class torch.optim.SGD(params, lr=<required parameter>, momentum=0, dampening=0, weight_decay=0, nesterov=False, *, maximize=False)[source]

Implements stochastic gradient descent (optionally with momentum).

input:γ (lr),θ0 (params),f(θ) (objective),λ (weight decay),μ (momentum),τ (dampening), nesterov, maximizefort=1todogtθft(θt1)ifλ0gtgt+λθt1ifμ0ift>1btμbt1+(1τ)gtelsebtgtifnesterovgtgt1+μbtelsegtbtifmaximizeθtθt1+γgtelseθtθt1γgtreturnθt\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-1} + \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.

  • 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)


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


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

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

where pp, gg, vv and μ\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

vt+1=μvt+lrgt+1,pt+1=ptvt+1.\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.


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


Loads the optimizer state.


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


Performs a single optimization step.


closure (callable, optional) – A closure that reevaluates the model and returns the loss.


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


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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