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Convergence of stochastic gradient descent in machine learning problems
Jelínková, Marie ; Branda, Martin (advisor) ; Kozmík, Karel (referee)
The aim of this thesis is solving minimization problems where the objective function is a sum of a differentiable (yet possibly non-convex) and general convex function. We focus on methods of stochastic and projected gradient descent from machine learning. By combining those two approaches we introduce an algorithm for solving such problems. The work is composed in a gradual manner where we firstly define necessary concepts needed for describing RSPG algorithm. Then we proceed to show the convergence of the algorithm for both convex and non-convex objective functions. A short numerical study is also included at the end. 1
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Robust approaches in portfolio optimization with stochastic dominance
Kozmík, Karel ; Kopa, Miloš (advisor)
We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...
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First order optimization methods in machine learning problems
Janáček, Patrik ; Branda, Martin (advisor) ; Kozmík, Karel (referee)
The goal of the thesis is to introduce the stochastic gradient method for optimizing differentiable objective function and discuss its convergence. First, supervised learning and empirical risk minimization (ERM) are explained. Then stochastic gradient descent (SG) is itroduced and analysed, first in the context of strictly convex objective function and then for the general non-convex function. In the last part, the classification of email spam is practically solved. 1
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Robust approaches in portfolio optimization with stochastic dominance
Kozmík, Karel ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...
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Monte Carlo Simulation of Swiss Franc LIBOR Using The Vasicek Model
Kozmík, Karel ; Teplý, Petr (advisor) ; Maršál, Aleš (referee)
We analyzed Swiss Franc LIBOR using R software and the Vasicek model. We utilized OLS, ML, bootstrap or simulations to test our hypotheses. The random walk hypothesis was not rejected, when we considered all the historical data. To get reasonable estimators, we used only data from the last adjustment of interest rates by the central bank and rejected the random walk hypothesis for all maturities but 12M. The difference in the results for OLS and ML estimates was negligible, so we did not reject the hypothesis that both methods give almost the same results. Performing a simulation study, we did not find any significant difference in the estimates for the Euler approximation for small values of the parameter a, but for larger values of a, the approximation led to biased results. All the hypotheses testing led the construction of confidence intervals for the estimated parameters, which are omitted in many papers and only point estimates are provided. We created confidence intervals for parameters of the Vasicek model for all the maturities but 12M. Extensive numerical simulations were run to explore the attributes of bootstrap estimates. We used an innovative approach of utilizing the logarithmic transformation to achieve a distribution closer to normal (which was necessary, because the intervals contained...
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Analysis of Profitability of Major World Lotteries
Kozmík, Karel ; Večeř, Jan (advisor) ; Lachout, Petr (referee)
Lottery tickets cost the same for every given jackpot, which might present an opportunity to make a profitable bet for very high jackpots. This work analyses whether buying a lottery ticket might be profitable in the mean value, for a given number of tickets sold, for four major American and European lotteries: Mega Millions, Powerball, EuroJackpot, Euro Millions. A regression of the sales on the jackpot is carried out for the American lotteries to find out whether some combination of the jackpot and the tickets sold, which was determined to be profitable, can be expected to happen. 1
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