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solve_cov_est.jl
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solve_cov_est.jl
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# ---------------------------------------------------------------------------------------------
# Solve covariance estimation problem
# min_{X \in \R^{d x r}} F(X) := E_{a,b} | <XX^T, aa^T> - b |
# using either 1) subgradient method or
# 2) mirror descent (with a polynomial mirror map)
#
# Inputs: X0 = initialization (d x r matrix)
# Xtrue = true matrix (used to plot dist. to solution)
# stochErr = variance of stochastic errors in residuals <XX^T,aa^T> - b
# maxIter = maximum no. iterations
# stepSizes = one-dim array of stepsizes (length maxIter)
# method = "subgradient" or "mirror"
# clipped = whether to use clipping (true/false)
# verbose = whether to print output (true/false)
#
# Outputs: (err_hist, fun_hist) = (array of normalized distances to solution,
# array of absolute errors in empirical function values)
# ---------------------------------------------------------------------------------------------
include("func.jl");
using Printf
function solve_cov_est(X0::Array{Float64,2},
Xtrue::Array{Float64,2},
stochErr::Float64,
maxIter::Int64,
stepSizes::Array{Float64,1};
method::String="subgradient",
clipped::Bool=false,
verbose::Bool=true
)
# Extract basic data
(d,r) = size(X0);
XTtrue = Xtrue';
sqnrmXtrue = sum(abs2, Xtrue);
# Initialize
X = copy(X0);
a = zeros(d);
XTa = zeros(r);
b = 0;
res = 0; # residual <XX^T, aa^T> - b
G = zeros(d,r); # subgradient at X
η = 0; # stepsize
if method=="mirror"
# Set coefficients for the Bregman divergence polynomials
# p(u) = a0 + a1 * u + a2 * u²
# Φ(x) = c0 ||x||₂² + c1 ||x||₂³ + c2 ||x||₂⁴ (mirror map)
(a0, a1, a2) = (1.0, 0.0, 1.0); # chosen by us
(c0, c1, c2) = (a0*7/2, a1*10/3, a2*13/4); # set according to theorem
# Initialize more quantities
Y = copy(X0); # need an extra copy of current iterate
V = zeros(d,r); # V = ∇Φ(X) - η * G
λ = 0; # λ = ||X||
end
# Draw stochastic a, b for all iterations at once
(A,B) = get_ab(XTtrue, stochErr, maxIter);
true_empirical_value = compute_empirical_function(Xtrue, A, B);
# Initialize vectors for tracking progress
err = NaN; # distance to solution
err_hist = fill(NaN, maxIter); # distances to solution
fun_val = compute_empirical_function(X, A, B); # empirical function value
fun_hist = fill(NaN, maxIter); # empirical function values
# Run the method (stochatic subgradient or stochastic mirror)
for k=1:maxIter
# Get stochastic a, b
a = A[:,k];
b = B[k];
XTa = X'*a;
# Update subgradient in place
subgrad!(G, XTa, a, b);
thresh = dot(G, X) - fun_val; # clipping threshold
# Update X in place
if method=="subgradient"
# Get step size
if clipped
η = min(stepSizes[k], fun_val/sum(abs2,G));
else
η = stepSizes[k];
end
BLAS.axpy!(-η,G,X);
elseif method=="mirror"
# Update V = ∇Φ(X) - η * G
η = stepSizes[k];
gradPhi = ( 2*c0 + 3*c1*norm(X,2) + 4*c2*sum(abs2, X) ) * X;
V = gradPhi - η*G;
nrmV = norm(V,2);
# Note that direction of X is identical to direction of V.
# Root finding problem to find λ = ||X|| is
# (2*c0) * λ + (3*c1) * λ^2 + (4*c2) * λ^3 = nrmV
λ = get_root([-nrmV, 2*c0, 3*c1, 4*c2]);
# Record intermediate matrix Y
lmul!(0.0, Y) # reset to zero
BLAS.axpy!(λ/nrmV, V, Y)
if (! clipped) | (dot(G,Y) >= thresh)
X = Y;
else # Need to find a better Y.
# First try: ∇ϕ(Y) = ∇ϕ(X_k), i.e. Lagrange multiplier is zero
nrmGrad= norm(gradPhi,2);
λ = get_root([-nrmGrad, 2*c0, 3*c1, 4*c2]);
lmul!(0.0, Y) # reset to zero
BLAS.axpy!(λ/nrmGrad, gradPhi, Y)
if (dot(G,Y) <= thresh)
X = Y;
else # Y still doesn't satisfy clipped constraint
# Second try: use gradPhi = W + α*G, W ⟂ G, and ∇Φ(Y) = W + (α-μ)*G
sqnormG = sum(abs2,G);
α = dot(G, gradPhi) / sqnormG;
W = gradPhi - α*G;
# Inner product with G + norm squared gives coefficients
d0, d1, d2 = 2*c0, 3*c1, 4*c2;
d3 = thresh^2/sqnormG;
# Now solve for λ=||Y|| using (a + bλ + cλ^2)^2 (λ^2 - d) - ||W||^2 = 0
λ = get_root([ -d0^2*d3 - sum(abs2,W),
-(2*d0*d1)*d3,
d0^2 - (2*d0*d2-d1^2)*d3,
(2*d0*d1) - (2*d1*d2)*d3,
(2*d0*d2 + d1^2) - d2^2*d3,
2*d1*d2,
d2^2
]);
# Get μ from λ
μ = α - (d0+d1*λ+d2*λ^2) * thresh / sqnormG ;
if μ > 0
V = W + (α-μ)*G;
Y = (λ/nrmV) * V;
X = Y;
else
# Can't find a good Y, so don't take a step at all. Keep X as is.
end
end
end
end
# record error status and print to console
err = sqnrmXtrue + sum(abs2, X) - 2 * sum(svdvals(XTtrue * X));
normalized_err = err / sqnrmXtrue;
err_hist[k] = normalized_err;
fun_val = compute_empirical_function(X, A, B);
abs_value_error = fun_val - true_empirical_value ;
fun_hist[k] = abs_value_error;
if verbose
@printf("iter %3d: function err = %1.2e, dist err = %1.2e, stepsize = %1.2e\n",
k, value_error, normalized_err, η);
end
end
return (err_hist, fun_hist)
end