Batch-and-Match algorithm for minimizing the covariance-weighted Fisher divergence

HISTORY.md

@@ -8,6 +8,7 @@ Specifically, the following measure-space optimization algorithms have been adde
   - `KLMinWassFwdBwd`
   - `KLMinNaturalGradDescent`
   - `KLMinSqrtNaturalGradDescent`
+  - `FisherMinBatchMatch`
 
 ## Interface Change
 

docs/make.jl

@@ -29,6 +29,7 @@ makedocs(;
             "`KLMinWassFwdBwd`" => "klminwassfwdbwd.md",
             "`KLMinNaturalGradDescent`" => "klminnaturalgraddescent.md",
             "`KLMinSqrtNaturalGradDescent`" => "klminsqrtnaturalgraddescent.md",
+            "`FisherMinBatchMatch`" => "fisherminbatchmatch.md",
         ],
         "Variational Families" => "families.md",
         "Optimization" => "optimization.md",

docs/src/fisherminbatchmatch.md

@@ -0,0 +1,61 @@
+# [`FisherMinBatchMatch`](@id fisherminbatchmatch)
+
+## Description
+
+This algorithm, known as batch-and-match (BaM) aims to minimize the covariance-weighted 2nd-order Fisher divergence by running a proximal point-type method[^CMPMGBS24].
+On certain low-dimensional problems, BaM can converge very quickly without any tuning.
+Since `FisherMinBatchMatch` is a measure-space algorithm, its use is restricted to full-rank Gaussian variational families (`FullRankGaussian`) that make the measure-valued operations tractable.
+
+```@docs
+FisherMinBatchMatch
+```
+
+The associated objective value can be estimated through the following:
+
+```@docs; canonical=false
+estimate_objective(
+    ::Random.AbstractRNG,
+    ::KLMinWassFwdBwd,
+    ::MvLocationScale,
+    ::Any;
+    ::Int,
+)
+```
+
+[^CMPMGBS24]: Cai, D., Modi, C., Pillaud-Vivien, L., Margossian, C. C., Gower, R. M., Blei, D. M., & Saul, L. K. (2024). Batch and match: black-box variational inference with a score-based divergence. In *Proceedings of the International Conference on Machine Learning*.
+## [Methodology](@id fisherminbatchmatch_method)
+
+This algorithm aims to solve the problem
+
+```math
+  \mathrm{minimize}_{q \in \mathcal{Q}}\quad \mathrm{F}_{\mathrm{cov}}(q, \pi),
+```
+
+where $\mathcal{Q}$ is some family of distributions, often called the variational family, and $\mathrm{F}_{\mathrm{cov}}$ is a divergence defined as
+
+```math
+\mathrm{F}_{\mathrm{cov}}(q, \pi) = \mathbb{E}_{z \sim q} {\left\lVert \nabla \log \frac{q}{\pi} (z) \right\rVert}_{\mathrm{Cov}(q)}^2 ,
+```
+
+where ${\lVert x \rVert}_{A}^2 = x^{\top} A x $ is a weighted norm.
+$\mathrm{F}_{\mathrm{cov}}$ can be viewed as a variant of the canonical 2nd-order Fisher divergence defined as
+
+```math
+\mathrm{F}_{2}(q, \pi) = \sqrt{ \mathbb{E}_{z \sim q} {\left\lVert \nabla \log \frac{q}{\pi} (z) \right\rVert}^2 }.
+```
+
+The use of the weighted norm ${\lVert \cdot \rVert}_{\mathrm{Cov}(q)}^2$ facilitates the use of a proximal point-type method for minimizing $\mathrm{F}_{2}(q, \pi)$.
+In particular, BaM iterates the update
+
+```math
+  q_{t+1} = \argmin_{q \in \mathcal{Q}} \left\{ \mathrm{F}_{\mathrm{cov}}(q, \pi) + \frac{2}{\lambda_t} \mathrm{KL}\left(q_t, q\right) \right\} .
+```
+
+Since $\mathrm{F}(q, \pi)$ is intractable, it is replaced with a Monte Carlo approximation with a number of samples `n_samples`.
+Furthermore, by restricting $\mathcal{Q}$ to a Gaussian variational family, the update rule admits a closed form solution[^CMPMGBS24].
+Notice that the update does not involve the parameterization of $q_t$, which makes `FisherMinBatchMatch` a measure-space algorithm.
+
+Historically, the idea of using a proximal point-type update for minimizing a Fisher divergence-like objective was initially coined as Gaussian score matching[^MGMYBS23].
+BaM can be viewed as a successor to this algorithm.
+
+[^MGMYBS23]: Modi, C., Gower, R., Margossian, C., Yao, Y., Blei, D., & Saul, L. (2023). Variational inference with Gaussian score matching. In *Advances in Neural Information Processing Systems*, 36.

