Universum data that do not belong to any class of a classification problem can be exploited to ut... more Universum data that do not belong to any class of a classification problem can be exploited to utilize prior knowledge to improve generalization performance. In this paper, we design a novel parametric ν-support vector machine with universum data ( $ \mathfrak {U} $ Par-ν-SVM). Unlabeled samples can be integrated into supervised learning by means of $ \mathfrak {U} $ Par-ν-SVM. We propose a fast method to solve the suggested problem of $ \mathfrak {U} $ Par-ν-SVM. The primal problem of $ \mathfrak {U} $ Par-ν-SVM, which is a nonconvex optimization problem, is transformed into an unconstrained optimization problem so that the objective function can be treated as a difference of two convex functions (DC). To solve this unconstrained problem, a boosted difference of convex functions algorithm (BDCA) based on a generalized Newton method is suggested (named DC- $\mathfrak {U} $ Par-ν-SVM). We examined our approach on UCI benchmark data sets, NDC data sets, a handwritten digit recognition data set, and a landmine detection data set. The experimental results confirmed the effectiveness and superiority of the proposed method for solving classification problems in comparison with other methods.
Proceedings of the AAAI Conference on Artificial Intelligence
The leading approach to solving large imperfect information games is to pre-calculate an approxim... more The leading approach to solving large imperfect information games is to pre-calculate an approximate solution using a simplified abstraction of the full game; that solution is then used to play the original, full-scale game. The abstraction step is necessitated by the size of the game tree. However, as the original game progresses, the remaining portion of the tree (the subgame) becomes smaller. An appealing idea is to use the simplified abstraction to play the early parts of the game and then, once the subgame becomes tractable, to calculate a solution using a finer-grained abstraction in real time, creating a combined final strategy. While this approach is straightforward for perfect information games, it is a much more complex problem for imperfect information games. If the subgame is solved locally, the opponent can alter his play in prior to this subgame to exploit our combined strategy. To prevent this, we introduce the notion of subgame margin, a simple value with appealing p...
Proceedings of the AAAI Conference on Artificial Intelligence
It is a well known fact that in extensive form games with perfect information, there is a Nash eq... more It is a well known fact that in extensive form games with perfect information, there is a Nash equilibrium with support of size one. This doesn't hold for games with imperfect information, where the size of minimal support can be larger. We present a dependency between the level of uncertainty and the minimum support size. For many games, there is a big disproportion between the game uncertainty and the number of actions available. In Bayesian extensive games with perfect information, the only uncertainty is about the type of players. In card games, the uncertainty comes from dealing the deck. In these games, we can significantly reduce the support size. Our result applies to general-sum extensive form games with any finite number of players.
We propose a novel DEA ranking based on a robust optimization viewpoint: the higher ranking for t... more We propose a novel DEA ranking based on a robust optimization viewpoint: the higher ranking for those DMU's that remain efficient even for larger variations of data and vice versa. This ranking can be computed by solving generalized linear fractional programming problems, but we also present a tight linear programming approximation that preserves the order of rankings. We show some remarkable properties of our approach: It preserves the order of rankings compared to the classical approach. It is naturally normalized, so it can be used as universal ranking of DMU's of unrelated models. It gives ranking not only for inefficient, but also for efficient decision making units. It can also be easily extended to generalized model, for instance to deal with interval data. We present several examples confirming the desirable properties of the method.
Due to their relation to the linear complementarity problem, absolute value equations have been i... more Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.
Annals of Mathematics and Artificial Intelligence, 2022
The Universum provides prior knowledge about data in the mathematical problem to improve the gene... more The Universum provides prior knowledge about data in the mathematical problem to improve the generalization performance of the classifiers. Several works have shown that the Universum twin support vector machine ( U $ \mathfrak {U} $ -TSVM) is an efficient method for binary classification problems. In this paper, we improve the U $ \mathfrak {U} $ -TSVM method and propose an improved Universum twin bounded support vector machine (named as IUTBSVM). Indeed, by introducing different Lagrangian functions for the primal problems, we obtain new dual formulations of U $ \mathfrak {U} $ -TSVM so that we do not need to compute inverse matrices. To reduce the computational time of the proposed method, we suggest a smaller size of the rectangular kernel matrices than the other methods. Numerical experiments on gender classification of human faces, handwritten digits recognition, and several UCI benchmark data sets indicate that the IUTBSVM is more efficient than the other four algorithms, namely U $\mathfrak {U}$ -SVM, TSVM, U $\mathfrak {U}$ -TSVM, and IUTSVM in the sense of the classification accuracy.
We discuss some basic concepts and present a numerical procedure for finding the minimum-norm sol... more We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be reduced to a simple constrained minimization problem with a once-differentiable convex objective function. We use finite termination of an appropriate Newton’s method to solve this problem. Numerical results show that the proposed method is efficient.
Although techniques for finding Nash equilibria in extensive form games have become more powerful... more Although techniques for finding Nash equilibria in extensive form games have become more powerful in recent years, many games that model real world interactions remain too large to be solved directly. The current approach is to create a smaller abstracted game, allowing the computation of an optimal solution. The strategy can then be used in the original game. Considering public information to create the abstraction can be strategically important, yet very few of the previous abstraction algorithms specifically consider public information or use an expert approach. In this paper, we show that the public information can be crucial, and we present a new, automatic technique for abstracting the public state space. We also present an experimental evaluation in the domain of Texas Hold’em poker and show that it outperforms state-of-the-art abstraction algorithms.
