  Natural gradient (NG) is an information-geometric optimization method that
plays a crucial role, especially in the estimation of parameters for machine
learning models like neural networks. To apply NG to quantum systems, the
quantum natural gradient (QNG) was introduced and utilized for noisy
intermediate-scale devices. Additionally, a mathematically equivalent approach
to QNG, known as the stochastic reconfiguration method, has been implemented to
enhance the performance of quantum Monte Carlo methods. It is worth noting that
these methods are based on the symmetric logarithmic derivative (SLD) metric,
which is one of the monotone metrics. So far, monotonicity has been believed to
be a guiding principle to construct a geometry in physics. In this paper, we
propose generalized QNG by removing the condition of monotonicity. Initially,
we demonstrate that monotonicity is a crucial condition for conventional QNG to
be optimal. Subsequently, we provide analytical and numerical evidence showing
that non-monotone QNG outperforms conventional QNG based on the SLD metric in
terms of convergence speed.
