訓練和測試數據具有不同機率分布的環境，是機器學習的難題之一，解決這個難題的研究方向稱為轉移學習。我們在共變量偏移假設的環境中討論轉移學習演算法，希望能夠提高演算法的準確性。對訓練和測試數據集的輸入數據的機率密度函數進行估計後，我們將兩函數的商稱為重要性加權函數。重要性加權函數會作為訓練權重，修正經驗風險最小化模式的演算法計算。在我們設計的實驗中，已經證明了重要性加權函數能夠使演算法的準確率明顯上升。為了更好的測量重要性加權函數，我們使用兩種演算法，改進了測量機率密度函數的演算法會遇到的缺點。

In the training of machine learning algorithms, one must confront scenarios where the training and testing data possess different probability distributions. The purpose of this study is to research transfer learning algorithms in the environment of covariate shift. After estimating the probability density functions of the input data for both the training and testing datasets, the ratio of these two functions is referred to as the importance weighting function. In the computation process of the algorithm based on empirical risk minimization, the importance weighting function is used as the training weight. In our experiment, it has been demonstrated that the importance weighting function significantly improves the accuracy of the algorithm. To better measure the importance weighting function, we utilize two algorithms that address the drawbacks encountered in improving the algorithm for measuring probability density functions.

[1] Tom Dietterich. Overfitting and undercomputing in machine learning. ACM computing surveys (CSUR), 27(3):326–327, 1995.
[2] Gene H Golub and Charles F Van Loan. Matrix computations. JHU press, 2013.
[3] Wolfgang H¨ardle, Marlene M¨uller, Stefan Sperlich, Axel Werwatz, et al. Nonparametric and semiparametric models, volume 1. Springer, 2004.
[4] Takafumi Kanamori, Shohei Hido, and Masashi Sugiyama. A least-squares approach to direct importance estimation. The Journal of Machine Learning Research, 10:1391–1445, 2009.
[5] Solomon Kullback and Richard A Leibler. On information and sufficiency. The annals of mathematical statistics, 22(1):79–86, 1951.
[6] Emanuel Parzen. On estimation of a probability density function and mode. The annals of mathematical statistics, 33(3):1065–1076, 1962.
[7] Claude Elwood Shannon. A mathematical theory of communication. ACM SIGMOBILE mobile computing and communications review, 5(1):3–55, 2001.
[8] Hidetoshi Shimodaira. Improving predictive inference under covariate shift by weighting the log-likelihood function. Journal of statistical planning and inference, 90(2):227–244, 2000.
[9] Masashi Sugiyama and Motoaki Kawanabe. Machine learning in non-stationary environments: Introduction to covariate shift adaptation. MIT press, 2012.
10] Masashi Sugiyama, Shinichi Nakajima, Hisashi Kashima, Paul Buenau, and Motoaki Kawanabe. Direct importance estimation with model selection and its application to covariate shift adaptation. Advances in neural information processing systems, 20, 2007.
[11] Masashi Sugiyama, Taiji Suzuki, Shinichi Nakajima, Hisashi Kashima, Paul Von Buenau, and Motoaki Kawanabe. Direct importance estimation for covariate shift adaptation. Annals of the Institute of Statistical Mathematics, 60:699–746,2008.
[12] N Vapnik Vladimir and Vlamimir Vapnik. Statistical learning theory. Xu JH and Zhang XG. translation. Beijing: Publishing House of Electronics Industry, 2O04, 1998.
[13] Xue Ying. An overview of overfitting and its solutions. In Journal of physics: Conference series, volume 1168, page 022022. IOP Publishing, 2019.