Torch.Tutorial

Description

Hasktorch is a library for scientific computing and differentiable programming.

Synopsis

# Documentation

Hasktorch is a Haskell library for scientific computing and differentiable programming. It leverages libtorch (the backend library powering PyTorch) for efficient tensor manipulation and automatic differentiation, while bringing to bear Haskell's expressive type system and first-class support for for the functional programming paradigm.

## Goal of this tutorial

The sequence of topics and examples here is loosely based on the PyTorch tutorial Deep Learning with PyTorch: A 60 Minute Blitz by Soumith Chintala.

In this tutorial we will implement a simple machine learning model. Along the way, you will learn to

• create and manipulate tensors
• build computation graphs from tensors and compute gradients
• optimize parameters with respect to an objective function

# Usage

The reader is encouraged to follow along with the examples in a GHCi session.

To start, import Torch:

>>> import Torch


## Tensors

A Tensor in Hasktorch is multidimensional array with a fixed shape and element type.

For example, we can initialize a tensor with shape [3, 4] and filled with zeros using

>>> Torch.zeros' [3, 4]
Tensor Float [3,4] [[ 0.0000,  0.0000,  0.0000,  0.0000],
[ 0.0000,  0.0000,  0.0000,  0.0000],
[ 0.0000,  0.0000,  0.0000,  0.0000]]


We can also initialize a tensor from a Haskell list using asTensor:

>>> asTensor ([[4, 3], [2, 1]] :: [[Float]])
Tensor Float [2,2] [[ 4.0000   ,  3.0000   ],
[ 2.0000   ,  1.0000   ]]


Note that the numerical type of the tensor is inferred from the types of the values in the list.

Scalar values are represented in Hasktorch as tensors with shape []:

>>> asTensor 3.5
Tensor Double []  3.5000


We can get the scalar value back out using asValue:

>>> asValue (asTensor 3.5)
3.5


### Specifying Tensor parameters

In the previous section we initialized a tensor filled with zeros using zeros' (note the prime suffix). Hasktorch functions use a convention where default versions of functions use a prime suffix. The unprimed versions of these functions expect an additional parameter specifying tensor parameters. For example:

  zeros :: [Int] -> TensorOptions -> Tensor


TensorOptions are typically specified by starting with defaultOpts and modifying using one or more of the following:

• withDType configures the data type of the elements
• withDevice configures on which device the tensor is to be used
• others (see TensorOptions)

For example, to construct a matrix filled with zeros of dtype Int64:

>>> zeros [4, 4] (withDType Int64 defaultOpts)
Tensor Int64 [4,4] [[ 0,  0,  0,  0],
[ 0,  0,  0,  0],
[ 0,  0,  0,  0],
[ 0,  0,  0,  0]]


### Tensor factories

Hasktorch comes with many "factory" functions similar to zeros and zeros' useful for initializing common kinds of tensors. For example, ones, full, eye, and the primed versions of these. See TensorFactories for a complete list.

One useful class of factory functions are those suffixed with "-like" (e.g. onesLike), which initialize a tensor with the same dimensions as their argument. For example:

>>> let x = zeros' [3, 2]
>>> onesLike x
Tensor Float [3,2] [[ 1.0000   ,  1.0000   ],
[ 1.0000   ,  1.0000   ],
[ 1.0000   ,  1.0000   ]]


## Operations

Most operations are pure functions, similar to Haskell standard library math operations.

