Abstract
This guide provides tips for improving the performance of fullyconnected (or linear) layers and an example of the impact of parameter choice with layers in the Transformer network.
1. QuickStart Checklist
The following quickstart checklist provides specific tips for fullyconnected layers.

Choose the batch size and the number of inputs and outputs to be divisible by 8 (FP16) / 16 (INT8) to enable Tensor Cores; see the Tensor Core Requirements section in the Matrix Multiplication Background User Guide.

Especially when one or more parameters are small, choosing the batch size and the number of inputs and outputs to be divisible by at least 64 and ideally 256 can streamline tiling and reduce overhead; see the Dimension Quantization Effects section in the Matrix Multiplication Background User Guide.

Larger values for batch size and the number of inputs and outputs improve parallelization and efficiency; see Performance and its subsections.

As a rough guideline, choose batch sizes and neuron counts greater than 128 to avoid being limited by memory bandwidth (Tesla V100); see Batch Size.
2. FullyConnected Layer
Fullyconnected layers, which connect every input neuron to every output neuron, are commonly used in neural networks.
Three parameters define a fullyconnected layer: batch size, number of inputs, and number of outputs. Forward propagation, activation gradient computation, and weight gradient computation are directly expressed as matrixmatrix multiplications. How the three parameters map to GEMM dimensions (General Matrix Multiplication, background in the Matrix Multiplication Background User Guide) varies among frameworks, but the underlying principles are the same. For the purposes of the discussion, we adopt the convention used by PyTorch and Caffe where A contains the weights and B the activations. In TensorFlow, matrices take the opposite roles, but the performance principles are the same.
The compositions of the matrices in the GEMM are shown in Figure 2.
3. Performance
3.1. Input Features And Output Neuron Counts
As fullyconnected layers directly correspond to GEMMs, their performance trends are identical to those described in the Typical Tile Dimensions In cuBLAS And Performance section in the Matrix Multiplication Background User Guide. Larger parameters tend to allow better parallelization and efficiency; a GEMM that is twice the size often takes less than twice the time to calculate.
3.2. Batch Size
The batch size directly contributes to the tiling strategy for two out of three training phases  forward pass and activation gradient computation. For these phases, the output matrix dimension includes batch size, so larger batch sizes result in more tiles. Training with larger batch sizes is one option to extract more performance when model size is too small to fully utilize a GPU.
For weight gradient computation, the output matrix has the same dimensions as the weights, thus batch size does not affect the tile count directly. Instead, batch size here maps to the K dimension of the GEMM; larger batch size enables more efficient computation per tile of weight gradients. Figure 15 shows the performance impact of varying batch size on forward, activation gradient, and weight gradient computations for a fullyconnected layer with 4096 inputs and 1024 outputs. The larger batch sizes exceed 90 TFLOPS delivered performance.
Of particular interest are GEMMs where one dimension is very small. For example, on Tesla V100 and for a fullyconnected layer with 4096 inputs and 4096 outputs, forward propagation, activation gradient computation, and weight gradient computation are estimated to be memorybound for batch sizes below 128 (see Figure 5).
Larger numbers of inputs and outputs improve performance somewhat, but the computation will always be bandwidthlimited for very small batch sizes, for example, 8 and below. For a discussion of math and bandwidthlimited computations, see the Math And Memory Bounds section in the Matrix Multiplication Background User Guide.
4. Transformer Case Study
4.1. Basics
Transformers are a popular neural network architecture used for sequencetosequence mapping tasks, for example for natural language translation. They use an encoderdecoder architecture making heavy use of attention, both to “selfattend” over input sequences, as well as to give the decoder access to the encoder’s context. Figure 6 shows the complete neural network architecture (Attention Is All You Need 2017 paper, page 3).
