  Recent work [4] analyses the local convergence of Adam in a neighbourhood of
an optimal solution for a twice-differentiable function. It is found that the
learning rate has to be sufficiently small to ensure local stability of the
optimal solution. The above convergence results also hold for AdamW. In this
work, we propose a new adaptive optimisation method by extending AdamW in two
aspects with the purpose to relax the requirement on small learning rate for
local stability, which we refer to as Aida. Firstly, we consider tracking the
2nd moment r_t of the pth power of the gradient-magnitudes. r_t reduces to v_t
of AdamW when p=2. Suppose {m_t} is the first moment of AdamW. It is known that
the update direction m_{t+1}/(v_{t+1}+epsilon)^0.5 (or
m_{t+1}/(v_{t+1}^0.5+epsilon) of AdamW (or Adam) can be decomposed as the sign
vector sign(m_{t+1}) multiplied elementwise by a vector of magnitudes
|m_{t+1}|/(v_{t+1}+epsilon)^0.5 (or |m_{t+1}|/(v_{t+1}^0.5+epsilon)). Aida is
designed to compute the qth power of the magnitude in the form of
|m_{t+1}|^q/(r_{t+1}+epsilon)^(q/p) (or |m_{t+1}|^q/((r_{t+1})^(q/p)+epsilon)),
which reduces to that of AdamW when (p,q)=(2,1).
  Suppose the origin 0 is a local optimal solution of a twice-differentiable
function. It is found theoretically that when q>1 and p>1 in Aida, the origin 0
is locally stable only when the weight-decay is non-zero. Experiments are
conducted for solving ten toy optimisation problems and training Transformer
and Swin-Transformer for two deep learning (DL) tasks. The empirical study
demonstrates that in a number of scenarios (including the two DL tasks), Aida
with particular setups of (p,q) not equal to (2,1) outperforms the setup
(p,q)=(2,1) of AdamW.
