Linearization of Exp(x)

Linearization of Exp(x)

def exp_lb(lb, ub):
    center = (lb + ub) / 2
    return (exp(center), exp(center) * (1 - center))

def exp_ub(lb, ub):
    k = (exp(ub) - exp(lb)) / (ub - lb)
    b = exp(lb) - k * lb
    return (k, b)

Selecting $\alpha$

def exp_alpha(lx, ux, ly, uy, ld, ud):
    return (exp(uy) - exp(ly)) / (exp(ux) - exp(lx))

Linearization of Exp(y) - 𝛼 Exp(x)

We only consider the area of lx ≤ x ≤ ux, ld ≤ y - 𝛽 x ≤ ud, we have the following formulas for the upper and lower bounds of $\mathrm{exp}(y) - 𝛼~ \mathrm{exp}(x)$:

# Upper bound for `exp(𝛽x + d) - 𝛼 exp(x)`
def exp_diff_ub(lx, ux, ld, ud, 𝛼, 𝛽):
    fudux = exp(𝛽 ux + ud) - 𝛼 exp(ux)
    fudlx = exp(𝛽 lx + ld) - 𝛼 exp(lx)
    k = (fudux - fudlx) / (ux - lx)
    b = fudlx - k * lx
    return k, 0, b

def point2line(x1, y1, x2, y2):
    k = (y2 - y1) / (x2 - x1)
    b = y1 - k * x1
    return (k, b)

# Lower bound for `relu(𝛽x + d) - 𝛼 relu(x)`
def relu_diff_lb(lx, ux, ld, ud, 𝛼, 𝛽):
    return (0,0,0) # TODO

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