Generalized leapfrog integrator (#370)

* feat: port generalized leapfrog

Signed-off-by: Kai Xu <[email protected]>

* refactor: remove copy

Signed-off-by: Kai Xu <[email protected]>

* format: add emplty line

Signed-off-by: Kai Xu <[email protected]>

* Update src/riemannian/integrator.jl

Co-authored-by: Hong Ge <[email protected]>

* chore: add warning for using generalized leapfrog with vectorization

Signed-off-by: Kai Xu <[email protected]>

* fix: type order

Signed-off-by: Kai Xu <[email protected]>

---------

Signed-off-by: Kai Xu <[email protected]>
Co-authored-by: Hong Ge <[email protected]>

src/AdvancedHMC.jl

@@ -52,6 +52,8 @@ export Hamiltonian
 
 include("integrator.jl")
 export Leapfrog, JitteredLeapfrog, TemperedLeapfrog
+include("riemannian/integrator.jl")
+export GeneralizedLeapfrog
 
 include("trajectory.jl")
 export Trajectory,

src/riemannian/integrator.jl

@@ -0,0 +1,102 @@
+"""
+$(TYPEDEF)
+
+Generalized leapfrog integrator with fixed step size `ϵ`.
+
+# Fields
+
+$(TYPEDFIELDS)
+
+
+## References 
+
+1. Girolami, Mark, and Ben Calderhead. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods." Journal of the Royal Statistical Society Series B: Statistical Methodology 73, no. 2 (2011): 123-214.
+"""
+struct GeneralizedLeapfrog{T<:AbstractScalarOrVec{<:AbstractFloat}} <: AbstractLeapfrog{T}
+    "Step size."
+    ϵ::T
+    n::Int
+end
+Base.show(io::IO, l::GeneralizedLeapfrog) =
+    print(io, "GeneralizedLeapfrog(ϵ=$(round.(l.ϵ; sigdigits=3)), n=$(l.n))")
+
+# fallback to ignore return_cache & cache kwargs for other ∂H∂θ
+function ∂H∂θ_cache(h, θ, r; return_cache = false, cache = nothing)
+    dv = ∂H∂θ(h, θ, r)
+    return return_cache ? (dv, nothing) : dv
+end
+
+# TODO(Kai) make sure vectorization works
+# TODO(Kai) check if tempering is valid
+# TODO(Kai) abstract out the 3 main steps and merge with `step` in `integrator.jl` 
+function step(
+    lf::GeneralizedLeapfrog{T},
+    h::Hamiltonian,
+    z::P,
+    n_steps::Int = 1;
+    fwd::Bool = n_steps > 0,  # simulate hamiltonian backward when n_steps < 0
+    full_trajectory::Val{FullTraj} = Val(false),
+) where {T<:AbstractScalarOrVec{<:AbstractFloat},TP,P<:PhasePoint{TP},FullTraj}
+    n_steps = abs(n_steps)  # to support `n_steps < 0` cases
+
+    ϵ = fwd ? step_size(lf) : -step_size(lf)
+    ϵ = ϵ'
+
+    if !(T <: AbstractFloat) || !(TP <: AbstractVector)
+        @warn "Vectorization is not tested for GeneralizedLeapfrog."
+    end
+
+    res = if FullTraj
+        Vector{P}(undef, n_steps)
+    else
+        z
+    end
+
+    for i = 1:n_steps
+        θ_init, r_init = z.θ, z.r
+        # Tempering
+        #r = temper(lf, r, (i=i, is_half=true), n_steps)
+        # eq (16) of Girolami & Calderhead (2011)
+        r_half = r_init
+        local cache
+        for j = 1:lf.n
+            # Reuse cache for the first iteration
+            if j == 1
+                @unpack value, gradient = z.ℓπ
+            elseif j == 2 # cache intermediate values that depends on θ only (which are unchanged)
+                retval, cache = ∂H∂θ_cache(h, θ_init, r_half; return_cache = true)
+                @unpack value, gradient = retval
+            else # reuse cache
+                @unpack value, gradient = ∂H∂θ_cache(h, θ_init, r_half; cache = cache)
+            end
+            r_half = r_init - ϵ / 2 * gradient
+        end
+        # eq (17) of Girolami & Calderhead (2011)
+        θ_full = θ_init
+        term_1 = ∂H∂r(h, θ_init, r_half) # unchanged across the loop
+        for j = 1:lf.n
+            θ_full = θ_init + ϵ / 2 * (term_1 + ∂H∂r(h, θ_full, r_half))
+        end
+        # eq (18) of Girolami & Calderhead (2011)
+        @unpack value, gradient = ∂H∂θ(h, θ_full, r_half)
+        r_full = r_half - ϵ / 2 * gradient
+        # Tempering
+        #r = temper(lf, r, (i=i, is_half=false), n_steps)
+        # Create a new phase point by caching the logdensity and gradient
+        z = phasepoint(h, θ_full, r_full; ℓπ = DualValue(value, gradient))
+        # Update result
+        if FullTraj
+            res[i] = z
+        else
+            res = z
+        end
+        if !isfinite(z)
+            # Remove undef
+            if FullTraj
+                res = res[isassigned.(Ref(res), 1:n_steps)]
+            end
+            break
+        end
+    end
+    return res
+end