# Operations in the expression graph¶

Operations are responsible for manipulating the elements of an expression graph. In Marian, many useful operations have already been implemented and can be found the code documentation. The provided operations cover simple arithmetic, logical comparisons and common mathematical functions; as well as tensor manipulation, for example slice or reshape, and aggregations such as sum or minimum. Finally, other routines, such as activation functions, useful in building neutral networks are also available.

There are several necessary components required to implement an operation in Marian’s expression graph. The highest-level component is the Expression Operator, responsible for setting up the Node Operator and adding it to the graph. Next, this Node Operator describes the nature of the forward and backward operation to be performed. These operations are implemented using some combination of Functional Operators (element wise), and Tensor Operators.

This overview aims to provide information about what each of the different operator components does, how they fit together and where to go to make changes. Then, equipped with this knowledge, to be able to add new functionality to Marian.

## Operator Structure¶

The central component in the graph is the Chainable<Tensor> object. This object provides the abstract interface necessary to interact with elements in the computation graph. The details of this interface can be found in /src/graph/chainable.h. Note that the template parameter corresponds to the underlying data structure, which in Marian is the Tensor. Therefore, for convenience, the type Expr is defined:

typedef IPtr<Chainable<Tensor>> Expr;


The implementation of the different operator components are divided across several files:

• Expression Operator

• Node Operator

• Functional Operator

• Tensor operation

• Declared Specialization

To understand how the different components are inter-linked, we’ll look at each of them in turn.

## Expression Operator¶

The expression operator is the user-facing method used when building a graph. It is responsible for constructing the corresponding Node Operation and inserting it into the expression graph. To accommodate these core requirements, the function Expression is able to perform both actions in generality:

template <class T, typename... Args>
Expr Expression(Args&&... args) {
auto e = Expr(new T(std::forward<Args>(args)...));
}


This helper-function simplifies the definition of many expression operators. For example, the implementation of the expression operator sin(x) is simply:

// src/graph/expression_operators.h
Expr sin(Expr x);

// src/graph/expression_operators.cpp
Expr sin(Expr x) {
return Expression<SinNodeOp>(x);
}


However, implementations may perform actions beyond the core functionality alone. Taking sum as an example

Expr sum(Expr a, int ax) {
if(a->shape()[ax] == 1) {
return a;
}
return Expression<ReduceNodeOp>(a, ax, ReduceNodeOpCode::sum);
}


The trivial operation is handled without needing to construct a node operation. This example also demonstrates a non-trivial construction of ReduceNodeOp, which is capable of performing differing reduction operations depending on instantiation.

Going further, an expression operator may be defined in terms of existing expressions. Operators such as weighted_average are composed of three different expression operator calls: scalar_product, sum, and operator/.

Expr weighted_average(Expr in, Expr weights, int ax) {
auto p = scalar_product(in, weights, ax);
auto s = sum(weights, ax);
return p / s;
}


While useful, composition at this level may be less efficient than lower-level implementations.

## Node Operator¶

The Node subclass of Chainable<Tensor> provides concrete implementations for much of the abstract interface, while subclasses of Node enable different node behaviours. In the context of operations, the relevant derived class is NaryNodeOp and is base class used for Node Operators. This subclass provides implementation focused on performing general N-arity operations. However, many common operations are unary and, for convenience, a further specialization, UnaryNodeOp, exists to simplify their definition.

The purpose of the Node Operator is to define the forward and backward behaviour of the operation. The forward operation performs the desired operation while the backward operation updates the gradients. These behaviours are written in terms of NodeOps, where a NodeOp is a wrapper to define a capturing lambda function. Explicitly these are defined as:

// src/graph/chainable.h
#define NodeOp(op) [=]() { op; }
typedef std::vector<std::function<void()>> NodeOps;


Each NodeOp is written as a function in terms of the value (val_), gradient (adj_) of the current node, and its children, via child(). The values and gradients the nth child node are accessed via the interfaces child(n)->val() and child(n)->grad(), respectively. NodeOps are executed in order when running the graph forwards and backwards, as this snippet from Node demonstrates

// Node in src/graph/node.h
virtual void runForward(const NodeOps& ops) {
for(auto&& op : ops)
op();
}

virtual void runBackward(const NodeOps& ops) {
size_t i = 0;
for(auto&& op : ops)
if(child(i++)->trainable())
op();
}


In backwards operation it is crucial that the NopeOp responsible for propagating a gradient to child(i) is the ith element of the NodeOps vector. The requirement that the child associated with the NodeOp be trainable means that an out-of-position NodeOp may not be run. To represent no operation a nullptr can be passed as a NodeOp.

