In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Ex Input:
We present a method for solving implicit (factored) Markov decision processes (MDPs) with very large state spaces. We intro duce a property of state space partitions which we call f-homogeneity. Intuitively, an f-homogeneous partition groups together states that behave approximately the same under all or some subset of policies. Borrow ing from recent work on model minimization in computer-aided software verification, we present an algorithm that takes a factored representation of an MDP and an 0 f I and computes a factored f-homogeneous par tition of the state space. This partition defines a family of related MOPs-those MOP's with state space equal to the blocks of the partition, and transition probabilities "appro:X:imately" like those of any (original MDP) state in the source block. To formally study such families of MDPs, we introduce the new notion of a "bounded parameter MDP" (BMDP), which is a fam ily of (traditional) MOPs defined by speci fying upper and lower bounds on the transi tion probabilities and rewards. We describe algorithms that operate on BMDPs to find policies that are approximately optimal with respect to the original MDP. In combination, our method for reducing a large implicit MDP to a possibly much smaller BMDP using an f-homogeneous par tition, and our methods for selecting actions in BMDP's constitute a new approach for an alyzing large implicit MOP's. Among its ad vantages, this new approach provides insight into existing algorithms to solving implicit MDPs, provides useful connections to work in automata theory and model minimization, and suggests methods, which involve vary ing f, to trade time and space (specifically in terms of the size of the corresponding state space) for solution quality.

Ex Output:
Model Reduction Techniques for Computing Approximately Optimal Solutions for Markov Decision Processes

Ex Input:
This paper presents a theoretical, idealized model of the thinking process with the following characteristics: 1) the model can produce complex thought sequences and can be generalized to new inputs, 2) it can receive and maintain input information indefinitely for the generation of thoughts and later use, and 3) it supports learning while executing. The crux of the model lies within the concept of internal consistency, or the generated thoughts should always be consistent with the inputs from which they are created. Its merit, apart from the capability to generate new creative thoughts from an internal mechanism, depends on the potential to help training to generalize better. This is consequently enabled by separating input information into several parts to be handled by different processing components with a focus mechanism to fetch information for each. This modularized view with the focus binds the model with the computationally capable Turing machines. And as a final remark, this paper constructively shows that the computational complexity of the model is at least, if not surpass, that of a universal Turing machine.

Ex Output:
(Yet) Another Theoretical Model of Thinking

Ex Input:
<lb>This paper introduces AdaSDCA: an adap-<lb>tive variant of stochastic dual coordinate as-<lb>cent (SDCA) for solving the regularized empir-<lb>ical risk minimization problems. Our modifica-<lb>tion consists in allowing the method adaptively<lb>change the probability distribution over the dual<lb>variables throughout the iterative process. AdaS-<lb>DCA achieves provably better complexity bound<lb>than SDCA with the best fixed probability dis-<lb>tribution, known as importance sampling. How-<lb>ever, it is of a theoretical character as it is expen-<lb>sive to implement. We also propose AdaSDCA+:<lb>a practical variant which in our experiments out-<lb>performs existing non-adaptive methods.

Ex Output:
Stochastic Dual Coordinate Ascent with Adaptive Probabilities