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[docs]class AdamOptimizer(GradientDescentOptimizer): r"""Gradient-descent optimizer with adaptive learning rate, first and second moment. Adaptive Moment Estimation uses a step-dependent learning rate, a first moment :math:a and a second moment :math:b, reminiscent of the momentum and velocity of a particle: .. math:: x^{(t+1)} = x^{(t)} - \eta^{(t+1)} \frac{a^{(t+1)}}{\sqrt{b^{(t+1)}} + \epsilon }, where the update rules for the three values are given by .. math:: a^{(t+1)} &= \frac{\beta_1 a^{(t)} + (1-\beta_1)\nabla f(x^{(t)})}{(1- \beta_1)},\\ b^{(t+1)} &= \frac{\beta_2 b^{(t)} + (1-\beta_2) ( \nabla f(x^{(t)}))^{\odot 2} }{(1- \beta_2)},\\ \eta^{(t+1)} &= \eta^{(t)} \frac{\sqrt{(1-\beta_2)}}{(1-\beta_1)}. Above, :math:( \nabla f(x^{(t-1)}))^{\odot 2} denotes the element-wise square operation, which means that each element in the gradient is multiplied by itself. The hyperparameters :math:\beta_1 and :math:\beta_2 can also be step-dependent. Initially, the first and second moment are zero. The shift :math:\epsilon avoids division by zero. For more details, see :cite:kingma2014adam. Args: stepsize (float): the user-defined hyperparameter :math:\eta beta1 (float): hyperparameter governing the update of the first and second moment beta2 (float): hyperparameter governing the update of the first and second moment eps (float): offset :math:\epsilon added for numerical stability """ def __init__(self, stepsize=0.01, beta1=0.9, beta2=0.99, eps=1e-8): super().__init__(stepsize) self.beta1 = beta1 self.beta2 = beta2 self.eps = eps self.fm = None self.sm = None self.t = 0
[docs] def apply_grad(self, grad, x): r"""Update the variables x to take a single optimization step. Flattens and unflattens the inputs to maintain nested iterables as the parameters of the optimization. Args: grad (array): The gradient of the objective function at point :math:x^{(t)}: :math:\nabla f(x^{(t)}) x (array): the current value of the variables :math:x^{(t)} Returns: array: the new values :math:x^{(t+1)} """ self.t += 1 grad_flat = list(_flatten(grad)) x_flat = _flatten(x) # Update first moment if self.fm is None: self.fm = grad_flat else: self.fm = [self.beta1 * f + (1 - self.beta1) * g for f, g in zip(self.fm, grad_flat)] # Update second moment if self.sm is None: self.sm = [g * g for g in grad_flat] else: self.sm = [self.beta2 * f + (1 - self.beta2) * g * g for f, g in zip(self.sm, grad_flat)] # Update step size (instead of correcting for bias) new_stepsize = self._stepsize*np.sqrt(1-self.beta2**self.t)/(1-self.beta1**self.t) x_new_flat = [e - new_stepsize * f / (np.sqrt(s)+self.eps) for f, s, e in zip(self.fm, self.sm, x_flat)] return unflatten(x_new_flat, x)