Tensors implement the Num typeclass:

>>> let x = ones' 
>>> x + x
Tensor Float  [ 2.0000   ,  2.0000   ,  2.0000   ,  2.0000   ]


Some operations transform a tensor:

>>> Torch.relu (asTensor ([-1.0, -0.5, 0.5, 1] :: [Float]))
Tensor Float  [ 0.0000,  0.0000,  0.5000   ,  1.0000   ]


selectDim slices out a selection by specifying a dimension and index:

>>> let x = asTensor [[[1, 2, 3]], [[4, 5, 6]], [[7, 8, 9]], [[10, 11, 12]]]
>>> shape x
[4,1,3]

>>> select 2 1 x
Tensor Double [4,1] [[ 2.0000   ],
[ 5.0000   ],
[ 8.0000   ],
[ 11.0000   ]]

>>> let y = asTensor [1, 2, 3]
>>> Torch.select 0 1 y
Tensor Double []  2.0000


Values can be extracted from a tensor using asValue so long as the dtype matches the Haskell type:

>>> let x = asTensor ( :: [Int])
>>> let y = asValue x :: Int
>>> y
2


## Randomness

Create a randomly initialized matrix:

>>> x <-randIO' [2, 2]
>>> x
Tensor Float [2,2] [[ 0.2961   ,  0.5166   ],
[ 0.2517   ,  0.6886   ]]


Note that since random initialization returns a different result each time, unlike other tensor constructors, is monadic reflecting the context of an underlying random number generator (RNG) changing state.

Hasktorch includes variations of random tensor initializers in which the RNG object is threaded explicitly rather than implicitly. See the Torch.Random module functions for details and variations. Samplers for which the RNG is not explicit such as randIO' example above use the -IO suffix.

## Automatic Differentiation

Automatic differentiation is achieved through the use of two primary functions in the Autograd module, makeIndependent and grad.

### Independent Tensors

makeIndependent is used to instantiate an independent tensor variable from which a compute graph is constructed for differentiation, while grad uses compute graph to compute gradients.

makeIndependent takes a tensor as input and returns an IO action which produces an IndependentTensor:

 makeIndependent :: Tensor -> IO IndependentTensor

What is the definition of the IndependentTensor type produced by the makeIndependent action? It’s defined in the Hasktorch library as:

 newtype IndependentTensor = IndependentTensor { toDependent :: Tensor }
deriving (Show)

Thus IndependentTensor is simply a wrapper around the underlying Tensor that is passed in as the argument to makeIndependent. Building up computations using ops applied to the toDependent tensor of an IndependentTensor will implicitly construct a compute graph to which grad can be applied.

All tensors have an underlying property that can be retrieved using the requiresGrad function which indicates whether they are a differentiable value in a compute graph. 

>>> let x = asTensor [1, 2, 3]
>>> y <- makeIndependent (asTensor [4, 5, 6])
>>> let y' = toDependent y
>>> let z = x + y'
>>> requiresGrad x
False

>>> requiresGrad y'
True

>>> requiresGrad z
True


In summary, tensors that are computations of values derived from tensor constructors (e.g. ones, zeros, fill, randIO etc.) outside the context of a IndependentTensor are not differentiable. Tensors that are derived from computations on the toDependent value of an IndependentTensor are differentiable, as the above example illustrates.

Once a computation graph is constructed by applying ops and computing derived quantities stemming from a toDependent value of an IndependentTensor, a gradient can be taken by using the grad function specifying in the first argument tensor corresponding to function value of interest and a list of Independent tensor variables that the the derivative is taken with respect to:

 grad :: Tensor -> [IndependentTensor] -> [Tensor]

Let’s demonstrate this with a concrete example. We create a tensor and derive an IndependentTensor from it:

>>> x <- makeIndependent (ones' [2, 2])
>>> let x' = toDependent x
>>> x'
Tensor Float [2,2] [[ 1.0000   ,  1.0000   ],
[ 1.0000   ,  1.0000   ]]


Now do some computations on the dependent tensor:

>>> let y = x' + 2
>>> y
Tensor Float [2,2] [[ 3.0000   ,  3.0000   ],
[ 3.0000   ,  3.0000   ]]


Since y is dependent on the x independent tensor, it is differentiable:

>>> requiresGrad y
True


Applying more operations:

>>> let z = y * y * 3
>>> let out = mean z
>>> z
Tensor Float [2,2] [[ 27.0000   ,  27.0000   ],
[ 27.0000   ,  27.0000   ]]


>>> grad out [x]
[Tensor Float [2,2] [[ 4.5000   ,  4.5000   ],
[ 4.5000   ,  4.5000   ]]]


## Differentiable Programs (Neural Networks)

From a functional programming perspective, a neural network is represented by data and functions, much like any other functional program. The only distinction that differentiates neural networks from any other functional program is that it implements a small interface surface to support differentiation. Thus, we can consider neural networks to be "differentiable functional programming".

