instruction:
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.
question:
Empirical risk minimization (ERM), with proper loss function and regularization, is the common practice of supervised classification. In this paper, we study training arbitrary (from linear to deep) binary classifier from only unlabeled (U) data by ERM. We prove that it is impossible to estimate the risk of an arbitrary binary classifier in an unbiased manner given a single set of U data, but it becomes possible given two sets of U data with different class priors. These two facts answer a fundamental question---what the minimal supervision is for training any binary classifier from only U data. Following these findings, we propose an ERM-based learning method from two sets of U data, and then prove it is consistent. Experiments demonstrate the proposed method could train deep models and outperform state-of-the-art methods for learning from two sets of U data.
answer:
Three class priors are all you need to train deep models from only U data, while any two should not be enough.


question:
Statistical inference methods are fundamentally important in machine learning. Most state-of-the-art inference algorithms are 
 variants of Markov chain Monte Carlo (MCMC) or variational inference (VI). However, both methods struggle with limitations in practice: MCMC methods can be computationally demanding; VI methods may have large bias. 
 In this work, we aim to improve upon MCMC and VI by a novel hybrid method based on the idea of reducing simulation bias of finite-length MCMC chains using gradient-based optimisation. The proposed method can generate low-biased samples by increasing the length of MCMC simulation and optimising the MCMC hyper-parameters, which offers attractive balance between approximation bias and computational efficiency. We show that our method produces promising results on popular benchmarks when compared to recent hybrid methods of MCMC and VI.
answer:
In this work, we aim to improve upon MCMC and VI by a novel hybrid method based on the idea of reducing simulation bias of finite-length MCMC chains using gradient-based optimisation.


question:
Infinite-width neural networks have been extensively used to study the theoretical properties underlying the extraordinary empirical success of standard, finite-width neural networks. Nevertheless, until now, infinite-width networks have been limited to at most two hidden layers. To address this shortcoming, we study the initialisation requirements of these networks and show that the main challenge for constructing them is defining the appropriate sampling distributions for the weights. Based on these observations, we propose a principled approach to weight initialisation that correctly accounts for the functional nature of the hidden layer activations and facilitates the construction of arbitrarily many infinite-width layers, thus enabling the construction of arbitrarily deep infinite-width networks. The main idea of our approach is to iteratively reparametrise the hidden-layer activations into appropriately defined reproducing kernel Hilbert spaces and use the canonical way of constructing probability distributions over these spaces for specifying the required weight distributions in a principled way. Furthermore, we examine the practical implications of this construction for standard, finite-width networks. In particular, we derive a novel weight initialisation scheme for standard, finite-width networks that takes into account the structure of the data and information about the task at hand. We demonstrate the effectiveness of this weight initialisation approach on the MNIST, CIFAR-10 and Year Prediction MSD datasets.
answer:

We propose a method for the construction of arbitrarily deep infinite-width networks, based on which we derive a novel weight initialisation scheme for finite-width networks and demonstrate its competitive performance.