I came across an interesting book about neural network basics, and the formula for gradient descent from one of the first chapters says:

Gradient descent: For each layer update the weights according to the rule

$w^l \rightarrow w^l-\frac{\eta}{m} \sum_x \delta^{x,l} (a^{x,l-1})^T$

where $w^l$ is the weights vector in layer $l$, and $x$ is the index of a specific training example.

I don't want to rewrite all formulas from the chapter, but the important part one is BP4 - equation for the rate of change of the cost with respect to any weight in the network:

$\frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j$

Am I missing something or the first formula is incorrect? Shouldn't we use Hadamard product instead, like this?

$w^l \rightarrow w^l-\frac{\eta}{m} \sum_x \delta^{x,l} \odot a^{x,l-1}$

Thanks for help.

I came across an interesting book about neural network basics, and the formula for gradient descent from one of the first chapters says:

Gradient descent: For each layer update the weights according to the rule

$w^l \rightarrow w^l-\frac{\eta}{m} \sum_x \delta^{x,l} (a^{x,l-1})^T$

where $w^l$ is the weights matrix in layer $l$, and $x$ is the index of a specific training example.

I don't want to rewrite all formulas from the chapter, but the important part one is BP4 - equation for the rate of change of the cost with respect to any weight in the network:

$\frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j$

Am I missing something or the first formula is incorrect? Shouldn't we use Hadamard product instead, like this?

$w^l \rightarrow w^l-\frac{\eta}{m} \sum_x \delta^{x,l} \odot a^{x,l-1}$

Thanks for help.