Exponentially weighted gradient history ($b_k={\mu }^k{b}_0+{\sum }_{j=0}^{k-1}{\mu }^{k-1-j}{g}_j$)

An exponentially weighted gradient history assigns geometrically decreasing coefficients to older supplied gradients. For constant $0\le \mu <1$ and momentum buffer recurrence $b_{k+1}=\mu b_k+g_k$,Each additional update multiplies an old contribution by $\mu$. The formula follows by substituting the expression for $b_k$ into the recurrence and adding the newest exponent-zero term. The same undampened recurrence, including its initialization convention, appears in the primary stochastic gradient descent documentation.^[ref]

For an original example, choose $b_0=0$, $\mu =1/2$ and gradients $g_0=2$, $g_1=-2$, $g_2=4$. Successive buffers are 2, −1, 3.5. Directly expanding the third buffer givesThe newest contribution has coefficient 1; the oldest has coefficient 1/4. The buffer is a weighted sum, not yet a weighted mean. At $b_0=0$, its coefficient sum is the geometric seriesHere $W_3=1.75$, so the normalized weighted mean is $b_3/W_3=2$, rather than the ordinary mean $4/3$.

The coefficients also give a concrete memory scale. In the infinite geometric sequence, the fraction assigned to the newest $m$ gradients is $1-{\mu }^m$. At $\mu =0.9$, 29 gradients account for at least 95% of that coefficient mass, because $0.9^{29}<0.05$ whereas $0.9^{28}>0.05$. This is a statement about coefficients, not a guarantee that gradients at different times are independent or that the optimizer averages a stationary process.