Universum data that do not belong to any class of a classification problem can be exploited to ut... more Universum data that do not belong to any class of a classification problem can be exploited to utilize prior knowledge to improve generalization performance. In this paper, we design a novel parametric ν-support vector machine with universum data ( $ \mathfrak {U} $ Par-ν-SVM). Unlabeled samples can be integrated into supervised learning by means of $ \mathfrak {U} $ Par-ν-SVM. We propose a fast method to solve the suggested problem of $ \mathfrak {U} $ Par-ν-SVM. The primal problem of $ \mathfrak {U} $ Par-ν-SVM, which is a nonconvex optimization problem, is transformed into an unconstrained optimization problem so that the objective function can be treated as a difference of two convex functions (DC). To solve this unconstrained problem, a boosted difference of convex functions algorithm (BDCA) based on a generalized Newton method is suggested (named DC- $\mathfrak {U} $ Par-ν-SVM). We examined our approach on UCI benchmark data sets, NDC data sets, a handwritten digit recognition data set, and a landmine detection data set. The experimental results confirmed the effectiveness and superiority of the proposed method for solving classification problems in comparison with other methods.
Proceedings of the AAAI Conference on Artificial Intelligence
The leading approach to solving large imperfect information games is to pre-calculate an approxim... more The leading approach to solving large imperfect information games is to pre-calculate an approximate solution using a simplified abstraction of the full game; that solution is then used to play the original, full-scale game. The abstraction step is necessitated by the size of the game tree. However, as the original game progresses, the remaining portion of the tree (the subgame) becomes smaller. An appealing idea is to use the simplified abstraction to play the early parts of the game and then, once the subgame becomes tractable, to calculate a solution using a finer-grained abstraction in real time, creating a combined final strategy. While this approach is straightforward for perfect information games, it is a much more complex problem for imperfect information games. If the subgame is solved locally, the opponent can alter his play in prior to this subgame to exploit our combined strategy. To prevent this, we introduce the notion of subgame margin, a simple value with appealing p...
Proceedings of the AAAI Conference on Artificial Intelligence
It is a well known fact that in extensive form games with perfect information, there is a Nash eq... more It is a well known fact that in extensive form games with perfect information, there is a Nash equilibrium with support of size one. This doesn't hold for games with imperfect information, where the size of minimal support can be larger. We present a dependency between the level of uncertainty and the minimum support size. For many games, there is a big disproportion between the game uncertainty and the number of actions available. In Bayesian extensive games with perfect information, the only uncertainty is about the type of players. In card games, the uncertainty comes from dealing the deck. In these games, we can significantly reduce the support size. Our result applies to general-sum extensive form games with any finite number of players.
We propose a novel DEA ranking based on a robust optimization viewpoint: the higher ranking for t... more We propose a novel DEA ranking based on a robust optimization viewpoint: the higher ranking for those DMU's that remain efficient even for larger variations of data and vice versa. This ranking can be computed by solving generalized linear fractional programming problems, but we also present a tight linear programming approximation that preserves the order of rankings. We show some remarkable properties of our approach: It preserves the order of rankings compared to the classical approach. It is naturally normalized, so it can be used as universal ranking of DMU's of unrelated models. It gives ranking not only for inefficient, but also for efficient decision making units. It can also be easily extended to generalized model, for instance to deal with interval data. We present several examples confirming the desirable properties of the method.
Due to their relation to the linear complementarity problem, absolute value equations have been i... more Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.
Annals of Mathematics and Artificial Intelligence, 2022
The Universum provides prior knowledge about data in the mathematical problem to improve the gene... more The Universum provides prior knowledge about data in the mathematical problem to improve the generalization performance of the classifiers. Several works have shown that the Universum twin support vector machine ( U $ \mathfrak {U} $ -TSVM) is an efficient method for binary classification problems. In this paper, we improve the U $ \mathfrak {U} $ -TSVM method and propose an improved Universum twin bounded support vector machine (named as IUTBSVM). Indeed, by introducing different Lagrangian functions for the primal problems, we obtain new dual formulations of U $ \mathfrak {U} $ -TSVM so that we do not need to compute inverse matrices. To reduce the computational time of the proposed method, we suggest a smaller size of the rectangular kernel matrices than the other methods. Numerical experiments on gender classification of human faces, handwritten digits recognition, and several UCI benchmark data sets indicate that the IUTBSVM is more efficient than the other four algorithms, namely U $\mathfrak {U}$ -SVM, TSVM, U $\mathfrak {U}$ -TSVM, and IUTSVM in the sense of the classification accuracy.
We discuss some basic concepts and present a numerical procedure for finding the minimum-norm sol... more We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be reduced to a simple constrained minimization problem with a once-differentiable convex objective function. We use finite termination of an appropriate Newton’s method to solve this problem. Numerical results show that the proposed method is efficient.
Although techniques for finding Nash equilibria in extensive form games have become more powerful... more Although techniques for finding Nash equilibria in extensive form games have become more powerful in recent years, many games that model real world interactions remain too large to be solved directly. The current approach is to create a smaller abstracted game, allowing the computation of an optimal solution. The strategy can then be used in the original game. Considering public information to create the abstraction can be strategically important, yet very few of the previous abstraction algorithms specifically consider public information or use an expert approach. In this paper, we show that the public information can be crucial, and we present a new, automatic technique for abstracting the public state space. We also present an experimental evaluation in the domain of Texas Hold’em poker and show that it outperforms state-of-the-art abstraction algorithms.
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Papers by Milan Hladík