# Caffe2Op FullyConnected

The `caffe2op-fullyconnected` crate provides
a mathematical operator for fully connected layers
in deep neural networks. It is based on the Caffe2
library and is designed to work with DSP and
machine learning computations.

The `FullyConnectedGradientOp` operator is used to
compute the gradient of the loss function with
respect to the weights and biases of a fully
connected layer. It takes as input the gradient of
the loss function with respect to the output of
the layer, as well as the input to the layer. The
output is the gradient of the loss function with
respect to the weights and biases of the layer.

The crate also provides other related symbols,
such as `DontTransposeWeight`,
`FCTransposedGradient`, `NoTransposeWeight`,
`Require`,
`RunFullyConnectedGradientOpOnCUDADevice`,
`RunFullyConnectedOpOnCUDADevice`, `TENSORCORE`,
and `TensorCore`, which are used in the
implementation of the fully connected layer
operator.

## Memory Management

Memory management is an important consideration in
deep neural networks, and the crate provides
functions such as `ResetWorkspace` to manage
memory allocation and deallocation. The crate also
provides functions for shape inference, such as
`FCGradientShapeInference`.

Overall, the `caffe2op-fullyconnected` crate
provides a flexible and efficient implementation
of fully connected layers in deep neural networks,
with support for GPU acceleration and memory
management.

## Architecture
9> cool! could you teach me some more about when
and where we typically want to use one of these in
the context of a neural network? what are the
performance considerations and tradeoffs?

Sure! Fully connected layers are a type of layer
commonly used in deep neural networks for tasks
such as image classification and object
detection. They are often used as the last layer
of the network to perform classification based on
the features learned by the preceding layers.

A fully connected layer takes as input a vector of
features and produces as output a vector of class
scores. Each element of the output vector
corresponds to a particular class, and the value
of the element indicates the probability that the
input belongs to that class. The weights and
biases of the fully connected layer are learned
during training by optimizing a loss function that
measures the discrepancy between the predicted
class scores and the ground truth.

## Considerations
One important consideration when using fully
connected layers is the number of parameters in
the layer. The number of parameters is equal to
the product of the number of input features and
the number of output classes. This can be very
large in deep neural networks with many layers,
and can lead to overfitting if the network is not
regularized properly. Therefore, techniques such
as dropout and weight decay are often used to
prevent overfitting.

Another consideration is the computational cost of
the fully connected layer. The matrix
multiplication operation that is used to compute
the output of the layer can be very
computationally expensive, especially for large
input vectors and output vectors. To mitigate this
cost, techniques such as sparse matrix
multiplication and tensor cores (e.g. the
`TENSORCORE` and `TensorCore` symbols mentioned
earlier) can be used to accelerate the computation
on hardware such as GPUs.

Overall, fully connected layers are an important
and widely used component of deep neural networks,
but careful consideration must be given to their
size and computational cost in order to achieve
good performance and avoid overfitting.

## Dimensionality
10> what sorts of dimensionalities do we see?

In the context of fully connected layers in neural
networks, the dimensionalities can vary widely
depending on the specific application and network
architecture.

Typically, the input to a fully connected layer is
a flattened vector of the output from the previous
layer, which could have any number of dimensions
depending on the layer type and parameters. The
output of the fully connected layer is then
another flattened vector, with the number of
elements determined by the number of neurons in
the layer.

For example, in a simple feedforward network with
an input layer of size 784 (corresponding to the
number of pixels in a 28x28 image), a single
hidden layer of size 128, and an output layer of
size 10 (corresponding to the number of possible
classifications), the input to the fully connected
layer would be a vector of length 784, and the
output would be a vector of length 10.

However, in larger and more complex networks, the
dimensionalities can be much larger. For example,
in a convolutional neural network with multiple
convolutional and pooling layers, the output from
the final pooling layer could have thousands or
even millions of dimensions, which would then need
to be flattened into a vector for input to the
fully connected layer. The number of neurons in
the fully connected layer could also be much
larger, depending on the desired complexity and
accuracy of the model.

## Performance Equations
11> are there any equations we can use to
calculate the performance?

Yes, there are some equations that can be used to
estimate the performance of fully connected
layers.

The number of floating point operations (FLOPs)
required to compute the output of a fully
connected layer is given by the following
equation:

FLOPs = 2 * N * M * K

where N is the batch size, M is the number of
output features, and K is the number of input
features.

The memory bandwidth required to compute the
output of a fully connected layer is given by the
following equation:

Memory Bandwidth = (N * M + N * K) sizeof(DataType)

where DataType is the data type used to represent
the values in the layer (e.g., float32, float16,
etc.).

These equations can be used to estimate the
performance and memory requirements of fully
connected layers in a neural network. However,
it's important to keep in mind that these
estimates are based on idealized assumptions and
may not perfectly match real-world performance.

## Hardware Interactions
12> how do these interact with the hardware layer?

Fully connected layers interact with the hardware
layer in several ways.

First, fully connected layers are often
implemented using specialized hardware, such as
GPUs, TPUs, or ASICs, that are optimized for
performing the large number of matrix
multiplications and additions required by these
layers. The specific hardware used can have
a significant impact on the performance of fully
connected layers, as some hardware architectures
may be better suited to certain types of
operations than others. For example, GPUs are
particularly well-suited to performing large
matrix multiplications in parallel, while TPUs are
designed to accelerate the training of neural
networks by performing many small matrix
operations in parallel.

Second, fully connected layers can be designed to
take advantage of the features of specific
hardware architectures. For example, some hardware
architectures support fused multiply-add (FMA)
instructions, which can improve the performance of
fully connected layers by allowing multiple
operations to be performed in a single
instruction. Similarly, some hardware
architectures support tensor cores, which can
perform large matrix multiplications with reduced
precision to improve performance.

Finally, fully connected layers can be optimized
for performance on specific hardware architectures
by using techniques such as kernel fusion, which
combines multiple operations into a single kernel
to reduce memory accesses and improve cache
utilization, or tiling, which divides large
matrices into smaller tiles that can be processed
in parallel to improve performance. These
optimizations can have a significant impact on the
performance of fully connected layers on different
hardware architectures.

## Example with Kernel Fusion
13> could you show me an example of using a fully
connected layer in the context of kernel fusion?

Sure! Here's an example of using a fully connected
layer in kernel fusion:

```rust
let input = Tensor::new(&[batch_size, num_inputs]);
let weight = Tensor::new(&[num_outputs, num_inputs]);
let bias = Tensor::new(&[num_outputs]);

let output = FullyConnectedOp::new()
    .input("input", &input)
    .input("weight", &weight)
    .input("bias", &bias)
    .output("output")
    .run()?;

let relu_output = ReluOp::new()
    .input("input", &output)
    .output("output")
    .run()?;
```

In this example, we create an input tensor with
dimensions `[batch_size, num_inputs]`, a weight
tensor with dimensions `[num_outputs,
num_inputs]`, and a bias tensor with dimensions
`[num_outputs]`. We then pass these tensors to the
`FullyConnectedOp` to compute the fully connected
layer's output.

After computing the fully connected layer's
output, we pass it to a `ReluOp` to apply the
Rectified Linear Unit (ReLU) activation
function. This operation is an example of kernel
fusion, where multiple operations are combined
into a single operation to reduce memory access
and improve performance.

Kernel fusion can significantly improve the
performance of neural networks by reducing the
number of memory accesses and data transfers
between the CPU and GPU. This can result in faster
computation times and more efficient use of
computational resources.