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# SGD¶

class torch.optim.SGD(params, lr=<required parameter>, momentum=0, dampening=0, weight_decay=0, nesterov=False, *, maximize=False, foreach=None, differentiable=False)[source]

Implements stochastic gradient descent (optionally with momentum).

\begin{aligned} &\rule{110mm}{0.4pt} \\ &\textbf{input} : \gamma \text{ (lr)}, \: \theta_0 \text{ (params)}, \: f(\theta) \text{ (objective)}, \: \lambda \text{ (weight decay)}, \\ &\hspace{13mm} \:\mu \text{ (momentum)}, \:\tau \text{ (dampening)}, \:\textit{ nesterov,}\:\textit{ maximize} \\[-1.ex] &\rule{110mm}{0.4pt} \\ &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do} \\ &\hspace{5mm}g_t \leftarrow \nabla_{\theta} f_t (\theta_{t-1}) \\ &\hspace{5mm}\textbf{if} \: \lambda \neq 0 \\ &\hspace{10mm} g_t \leftarrow g_t + \lambda \theta_{t-1} \\ &\hspace{5mm}\textbf{if} \: \mu \neq 0 \\ &\hspace{10mm}\textbf{if} \: t > 1 \\ &\hspace{15mm} \textbf{b}_t \leftarrow \mu \textbf{b}_{t-1} + (1-\tau) g_t \\ &\hspace{10mm}\textbf{else} \\ &\hspace{15mm} \textbf{b}_t \leftarrow g_t \\ &\hspace{10mm}\textbf{if} \: \textit{nesterov} \\ &\hspace{15mm} g_t \leftarrow g_{t} + \mu \textbf{b}_t \\ &\hspace{10mm}\textbf{else} \\[-1.ex] &\hspace{15mm} g_t \leftarrow \textbf{b}_t \\ &\hspace{5mm}\textbf{if} \: \textit{maximize} \\ &\hspace{10mm}\theta_t \leftarrow \theta_{t-1} + \gamma g_t \\[-1.ex] &\hspace{5mm}\textbf{else} \\[-1.ex] &\hspace{10mm}\theta_t \leftarrow \theta_{t-1} - \gamma g_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] &\bf{return} \: \theta_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] \end{aligned}

Nesterov momentum is based on the formula from On the importance of initialization and momentum in deep learning.

Parameters:
• params (iterable) – iterable of parameters to optimize or dicts defining parameter groups

• lr (float) – learning rate

• momentum (float, optional) – momentum factor (default: 0)

• weight_decay (float, optional) – weight decay (L2 penalty) (default: 0)

• dampening (float, optional) – dampening for momentum (default: 0)

• nesterov (bool, optional) – enables Nesterov momentum (default: False)

• maximize (bool, optional) – maximize the params based on the objective, instead of minimizing (default: False)

• foreach (bool, optional) – whether foreach implementation of optimizer is used (default: None)

Example

>>> optimizer = torch.optim.SGD(model.parameters(), lr=0.1, momentum=0.9)
>>> loss_fn(model(input), target).backward()
>>> optimizer.step()


Note

The implementation of SGD with Momentum/Nesterov subtly differs from Sutskever et. al. and implementations in some other frameworks.

Considering the specific case of Momentum, the update can be written as

\begin{aligned} v_{t+1} & = \mu * v_{t} + g_{t+1}, \\ p_{t+1} & = p_{t} - \text{lr} * v_{t+1}, \end{aligned}

where $p$, $g$, $v$ and $\mu$ denote the parameters, gradient, velocity, and momentum respectively.

This is in contrast to Sutskever et. al. and other frameworks which employ an update of the form

\begin{aligned} v_{t+1} & = \mu * v_{t} + \text{lr} * g_{t+1}, \\ p_{t+1} & = p_{t} - v_{t+1}. \end{aligned}

The Nesterov version is analogously modified.

Add a param group to the Optimizer s param_groups.

This can be useful when fine tuning a pre-trained network as frozen layers can be made trainable and added to the Optimizer as training progresses.

Parameters:

param_group (dict) – Specifies what Tensors should be optimized along with group specific optimization options.

Loads the optimizer state.

Parameters:

state_dict (dict) – optimizer state. Should be an object returned from a call to state_dict().

state_dict()

Returns the state of the optimizer as a dict.

It contains two entries:

• state - a dict holding current optimization state. Its content

differs between optimizer classes.

• param_groups - a list containing all parameter groups where each

parameter group is a dict

Sets the gradients of all optimized torch.Tensor s to zero.

Parameters:

set_to_none (bool) – instead of setting to zero, set the grads to None. This will in general have lower memory footprint, and can modestly improve performance. However, it changes certain behaviors. For example: 1. When the user tries to access a gradient and perform manual ops on it, a None attribute or a Tensor full of 0s will behave differently. 2. If the user requests zero_grad(set_to_none=True) followed by a backward pass, .grads are guaranteed to be None for params that did not receive a gradient. 3. torch.optim optimizers have a different behavior if the gradient is 0 or None (in one case it does the step with a gradient of 0 and in the other it skips the step altogether).

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