Why does the learning rate increase in Adam?

Question

I have been using the following piece of code to print the lr_t learning_rate in Adam() optimizer for my trainable_model.

if(np.random.uniform()*100 < 3 and self.training):
    model = self.trainable_model
    _lr    = tf.to_float(model.optimizer.lr, name='ToFloat')
    _decay = tf.to_float(model.optimizer.decay, name='ToFloat')
    _beta1 = tf.to_float(model.optimizer.beta_1, name='ToFloat')
    _beta2 = tf.to_float(model.optimizer.beta_2, name='ToFloat')
    _iterations = tf.to_float(model.optimizer.iterations, name='ToFloat')
    t = K.cast(_iterations, K.floatx()) + 1
    _lr_t = lr * (K.sqrt(1. - K.pow(_beta2, t)) /  (1. - K.pow(_beta1, t)))
    print(" - LR_T: "+str(K.eval(_lr_t)))

What I don't understand is that this learning rate increases. (with decay at default value of 0).

If we look at the learning_rate equation in Adam, we find this:

 lr_t = lr * (K.sqrt(1. - K.pow(self.beta_2, t)) /
                 (1. - K.pow(self.beta_1, t)))

which corresponds to the equation (with default values for parameters):

= 0.001*sqrt(1-0.999^x)/(1-0.99^x)

If we print this equation we obtain :

which clearly shows that the learning_rate is increasing exponentially over time (since t starts at 1)

can someone explain why this is the case ? I have read everywhere that we should use a learning_rate that decays over time, not increase.

Does it means that my neural network makes bigger updates over time as Adam's learning_rate increases ?

Answer 1

Looking at the source code of the Adam optimizer in Keras, it looks like the actual "decay" is performed at: this line. The code you reported is executed only after and is not the decay itself.
If the question is "why it is like that" I would suggest you to read some theory about Adam like the original paper.

EDIT
It should be clear that the update equation of the Adam optimizer does NOT include a decay by itself. The decay should be applied separately.