also show that our suboptimal solutions can be interpreted as the solution of a \emph{perturbed optimization problem} from the original one. We provide some theoretical analyses of our algorithm based on this novel interpretation. The experimental results also demonstrate the effectiveness of our algorithm.Comment: A shorter version of this paper is submitted to ICML 201

Efficient Data Representation by Selecting Prototypes with Importance Weights

Prototypical examples that best summarizes and compactly represents an underlying complex data distribution communicate meaningful insights to humans in domains where simple explanations are hard to extract. In this paper we present algorithms with strong theoretical guarantees to mine these data sets and select prototypes a.k.a. representatives that optimally describes them. Our work notably generalizes the recent work by Kim et al. (2016) where in addition to selecting prototypes, we also associate non-negative weights which are indicative of their importance. This extension provides a single coherent framework under which both prototypes and criticisms (i.e. outliers) can be found. Furthermore, our framework works for any symmetric positive definite kernel thus addressing one of the key open questions laid out in Kim et al. (2016). By establishing that our objective function enjoys a key property of that of weak submodularity, we present a fast ProtoDash algorithm and also derive approximation guarantees for the same. We demonstrate the efficacy of our method on diverse domains such as retail, digit recognition (MNIST) and on publicly available 40 health questionnaires obtained from the Center for Disease Control (CDC) website maintained by the US Dept. of Health. We validate the results quantitatively as well as qualitatively based on expert feedback and recently published scientific studies on public health, thus showcasing the power of our technique in providing actionability (for retail), utility (for MNIST) and insight (on CDC datasets) which arguably are the hallmarks of an effective data mining method.Comment: Accepted for publication in International Conference on Data Mining (ICDM) 201

An incremental dual nu-support vector regression algorithm

© 2018, Springer International Publishing AG, part of Springer Nature. Support vector regression (SVR) has been a hot research topic for several years as it is an effective regression learning algorithm. Early studies on SVR mostly focus on solving large-scale problems. Nowadays, an increasing number of researchers are focusing on incremental SVR algorithms. However, these incremental SVR algorithms cannot handle uncertain data, which are very common in real life because the data in the training example must be precise. Therefore, to handle the incremental regression problem with uncertain data, an incremental dual nu-support vector regression algorithm (dual-v-SVR) is proposed. In the algorithm, a dual-v-SVR formulation is designed to handle the uncertain data at first, then we design two special adjustments to enable the dual-v-SVR model to learn incrementally: incremental adjustment and decremental adjustment. Finally, the experiment results demonstrate that the incremental dual-v-SVR algorithm is an efficient incremental algorithm which is not only capable of solving the incremental regression problem with uncertain data, it is also faster than batch or other incremental SVR algorithms

Regularized approximate policy iteration using kernel for on-line reinforcement learning