A typical node operator has the functionality demonstrated in the following snippet.

// outline of a node op
struct MyNodeOp : public NaryNodeOp {
MyNodeOp(Expr a)
: NaryNodeOp({a}, newShape(...), newType(...)) {}

Shape newShape(...) {}  // optional
Type newType(...) {}    // optional

const std::string type() override { return "my_node_op"; }
virtual size_t hash() override {}          // potentially required
virtual bool equal(Expr node) override {}  // potentially required

NodeOps forwardOps() override {}
NodeOps backwardOps() override {}


This outline describes a node operator that takes a single argument a. The shape and type of the node would be determined by the result of newShape and newType when constructing the NaryNodeOp. These functions represent any custom logic used to determine the shape and type of the node. As indicated in this example code, these are optional and, when omitted, calling NaryNodeOp({a}) would result in a node with the same shape and type as a. The type() method returns the friendly name for the node. Note that the ONNX interface maintains a mapping of these friendly names to their ONNX representation. In the absence of any member variables the hash() and equal() methods can be omitted, and defer to their NaryNodeOp definition. However, if such variables exist then hash() should implement a hashed representation and equal() should provide the necessary conditions to consider nodes equivalent. Finally, the operations of the node are defined in forwardOps() and backwardOps().

Continuing with the example of sin(x), the code responsible for implementing the behaviour is

// src/graph/node_operators_unary.h
struct SinNodeOp : public UnaryNodeOp {
SinNodeOp(Expr x) : UnaryNodeOp(x) {}

NodeOps forwardOps() override {
using namespace functional;
return {NodeOp(Element(_1 = sin(_2), val_, child(0)->val()))};
}

NodeOps backwardOps() override {
using namespace functional;
}

const std::string type() override { return "sin"; }
};


In this code, the constructor trivially initialises the UnaryNodeOp, passing the expression x as its input. This propagates up to NaryNodeOp and becomes child(0) of the node. The size and type of the SinNodeOp are equivalent to that of x. The lack of any member variables allows the hash() and equal() methods to be omitted. The friendly name for this node is the string sin. The forward and backward implementation are accomplished using a single NodeOp each.

### Forward operation¶

The forward NodeOp calls the tensor operation Element, that execute the element-wise operation described by the functor:

_1 = sin(_2)


The placeholders _1, _2 are enabled by code in /src/functional and interoperate with the functional operators. In the call to Element, val_ is assigned to _1 and child(0)->val() to _2. Therefore, this has the action of setting the elements of this node to the result obtained by applying sin to the elements of child(0).

### Backward Operation¶

The backward NodeOp is responsible for backpropagation of the gradients via reverse-mode automatic differentiation. In this example, where y = sin(x), this corresponds to evaluating

dJ/dx += dJ/dy * dy/dx, dy/dx = cos(x)


This is realised using the tensor operator Add with the functor

_1 * cos(_2)


In the call to Add, adj_ is assigned to _1 and child(0)->val() to _2. Therefore, this functor represents dJ/dy * dy/dx: the product of the gradient at the current node and the gradient of the operation. This value is then added to the gradient of the child child(0)->grad() as required.

### Shape and Type Changes¶

The newShape and newType methods are just a suggestion of how custom logic may be encapsulated where needed. However, in practice, many operations do not require a change in shape or type. In these instances, the node inherits the broadcasted shape of its children as well as their common type. An important feature of the type deduction in NaryNodeOp::commonType() is that it guarantees that all child nodes are of the same type.

There are few operations in Marian that require a type specification. Where they do exist, they are often simple as the desired type is explicitly provided, or is trivially deduced. An example of this is CastNodeOp

// CastNodeOp in src/graph/node_operators_unary.h
CastNodeOp(Expr a, Type type) : UnaryNodeOp(a, type) {}


The desired type is set explicitly in construction. A slightly different example is that of CSRDotNodeOp. It has several child nodes which are a mixture of DataType and IndexType and therefore do not share a common type. The solution is to explicitly specify the relevant children to NaryNodeOp::commonType({...}).

