An overview of gradient descent optimization algorithms

Reference from An overview of gradient descent optimization algorithms

Batch gradient descent

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for i in range(nb_epochs):
params_grad = evaluate_gradient(loss_function, data, params)
params = params - learning_rate * params_grad
  • Batch gradient descent is guaranteed to converge to the global minimum for convex error surfaces and to a local minimum for non-convex surfaces.

Stochastic gradient descent

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for i in range(nb_epochs):
np.random.shuffle(data)
for example in data:
params_grad = evaluate_gradient(loss_function, example, params)
params = params - learning_rate * params_grad
  • SGD performs frequent updates with a high variance that cause the objective function to fluctuate heavily as in Image 1.
  • image1.png
  • SGD shows the same convergence behaviour as batch gradient descent, almost certainly converging to a local or the global minimum for non-convex and convex optimization respectively.

Mini-batch gradient descent

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for i in range(nb_epochs):
np.random.shuffle(data)
for batch in get_batches(data, batch_size=50):
params_grad = evaluate_gradient(loss_function, batch, params)
params = params - learning_rate * params_grad
  • Common mini-batch sizes range between 50 and 256, but can vary for different applications
  • Mini-batch gradient descent is typically the algorithm of choice when training a neural network and the term SGD usually is employed also when mini-batches are used

Challenges

  • Choosing a proper learning rate can be difficult
  • the same learning rate applies to all parameter updates
  • Learning rate schedules
    • ry to adjust the learning rate during training by e.g. annealing
    • reducing the learning rate according to a pre-defined schedule or when the change in objective between epochs falls below a threshold
  • Another key challenge of minimizing highly non-convex error functions common for neural networks is avoiding getting trapped in their numerous suboptimal local minima

Momentum

Momentum is a method that helps accelerate SGD in the relevant direction and dampens oscillations as can be seen in Image 3. It does this by adding a fraction of the update vector of the past time step to the current update vector:

\begin{align}
& v_{t} = \gamma v_{t-1} + \eta
\triangledown_{\theta}J(\theta) \\
& \theta = \theta - v_{t}\\
\end{align*}

  • The momentum term $\gamma$ is usually set to 0.9 or a similar value.
  • The ball accumulates momentum as it rolls downhill, becoming faster and faster on the way

Nesterov accelerated gradient

\begin{align}
& v_{t} = \gamma v_{t-1} + \eta
\triangledown_{\theta}J(\theta - \gamma m) \\
& \theta = \theta - v_{t}\\
\end{align
}

  • 既然参数要沿着 $\theta - \gamma * m$更新,那就先先计算未来位置的梯度
  • This anticipatory update prevents us from going too fast and results in increased responsiveness, which has significantly increased the performance of RNNs on a number of tasks

Adagrad

\begin{align}
& s = s + \triangledown J(\theta) \bigodot \triangledown J(\theta) \\
& \theta = \theta - \frac{\eta}{\sqrt{s + \epsilon}} \bigodot \triangledown J(\theta) \\
\end{align
}

  • One of Adagrad’s main benefits is that it eliminates the need to manually tune the learning rate
  • Adagrad modifies the general learning rate $\gamma$ at each time step t for every parameter $\theta_{i}$ based on the past gradients that have been computed for $\theta_{i}$

RMSprop

\begin{align}
& v_{t} = \gamma v_{t-1} + (1-\gamma)
\triangledown J(\theta) \bigodot \triangledown J(\theta) \\
& \theta = \theta - v_{t} \\
\end{align*}

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tf.train.RMSPropOptimizer(learning_rate=learning_rate, momentum=0.9, decay=0.9, epsilon=1e-10)
  • 加入Momentum,主要是解决学习速率过快衰减的问题
  • RMSprop as well divides the learning rate by an exponentially decaying average of squared gradients. Hinton suggests $\gamma$ to be set to 0.9, while a good default value for the learning rate $\eta$ is 0.001.

Adaptive moment estimation (Adam)

\begin{align}
& m = \beta_{1}
m + (1-\beta_{1}) \triangledown J(\theta) \\
& s = \beta_{2}
s + (1-\beta_{2}) \triangledown J(\theta) \bigodot \triangledown J(\theta) \\
& m = \frac{m}{1-\beta^{t}_{1}} \\
& s = \frac{s}{1-\beta^{t}_{2}} \\
& \theta = \theta - \frac{\eta}{\sqrt{s + \epsilon}} \bigodot m
\end{align
}

  • 其结合了Momentum和RMSprop算法的思想。相比Momentum算法,其学习速率是自适应的,而相比RMSprop,其增加了冲量项, 第三和第四项主要是为了放大它们
  • The authors propose default values of 0.9 for $\beta1$, 0.9999 for $\beta2$ and $10^{-8}$ for $\epsilon$
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