In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Q: Clustering is a fundamental machine learning method. The quality of its results is dependent on the data distribution. For this reason, deep neural networks can be used for learning better representations of the data. In this paper, we propose a systematic taxonomy for clustering with deep learning, in addition to a review of methods from the field. Based on our taxonomy, creating new methods is more straightforward. We also propose a new approach which is built on the taxonomy and surpasses some of the limitations of some previous work. Our experimental evaluation on image datasets shows that the method approaches state-of-the-art clustering quality, and performs better in some cases.

A: Unifying framework to perform clustering using deep neural networks
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Q: Social dilemmas are situations where individuals face a temptation to increase their payoffs at a cost to total welfare. Building artificially intelligent agents that achieve good outcomes in these situations is important because many real world interactions include a tension between selfish interests and the welfare of others. We show how to modify modern reinforcement learning methods to construct agents that act in ways that are simple to understand, nice (begin by cooperating), provokable (try to avoid being exploited), and forgiving (try to return to mutual cooperation). We show both theoretically and experimentally that such agents can maintain cooperation in Markov social dilemmas. Our construction does not require training methods beyond a modification of self-play, thus if an environment is such that good strategies can be constructed in the zero-sum case (eg. Atari) then we can construct agents that solve social dilemmas in this environment.

A: How can we build artificial agents that solve social dilemmas (situations where individuals face a temptation to increase their payoffs at a cost to total welfare)?
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Q: In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to {\em global optimality} with runtime polynomial in the data size albeit exponential in the input dimension. Further, we improve on the known lower bounds on size (from exponential to super exponential) for approximating a ReLU deep net function by a shallower ReLU net. Our gap theorems hold for smoothly parametrized families of ``hard'' functions, contrary to countable, discrete families known in the literature.   An example consequence of our gap theorems is the following: for every natural number $k$ there exists a function representable by a ReLU DNN with $k^2$ hidden layers and total size $k^3$, such that any ReLU DNN with at most $k$ hidden layers will require at least $\frac12k^{k+1}-1$ total nodes. Finally, for the family of $\R^n\to \R$ DNNs with ReLU activations, we show a new lowerbound on the number of affine pieces, which is larger than previous constructions in certain regimes of the network architecture and most distinctively our lowerbound is demonstrated by an explicit construction of a \emph{smoothly parameterized} family of functions attaining this scaling. Our construction utilizes the theory of zonotopes from polyhedral theory.

A:
This paper 1) characterizes functions representable by ReLU DNNs, 2) formally studies the benefit of depth in such architectures,  3) gives an algorithm to implement empirical risk minimization to global optimality for two layer ReLU nets.
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