IEEE Circuits and Systems Magazine - Q3 2023 - 26

Figure 15. Algorithm and system co-design reduces the training memory from 303 MB (PyTorch) to 149KB with the same transfer
learning accuracy, leading to 2077× reduction. The numbers are measured with MobilenetV2-w0.35 [4], batch size 1 and
resolution 128×128. It can be deployed to a microcontroller with 256 KB SRAM.
Figure 16. The quantized model has a very different weight/gradient norm ratio (i.e.,  W G/) compared to the floating-point
model at training time (left). QAS stabilizes the  
WG/
QAS. QAS effectively helps convergence, leading to better accuracy (right). The results are from updating the last two blocks of
the MCUNet model on the Cars dataset. (a) Weight/gradient norm ratio. (b) Training convergence.
ratio and helps optimization. The validation loss curves w/ and w/o
distorts the gradient update. To verify the idea, Figure
16 plot the ratio between weight norm and gradient
norm (i.e.,  
WG
/) for each tensor at the beginning
of the training on the CIFAR dataset [147]. The ratio
curve is very different after quantization: (1) the ratio is
much larger (could be addressed by adjusting the learning
rate) and (2) the ratio has a different pattern after
quantization. Take the highlighted area (red box) as an
example, the quantized ratios have a zigzag pattern, differing
from the floating-point curve. If all the tensors
are updated with a fixed learning rate, then the update
speed of each tensor would be very different compared
to the floating-point case, leading to inferior accuracy.
Even adaptive-learning rate optimizers like Adam [148]
cannot fully address the issue, as shown in Table 7.
2) Hyperparameter-Free Gradient Scaling. To address
the problem, a hyper-parameter-free learning rate scaling
rule, QAS, is proposed. Consider a 2D weight matrix
of a linear layer W∈
 ×cc
26
IEEE CIRCUITS AND SYSTEMS MAGAZINE
output channel. To perform per-tensor quantization4 a
scaling rate sW ∈ R is computed, such that W′s largest
magnitude is 27 − 1 = 127:
WW WGW WW (2)
=⋅() ≈⋅ ≈⋅G
ss ssWW W
/,
quantize
The process (roughly) preserves the mathematical
functionality during the forward (Equation 1)], but it
distorts the magnitude ratio between the weight and its
corresponding gradient:
    
WG WG WG (3)// .
//W WW WW≈⋅ =⋅−
ss s 2
The weight and gradient ratios are off by sW
−2, leading
to the distorted pattern in Figure 16: (1) the scaling factor
is far smaller than 1, making the weight-gradient ratio
much larger and (2) weights and biases have different
12 , where c1,c2 are the input and
4For simplicity. In practice, per-channel quantization [146] is used and
the scaling factor is a vector of size C2.
THIRD QUARTER 2023

IEEE Circuits and Systems Magazine - Q3 2023

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