func (h *QuasiNewtonHelper) BackTrackingLineSearch(cost float64, pos *core.Vector, grad *core.Vector, dir *core.Vector, isInit bool) (nextCost float64, nextPos *core.Vector) {
dotGradDir := grad.Dot(dir)
if dotGradDir == 0 {
return cost, pos
}
if dotGradDir > 0 {
panic("BackTracking: to the opposite direction of grad")
}
alpha := 1.0
backoff := 0.5
if isInit {
normDir := math.Sqrt(dir.Dot(dir))
alpha = (1 / normDir)
backoff = 0.1
}
var c1 float64 = 1e-4
for cntItr := 0; cntItr <= MAX_BACKTRACKING_ITER; cntItr++ {
nextPos = h.minimizer.NextPoint(pos, dir, alpha)
nextCost = h.minimizer.Evaluate(nextPos)
if nextCost <= cost+c1*dotGradDir*alpha {
break
}
alpha *= backoff
}
return nextCost, nextPos
}
// Description: Update the dir from -grad to optimal direction
// Dir will be modified directly
func (h *QuasiNewtonHelper) ApplyQuasiInverseHession(dir *core.Vector) {
count := len(h.sList)
if count == 0 {
return
}
alphas := make([]float64, count, count)
for n := count - 1; n >= 0; n-- {
alphas[n] = -dir.Dot(h.sList[n]) / h.roList[n]
dir.ApplyElemWiseMultiplyAccumulation(h.yList[n], alphas[n])
}
lastY := h.yList[count-1]
yDotY := lastY.Dot(lastY)
scalar := h.roList[count-1] / yDotY
dir.ApplyScale(scalar)
for n := 0; n < count; n++ {
beta := dir.Dot(h.yList[n]) / h.roList[n]
dir.ApplyElemWiseMultiplyAccumulation(h.sList[n], -alphas[n]-beta)
}
return
}
/*
Given matrix m and vector v, compute inv(m)*v.
Based on Gibbs and MacKay 1997, and Mark N. Gibbs's PhD dissertation
Details:
A - positive seminidefinite matrix
u - a vector
theta - positive number
C = A + I*theta
Returns inv(C)*u - So you need the diagonal noise term for covariance matrix in a sense.
However, this algorithm is numerically stable, the noise term can be very small and the inversion can still be calculated...
*/
func (algo *GaussianProcess) ApproximateInversion(A *core.Matrix, u *core.Vector, theta float64, dim int64) *core.Vector {
max_itr := 500
tol := 0.01
C := core.NewMatrix()
for key, val := range A.Data {
C.Data[key] = val.Copy()
}
// Add theta to diagonal elements
for i := int64(0); i < dim; i++ {
_, ok := C.Data[i]
if !ok {
C.Data[i] = core.NewVector()
}
C.Data[i].Data[i] = C.Data[i].Data[i] + theta
}
var Q_l float64
var Q_u float64
var dQ float64
u_norm := u.Dot(u) / 2
// Lower bound
y_l := core.NewVector()
g_l := u.Copy()
h_l := u.Copy()
lambda_l := float64(0)
gamma_l := float64(0)
var tmp_f1 float64
var tmp_f2 float64
var tmp_v1 *core.Vector
tmp_f1 = g_l.Dot(g_l)
tmp_v1 = C.MultiplyVector(h_l)
// Upper bound
y_u := core.NewVector()
g_u := u.Copy()
h_u := u.Copy()
lambda_u := float64(0)
gamma_u := float64(0)
var tmp_f3 float64
var tmp_f4 float64
var tmp_v3 *core.Vector
var tmp_v4 *core.Vector
tmp_v3 = g_u.MultiplyMatrix(A)
tmp_v4 = C.MultiplyVector(h_u)
tmp_f3 = tmp_v1.Dot(g_u)
for i := 0; i < max_itr; i++ {
// Lower bound
lambda_l = tmp_f1 / h_l.Dot(tmp_v1)
y_l.AddVector(h_l, lambda_l) //y_l next
Q_l = y_l.Dot(u) - 0.5*(y_l.MultiplyMatrix(C)).Dot(y_l)
// Upper bound
lambda_u = tmp_f3 / tmp_v3.Dot(tmp_v4)
y_u.AddVector(h_u, lambda_u) //y_u next
Q_u = (y_u.MultiplyMatrix(A)).Dot(u) - 0.5*((y_u.MultiplyMatrix(C)).MultiplyMatrix(A)).Dot(y_u)
dQ = (u_norm-Q_u)/theta - Q_l
if dQ < tol {
break
}
// Lower bound var updates
g_l.AddVector(tmp_v1, -lambda_l) //g_l next
tmp_f2 = g_l.Dot(g_l)
gamma_l = tmp_f2 / tmp_f1
for key, val := range h_l.Data {
h_l.SetValue(key, val*gamma_l)
}
h_l.AddVector(g_l, 1) //h_l next
tmp_f1 = tmp_f2 //tmp_f1 next
tmp_v1 = C.MultiplyVector(h_l) //tmp_v1 next
// Upper bound var updates
g_u.AddVector(tmp_v4, -lambda_u) //g_u next
tmp_v3 = g_u.MultiplyMatrix(A) //tmp_v3 next
tmp_f4 = tmp_v3.Dot(g_u)
gamma_u = tmp_f4 / tmp_f3
for key, val := range h_u.Data {
h_u.SetValue(key, val*gamma_u)
}
h_u.AddVector(g_u, 1) //h_u next
tmp_v4 = C.MultiplyVector(h_u) //tmp_v4 next
tmp_f3 = tmp_f4 // tmp_f3 next
}
return y_l
}