HuggingFace.Quantization_in_Depth/Transcript/3_Get the Scale and Zero Point_HFQD.txt

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As we saw in the notebook, the last piece
we are missing is how to determine

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the optimal s and z. To obtain the scale
and the zero point,

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we need to look at the extreme values
r min should mapped q, min

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and r max should map to q max,

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and we get the following two equation.

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Since we have two unknowns s
and z, we can solve this equation.

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If we subtract the first equation
from the second one, we can get the scale.

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So this equation minus
this one will give us the scale.

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And for the zero point.

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Since we always determine s,

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we just need for example,
to use the first equation and replace

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s by the value
we got before to get the zero point.

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And at the end
we end up with this specific formula.

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We also need to round the value
and to cast it to the correct d-type

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since we saw that z has the same d-type
as the quantized value.

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If you want to have a look at the details
of how we derived the scale

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and the zero point,
I invite you to pause the video

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and take a screenshot
at the following slides.

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So this one is for the scales derivation,
and this one is for

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So this one is for the scales derivation,
and this one is for

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the zero point derivation.

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As you saw previously, we

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make z as the same
the d-type as the quantized tensor.

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For example, as an integer and
this is not the same d-type as the scale.

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The goal behind
this choice is to represent zero

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in the original range
as an integer in the quantized range.

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So thanks to that,
when you quantize the value zero,

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it will take the value
z in the quantized range

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and what is great
is that if you'd dequantize the value z,

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it will become zero again.

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Now let's have a quick
look at how we calculate the scale

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and the zero point on this example
that you saw in the previous slides.

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So first, we need to get

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the maximum
and minimum range of the original tensor.

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So we have -184 and 728.6.

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So this is the maximum value
and this is the minimum value.

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And for the range of the quantized value
since we are quantize it in torch.int8.

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In it the minimum value is -128
and the maximum value is 127.

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So if you take the formula
we learned before, you get that

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the scale is equal to 3.58

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and the zero point is equal to -77.

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The last case

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we need to figure out is what happens
when the zero point is out of range.

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For example, since we need to cast
z to the quantized datatype,

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such as int8,
what should we do when z is out of range?

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So if z_min less than q_min,
we set z equal to q_min,

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and if z is superior to q_max,
we set that z to be equal to q_max.

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So this way
we don't have overflow and underflow.

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Now we have everything to code
how to get the scale and the zero point.

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Let's do that.

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And don't worry about the slide.

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Will code it directly in the notebook.

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Now, let's
get the scale and the zero point.

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Let's first start with the scale.

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As you saw in the formula
we need r_max , r_min.

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Q_max and q_min. We already saw
how to get the q_max in the q_min.

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So, I'll just copy-paste the code.

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So q_mean will be equal to the minimum

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value of the torch.int8 information.

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And the same for q_max
where we have max here.

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And as you saw in the example
q_min should be equal to

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minus 128 and q_max should be equal to 127

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Let's have a look.

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And we indeed have the same results.

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Now we need to get r_min and r_max,

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to get the minimum value of the tensor.

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We can just use the min methods.

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And we also need to call item
to get the value and not the tensor.

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As you can see here we have the tensor.

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But we need to call
also item to only get the value.

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We do the same thing for r_max,

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but this time we can get the maximum value

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by calling max.

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Now we have everything to get the scale.

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As we said earlier, the scale is equal to

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(r_max-r_min)

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/(q_max-q_min).

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And if you remember the example
we just saw before,

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we have the right scale around 3.58.

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As you can see here.

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Now let's get the zero point.

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To get the zero point.

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We just use the formula.

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So zero_point=q_min-
(r_min/scale)

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And let's have a look at the zero point.

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We have -76.5 around.

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So we need to run and cast it to int.

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And we get that the zero point is equal
to -77.

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As we saw before,

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if the zero point was inferior to q_min
we will set it to q_min.

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And if the zero point was superior
q_max we will set it to q_max.

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Now let's define the general function
to get the scale and the zero point.

