  Nonmonotone gradient methods generally perform better than their monotone
counterparts especially on unconstrained quadratic optimization. However, the
known convergence rate of the monotone method is often much better than its
nonmonotone variant. With the aim of shrinking the gap between theory and
practice of nonmonotone gradient methods, we introduce a property for
convergence analysis of a large collection of gradient methods. We prove that
any gradient method using stepsizes satisfying the property will converge
$R$-linearly at a rate of $1-\lambda_1/M_1$, where $\lambda_1$ is the smallest
eigenvalue of Hessian matrix and $M_1$ is the upper bound of the inverse
stepsize. Our results indicate that the existing convergence rates of many
nonmonotone methods can be improved to $1-1/\kappa$ with $\kappa$ being the
associated condition number.
