Multi-task learning (MTL) improves the performance achieved on each task by exploiting the relevant information between tasks. At present, most of the mainstream deep MTL models are based on hard parameter sharing mechanisms, which can reduce the risk of model overfitting. However, negative knowledge transfer may occur, which hinders the performance improvement achieved for each task. In this paper, for situations when multiple tasks are jointly trained, we propose the adaptive hard parameter sharing method. On the basis of the adaptive hard parameter sharing method, the number of nodes in the network is dynamically updated by setting a continuous gradient difference-based sign threshold and a warm-up training iteration threshold through the relationships between the parameters and the loss function. After each task fully utilizes the shared information, adaptive nodes are used to further optimize each task, reducing the impact of negative migration. By using simulation studies and instance analyses, we demonstrate theoretical proof that the performance of the proposed method is better than that of the competing method.

To describe the stylized features of volatility comprehensively, this paper embeds the time-varying leverage effect of volatility into the Realized Generalized AutoRegressive Conditional Heteroskedasticity (RG) model and proposes a new volatility model with a time-varying leverage effect. The Quasi-Maximum Likelihood-Kernel (QML-K) method is proposed to approximate the density function of returns and to estimate the parameters in the new model. Under some mild regularity conditions, the asymptotic properties of the resulting estimators are achieved. Simulation studies demonstrate that the proposed model yields better performances than traditional RG models under different situations. Finally, the empirical analysis shows better finite sample performance of the estimation method and the new model on real data compared with existing methods.

With high dimensionality and dependence in spatial data, traditional parametric methods suffer from the curse of dimensionality problem. The theoretical properties of deep neural network estimation methods for high-dimensional spatial models with dependence and heterogeneity have been investigated only in a few studies. In this paper, we propose a deep neural network with a ReLU activation function to estimate unknown trend components, considering both spatial dependence and heterogeneity. We prove the compatibility of the estimated components under spatial dependence conditions and provide an upper bound for the mean squared error ($MSE$). Simulations and empirical studies demonstrate that the convergence speed of neural network methods is significantly better than that of local linear methods.

To better fit the actual data, this paper will consider both spatio-temporal correlation and heterogeneity to build the model. In order to overcome the “curse of dimensionality” problem in the nonparametric method, we improve the estimation method of the single-index model and combine it with the correlation and heterogeneity of the spatio-temporal model to obtain a good estimation method. In this paper, assuming that the spatio-temporal process obeys the $\alpha$ mixing condition, a nonparametric procedure is developed for estimating the variance function based on a fully nonparametric function or dimensional reduction structure, and the resulting estimator is consistent. Then, a reweighting estimation of the parametric component can be obtained via taking the estimated variance function into account. The rate of convergence and the asymptotic normality of the new estimators are established under mild conditions. Simulation studies are conducted to evaluate the efficacy of the proposed methodologies, and a case study about the estimation of the air quality evaluation index in Nanjing is provided for illustration.

We consider the transfer learning problem in the linear regression model, where the source domain and target domain have different features and the model exhibits heteroscedasticity. The existing homogeneous transfer learning methods cannot yet handle this type of problem. In this work, a transfer learning algorithm is proposed, which integrates data of varying dimensions and accounts for heteroscedasticity, thereby yielding a data-pooling estimator. The algorithm is simple to implement and easy to operate. The theoretical properties of the proposed estimator are established, including an upper bound on the generalization error and robustness against negative transfer. Simulation studies indicate that the proposed method performs well in terms of parameter estimation. The effectiveness of the proposed method is also validated using real-life datasets in UCI public repository, demonstrating favorable performance compared to the existing methods.