From a performance standpoint, Transformers fundamentally process all the tokens in an input sequence in parallel, unlike  for example  RNN architectures with their sequential dependency. That makes Transformers very amenable to highly parallel architectures such as GPUs, and leads to large GEMMs that, with a few simple guidelines, can take great advantage of Tensor Core acceleration.
4.2. Applying Tensor Core Guidelines
4.2.1. Step 1: Padding The Vocabulary Size
Consider the final linear layer in the Transformer network, whose number of outputs is equal to the vocabulary size, as it is feeding the final SoftMax layer in the network to produce a probability distribution across tokens in the vocabulary.
This linear layer, as discussed in the Optimizing FullyConnected Layers User Guide, has M equal to the vocabulary size, N equal to the batch size, and K equal to the input feature size (all in the forward pass). Because the vocabulary is usually large, this is a heavyweight computation, and it is important to ensure Tensor Cores are being used effectively.
Figure 7 shows what happens when the vocabulary size is chosen without regard to the multipleof8 rule (V=33708): Tensor Cores cannot be applied, and performance reduces drastically to the levels sustained by the CUDA cores. Simply adding four padding tokens (to reach V=33712) to ensure a multipleof8 size can dramatically accelerate the overall computation.
4.2.2. Step 2: Choosing MultipleOf8 Batch Sizes
Besides the projection layer near the end of the network, fullyconnected layers are a major Transformer building block in all other parts of the network as well, including the big selfattention and feedforward blocks. As described before, batch size directly maps to one of the GEMM dimensions in such layers  N in the forward and activation gradient passes, K in the weight gradient pass  and therefore, the guideline to enable Tensor Cores by padding to a multiple of 8 applies to batch size as well.
The effect from padding batch size on one of the fullyconnected layers in the network is shown in Figure 8. Here, we’ve picked the first layer in the feedforward block, which is an fullyconnected layer with 1024 inputs and 4096 outputs. As the chart shows, this is an example where the multipleof8 rule does not necessarily need to be applied to all three GEMM dimensions; both forward and activation gradient passes perform the same with and without padding. The weight gradient pass, on the other hand, shows the same dramatic performance difference we saw on the projection GEMM earlier: With a batch size of 4095 tokens, CUDA cores are used as a fallback, whereas a batch size of 4096 tokens enables Tensor Core acceleration. It is still recommended to ensure all three GEMM dimensions are multiples of 8 when training in FP16 so that all passes will benefit from Tensor Cores.
4.2.3. Step 3: Avoiding Wave Quantization Through Batch Size Choice
Because batch size directly controls the shape of the MxN output matrix and Tensor Core GEMMs are parallelized by tiling the output matrix, choosing batch size appropriately can be used to reduce tile and wave quantization effects.
For Transformer, let us consider the first layer in the feedforward block again (4096 outputs, 1024 inputs). In this layer, the output matrix is of shape 4096 x batch size. Assuming a tile size of 256x128 as an example, the M=4096 dimension results in 4096/256=16 thread block tiles stacked vertically. On a Tesla V100 GPU with 80 SMs, wave quantization is minimal if the total number of thread blocks is a multiple of 80 (or just below). Therefore, choosing the batch size to result in n*80/16=n*5 thread block tiles in the N dimension achieves optimal wave quantization. With 256x128 thread blocks, this is achieved by choosing batch sizes of N=1*5*128=640, N=2*5*128=1280, and so on. Figure 9 illustrates the effect this has using two common batch sizes, 2048 and 4096.
The chart shows that choosing a quantizationfree batch size (2560 instead of 2048, 5120 instead of 4096) can noticeably improve performance. In particular, it is noteworthy that batch size 2560 (resulting in 4 waves of 80 thread block tiles each, assuming 256x128 tile size) achieves higher throughput than the larger batch size of 4096 (512 thread block tiles, 6.4 waves with 256x128 tile size). The activation gradient with batch size 5120 achieves about 100 TFLOPS delivered performance. For the weight gradient computation, batch size maps to the K parameter of the GEMM, and hence does not directly influence the size and shape of the output matrix or the number of thread block tiles that are created.
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