.. _sec_adam:
Adam
====
In the discussions leading up to this section we encountered a number of
techniques for efficient optimization. Let’s recap them in detail here:
- We saw that :numref:`sec_sgd` is more effective than Gradient
Descent when solving optimization problems, e.g., due to its inherent
resilience to redundant data.
- We saw that :numref:`sec_minibatch_sgd` affords significant
additional efficiency arising from vectorization, using larger sets
of observations in one minibatch. This is the key to efficient
multi-machine, multi-GPU and overall parallel processing.
- :numref:`sec_momentum` added a mechanism for aggregating a history
of past gradients to accelerate convergence.
- :numref:`sec_adagrad` used per-coordinate scaling to allow for a
computationally efficient preconditioner.
- :numref:`sec_rmsprop` decoupled per-coordinate scaling from a
learning rate adjustment.
Adam :cite:`Kingma.Ba.2014` combines all these techniques into one
efficient learning algorithm. As expected, this is an algorithm that has
become rather popular as one of the more robust and effective
optimization algorithms to use in deep learning. It is not without
issues, though. In particular, :cite:`Reddi.Kale.Kumar.2019` show that
there are situations where Adam can diverge due to poor variance
control. In a follow-up work :cite:`Zaheer.Reddi.Sachan.ea.2018`
proposed a hotfix to Adam, called Yogi which addresses these issues.
More on this later. For now let’s review the Adam algorithm.
The Algorithm
-------------
One of the key components of Adam is that it uses exponential weighted
moving averages (also known as leaky averaging) to obtain an estimate of
both the momentum and also the second moment of the gradient. That is,
it uses the state variables
.. math::
\begin{aligned}
\mathbf{v}_t & \leftarrow \beta_1 \mathbf{v}_{t-1} + (1 - \beta_1) \mathbf{g}_t, \\
\mathbf{s}_t & \leftarrow \beta_2 \mathbf{s}_{t-1} + (1 - \beta_2) \mathbf{g}_t^2.
\end{aligned}
Here :math:`\beta_1` and :math:`\beta_2` are nonnegative weighting
parameters. Common choices for them are :math:`\beta_1 = 0.9` and
:math:`\beta_2 = 0.999`. That is, the variance estimate moves *much more
slowly* than the momentum term. Note that if we initialize
:math:`\mathbf{v}_0 = \mathbf{s}_0 = 0` we have a significant amount of
bias initially towards smaller values. This can be addressed by using
the fact that
:math:`\sum_{i=0}^t \beta^i = \frac{1 - \beta^t}{1 - \beta}` to
re-normalize terms. Correspondingly the normalized state variables are
given by
.. math:: \hat{\mathbf{v}}_t = \frac{\mathbf{v}_t}{1 - \beta_1^t} \text{ and } \hat{\mathbf{s}}_t = \frac{\mathbf{s}_t}{1 - \beta_2^t}.
Armed with the proper estimates we can now write out the update
equations. First, we rescale the gradient in a manner very much akin to
that of RMSProp to obtain
.. math:: \mathbf{g}_t' = \frac{\eta \hat{\mathbf{v}}_t}{\sqrt{\hat{\mathbf{s}}_t} + \epsilon}.
Unlike RMSProp our update uses the momentum :math:`\hat{\mathbf{v}}_t`
rather than the gradient itself. Moreover, there’s a slight cosmetic
difference as the rescaling happens using
:math:`\frac{1}{\sqrt{\hat{\mathbf{s}}_t} + \epsilon}` instead of
:math:`\frac{1}{\sqrt{\hat{\mathbf{s}}_t + \epsilon}}`. The former works
arguably slightly better in practice, hence the deviation from RMSProp.
Typically we pick :math:`\epsilon = 10^{-6}` for a good trade-off
between numerical stability and fidelity.
Now we have all the pieces in place to compute updates. This is slightly
anticlimactic and we have a simple update of the form
.. math:: \mathbf{x}_t \leftarrow \mathbf{x}_{t-1} - \mathbf{g}_t'.
Reviewing the design of Adam its inspiration is clear. Momentum and
scale are clearly visible in the state variables. Their rather peculiar
definition forces us to debias terms (this could be fixed by a slightly
different initialization and update condition). Second, the combination
of both terms is pretty straightforward, given RMSProp. Last, the
explicit learning rate :math:`\eta` allows us to control the step length
to address issues of convergence.
Implementation
--------------
Implementing Adam from scratch isn’t very daunting. For convenience we
store the timestep counter :math:`t` in the ``hyperparams`` dictionary.
Beyond that all is straightforward.