docs/src/index.md

@@ -20,3 +20,4 @@ For using the algorithms implemented in `AdvancedVI`, refer to the corresponding
   - [KLMinNaturalGradDescent](@ref klminnaturalgraddescent)
   - [KLMinSqrtNaturalGradDescent](@ref klminsqrtnaturalgraddescent)
   - [KLMinWassFwdBwd](@ref klminwassfwdbwd)
+  - [FisherMinBatchMatch](@ref fisherminbatchmatch)

src/AdvancedVI.jl

@@ -358,7 +358,9 @@ include("algorithms/gauss_expected_grad_hess.jl")
 include("algorithms/klminwassfwdbwd.jl")
 include("algorithms/klminsqrtnaturalgraddescent.jl")
 include("algorithms/klminnaturalgraddescent.jl")
+include("algorithms/fisherminbatchmatch.jl")
 
-export KLMinWassFwdBwd, KLMinSqrtNaturalGradDescent, KLMinNaturalGradDescent
+export KLMinWassFwdBwd,
+    KLMinSqrtNaturalGradDescent, KLMinNaturalGradDescent, FisherMinBatchMatch
 
 end

src/algorithms/fisherminbatchmatch.jl

@@ -0,0 +1,195 @@
+
+"""
+    FisherMinBatchMatch(n_samples, subsampling)
+    FisherMinBatchMatch(; n_samples, subsampling)
+
+Covariance-weighted Fisher divergence minimization via the batch-and-match algorithm, which is a proximal point-type optimization scheme.
+
+# (Keyword) Arguments
+- `n_samples::Int`: Number of samples (batchsize) used to compute the moments required for the batch-and-match update. (default: `32`)
+- `subsampling::Union{Nothing,<:AbstractSubsampling}`: Optional subsampling strategy. (default: `nothing`)
+
+!!! warning
+    `FisherMinBatchMatch` with subsampling enabled results in a biased algorithm and may not properly optimize the covariance-weighted Fisher divergence.
+
+!!! note
+    `FisherMinBatchMatch` requires a sufficiently large `n_samples` to converge quickly.
+
+!!! note
+    The `subsampling` strategy is only applied to the target `LogDensityProblem` but not to the variational approximation `q`. That is, `FisherMinBatchMatch` does not support amortization or structured variational families.
+
+# Output
+- `q`: The last iterate of the algorithm.
+
+# Callback Signature
+The `callback` function supplied to `optimize` needs to have the following signature:
+
+    callback(; rng, iteration, q, info)
+
+The keyword arguments are as follows:
+- `rng`: Random number generator internally used by the algorithm.
+- `iteration`: The index of the current iteration.
+- `q`: Current variational approximation.
+- `info`: `NamedTuple` containing the information generated during the current iteration.
+
+# Requirements
+- The variational family is [`FullRankGaussian`](@ref FullRankGaussian).
+- The target distribution has unconstrained support.
+- The target `LogDensityProblems.logdensity(prob, x)` has at least first-order differentiation capability.
+"""
+@kwdef struct FisherMinBatchMatch{Sub<:Union{Nothing,<:AbstractSubsampling}} <:
+              AbstractVariationalAlgorithm
+    n_samples::Int = 32
+    subsampling::Sub = nothing
+end
+
+struct BatchMatchState{Q,P,Sigma,Sub,UBuf,GradBuf}
+    q::Q
+    prob::P
+    sigma::Sigma
+    iteration::Int
+    sub_st::Sub
+    u_buf::UBuf
+    grad_buf::GradBuf
+end
+
+function init(
+    rng::Random.AbstractRNG,
+    alg::FisherMinBatchMatch,
+    q::MvLocationScale{<:LowerTriangular,<:Normal,L},
+    prob,
+) where {L}
+    (; n_samples, subsampling) = alg
+    capability = LogDensityProblems.capabilities(typeof(prob))
+    if capability < LogDensityProblems.LogDensityOrder{1}()
+        throw(
+            ArgumentError(
+                "`FisherMinBatchMatch` requires at least first-order differentiation capability. The capability of the supplied `LogDensityProblem` is $(capability).",
+            ),
+        )
+    end
+    sub_st = isnothing(subsampling) ? nothing : init(rng, subsampling)
+    params, _ = Optimisers.destructure(q)
+    n_dims = LogDensityProblems.dimension(prob)
+    u_buf = Matrix{eltype(params)}(undef, n_dims, n_samples)
+    grad_buf = Matrix{eltype(params)}(undef, n_dims, n_samples)
