$b$.

\[col:1\] Fixing initial weights $w_0$, both $\var\left(g_t^b \, \middle| \calF_0 \right)$ and $\var\left(\grad L\pth{w_t^b} \, \middle| \calF_0 \right)$ are decreasing functions of mini-batch size $b$ for all $b \in [n]$ and $t \in \naturals$.

In conclusion, we provide a framework for calculating the explicit value of variance of the stochastic gradient estimators and the norm of any linear combination of sample-wise gradients. We further show that the variance of both the full gradient and the stochastic gradient estimator are a decreasing function of the mini-batch size $b$.

Two-layer Linear Network with Online Setting {#subsec:linearNetwork}
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In this section, we study the dynamics of SGD on deep linear networks. We consider the two-layer linear network while the results and proofs can be easily extended to deep linear network with any depth. We consider the population loss $$\label{eq:populationLoss} 
        \calL (w) = \Expect_{x\sim \calN(0, I_p)} \left[ \frac{1}{2} \norm{W_2 W_1 x - W_2^* W_1^* x}^2 \right]$$ under the teacher-student learning framework [@hinton2015distilling] with $w=(W_1,W_2)$ a tuple of two matrices. Here $W_1 \in \mathbb{R}^{p_1 \times p}$ and $W_2 \in \mathbb{R}^{p_2 \times p_1}$ are parameter matrices of the student network and $W_1^*$ and $W_2^*$ are the fixed ground-truth parameters of the teacher network. We use 