実装はこちらのGitHubリポジトリのdnn/optimizers.py中にあります．
はじめに
MNISTをベンチマークとして簡単なニューラルネットワークで以下の学習係数最適化を使ってみました．それぞれの評価用データセットに対するaccuracyを比較してみます．
- SGD
- Momentum SGD
- AdaGrad
- RMSprop
- AdaDelta
- Adam
実装
実装はTheanoを使いました．
今回はソースコードを私のGitHubリポジトリにアップロードしました．
学習係数最適化に関してはここら辺参考にしました．
https://gist.github.com/SnippyHolloW/67effa81dd1cd5a488b4
https://gist.github.com/skaae/ae7225263ca8806868cb
http://chainer.readthedocs.org/en/stable/reference/optimizers.html?highlight=optimizers
http://qiita.com/skitaoka/items/e6afbe238cd69c899b2a
下のコードではparams(or self.params)がネットワーク全体の重み，バイアスを保持しています．確率的勾配降下法で学習するので，誤差関数値lossをparamsで微分した値が必要ですが，gparams(or self.gparams)がそれに当たります．(T.grad(loss, param)で微分しています．) 微分して傾きを求めるところまでをOptimizerクラスとして，それを継承してSGD，Momentum SGDなどを実装していきます．
optimizers.py
class Optimizer(object):
def __init__(self, params=None):
if params is None:
return NotImplementedError()
self.params = params
def updates(self, loss=None):
if loss is None:
return NotImplementedError()
self.updates = OrderedDict()
self.gparams = [T.grad(loss, param) for param in self.params]
ちなみにここでのself.updatesは重みなどを更新するときに使います．
SGD
optimizers.py
class SGD(Optimizer):
def __init__(self, learning_rate=0.01, params=None):
super(SGD, self).__init__(params=params)
self.learning_rate = 0.01
def updates(self, loss=None):
super(SGD, self).updates(loss=loss)
for param, gparam in zip(self.params, self.gparams):
self.updates[param] = param - self.learning_rate * gparam
return self.updates
Momentum SGD
optimizers.py
class MomentumSGD(Optimizer):
def __init__(self, learning_rate=0.01, momentum=0.9, params=None):
super(MomentumSGD, self).__init__(params=params)
self.learning_rate = learning_rate
self.momentum = momentum
self.vs = [build_shared_zeros(t.shape.eval(), 'v') for t in self.params]
def updates(self, loss=None):
super(MomentumSGD, self).updates(loss=loss)
for v, param, gparam in zip(self.vs, self.params, self.gparams):
_v = v * self.momentum
_v = _v - self.learning_rate * gparam
self.updates[param] = param + _v
self.updates[v] = _v
return self.updates
AdaGrad
optimizers.py
class AdaGrad(Optimizer):
def __init__(self, learning_rate=0.01, eps=1e-6, params=None):
super(AdaGrad, self).__init__(params=params)
self.learning_rate = learning_rate
self.eps = eps
self.accugrads = [build_shared_zeros(t.shape.eval(),'accugrad') for t in self.params]
def updates(self, loss=None):
super(AdaGrad, self).updates(loss=loss)
for accugrad, param, gparam\
in zip(self.accugrads, self.params, self.gparams):
agrad = accugrad + gparam * gparam
dx = - (self.learning_rate / T.sqrt(agrad + self.eps)) * gparam
self.updates[param] = param + dx
self.updates[accugrad] = agrad
return self.updates
RMSprop
optimizers.py
class RMSprop(Optimizer):
def __init__(self, learning_rate=0.001, alpha=0.99, eps=1e-8, params=None):
super(RMSprop, self).__init__(params=params)
self.learning_rate = learning_rate
self.alpha = alpha
self.eps = eps
self.mss = [build_shared_zeros(t.shape.eval(),'ms') for t in self.params]
def updates(self, loss=None):
super(RMSprop, self).updates(loss=loss)
for ms, param, gparam in zip(self.mss, self.params, self.gparams):
_ms = ms*self.alpha
_ms += (1 - self.alpha) * gparam * gparam
self.updates[ms] = _ms
self.updates[param] = param - self.learning_rate * gparam / T.sqrt(_ms + self.eps)
return self.updates
AdaDelta
optimizers.py
class AdaDelta(Optimizer):
def __init__(self, rho=0.95, eps=1e-6, params=None):
super(AdaDelta, self).__init__(params=params)
self.rho = rho
self.eps = eps
self.accugrads = [build_shared_zeros(t.shape.eval(),'accugrad') for t in self.params]
self.accudeltas = [build_shared_zeros(t.shape.eval(),'accudelta') for t in self.params]
def updates(self, loss=None):
super(AdaDelta, self).updates(loss=loss)
for accugrad, accudelta, param, gparam\
in zip(self.accugrads, self.accudeltas, self.params, self.gparams):
agrad = self.rho * accugrad + (1 - self.rho) * gparam * gparam
dx = - T.sqrt((accudelta + self.eps)/(agrad + self.eps)) * gparam
self.updates[accudelta] = (self.rho*accudelta + (1 - self.rho) * dx * dx)
self.updates[param] = param + dx
self.updates[accugrad] = agrad
return self.updates
Adam
optimizers.py
class Adam(Optimizer):
def __init__(self, alpha=0.001, beta1=0.9, beta2=0.999, eps=1e-8, gamma=1-1e-8, params=None):
super(Adam, self).__init__(params=params)
self.alpha = alpha
self.b1 = beta1
self.b2 = beta2
self.gamma = gamma
self.t = theano.shared(np.float32(1))
self.eps = eps
self.ms = [build_shared_zeros(t.shape.eval(), 'm') for t in self.params]
self.vs = [build_shared_zeros(t.shape.eval(), 'v') for t in self.params]
def updates(self, loss=None):
super(Adam, self).updates(loss=loss)
self.b1_t = self.b1 * self.gamma ** (self.t - 1)
for m, v, param, gparam \
in zip(self.ms, self.vs, self.params, self.gparams):
_m = self.b1_t * m + (1 - self.b1_t) * gparam
_v = self.b2 * v + (1 - self.b2) * gparam ** 2
m_hat = _m / (1 - self.b1 ** self.t)
v_hat = _v / (1 - self.b2 ** self.t)
self.updates[param] = param - self.alpha*m_hat / (T.sqrt(v_hat) + self.eps)
self.updates[m] = _m
self.updates[v] = _v
self.updates[self.t] = self.t + 1.0
return self.updates
実験
MNISTを使って20epochずつ30シード平均とりました．学習係数やニューラルネットの細かい設定に関しては私のGitHubリポジトリを参照ください．
結果
うっ，上の方がごちゃごちゃしててわからない，拡大しよう．
SGDさんが消えてしまった．
おわりに
誤差関数値もとっておけばよかった．．．．
今回のGitHubリポジトリに畳み込みニューラルネットやらStacked Denoising Autoencodersやら今後追加していく予定です．
変なところとかありましたらご指摘いただけるとありがたいです．