.. code:: python
%matplotlib inline
import d2l
from mxnet import np, npx
npx.set_np()
def init_adam_states(feature_dim):
v_w, v_b = np.zeros((feature_dim, 1)), np.zeros(1)
s_w, s_b = np.zeros((feature_dim, 1)), np.zeros(1)
return ((v_w, s_w), (v_b, s_b))
def adam(params, states, hyperparams):
beta1, beta2, eps = 0.9, 0.999, 1e-6
for p, (v, s) in zip(params, states):
v[:] = beta1 * v + (1 - beta1) * p.grad
s[:] = beta2 * s + (1 - beta2) * np.square(p.grad)
v_bias_corr = v / (1 - beta1 ** hyperparams['t'])
s_bias_corr = s / (1 - beta2 ** hyperparams['t'])
p[:] -= hyperparams['lr'] * v_bias_corr / (np.sqrt(s_bias_corr) + eps)
hyperparams['t'] += 1
We are ready to use Adam to train the model. We use a learning rate of
:math:`\eta = 0.01`.
.. code:: python
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(adam, init_adam_states(feature_dim),
{'lr': 0.01, 't': 1}, data_iter, feature_dim);
.. parsed-literal::
:class: output
loss: 0.243, 0.096 sec/epoch
.. figure:: output_adam_944a18_3_1.svg
A more concise implementation is straightforward since ``adam`` is one
of the algorithms provided as part of the Gluon ``trainer`` optimization
library. Hence we only need to pass configuration parameters for an
implementation in Gluon.
.. code:: python
d2l.train_gluon_ch11('adam', {'learning_rate': 0.01}, data_iter)
.. parsed-literal::
:class: output
loss: 0.243, 0.037 sec/epoch
.. figure:: output_adam_944a18_5_1.svg
Yogi
----
One of the problems of Adam is that it can fail to converge even in
convex settings when the second moment estimate in :math:`\mathbf{s}_t`
blows up. As a fix :cite:`Zaheer.Reddi.Sachan.ea.2018` proposed a
refined update (and initialization) for :math:`\mathbf{s}_t`. To
understand what’s going on, let’s rewrite the Adam update as follows:
.. math:: \mathbf{s}_t \leftarrow \mathbf{s}_{t-1} + (1 - \beta_2) \left(\mathbf{g}_t^2 - \mathbf{s}_{t-1}\right).
Whenever :math:`\mathbf{g}_t^2` has high variance or updates are sparse,
:math:`\mathbf{s}_t` might forget past values too quickly. A possible
fix for this is to replace :math:`\mathbf{g}_t^2 - \mathbf{s}_{t-1}` by
:math:`\mathbf{g}_t^2 \odot \mathop{\mathrm{sgn}}(\mathbf{g}_t^2 - \mathbf{s}_{t-1})`.
Now the magnitude of the update no longer depends on the amount of
deviation. This yields the Yogi updates
.. math:: \mathbf{s}_t \leftarrow \mathbf{s}_{t-1} + (1 - \beta_2) \mathbf{g}_t^2 \odot \mathop{\mathrm{sgn}}(\mathbf{g}_t^2 - \mathbf{s}_{t-1}).
The authors furthermore advise to initialize the momentum on a larger
initial batch rather than just initial pointwise estimate. We omit the
details since they are not material to the discussion and since even
without this convergence remains pretty good.
.. code:: python
def yogi(params, states, hyperparams):
beta1, beta2, eps = 0.9, 0.999, 1e-3
for p, (v, s) in zip(params, states):
v[:] = beta1 * v + (1 - beta1) * p.grad
s[:] = s + (1 - beta2) * np.sign(
np.square(p.grad) - s) * np.square(p.grad)
v_bias_corr = v / (1 - beta1 ** hyperparams['t'])
s_bias_corr = s / (1 - beta2 ** hyperparams['t'])
p[:] -= hyperparams['lr'] * v_bias_corr / (np.sqrt(s_bias_corr) + eps)
hyperparams['t'] += 1
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(yogi, init_adam_states(feature_dim),
{'lr': 0.01, 't': 1}, data_iter, feature_dim);
.. parsed-literal::
:class: output
loss: 0.243, 0.086 sec/epoch
.. figure:: output_adam_944a18_7_1.svg
Summary
-------
- Adam combines features of many optimization algorithms into a fairly
robust update rule.
- Created on the basis of RMSProp, Adam also uses EWMA on the minibatch
stochastic gradient
- Adam uses bias correction to adjust for a slow startup when
estimating momentum and a second moment.
- For gradients with significant variance we may encounter issues with
convergence. They can be amended by using larger minibatches or by
switching to an improved estimate for :math:`\mathbf{s}_t`. Yogi
offers such an alternative.
Exercises
---------
1. Adjust the learning rate and observe and analyze the experimental
results.
2. Can you rewrite momentum and second moment updates such that it
doesn’t require bias correction?
3. Why do you need to reduce the learning rate :math:`\eta` as we
converge?
4. Try to construct a case for which Adam diverges and Yogi converges?
`Discussions `__
-------------------------------------------------
|image0|
.. |image0| image:: ../img/qr_adam.svg