+    return BatchMatchState(q, prob, cov(q), 0, sub_st, u_buf, grad_buf)
+end
+
+output(::FisherMinBatchMatch, state) = state.q
+
+function rand_batch_match_samples_with_objective!(
+    rng::Random.AbstractRNG,
+    q::MvLocationScale,
+    n_samples::Int,
+    prob,
+    u_buf=Matrix{eltype(q)}(undef, LogDensityProblems.dimension(prob), n_samples),
+    grad_buf=Matrix{eltype(q)}(undef, LogDensityProblems.dimension(prob), n_samples),
+)
+    μ = q.location
+    C = q.scale
+    u = Random.randn!(rng, u_buf)
+    z = C*u .+ μ
+    logπ_sum = zero(eltype(μ))
+    for b in 1:n_samples
+        logπb, gb = LogDensityProblems.logdensity_and_gradient(prob, view(z, :, b))
+        grad_buf[:, b] = gb
+        logπ_sum += logπb
+    end
+    logπ_avg = logπ_sum/n_samples
+
+    # Estimate objective values
+    #
+    # F = E[| ∇log(q/π) (z) |_{CC'}^2] (definition)
+    #   = E[| C' (∇logq(z) - ∇logπ(z)) |^2] (Σ = CC')
+    #   = E[| C' ( -(CC')\((Cu + μ) - μ) - ∇logπ(z)) |^2] (z = Cu + μ)
+    #   = E[| C' ( -(CC')\(Cu) - ∇logπ(z)) |^2]
+    #   = E[| -u - C'∇logπ(z)) |^2]
+    fisher = sum(abs2, -u_buf - (C'*grad_buf))/n_samples
+
+    return u_buf, z, grad_buf, fisher, logπ_avg
+end
+
+function step(
+    rng::Random.AbstractRNG,
+    alg::FisherMinBatchMatch,
+    state,
+    callback,
+    objargs...;
+    kwargs...,
+)
+    (; n_samples, subsampling) = alg
+    (; q, prob, sigma, iteration, sub_st, u_buf, grad_buf) = state
+
+    d = LogDensityProblems.dimension(prob)
+    μ = q.location
+    C = q.scale
+    Σ = sigma
+    iteration += 1
+
+    # Maybe apply subsampling
+    prob_sub, sub_st′, sub_inf = if isnothing(subsampling)
+        prob, sub_st, NamedTuple()
+    else
+        batch, sub_st′, sub_inf = step(rng, subsampling, sub_st)
+        prob_sub = subsample(prob, batch)
+        prob_sub, sub_st′, sub_inf
+    end
+
+    u_buf, z, grad_buf, fisher, logπ_avg = rand_batch_match_samples_with_objective!(
+        rng, q, n_samples, prob_sub, u_buf, grad_buf
+    )
+
+    # BaM updates
+    zbar, C = mean_and_cov(z, 2)
+    gbar, Γ = mean_and_cov(grad_buf, 2)
+
+    μmz = μ - zbar
+    λ = convert(eltype(μ), d*n_samples / iteration)
+
+    U = Symmetric(λ*Γ + (λ/(1 + λ)*gbar)*gbar')
+    V = Symmetric(Σ + λ*C + (λ/(1 + λ)*μmz)*μmz')
+
+    Σ′ = Hermitian(2*V/(I + real(sqrt(I + 4*U*V))))
+    μ′ = 1/(1 + λ)*μ + λ/(1 + λ)*(Σ′*gbar + zbar)
+    q′ = MvLocationScale(μ′[:, 1], cholesky(Σ′).L, q.dist)
+
+    elbo = logπ_avg + entropy(q)
+    info = (iteration=iteration, covweighted_fisher=fisher, elbo=elbo)
+
+    state = BatchMatchState(q′, prob, Σ′, iteration, sub_st′, u_buf, grad_buf)
+
+    if !isnothing(callback)
+        info′ = callback(; rng, iteration, q, state)
+        info = !isnothing(info′) ? merge(info′, info) : info
+    end
+    state, false, info
+end
+
+"""
+    estimate_objective([rng,] alg, q, prob; n_samples)
+
+Estimate the covariance-weighted Fisher divergence of the variational approximation `q` against the target log-density `prob`.
+
+# Arguments
+- `rng::Random.AbstractRNG`: Random number generator.
+- `alg::FisherMinBatchMatch`: Variational inference algorithm.
+- `q::MvLocationScale{<:Any,<:Normal,<:Any}`: Gaussian variational approximation.
+- `prob`: The target log-joint likelihood implementing the `LogDensityProblem` interface.
+
+# Keyword Arguments
+- `n_samples::Int`: Number of Monte Carlo samples for estimating the objective. (default: Same as the the number of samples used for estimating the gradient during optimization.)
+
+# Returns
+- `obj_est`: Estimate of the objective value.
+"""
+function estimate_objective(
+    rng::Random.AbstractRNG,
+    alg::FisherMinBatchMatch,
+    q::MvLocationScale{S,<:Normal,L},
+    prob;
+    n_samples::Int=alg.n_samples,
+) where {S,L}
+    _, _, _, fisher, _ = rand_batch_match_samples_with_objective!(rng, q, n_samples, prob)
+    return fisher
+end