autograd-minimize is a wrapper around the minimize routine of scipy which uses the autograd capacities of tensorflow or pytorch to compute automatically the gradients, hessian vector products and hessians.
It also accepts functions of more than one variables as input.
pip install autograd-minimize
It uses tensorflow as the default backend:
import tensorflow as tf
from autograd_minimize import minimize
def rosen_tf(x):
return tf.reduce_sum(100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0)
res = minimize(rosen_tf, np.array([0.,0.]))
print(res.x)
>>> array([0.99999912, 0.99999824])
But you can also use pytorch:
import torch
from autograd_minimize import minimize
def rosen_torch(x):
return (100.0*(x[1:] - x[:-1]**2.0)**2.0 + (1 - x[:-1])**2.0).sum()
res = minimize(rosen_torch, np.array([0.,0.]), backend='torch')
print(res.x)
>>> array([0.99999912, 0.99999824])
You can also try other optimization methods such as Newton-CG which uses automatic computation of the hessian vector product (hvp). Let's as well increase the precision of hvp and gradient computation to float64 and the tolerance to 1e-8:
import numpy as np
res = minimize(rosen_tf, np.array([0.,0.]), method='Newton-CG', precision='float64', tol=1e-8)
print(np.mean(res.x-1))
>>> -2.6886433635020524e-09
Or we can use the trust-exact method (with automatic computation of the hessian):
import numpy as np
res = minimize(rosen_tf, np.array([0.,0.]), method='trust-exact', precision='float64', tol=1e-8)
print(np.mean(res.x-1))
>>> -1.6946999359390702e-12
Let's now try to do matrix factorization. In this case it is much easier to deal with a function with two inputs, where the input should be a dict or a list with a similar signature as the function:
shape = (10, 15)
inner_shape=3
from numpy.random import random
U = random((shape[0], inner_shape))
V = random((inner_shape, shape[1]))
prod = U@V
def mat_fac(U=None, V=None):
return tf.reduce_mean((U@V-tf.constant(prod, dtype=tf.float32))**2)
x0 = {'U': -random((shape[0], inner_shape)), 'V': random((inner_shape, shape[1]))}
tic = time()
res = minimize(mat_fac, x0)
print(res.fun)
>>> 6.136937713563384e-08
You can also set bounds (only for the methods: L-BFGS-B, TNC, SLSQP, Powell, and trust-constr):
If bounds is a tuple, the same bound is applied to all variables:
res = minimize(mat_fac, x0, bounds=(None, 0))
print(res.x['U'].mean())
>>> -0.6171053993128699
You can apply bounds only to a subset of variables by using a list or a dict (but it should be the same as the format of input x0):
res = minimize(mat_fac, x0, bounds={'U': (None, 0), 'V': (-1, None)})
print(res.x['U'].mean(), res.x['V'].mean())
>>> -0.8173837691822693 0.11222992115637932
Inside each variable of the dict/list, you can pass a numpy array or a list of bounds which the same shape or len as the variable to specify in more details the bounds:
res = minimize(mat_fac, x0, bounds={'U': (0, None), 'V': [(0, None)]*inner_shape*shape[1]})
You can also optimize keras models by transforming them into a function of their parameters, using
autograd_minimize.tf_wrapper.tf_function_factory:
import numpy as np
from tensorflow import keras
from tensorflow.keras import layers
from autograd_minimize.tf_wrapper import tf_function_factory
from autograd_minimize import minimize
import tensorflow as tf
#### Prepares data
X = np.random.random((200, 2))
y = X[:,:1]*2+X[:,1:]*0.4-1
#### Creates model
model = keras.Sequential([keras.Input(shape=2),
layers.Dense(1)])
# Transforms model into a function of its parameter
func, params = tf_function_factory(model, tf.keras.losses.MSE, X, y)
# Minimization
res = minimize(func, params, method='L-BFGS-B')
Note that you can do the same on torch models by replacing autograd_minimize.tf_wrapper.tf_function_factory by autograd_minimize.torch_wrapper.torch_function_factory.
And you can set constraints (with automatic computation of the jacobian). An example is given in examples/multiknapsack, where the (relaxed) multiknapsack problem is solved.
- Adds comparison with LBFGS from pytorch or keras