Shape modifying operations are more common. A simple example is the class of operations performed by ReduceNodeOp which involve an aggregation process along one axis of the Tensor. The output shape is determined by

// ReduceNodeOp in src/graph/node_operators_unary.h
Shape newShape(Expr a, int axis) {
Shape shape = a->shape();
axis_ = shape.axis(axis);

shape.set(axis_, 1);
return shape;
}


The output shape is the same as the input but with the processed axis is reduced to a single element. Other use cases include transpose and slicing operations, as well as tensor products.

## Functional Operator¶

As the NodeOp are evaluated, they encounter the underlying datatype of the Tensor. At this stage, type-specific intrinsic functions are required. These intrinsics are implemented in the templated struct Ops<ElementType>, with a specialization required for each type. The current required types are:

• float

• double

• float32x4 (see src/3rd_party/sse_mathfun.h)

• float32x8 (see src/3rd_party/avx_mathfun.h)

• half (see cuda_fp16.h in the CUDA Math API)

Further details are available in /src/common/types.h.

Returning to the example of sin(x), the specialization for float and double requires

// src/functional/operators.h
// in namespace marian::functional
template <typename T>
struct Ops {
static HOST_DEVICE_INLINE T sin(const T&)  { ABORT("Unknown type"); }
};

// Specialization for float
template <>
struct Ops<float> {
static HOST_DEVICE_INLINE float sin(const float& x)  { return sinf(x); }
};

// Specialization for double
template <>
struct Ops<double> {
static HOST_DEVICE_INLINE double sin(const double& x)  { return std::sin(x); }
};


The remaining specializations can be seen in /src/functional/operators.h. Note that the general template must produce a runtime abort.

The final component of the functional operator is to call the macro that enables interoperability with the framework of /src/functional. For a unary operator, this is the macro UNARY.

UNARY(Sin,     sin,        Ops<ElementType>::sin(x));


where template parameter ElementType must be used. There are equivalent macros for BINARY and TERNARY Ops.

## Tensor Operator¶

Tensor operations use less abstracted interfaces to interact with the Tensors, often working with the Tensor data directly. They also rely on BLAS (Basic Linear Algebra Subprograms) libraries to accelerate these operations. As well as libraries containing device-specific optimisations. These libraries include:

• CPU

• CBLAS / OpenBLAS

• FBGEMM

• INTGEMM

• MKL

• GPU

• CUDA (cuBLAS)

An important subtlety is that while the CPU focused libraries use a row-major representation, the cuBLAS library (GPU) instead uses a column-major representation.

Furthermore, the OpenMPI and OpenMP libraries are employed for parallelisation. While macros provided in /src/common/definitions.h locally enable faster floating-point math in supported compilers.

MARIAN_FFAST_MATH_BEGIN
// ffmath code
MARIAN_FFAST_MATH_END


The usual caveats apply when enabling fast_math, and can be found in /src/common/definitions.h

Tensor operators are declared in /src/tensors/tensor_operators.h, these are device-agnostic function that call the relevant device-specific implementation. The CPU- and GPU-specific implementation are defined in cpu namespace in /src/tensors/cpu/ and the gpu namespace /src/tensors/gpu/. Therefore a typical operator defers to an implementation in the device-specific namespace.

void TensorOp(marian::Tensor out, marian::Tensor in) {
#ifdef CUDA_FOUND
if(out->getBackend()->getDeviceId().type == DeviceType::gpu)
gpu::TensorOp(out, in);
else
#endif
cpu::TensorOp(out, in);
}


When compiled with GPU support, this function dispatches a call to the implementation that corresponds to the backend device type configured in the graph (either GPU or CPU). Without GPU support, only the CPU implementation is available.

Many operations are covered by three general tensor operators: Element, Aggregate and Prod. The Element operator applies a function element-wise across an arbitrary number of input tensors and stores the result in the output tensor. The Aggregate operator also applies a function element-wise across its inputs, but instead aggregates the results in the output via a given aggregation function. A common aggregation function used is addition, which is the basis of the Add and Reduce operators. Finally, Prod deals with products of tensors. This operator performs a general matrix multiplication with the underlying implementation relying on the libraries mentioned above.

