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．