Title: DIMENSION-REDUCED ADAPTIVE GRADIENT METHOD. Abstract: Adaptive gradient methods, such as Adam, have shown faster convergence speed than  SGD across various kinds of network models. However, adaptive algorithms often suffer from inferior generalization performance than SGD. Though much effort via combining Adam and SGD have been invested to solve this issue, adaptive methods still fail to attain as good generalization as SGD. In this work, we proposed a Dimension-Reduced Adaptive Gradient Method (DRAG) to eliminate the generalization gap. DRAG makes an elegant combination of SGD and Adam by adopting a trust-region like framework. We observe that 1) Adam adjusts stepsizes for each gradient coordinate according to some loss curvature, and indeed decomposes the $n$-dimensional gradient into $n$ independent directions to search, in which each direction inherits one coordinate element from the gradient and sets the remaining coordinate positions as zeros; 2) SGD uniformly scales gradient for all gradient coordinates and actually has only one descent direction to minimize. Accordingly, DRAG reduces the high degree of freedom of Adam and also improves the flexibility of SGD via optimizing the loss along $k\  (\ll \! n)$ descent directions, e.g. the gradient direction and momentum direction used in this work. Then per iteration, DRAG finds the best stepsizes for $k$ descent directions by solving a trust-region subproblem whose computational overhead is negligible since the trust-region subproblem is low-dimensional, e.g. $k=2$ in this work. DRAG is compatible with the common deep learning training pipeline without introducing extra hyper-parameters and with negligible extra computation. Moreover, we prove the convergence property of DRAG for non-convex stochastic problems that often occur in deep learning training. Experimental results on representative benchmarks testify the fast convergence speed and also superior generalization of DRAG.