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Overfitting commonly occurs when applying deep neural networks (DNNs) on small-scale datasets, where DNNs do not generalize well from existing data to unseen data. The main reason resulting in overfitting is that small-scale datasets cannot reflect the situations of the real world. Label smoothing (LS) is an effective regularization method to prevent overfitting, avoiding it by mixing one-hot labels with uniform label vectors. However, LS only focuses on labels while ignoring the distribution of existing data. In this paper, we introduce the distributionally robust optimization (DRO) to LS, achieving shift the existing data distribution flexibly to unseen domains when training DNNs. Specifically, we prove that the regularization of LS can be extended to a regularization term for the DNNs parameters when integrating DRO. The regularization term can be utilized to shift existing data to unseen domains and generate new data. Furthermore, we propose an approximate gradient-iteration label smoothing algorithm (GI-LS) to achieve the findings and train DNNs. We prove that the shift for the existing data does not influence the convergence of GI-LS. Since GI-LS incorporates a series of hyperparameters, we further consider using Bayesian optimization (BO) to find the relatively optimal combinations of these hyperparameters. Taking small-scale anomaly classification tasks as a case, we evaluate GI-LS, and the results clearly demonstrate its superior performance.

This paper focuses on the contextual optimization problem where a decision is subject to some uncertain parameters and covariates that have some predictive power on those parameters are available before the decision is made. More specifically, we focus on solving the Wasserstein-distance-based distributionally robust optimization (DRO) model for the problem, which maximizes the worst-case expected objective over an uncertainty set including all distributions closed enough to a nominal distribution with respect to the Wasserstein distance. We develop a stochastic gradient descent algorithm based on the idea of data augmentation to solve the model efficiently. The algorithm iteratively a) does a bootstrapping sample from the nominal distribution; b) perturbs the adversarially and c) updates decisions. Accordingly, the computational time of the algorithm is only determined by the number of iterations and the complexity of computing the gradient of a single sample. Except for efficiently solving the model, the algorithm provide additional advantages that the proposed algorithm can cope with any nominal distributions and therefore is extendable to solve the problem in an online setting. We also prove that the algorithm converges to the optimal solution of the DRO model at a rate of a $O(1/\sqrt{T})$, where $T$ is the number of iterations of bootstrapping. Consequently, the performance guarantee of the algorithm is that of the DRO model plus $O(1/\sqrt{T})$. Through extensive numerical experiments, we demonstrate the superior performance of the proposed algorithm to several benchmarks.

We propose a method to analytically solve the one-dimensional coupled nonlinear Gross-Pitaevskii equations which govern the motion of the spinor Bose-Einstein condensates. In a uniform external potential, the Hamiltonian comprises the kinetic energy, the linear and the quadratic Zeeman energies. Several classes of exact periodic and solitonic solutions, either in real or in complex forms, are obtained for both the F=1 and F=2 condensates. These solutions are general that contain neither approximations nor constraints on the system parameters., Comment: 5 figures

We construct exact stationary solutions to the one-dimensional coupled Gross-Pitaevskii equations for the two-species Bose-Einstein condensates with equal intraspecies and interspecies interaction constants. Three types of complex solutions as well as their soliton limits are derived. By making use of the SU(2) unitary symmetry, we further obtain analytical time-evolving solutions. These solutions exhibit spatiotemporal periodicity., Comment: 3 figures

We present several exact solutions to the coupled nonlinear Gross-Pitaevskii equations which describe the motion of the one-dimensional spin-2 Bose-Einstein condensates. The nonlinear density-density interactions are decoupled by making use of the properties of Jacobian elliptical functions. The distinct time factors in each hyperfine state implies a "Lamor" procession in these solutions. Furthermore, exact time-evolving solutions to the time-dependent Gross-Pitaevskii equations are constructed through the spin-rotational symmetry of the Hamiltonian. The spin-polarizations and density distributions in the spin-space are analyzed., Comment: 6 figures

The impurity effects on the quantum coherence of a few-boson system are studied within the two-site Hubbard model. Periodical collapses and revivals of coherence occur in the presence of either polarized or unpolarized fermionic impurities. The relative strength $U_{BF}/U_{BB}$ of the boson-fermion versus the boson-boson interactions plays a key role in the coherence revivals. As the average filling of the impurity increases, the coherence revivals remain nearly unaffected for $U_{BF}/U_{BB}=z$ ($z\in Z$ is an integer) while the odd revival peaks are damped for $U_{BF}/U_{BB}=z+1/2$, in agreement with the experimental observations. For unpolarized fermionic impurities, the coherence revivals are irrelevant to the strength of the fermion-fermion interactions., Comment: 5 pages, 4 figures

We analytically solve the one-dimensional coupled Gross-Pitaevskii equations which govern the motion of F=1 spinor Bose-Einstein condensates. The nonlinear density-density interactions are decoupled by making use of the unique properties of the Jacobian elliptical functions. Several types of complex stationary solutions are deduced. Furthermore, exact non-stationary solutions to the time-dependent Gross-Pitaevskii equations are constructed by making use of the spin-rotational symmetry of the Hamiltonian. The spin-polarizations exhibit kinked configurations. Our method is applicable to other coupled nonlinear systems., Comment: 12 figures

The effects of disorders on the quantum coherence for many-bosons are studied in a double well model. For the ground state, the disorder enhances the quantum coherence. In the deep Mott regime, dynamical evolution reveals periodical collapses and revivals of the quantum coherence which is robust against the disorder. The average over variations in both the on-site energy and the interaction reveals a beat phenomenon of the coherence-decoherence oscillation in the temporal evolution., Comment: 4 figures