The data in neural networks are the values to be fitted that parameterize the functions which carry out the inference operation and are modified based on gradients of through those functions.

As with a regular Haskell program, this data is represented by an algebraic data type (ADT). The ADT can take on any shape that’s needed to model the domain of interest, allowing a great deal of flexibility and enabling all of Haskell’s strenghts in data modeling - can use sum or product types, nest types, etc. The ADT can implement various typeclasses to take on other functionality.

The core interface that defines capability specific to differentiable programming is the Parameterized typeclass:

 class Parameterized f where
flattenParameters :: f -> [Parameter]
default flattenParameters :: (Generic f, Parameterized' (Rep f)) => f -> [Parameter]
flattenParameters f = flattenParameters' (from f)

replaceOwnParameters :: f -> ParamStream f
default replaceOwnParameters :: (Generic f, Parameterized' (Rep f)) => f -> ParamStream f
replaceOwnParameters f = to <$> replaceOwnParameters' (from f) Note Parameter is simply a type alias for IndependentTensor in the context of neural networks (i.e. type Parameter = IndependentTensor). The role of flattenParameters is to unroll any arbitrary ADT representation of a neural network into a standard flattened representation consisting a list of IndependentTensor which is used to compute gradients. replaceOwnParameters is used to update parameters. ParamStream is a type alias for a State type with state represented by a Parameter list and a value parameter corresponding to the ADT defining the model.  type ParamStream a = State [Parameter] a Note the use of generics. Generics allow the compiler to usually automatically derive flattenParameters and replaceOwnParameter instances without any code if your type is built up on tensors, containers of tensors, or other types that are built from tensor values (for example, layer modules provided in Torch.NN. In many cases, as you’ll see in the following examples, you will only need to add  instance Parameterized MyNeuralNetwork (where MyNeuralNetwork is an ADT definition for your model) and the compiler will derive implementations for the flattenParameters and replaceOwnParameters. ### Linear Regression Lets start with a simple example of linear regression. Here we generate random data with an underlying affine relationship between the inputs and outputs, then fit a linear regression to reproduce that relationship. This example is adapted from https://github.com/hasktorch/hasktorch/tree/master/examples/regression. In a standard supervised learning model, the neural network is initialized using a randomized initialization scheme. An iterative optimization is performed such that at each iteration a batch.  module Main where import Control.Monad (when) import Torch groundTruth :: Tensor -> Tensor groundTruth t = squeezeAll$ matmul t weight + bias
where
weight = asTensor ([42.0, 64.0, 96.0] :: [Float])
bias = full'  (3.14 :: Float)

