How to compute the hessian in derivatives in theano?

This recipe helps you to compute the hessian in derivatives in theano.

Recipe Objective - How to compute the hessian in derivatives in theano?

It is the matrix in Theano that contains the second-order partial derivative of a scalar function with vector input and scalar output. Theano has a macro called theano.gradient.hessian() that does everything needed to compute the Hessian. The instructions in the following text will show you how to accomplish it manually. It resembles the Jacobian period. The only difference is that we now compute the Jacobian of tt.grad(cost,x), where cost is a scalar, rather than the Jacobian of some expression y.

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Example -

# Import libraries
import theano
from theano import tensor

# Creating a vector
x = tensor.dvector('x')

# Creating 'y' expression
y = (2 * x ** 3)

# Calculating cost
cost = y.sum()

# Computing derivative
derivative = tensor.grad(cost, x)

output, updates = theano.scan(lambda i, derivative,x : tensor.grad(derivative[i], x), sequences=tensor.arange(derivative.shape[0]), non_sequences=[derivative, x])

# Creating function
fun = theano.function([x], output, updates=updates)

# Calling function
fun([3,3])

Output -
array([[36.,  0.],
       [ 0., 36.]])

In this way, we can compute the hessian in derivatives in theano.

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