Entscheidungen mathematisch treffen : Approximative Dynamische Programmierung mit Tiefen Neuronalen Netzwerken

Belomestny, Denis GND

Entscheidungen begleiten uns im Alltag. Wenn Sie zum Beispiel eine Ware im Laden kaufen, entscheiden Sie sich für eine Marke. Wenn Sie das Haus verlassen, entscheiden Sie darüber, was Sie anziehen sollen. Oft treffen die Leute Entscheidungen anhand der unvollständigen oder unsicheren Information (zum Beispiel anhand der Information auf Etiketten oder der Wettervorhersage). Was ist eine optimale Entscheidung unter diesen Umständen? Wie soll man handeln um diese optimale Entscheidung zu treffen? All diese Fragen beantwortet die mathematische Entscheidungstheorie.

Decision problems are part of our everyday life. For example, if we want to buy a product in the store, we choose a brand. Before leaving the house, we decide what to wear. Often, people make decisions based on incomplete or uncertain information (for example etiquette or the weather forecast). What is an optimal decision under these circumstances? How should we act in order to make this optimal decision? These questions are answered using mathematical decision theory. An important class of decision problems are those that are time-dependent (dynamic) and irreversible. Such problems are quite natural, as every real decision must take into account the time-related changes the decision-maker has to adjust to. These types of decision problems can be extremely complex, involving changes in preferences, technologies, and resources. To solve a dynamic decision problem, we must make assumptions about how the decision-maker evaluates alternative strategies. The standard assumption is that the decision-maker maximizes the expected benefit. Furthermore, in order to calculate the best decision for today, the decision-maker must make the optimal decision for the future too, i.e. for any future scenarios. The search for the optimal decision should therefore not be made chronologically, but in reverse order, i.e. backwards in time, because the present optimum depends on the future optimum. This fundamental consideration forms the basis of the principle of dynamic programming developed by R. Bellman. The complexity of dynamic programming results primarily from the exponential growth in the number of possible scenarios with the increase in the number of possible values for the state variables, decision variables and/or the number of time periods. This exponential growth becomes a major challenge in dynamic programming applications. In this article I describe different strategies for dealing with this problem.

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Belomestny, D., 2019. Entscheidungen mathematisch treffen: Approximative Dynamische Programmierung mit Tiefen Neuronalen Netzwerken. Mathematik - Herausforderung des Nichtlinearen. https://doi.org/10.17185/duepublico/70317
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