AdamW

AdamW combines the adaptive update of Adam with decoupled weight decay. It maintains exponentially weighted histories of the supplied data gradient and its componentwise square, while applying parameter shrinkage separately. Ilya Loshchilov and Frank Hutter presented the method in their work on decoupling decay from gradient-based regularization.^[ref]

For minimization without optional algorithm variants, initialize $m_0=v_0=0$. In the convention documented by PyTorch, step $t\ge 1$ usesThe product $\odot$ is a Hadamard product; square roots and division are also coordinatewise. Assume $0\le {\beta }_1,{\beta }_2<1$, $ϵ>0$ and the stated learning rate ${\eta }_t$. The decay coefficient is $\lambda$. This explicitly fixes epsilon placement and the decay convention.^[ref]

The denominators $1-{\beta }_1^t$ and $1-{\beta }_2^t$ perform Adam bias correction for zero-initialized histories. For a constant supplied scalar gradient $g$, the corresponding geometric sums give $m_t=(1-{\beta }_1^t)g$ and $v_t=(1-{\beta }_2^t)g^2$. Their corrected values are therefore $g$ and $g^2$. The squared-gradient quantity concerns a raw second moment, rather than a centered variance: a constant gradient of 2 has square 4 even though its variance is zero.

For an original first-step example, choose $f(\theta )={\theta }^2$, ${\theta }_0=1$, $\eta =0.1$, $\lambda =0.2$, ${\beta }_1=0.9$, ${\beta }_2=0.99$ and $ϵ=0.1$. ThenandThe constants are chosen for transparent arithmetic. Feeding $g_1+\lambda {\theta }_0=2.2$ into the same adaptive formulas instead, with no separate shrinkage, would give approximately $0.904348$. That coupled L2 regularization also changes the stored histories. Resuming AdamW therefore requires the relevant optimizer state and step count as well as the parameter values and settings.