ENH: Add ADAM solver, see #984

Author: adler-j

odl/solvers/smooth/gradient.py

@@ -27,7 +27,7 @@
 from odl.solvers.util import ConstantLineSearch
 
 
-__all__ = ('steepest_descent',)
+__all__ = ('steepest_descent', 'adam')
 
 
 # TODO: update all docs
@@ -110,6 +110,77 @@ def steepest_descent(f, x, line_search=1.0, maxiter=1000, tol=1e-16,
             callback(x)
 
 
+def adam(f, x, learning_rate=1e-3, beta1=0.9, beta2=0.999, eps=1e-8,
+         maxiter=1000, tol=1e-16, callback=None):
+    """ADAM method to minimize an objective function.
+
+    General implementation of ADAM for solving
+
+    .. math::
+        \min f(x)
+
+    The algorithm is intended for unconstrained problems.
+
+    The algorithm is described in
+    `Adam: A Method for Stochastic Optimization
+    <https://arxiv.org/abs/1412.6980>`_. All parameter names are taken from
+    that article.
+
+    Parameters
+    ----------
+    f : `Functional`
+        Goal functional. Needs to have ``f.gradient``.
+    x : ``f.domain`` element
+        Starting point of the iteration
+    learning_rate : float, optional
+        Step length of the method.
+    beta1 : float, optional
+        Update rate for first order moment estimate.
+    beta2 : float, optional
+        Update rate for second order moment estimate.
+    eps : float, optional
+        A small constant for numerical stability.
+    maxiter : int, optional
+        Maximum number of iterations.
+    tol : float, optional
+        Tolerance that should be used for terminating the iteration.
+    callback : callable, optional
+        Object executing code per iteration, e.g. plotting each iterate
+
+    See Also
+    --------
+    odl.solvers.smooth.gradient.steepest_descent : simple steepest descent
+    odl.solvers.iterative.iterative.landweber :
+        Optimized solver for the case ``f(x) = ||Ax - b||_2^2``
+    odl.solvers.iterative.iterative.conjugate_gradient :
+        Optimized solver for the case ``f(x) = x^T Ax - 2 x^T b``
+    """
+    grad = f.gradient
+    if x not in grad.domain:
+        raise TypeError('`x` {!r} is not in the domain of `grad` {!r}'
+                        ''.format(x, grad.domain))
+
+    m = grad.domain.zero()
+    v = grad.domain.zero()
+
+    grad_x = grad.range.element()
+    for _ in range(maxiter):
+        grad(x, out=grad_x)
+
+        if grad_x.norm() < tol:
+            return
+
+        m.lincomb(beta1, m, 1 - beta1, grad_x)
+        v.lincomb(beta2, v, 1 - beta2, grad_x ** 2)
+
+        step = learning_rate * np.sqrt(1 - beta2) / (1 - beta1)
+
+        x.lincomb(1, x, -step, m / np.sqrt(v + eps))
+
+        if callback is not None:
+            callback(x)
+
+
 if __name__ == '__main__':
     # pylint: disable=wrong-import-position
     from odl.util.testutils import run_doctests

odl/test/solvers/iterative/iterative_test.py

@@ -31,6 +31,7 @@
 # Find the valid projectors
 @pytest.fixture(scope="module",
                 params=['steepest_descent',
+                        'adam',
                         'landweber',
                         'conjugate_gradient',
                         'conjugate_gradient_normal',
@@ -47,6 +48,12 @@ def solver(op, x, rhs):
             func = odl.solvers.L2NormSquared(op.domain) * (op - rhs)
 
             odl.solvers.steepest_descent(func, x, line_search=0.5 / norm2)
+    elif solver_name == 'adam':
+        def solver(op, x, rhs):
+            norm2 = op.adjoint(op(x)).norm() / x.norm()
+            func = odl.solvers.L2NormSquared(op.domain) * (op - rhs)
+
+            odl.solvers.adam(func, x, learning_rate=4.0 / norm2, maxiter=150)
     elif solver_name == 'landweber':
         def solver(op, x, rhs):
             norm2 = op.adjoint(op(x)).norm() / x.norm()

odl/test/solvers/smooth/smooth_test.py

@@ -136,6 +136,15 @@ def test_steepest_descent(functional):
     assert functional(x) < 1e-3
 
 
+def test_adam(functional):
+    """Test the ``adam`` solver."""
+
+    x = functional.domain.one()
+    odl.solvers.adam(functional, x, tol=1e-2, learning_rate=0.5)
+
+    assert functional(x) < 1e-3
+
+
 def test_conjguate_gradient_nonlinear(functional, nonlinear_cg_beta):
     """Test the ``conjugate_gradient_nonlinear`` solver."""
     line_search = odl.solvers.BacktrackingLineSearch(functional)