PreallocationTools.jl

PreallocationTools.jl is a set of tools for helping build non-allocating pre-cached functions for high-performance computing in Julia. Its tools handle edge cases of automatic differentiation to make it easier for users to get high performance even in the cases where code generation may change the function that is being called.

DiffCache

DiffCache is a type for doubly-preallocated vectors which are compatible with non-allocating forward-mode automatic differentiation by ForwardDiff.jl. Since ForwardDiff.jl uses chunked duals in its forward pass, two vector sizes are required in order for the arrays to be properly defined. DiffCache creates a dispatching type to solve this, so that by passing a qualifier it can automatically switch between the required cache. This method is fully type-stable and non-dynamic, made for when the highest performance is needed.

The DiffCache also supports sparsity detection via SparseConnectivityTracer.jl. However, the implementation may allocate memory in this case since we assume that sparsity detection happens only once (or maybe a few times). Allocating memory allows to save memory in the long run since no additional cache needs to be stored forever.

Using DiffCache

julia DiffCache(u::AbstractArray, N::Int = ForwardDiff.pickchunksize(length(u)); levels::Int = 1) DiffCache(u::AbstractArray, N::AbstractArray{<:Int})

The DiffCache function builds a DiffCache object that stores both a version of the cache for u and for the Dual version of u, allowing use of pre-cached vectors with forward-mode automatic differentiation. Note that DiffCache, due to its design, is only compatible with arrays that contain concretely typed elements.

To access the caches, one uses:

julia get_tmp(tmp::DiffCache, u)

When u has an element subtype of Dual numbers, then it returns the Dual version of the cache. Otherwise it returns the standard cache (for use in the calls without automatic differentiation).

In order to preallocate to the right size, the DiffCache needs to be specified to have the correct N matching the chunk size of the dual numbers or larger. If the chunk size N specified is too large, get_tmp will automatically resize when dispatching; this remains type-stable and non-allocating, but comes at the expense of additional memory.

In a differential equation, optimization, etc., the default chunk size is computed from the state vector u, and thus if one creates the DiffCache via DiffCache(u) it will match the default chunking of the solver libraries.

DiffCache is also compatible with nested automatic differentiation calls through the levels keyword (N for each level computed using based on the size of the state vector) or by specifying N as an array of integers of chunk sizes, which enables full control of chunk sizes on all differentation levels.

DiffCache Example 1: Direct Usage

```julia using ForwardDiff, PreallocationTools randmat = rand(5, 3) sto = similar(randmat) stod = DiffCache(sto)

function claytonsample!(sto, τ, α; randmat = randmat) sto = get_tmp(sto, τ) sto .= randmat τ == 0 && return sto

n = size(sto, 1)
for i in 1:n
    v = sto[i, 2]
    u = sto[i, 1]
    sto[i, 1] = (1 - u^(-τ) + u^(-τ) * v^(-(τ / (1 + τ))))^(-1 / τ) * α
    sto[i, 2] = (1 - u^(-τ) + u^(-τ) * v^(-(τ / (1 + τ))))^(-1 / τ)
end
return sto

end

ForwardDiff.derivative(τ -> claytonsample!(stod, τ, 0.0), 0.3) ForwardDiff.jacobian(x -> claytonsample!(stod, x[1], x[2]), [0.3; 0.0]) ```

In the above, the chunk size of the dual numbers has been selected based on the size of randmat, resulting in a chunk size of 8 in this case. However, since the derivative is calculated with respect to τ and the Jacobian is calculated with respect to τ and α, specifying the DiffCache with stod = DiffCache(sto, 1) or stod = DiffCache(sto, 2), respectively, would have been the most memory efficient way of performing these calculations (only really relevant for much larger problems).

DiffCache Example 2: ODEs

julia using LinearAlgebra, OrdinaryDiffEq function foo(du, u, (A, tmp), t) mul!(tmp, A, u) @. du = u + tmp nothing end prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), zeros(5, 5))) solve(prob, TRBDF2())

fails because tmp is only real numbers, but during automatic differentiation we need tmp to be a cache of dual numbers. Since u is the value that will have the dual numbers, we dispatch based on that:

julia using LinearAlgebra, OrdinaryDiffEq, PreallocationTools function foo(du, u, (A, tmp), t) tmp = get_tmp(tmp, u) mul!(tmp, A, u) @. du = u + tmp nothing end chunk_size = 5 prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), DiffCache(zeros(5, 5), chunk_size))) solve(prob, TRBDF2(chunk_size = chunk_size))

or just using the default chunking:

julia using LinearAlgebra, OrdinaryDiffEq, PreallocationTools function foo(du, u, (A, tmp), t) tmp = get_tmp(tmp, u) mul!(tmp, A, u) @. du = u + tmp nothing end chunk_size = 5 prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), DiffCache(zeros(5, 5)))) solve(prob, TRBDF2())

DiffCache Example 3: Nested AD calls in an optimization problem involving a Hessian matrix