By using Reinforcement Learning (RL), an autonomous agent interacting with the environment can learn how to take adequate actions for every situation in order to optimally achieve its own goal. RL provides a general methodology able to solve uncertain and complex decision problems which may be present in many real-world applications. RL problems are usually modeled as a Markov Decision Processes (MDPs) deeply studied in the literature. The main peculiarity of a RL algorithm is that the RL agent is assumed to learn the optimal policies from its experiences without knowing the parameters of the MDP. The key element in solving the MDP is learning a value function which gives the expectation of total reward an agent might expect at its current state taking a given action. This value function allows to obtain the optimal policy. In this thesis we study the capacity of SVR using kernel methods to adapt and solve complex RL problems in large or continuous state space. SVR can be studied using a geometrical interpretation in terms of optimal margin or can be seen as a regularization problem given in a Reproducing Kernel Hilbert Space (RKHS) SVR have good properties over the generalization ability and as they are based a on convex optimization problem, they do not suffer from sub-optimality. SVR are non-parametric showing the ability to automatically adapt to the complexity of the problem. Accordingly, applying SVR to approximate value functions sounds to be a good approach. SVR can be solved both in batch mode when the whole set of training sample are at disposal of the learning agents or incrementally which enables the addition or removal of training samples very effectively. Incremental SVR finds the appropriate KKT conditions for new or updated data by modifying their influences into the regression function maintaining consistence in the KKT conditions for the rest of data used for learning. In RL problems an incremental SVR should be able to approximate the action value function leading to the optimal policy. Accordingly, computation load should be lower, learning speed faster and generalization more effective than other existing method The overall contribution coming from of our work is to develop, formalize, implement and study a new RL technique for generalization in discrete and continuous state spaces with finite actions. Our method uses the Approximate Policy Iteration (API) framework with the BRM criterion which allows to represent the action value function using SVR. This approach for RL is the first one we know using SVR compatible to the agent interaction- with-the-environment framework of RL which shows his power by solving a large number of benchmark problems, including very difficult ones, like the bicycle driving and riding control problem. In addition, unlike most RL approaches to generalization, we develop a proof finding theoretical bounds for the convergence of the method to the optimal solution under given conditions.Mediante el uso de aprendizaje por refuerzo (RL), un agente autónomo interactuando con el medio ambiente puede aprender a tomar adecuada acciones para cada situación con el fin de lograr de manera óptima su propia meta. RL proporciona una metodología general capaz de resolver problemas de decisión complejos que pueden estar presentes en muchas aplicaciones del mundo real. Problemas RL usualmente se modelan como una Procesos de Decisión de Markov (MDP) estudiados profundamente en la literatura. La principal peculiaridad de un algoritmo de RL es que el agente es asumido para aprender las políticas óptimas de sus experiencias sin saber los parámetros de la MDP. El elemento clave en resolver el MDP está en el aprender una función de valor que da la expectativa de recompensa total que un agente puede esperar en su estado actual para tomar una acción determinada. Esta función de valor permite obtener la política óptima. En esta tesis se estudia la capacidad del SVR utilizando núcleo métodos para adaptarse y resolver problemas RL complejas en el espacio estado grande o continua. RVS puede ser estudiado mediante un interpretación geométrica en términos de margen óptimo o puede ser visto como un problema de regularización dado en un Reproducing Kernel Hilbert Space (RKHS). SVR tiene buenas propiedades sobre la capacidad de generalización y ya que se basan en una optimización convexa problema, ellos no sufren de sub-optimalidad. SVR son no paramétrico que muestra la capacidad de adaptarse automáticamente a la complejidad del problema. En consecuencia, la aplicación de RVS para aproximar funciones de valor suena para ser un buen enfoque. SVR puede resolver tanto en modo batch cuando todo el conjunto de muestra de entrenamiento están a disposición de los agentes de aprendizaje o incrementalmente que permite la adición o eliminación de muestras de entrenamiento muy eficaz. Incremental SVR encuentra las condiciones adecuadas para KKT nuevas o actualizadas de datos modificando sus influencias en la función de regresión mantener consistencia en las condiciones KKT para el resto de los datos utilizados para el aprendizaje. En los problemas de RL una RVS elemental será capaz de aproximar la función de valor de acción que conduce a la política óptima. En consecuencia, la carga de cálculo debería ser menor, la velocidad de aprendizaje más rápido y generalización más efectivo que el otro método existente La contribución general que viene de nuestro trabajo es desarrollar, formalizar, ejecutar y estudiar una nueva técnica de RL para la generalización en espacio de estados discretos y continuos con acciones finitas. Nuestro método utiliza el marco de la Approximate Policy Iteration (API) con el criterio de BRM que permite representar la función de valor de acción utilizando SVR. Este enfoque de RL es el primero que conocemos usando SVR compatible con el marco de RL con agentes interaccionado con el ambiente que muestra su poder mediante la resolución de un gran número de problemas de referencia, incluyendo los muy difíciles, como la conducción de bicicletas y problema de control de conducción. Además, a diferencia de la mayoría RL se acerca a la generalización, desarrollamos un hallazgo prueba límites teóricos para la convergencia del método a la solución óptima en condiciones dadas.Postprint (published version