TASK DEFINITION: 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.
PROBLEM: Neural network training depends on the structure of the underlying loss landscape, i.e. local minima, saddle points, flat plateaus, and loss barriers. In relation to the structure of the landscape, we study the permutation symmetry of neurons in each layer of a deep neural network, which gives rise not only to multiple equivalent global minima of the loss function but also to critical points in between partner minima. In a network of $d-1$ hidden layers with $n_k$ neurons in layers $k = 1, \ldots, d$, we construct continuous paths between equivalent global minima that lead through a `permutation point' where the input and output weight vectors of two neurons in the same hidden layer $k$ collide and interchange. We show that such permutation points are critical points which lie inside high-dimensional subspaces of equal loss, contributing to the global flatness of the landscape. We also find that a permutation point for the exchange of neurons $i$ and $j$ transits into a flat high-dimensional plateau that enables all $n_k!$ permutations of neurons in a given layer $k$ at the same loss value. Moreover , we introduce higher-order permutation points by exploiting the hierarchical structure in the loss landscapes of neural networks, and find that the number of $K$-th order permutation points is much larger than the (already huge) number of equivalent global minima -- at least by a polynomial factor of order $K$. In two tasks, we demonstrate numerically with our path finding method that continuous paths between partner minima exist: first, in a toy network with a single hidden layer on a function approximation task and, second, in a multilayer network on the MNIST task. Our geometric approach yields a lower bound on the number of critical points generated by weight-space symmetries and provides a simple intuitive link between previous theoretical results and numerical observations.

SOLUTION: Weight-space symmetry in neural network landscapes gives rise to numerous number of saddles and flat high-dimensional subspaces.

PROBLEM: This paper proposes a method for efficient training of Q-function for continuous-state Markov Decision Processes (MDP), such that the traces of the resulting policies satisfy a Linear Temporal Logic (LTL) property. LTL, a modal logic, can express a wide range of time-dependent logical properties including safety and liveness. We convert the LTL property into a limit deterministic Buchi automaton with which a synchronized product MDP is constructed. The control policy is then synthesised by a reinforcement learning algorithm assuming that no prior knowledge is available from the MDP. The proposed method is evaluated in a numerical study to test the quality of the generated control policy and is compared against conventional methods for policy synthesis such as MDP abstraction (Voronoi quantizer) and approximate dynamic programming (fitted value iteration).

SOLUTION: As safety is becoming a critical notion in machine learning we believe that this work can act as a foundation for a number of research directions such as safety-aware learning algorithms.

PROBLEM: We propose a novel algorithm, Difference-Seeking Generative Adversarial Network (DSGAN), developed from traditional GAN. DSGAN considers the scenario that the training samples of target distribution, $p_{t}$, are difficult to collect.

 Suppose there are two distributions  $p_{\bar{d}}$ and $p_{d}$ such that the density of the target distribution can be the differences between the densities of $p_{\bar{d}}$ and $p_{d}$. We show how to learn the target distribution $p_{t}$ only via samples from $p_{d}$ and $p_{\bar{d}}$ (relatively easy to obtain).

 DSGAN has the flexibility to produce samples from various target distributions (e.g. the out-of-distribution). Two key applications, semi-supervised learning and adversarial training, are taken as examples to validate the effectiveness of DSGAN. We also provide theoretical analyses about the convergence of DSGAN.

SOLUTION:
We proposed "Difference-Seeking Generative Adversarial Network" (DSGAN) model to learn the target distribution which is hard to collect training data.