```julia using LinearAlgebra, OrdinaryDiffEq, PreallocationTools, Optimization, OptimizationOptimJL function foo(du, u, p, t) tmp = p[2] A = reshape(p[1], size(tmp.du)) tmp = get_tmp(tmp, u) mul!(tmp, A, u) @. du = u + tmp nothing end

coeffs = -collect(0.1:0.1:0.4) cache = DiffCache(zeros(2, 2), levels = 3) prob = ODEProblem(foo, ones(2, 2), (0.0, 1.0), (coeffs, cache)) realsol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

function objfun(x, prob, realsol, cache) prob = remake(prob, u0 = eltype(x).(prob.u0), p = (x, cache)) sol = solve(prob, TRBDF2(), saveat = 0.0:0.1:10.0, reltol = 1e-8)

ofv = 0.0
if any((s.retcode != :Success for s in sol))
    ofv = 1e12
else
    ofv = sum((sol .- realsol) .^ 2)
end
return ofv

end fn(x, p) = objfun(x, p[1], p[2], p[3]) optfun = OptimizationFunction(fn, Optimization.AutoForwardDiff()) optprob = OptimizationProblem(optfun, zeros(length(coeffs)), (prob, realsol, cache)) solve(optprob, Newton()) ```

Solves an optimization problem for the coefficients, coeffs, appearing in a differential equation. The optimization is done with Optim.jl's Newton() algorithm. Since this involves automatic differentiation in the ODE solver and the calculation of Hessians, three automatic differentiations are nested within each other. Therefore, the DiffCache is specified with levels = 3.

FixedSizeDiffCache

FixedSizeDiffCache is a lot like DiffCache, but it stores dual numbers in its caches instead of a flat array. Because of this, it can avoid a view, making it a little bit more performant for generating caches of non-Array types. However, it is a lot less flexible than DiffCache, and is thus only recommended for cases where the chunk size is known in advance (for example, ODE solvers) and where u is not an Array.

The interface is almost exactly the same, except with the constructor:

julia FixedSizeDiffCache(u::AbstractArray, chunk_size = Val{ForwardDiff.pickchunksize(length(u))}) FixedSizeDiffCache(u::AbstractArray, chunk_size::Integer)

Note that the FixedSizeDiffCache can support duals that are of a smaller chunk size than the preallocated ones, but not a larger size. Nested duals are not supported with this construct.

LazyBufferCache

julia LazyBufferCache(f::F = identity)

A LazyBufferCache is a Dict-like type for the caches which automatically defines new cache arrays on demand when they are required. The function f maps size_of_cache = f(size(u)), which by default creates cache arrays of the same size.

By default the created buffers are not initialized, but a function initializer! can be supplied which is applied to the buffer when it is created, for instance buf -> fill!(buf, 0.0).

Note that LazyBufferCache is type-stable and contains no dynamic dispatch. This gives it a ~15ns overhead. The upside of LazyBufferCache is that the user does not have to worry about potential issues with chunk sizes and such: LazyBufferCache is much easier!

Example

julia using LinearAlgebra, OrdinaryDiffEq, PreallocationTools function foo(du, u, (A, lbc), t) tmp = lbc[u] mul!(tmp, A, u) @. du = u + tmp nothing end prob = ODEProblem(foo, ones(5, 5), (0.0, 1.0), (ones(5, 5), LazyBufferCache())) solve(prob, TRBDF2())

GeneralLazyBufferCache

julia GeneralLazyBufferCache(f = identity)

A GeneralLazyBufferCache is a Dict-like type for the caches which automatically defines new caches on demand when they are required. The function f generates the cache matching for the type of u, and subsequent indexing reuses that cache if that type of u has already ben seen.

Note that GeneralLazyBufferCache's return is not type-inferred. This means it's the slowest of the preallocation methods, but it's the most general.

Example

In all of the previous cases our cache was an array. However, in this case we want to preallocate a DifferentialEquations ODEIntegrator object. This object is the one created via DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t), and we want to optimize p in a way that changes its type to ForwardDiff. Thus what we can do is make a GeneralLazyBufferCache which holds these integrator objects, defined by p, and indexing it with p in order to retrieve the cache. The first time it's called it will build the integrator, and in subsequent calls it will reuse the cache.

Defining the cache as a function of p to build an integrator thus looks like:

julia lbc = GeneralLazyBufferCache(function (p) DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t) end)

then lbc[p] (or, equivalently, get_tmp(lbc, p)) will be smart and reuse the caches. A full example looks like the following:

```julia using Random, DifferentialEquations, LinearAlgebra, Optimization, OptimizationNLopt, OptimizationOptimJL, PreallocationTools

lbc = GeneralLazyBufferCache(function (p) DifferentialEquations.init(ODEProblem(ode_fnc, y₀, (0.0, T), p), Tsit5(); saveat = t) end)

Random.seed!(2992999) λ, y₀, σ = -0.5, 15.0, 0.1 T, n = 5.0, 200 Δt = T / n t = [j * Δt for j in 0:n] y = y₀ * exp.(λ * t) yᵒ = y .+ [0.0, σ * randn(n)...] ode_fnc(u, p, t) = p * u function loglik(θ, data, integrator) yᵒ, n, ε = data λ, σ, u0 = θ integrator.p = λ reinit!(integrator, u0) solve!(integrator) ε = yᵒ .- integrator.sol.u ℓ = -0.5n * log(2π * σ^2) - 0.5 / σ^2 * sum(ε .^ 2) end θ₀ = [-1.0, 0.5, 19.73] negloglik = (θ, p) -> -loglik(θ, p, lbc[θ[1]]) fnc = OptimizationFunction(negloglik, Optimization.AutoForwardDiff()) ε = zeros(n) prob = OptimizationProblem(fnc, θ₀, (yᵒ, n, ε), lb = [-10.0, 1e-6, 0.5], ub = [10.0, 10.0, 25.0]) solve(prob, LBFGS()) ```

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