model :: Linear -> Tensor -> Tensor
model state input = squeezeAll $linear state input main :: IO () main = do init <- sample$ LinearSpec{in_features = numFeatures, out_features = 1}
randGen <- mkGenerator (Device CPU 0) 12345
(trained, _) <- foldLoop (init, randGen) 2000 \$ $$state, randGen) i -> do let (input, randGen') = randn' [batchSize, numFeatures] randGen (y, y') = (groundTruth input, model state input) loss = mseLoss y y' when (i mod 100 == 0)  do putStrLn  "Iteration: " ++ show i ++ " | Loss: " ++ show loss (newParam, _) <- runStep state GD loss 5e-3 pure (replaceParameters state newParam, randGen') pure () where batchSize = 4 numFeatures = 3 Note the expression of the architecture in the linear function (a single linear layer, or alternatively a neural network with zero hidden layers), does not require an explicit representation of the compute graph, but is simply a composition of tensor ops. Because of the autodiff mechanism described in the previous section, the graph is constructed automatically as pure functional ops are applied, given a context of a set of independent variables. The init variable is initialized as a Linear type (defined in Torch.NN) using sample which randomly initializes a Linear value. Linear is a built-in ADT implementing the Parameterized typeclass and representing a fully connected linear layer, equivalent to linear regression when no hidden layers are present. init is passed into the foldLoop as the state@ variable. A new list of Parameter values is passed back from runStep (which calls grad to retrieve gradients, given a loss function, learning rate, and optimizer) and the typeclass function replaceParameters is used to update the model at each iteration. Initialization is discussed in more detail in the following section ### Weight Initialization Random initialization of weights is not a pure function since two random initializations return different values. Initialization occurs by calling the sample function for an ADT (spec) implementing the Randomizable typeclass:  class Randomizable spec f | spec -> f where sample :: spec -> IO f In a typical (but not required) usage, f is an ADT that implements the Parameterized typeclass, so that there’s a pair of types - a specification type implementing the spec input to sample and a type implementing Parameterizable representing the model state. For example, a linear fully connected layer is provided by the Torch.NN module and defined therein as:  data Linear = Linear { weight :: Parameter, bias :: Parameter } deriving (Show, Generic) and is typically used with a specification type:  data LinearSpec = LinearSpec { in_features :: Int, out_features :: Int } deriving (Show, Eq) Putting this together, in untyped tensor usage, the user can implement custom models or layers implementing the Parameterizable typeclass built up from other ADTs implementing Parameterizable. The shape of the data required for initialization is described by a type implementing Randomizable’s spec parameter, and the sample implementation specifies the default weight initialization. Note this initialization approach is specific to untyped tensors. One consequence of using typed tensors is that the information in these spec types is reflected in the type itself and thus are not needed. What if you want to use a custom initialization that differs from the default? You can define an alternative function with the same signature spec -> IO f and use the alternative function instead of sample. ### Optimizers Optimization implementations are functions that take as input the current parameter values of a model, parameter gradient estimates of the loss function at those parameters for a single batch, and a characteristic learning describing how large a perturbation to make to the parameters in order to reduce the loss. Given those inputs, they output a new set of parameters. In the simple case of stochastic gradient descent, the function to output a new set of parameters is to subtract from the current parameter \(\theta$$, the gradient of the loss $$\nabla J$$ scaled by the learning rate $$\eta$$:

$\theta_{i+1} = \theta_i - \eta \nabla J(\theta)$

While stochastic gradient descent is a stateless function of the parameters, loss, and gradient, some optimizers have a notion of internal state that is propagated from one step to the step, for example, retaining and updating momentum between steps:

$\begin{gathered} \Delta \theta_i = \alpha \Delta \theta_{i-1} - \eta \nabla J(\theta) \\ \theta_{i+1} = \theta_i + \Delta \theta_i \end{gathered}$

In this case, the momentum term $$\Delta \theta_i$$ is carried forward as internal state of the optimizer that is propagated to the next step. $$\alpha$$ is an optimizer parameter which determines a weighting on the momentum term relative to the gradient.

Implementation of an optimizer consists of defining an ADT describing the optimizer state and a step function that implements a single step perturbation given the learning rate, loss gradients, current parameters, and optimizer state.

This function interface is described in the Optimizer typeclass interface:

 class Optimizer o where
step :: LearningRate -> Gradients -> [Tensor] -> o -> ([Tensor], o)

Gradients is a newtype wrapper around a list of tensors to make intent explicit: newtype Gradients = Gradients [Tensor].

Hasktorch provides built-in optimizer implementations in Torch.Optim. Some illustrative example implementations follow.