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We'll call it "get q scale and zero point."

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This function takes two arguments:
the tensor

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and the type.

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And we'll set it to torch.int8 by default.

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As we saw before,

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we need to define the q_min and the q_max.

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Then we need to define the r_min
and the max of the tensor.

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We then define the scale

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and the zero point.

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For the zero point.

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As we saw in the slide.

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There are three cases.

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Indicates
the zero point is less than q_mean.

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We set the zero point to be equal to
q_min.

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Is the zero point is superior to q_max.

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We set it to q_max.

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And the last case is we just run it

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and cast it to an integer.

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And we just return the scale
and the zero point.

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Now let's test this general function
with the test tensor

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we define earlier.

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You can see that

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indeed we get the same scale
and the same zero point

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as the one we saw in the lecture
and before.

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Now using these new scales and new zero
point let's quantize r tensor.

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So we will call the linear
q with scale and zero point function

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by passing
the new scale and the new zero point.

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So the quantized tensor is equals
to this function

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where we pass this time

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the test tensor.

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But with the new scale

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and the new zero point.

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And as we did earlier.

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Also, we also need to dequantize
r tensor to compare

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with the original tensor.

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So we call the linear
the dequantization function

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where we pass the quantized tensor and

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the new scale and the new zero points.

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To have a summary of what we just did.

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Let's call the plot quantization
error function with the test tensor,

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the quantized tensor
and the dequantized tensor.

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And as you can see this time,

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the original tensor and the dequantized
tensor are very similar,

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and the quantization error
tensor looks also way much better.

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Now let's also have a look

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at the quantization error
to see if it has decreased a lot or not.

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So if you remember well,
to get the quantization error,

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you subtract the dequantized tensor
and the test tensor.

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We take the square and you do the min.

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And this time, as

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you can see, compared with the

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quantization error of around 170.

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Now we only have a quantization
error of around one.

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Now let's put everything inside
a linear quantization function

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that will only take a tensor
and will return to you

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the quantized tensor,
the scale and the zero point.

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So we defined the linear quantization
function.

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It takes as input a tensor

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and a d-type that we will set to torch.int8
by default.

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In this function, we will use
the two function that we coded before.

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So the get q scales and zero point
to get the scales

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and the zero point.

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So we just call that function

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and we just pass the tensor.

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And also the d-type.

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Then after getting the scale
and the zero point

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we can perform
the quantization of the tensor.

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So we will get the quantized tensor.

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If we use the linear q

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scale and zero point function
we coded before

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where we passed the tensor and

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the scale,

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the zero point,

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and the d-type.

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We just return the quantized sensor,
the scale

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and the zero point.

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Now let's play

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with this linear
quantizer on a random matrix.

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So we'll define a tensor.

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Which will take random values.

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And it will be of size 4x4.

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As you can see,

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we do have a random tensor of size 4x4.

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And we can just call the linear

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quantization function on our random tensor

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to get the quantized tensor, the scale
and the zero point.

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Let's have a look at the quantized tensor.

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As you can see,

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the tensor was quantized
and we also have the following values

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for the scales and the zero points to have
the summary of the quantization process,

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let's also dequantize the tensor
by calling the linear dequantization

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and by passing the quantized tensor,
the scale and the zero point.

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And we can use the plot quantization
error function

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to have the summary
of the quantization process.

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We passed the random tensor,

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the quantized tensor,
and the dequantized tensor.

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Oh, and as you can see,

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the original tensor here

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is pretty much the same
as the dequantized tensor,

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and the quantization errror tensor
is very small.

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And we can also print

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the quantization error.

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Which is also pretty low.

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And now I invite you to pause

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the video
and try to play with this quantization

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with your own inputs
and see how it performs.

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In the next lesson,
we will dive deeper into linear

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quantization
by learning its symmetric variants.

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And we will also look into quantization
granularity, such as per tensor,

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per channel and group quantization.

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Finally, we will also look at how to
perform inference with quantized models.