In many applications, the pre-information on regression function is always unknown. Therefore, it is necessary to learn regression function by means of some valid tools. In this paper we investigate the regression problem in learning theory, i.e., convergence rate of regression learning algorithm with least square schemes in multi-dimensional polynomial space. Our main aim is to analyze the generalization error for multi-regression problems in learning theory. By using the famous Jackson operators in approximation theory, covering number, entropy number and relative probability inequalities, we obtain the estimates of upper and lower bounds for the convergence rate of learning algorithm. In particular, it is shown that for multi-variable smooth regression functions, the estimates are able to achieve almost optimal rate of convergence except for a logarithmic factor. Our results are significant for the research of convergence, stability and complexity of regression learning algorithm. In this paper we investigate the regression problem in learning theory, ie, convergence rate of regression learning algorithm with least square schemes in multi-dimensional polynomial space. Our main aim is to analyze the generalization error for multi-regression problems in learning theory. By using the famous Jackson operators in approximation theory, covering number, entropy number and relative probability inequalities, we obtained the estimates of upper and lower bounds for the convergence rate of learning algorithm. In particular, it is shown that for multi-variable smooth regression functions, the estimates are able to achieve almost optimal rate of convergence except for a logarithmic factor. results are significant for the research of convergence, stability and complexity of regression learning algorithm.