Being stateless, stochastic gradient descent has an ADT that has only one constructor value:

 data GD = GD

and implements the step function as:

 instance Optimizer GD where
where
step p dp = p - (lr * dp)
gd lr (Gradients gradients) parameters = zipWith step parameters gradients

The use of an optimizer was illustrated in the linear regression example using the function runStep

 (newParam, _) <- runStep state GD loss 5e-3

In this case the new optimizer state returned is ignored (as _) since gradient descent does not have any internal state. Under the hood, runStep does a little bookkeeping making independent variables from a model, computing gradients, and passing values to the step function. Usually a user can ignore the details and just pass model parameters and the optimizer to runStep as an abstracted interface which takes parameter values, the optimizer value, loss (a tensor), and learning rate as input and returns new parameters and an updated optimizer value.

 runStep :: (Parameterized p, Optimizer o) =>
p -> o -> Tensor -> LearningRate -> IO ([Parameter], o)

# Typed Tensors

Typed tensors provide an alternative API for which tensor characteristics are encoded in the type of the tensor. The Hasktorch library is layered such that [ TODO ]

Using typed tensors increases the expressiveness of program invariants that can be automatically checked by GHC at the cost of needing to be more explicit in writing code and also requiring working with Haskell’s type-level machinery. Type-level Haskell programming has arisen through progressive compiler iteration leading to mechanisms that have a higher degree of complexity compared to value-level Haskell code or other languages such as Idris which were designed with type-level computations from inception. In spite of these compromises, Haskell offers enough capability to express powerful type-level representations of models that is matched by few other languages used for production applications.

Things we can do in typed Hasktorch:

• specify, check, and infer tensor shapes at compile time
• specify, check, and infer tensor data types at compile time
• specify, check, and infer tensor compute devices at compile time

We can encode all Hasktorch tensor shapes on the type level using type-level lists and natural numbers:

 type EmptyShape = '[]
type OneDimensionalShape (a :: Nat) = '[a]
type TwoDimensionalShape (a :: Nat) (b :: Nat) = '[a, b]
...

Tensor data types and compute device types are lifted to the type level using the DataKinds language extension:

 type BooleanCPUTensor (shape :: [Nat]) = Tensor '(CPU,  0) 'Bool  shape
type IntCUDATensor    (shape :: [Nat]) = Tensor '(CUDA, 1) 'Int64 shape

Devices are represented as tuples consisting of a DeviceType (here CPU for the CPU and CUDA for a CUDA device, respectively) and a device id (here the Nats 0 and 1, respectively).

It is a common misconception that specifying tensor properties at compile time is only possible if all tensor properties are constants and are statically known. If this were the case, then we could only write functions over fully specified tensors, say,

 boring :: BooleanCPUTensor '[] -> BooleanCPUTensor '[]
boring = id

Fortunately, Haskell has the ability to reason about type variables. This feature is called parametric polymorphism. Consider this simple example of a function:

 tensorNoOp
:: forall (shape :: [Nat]) (dtype :: DType) (device :: (DeviceType, Nat))
. Tensor device dtype shape
-> Tensor device dtype shape
tensorNoOp = id

Here, shape, dtype, and device are type variables that have been constrained to be of kind shape, data type, and device, respectively. The universal quantifier forall implies that this function is well-defined for all inhabitants of the types that are compatible with the type variables and their constraints.

The tensorNoOp function may seem trivial, and that is because its type is very strongly constrained: Given any typed tensor, it must return a tensor of the same shape and with the same data type and on the same device. Besides by means of the identity, id, there are not many ways in which this function can be implemented.

There is a connection between a type signature of a function and mathematical proofs. We can say that the fact that a function exists is witnessed by its implementation. The implementation, tensorNoOp = id, is the proof of the theorem stated by tensorNoOp’s type signature. And here we have a proof that we can run this function on any device and for any data type and tensor shape.

This is the essence of typed Hasktorch. As soon as the compiler gives its OK, we hold a proof that our program will run without shape or CUDA errors.