Specialized operators exist to manipulation tensors beyond the cases covered above; such as under transposition and concatenation. These operators may even be expressed in terms of existing tensor operators.

Furthermore, for complicated multi-operation computations, performance gains and memory improvements may be realised by implementing a tensor operator for that specific purpose. An example of this is softmax, which could be implemented using multiple expression operators (exp, sum), but is instead implemented directly as a tensor operator. These optimized implementations may be device specific.

## Declared Specialization¶

The operations performed in the forward and backward methods of NodeOp require their GPU templates to be explicitly declared. When a new specialization is introduced without being explicitly instantiated it will cause a link error on compilation:

.../src/tensors/tensor_operators.h:41: undefined reference to void marian::gpu::Element<marian::functional::Assign< ... > ( ... )'


To fix these undefined references, we must explicitly add the specialization to the .inc files of /src/tensors/gpu/. Each .inc file is included at the end of its corresponding .cu file, ensuring that the specialization is compiled.

The undefined references should be added to the .inc file that corresponds to the header file in which contains the declaration of the missing functions.

The file element.inc contains the specializations of the function defined in element.h:

// src/tensors/gpu/element.h
template <class Functor, class... Tensors>
void Element(Functor functor, Tensor out, Tensors... tensors);


Similarly, add.inc contains the specializations for functions matching either of the two signatures in add.h:

// src/tensors/gpu/add.h
template <class Functor, class... Tensors>
void Add(Functor functor, float scale, marian::Tensor out, Tensors... tensors);

template <class Functor, class AggFunctor, class... Tensors>
void Aggregate(Functor functor, float initAgg, AggFunctor aggFunctor, float scale, marian::Tensor out, Tensors... tensors);


Finally add_all.inc contains the specializations for add_all.h, which are several versions of:

// src/tensors/gpu/add_all.h
template <typename T, typename AccType, class Functor, class AggFunctor>
void AggregateAll(Ptr<Allocator> allocator,
Functor functor,
AccType aggInit,
AggFunctor aggFunctor,
AccType scale,
Tensor out,
const Tensor in1);


However, for add_all.h, there is an additional type dependence in the first template parameter, which requires two entries:

marian::gpu::AggregateAll< float, ... >( ... );
marian::gpu::AggregateAll< __half, ... >( ... );  // for COMPILE_FP16


where the __half specialization is related to half-precision floats and should be added to the COMPILE_FP16 preprocessor block.

The simplest method to add the correct specialization is to take the compilation error output and extract the needed signature. To extract the signature:

1. Replace up to, and including, “undefined reference to ” with “template”

2. Replace the final ‘ with a semi-colon

To conform with definitions in the codebase, we should replace IntrusivePtr<marian::TensorBase> with its typedef marian::Tensor. Note that as these files are included in marian::gpu namespace, and explicitly use marian::functional namespace it is also possible to omit both of these prefixes. Typically, the namespace prefix of the specialized function is removed as well. Following these rules for the example of SinNodeOp results in the following entries:

element

template void Element<Assign<Var<1>, UnaryFunctor<elem::Sin, Assignee<2> > >, marian::Tensor >(Assign<Var<1>, UnaryFunctor<elem::Sin, Assignee<2> > >, marian::Tensor, marian::Tensor);


template void Add<BinaryFunctor<elem::Mult,Assignee<1>,UnaryFunctor<elem::Cos,Assignee<2> > >,class marian::Tensor,class marian::Tensor >(BinaryFunctor<elem::Mult,Assignee<1>,UnaryFunctor<elem::Cos,Assignee<2> > >,float,class marian::Tensor,class marian::Tensor,class marian::Tensor);

template void AggregateAll<float,float,BinaryFunctor<elem::Mult,Assignee<1>,UnaryFunctor<elem::Cos,Assignee<2> > >,BinaryFunctor<elem::Plus,Assignee<1>,Assignee<2> > >(std::shared_ptr<marian::Allocator>,BinaryFunctor<elem::Mult,Assignee<1>,UnaryFunctor<elem::Cos,Assignee<2> > >,float,BinaryFunctor<elem::Plus,Assignee<1>,Assignee<2> >,float,marian::Tensor,marian::Tensor,marian::Tensor);