Binarized neural network
I take a look at their github.

Here are the relevant parts:

local stepSize = lr/biasCorrection1 --math.sqrt(biasCorrection2)/biasCorrection1

stepSize=math.pow(2,torch.round(math.log(stepSize)/(math.log(2))))


and also

state.v:copy(torch.pow(2,torch.round(torch.log(state.v):div(math.log(2)))))
state.v:add(epsilon)
tmp:addcdiv(1, state.m, state.v)


It seems that for $\alpha \oslash (1-\beta_1^t)$, what is being done is $$2^{\lfloor \log_2 \frac{\alpha}{1-\beta_1^t} \rfloor}$$

which is a left shift if $\lfloor \log_2 \frac{\alpha}{1-\beta_1^t} \rfloor &gt; 0$ and a right shift  if $\lfloor \log_2 \frac{\alpha}{1-\beta_1^t} \rfloor &lt; 0$.

On the other hand, for $m_t \oslash v_t^{-1}$, what is being done is 

$$\frac{m_t}{2^{\lfloor \log_2 v_t\rfloor}}$$

which is a right shift if $\lfloor \log_2 v_t\rfloor&gt; 0$ and a left shift if $\lfloor \log_2 v_t\rfloor &lt; 0$.