Summary:

Classic settings, i.e. deep networks with convolutional layers and large batch sizes, almost always have backward-forward FLOP ratios close to 2:1.
Depending on the following criteria we can encounter ratios between 1:1 and 3:1
1. Type of layer: Passes through linear layers have as many FLOP as they use to do weight updates. Convolutional layers have many more FLOP for passes than for weight updates. Therefore, in CNNs, FLOP for weight updates basically play no role.
2. Batch size: Weights are updated after the gradients of the batch have been aggregated. Thus, FLOP for passes increase with batch size but stay constant for weight updates.
3. Depth: The first layer has a backward-forward ratio of 1:1 while all others have 2:1. Therefore, the overall ratio is influenced by the fraction of FLOP in first vs. FLOP in other layers.
We assume the network is being optimized by stochastic gradient descent (w += ɑ⋅dw) and count the weight update as part of the backward pass. Other optimizers would imply different FLOP counts and could create ratios even larger than 3:1 for niche settings (see appendix B). However, the ratio of 2:1 in the classic setting (see point 1) should still hold even when you use momentum or Adam.

Compute-intensity of the weight update	Most compute-intensive layers	Backward-forward ratio
Large batch size OR compute-intensive convolutional layer	First layer	1:1
Large batch size OR compute-intensive convolutional layer	Other layers	2:1
Small batch size AND no compute-intensive convolutional layers	First layer	2:1
	Other layers	3:1

Introduction:

How many more floating-point operations (FLOP) does it take to compute a backward pass than a forward pass in a neural network? We call this the backward-forward FLOP ratio.

This ratio is useful to estimate the total amount of training compute from the forward compute; something we are interested in the context of our study of Parameter, Compute and Data Trends in Machine Learning.

In this post, we first provide a theoretical analysis of the ratio, and we then corroborate our findings empirically.

Theory:

To understand where the differences in ratios come from, we need to look at the classical equations of backpropagation.

Let’s start with a simple example---a neural network with 2 hidden layers.

In this example, we have the following computations for forward and backward pass assuming linear layers with ReLU activations. The “@”-symbols denote matrix multiplications.

Operation

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H Jahani

H Jahani