Adam bias correction (${\hat{m}}_t=m_t/(1-{\beta }_1^t)$)

Adam bias correction compensates for the initial zero values in Adam's exponentially weighted gradient histories. The basic algorithm usesThese factors normalize the coefficient sums of the two running histories.^[ref]

For a scalar coordinate, suppose $m_0=0$ andExpanding the recurrence relation givesIf every $g_s$ has the same expectation $\mu$, linearity of expectation and the geometric series yieldThus ${\hat{m}}_t$ is an unbiased estimator of $\mu$ under these assumptions. Correlation between different times is allowed by this argument. Applying the same calculation to squared gradients with constant raw second moment $\nu =\mathbb{E}[g_s^2]$ gives $\mathbb{E}[v_t]=(1-{\beta }_2^t)\nu$ and $\mathbb{E}[{\hat{v}}_t]=\nu$.

For a constant supplied gradient of 2, choose ${\beta }_1=0.9$ and ${\beta }_2=0.99$. The first uncorrected histories are $m_1=0.2$ and $v_1=0.04$, while their corrected values are 2 and 4. The normalization removes the missing coefficient mass caused by initialization.

It does not make a changing history equal to the latest gradient. For an original deterministic example, let ${\beta }_1=1/2$, $g_1=1$ and $g_2=3$. Then $m_2=7/4$, sowhich differs from $g_2=3$. More generally, changing moment expectations produce a weighted average of past expectations. State the moment assumptions before interpreting “bias-corrected” as unbiased for a present distribution. The nonlinear parameter update built from the corrected histories is a separate object from either moment estimator.