The AdaBelief optimizer.
AdaBelief is an adaptive learning rate optimizer that focuses on fast convergence, generalization, and stability. It adapts the step size depending on its “belief” in the gradient direction — the optimizer adaptively scales the step size by the difference between the predicted and observed gradients. AdaBelief is a modified version of optax.adam() and contains the same number of parameters.
Let \(\alpha_t\) represent the learning rate and \(\beta_1, \beta_2\), \(\varepsilon\), \(\bar{\varepsilon}\) represent the arguments b1, b2, eps and eps_root respectively. The learning rate is indexed by \(t\) since the learning rate may also be provided by a schedule function.
The init function of this optimizer initializes an internal state \(S_0 := (m_0, s_0) = (0, 0)\), representing initial estimates for the first and second moments. In practice these values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\) and optimizer state \(S_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have,
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ s_t &\leftarrow \beta_2 \cdot s_{t-1} + (1-\beta_2) \cdot (g_t - m_t)^2 + \bar{\varepsilon} \\ \hat{m}_t &\leftarrow m_t / {(1-\beta_1^t)} \\ \hat{s}_t &\leftarrow s_t / {(1-\beta_2^t)} \\ u_t &\leftarrow -\alpha_t \cdot \hat{m}_t / \left(\sqrt{\hat{s}_t} + \varepsilon \right) \\ S_t &\leftarrow (m_t, s_t). \end{align*}\]
With the keyword argument nesterov=True, the optimizer uses Nesterov momentum, replacing the above \(\hat{m}_t\) with
\[\hat{m}_t \leftarrow \beta_1 m_t / {(1-\beta_1^{t+1})} + (1 - \beta_1) g_t / {(1-\beta_1^t)}. \]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – Term added to the denominator to improve numerical stability.
eps_root – Term added to the second moment of the prediction error to improve numerical stability. If backpropagating gradients through the gradient transformation (e.g. for meta-learning), this must be non-zero.
nesterov – Whether to use Nesterov momentum.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adabelief(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.38E+01
References
Zhuang, AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients, 2020
Note
The default epsilon values in the paper are eps=1e-8, eps_root=0..
The Adadelta optimizer.
Adadelta is a stochastic gradient descent method that adapts learning rates based on a moving window of gradient updates. Adadelta is a modification of Adagrad. It addresses the diminishing learning rates problem in Adagrad by maintaining running averages of squared gradients.
The weight update \(\Delta w_t\) for this optimizer is given as follows:
\[\begin{align*} &E[g^2]_t = \rho \cdot E[g^2]_{t-1} + (1-\rho) \cdot g_t^2 \\ &\Delta w_t = -\frac{\sqrt{E[\Delta w^2]_{t-1} + \epsilon}}{\sqrt{E[g^2]_t + \epsilon}} \cdot g_t \end{align*}\]
\(g_t\) is the gradient at time step \(t\),
\(E[g^2]_t\) is the running average of squared gradients,
\(E[\Delta w^2]_t\) is the running average of squared parameter updates,
\(\rho\) is the decay rate (typically 0.9),
\(\epsilon\) is a small constant for numerical stability.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
rho – A coefficient used for computing a running average of squared gradients.
eps – Term added to the denominator to improve numerical stability.
weight_decay – Optional rate at which to decay weights.
weight_decay_mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the transformation to, and False for those you want to skip.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> f = lambda x: jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adadelta(learning_rate=10.)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.36E+01
Objective function: 1.32E+01
Objective function: 1.29E+01
Objective function: 1.25E+01
Objective function: 1.21E+01
References
Zeiler, Adadelta: An Adaptive Learning Rate Optimizer, 2012
The ADAptive Nesterov momentum algorithm (Adan).
Adan first reformulates the vanilla Nesterov acceleration to develop a new Nesterov momentum estimation (NME) method, which avoids the extra overhead of computing gradient at the extrapolation point. Then Adan adopts NME to estimate the gradient’s first- and second-order moments in adaptive gradient algorithms for convergence acceleration.
The algorithm is as follows. First, we define the following parameters:
\(\eta > 0\): the step size.
\(\beta_1 \in [0, 1]\): the decay rate for the exponentially weighted average of gradients.
\(\beta_2 \in [0, 1]\): the decay rate for the exponentially weighted average of differences of gradients.
\(\beta_3 \in [0, 1]\): the decay rate for the exponentially weighted average of the squared term.
\(\varepsilon > 0\): a small constant for numerical stability.
\(\lambda > 0\): a weight decay.
Second, we define the following variables:
\(\theta_t\): the parameters.
\(g_t\): the incoming stochastic gradient.
\(m_t\): the exponentially weighted average of gradients.
\(v_t\): the exponentially weighted average of differences of gradients.
\(n_t\): the exponentially weighted average of the squared term.
\(u_t\): the outgoing update vector.
\(S_t\): the saved state of the optimizer.
Third, we initialize these variables as follows:
\(m_0 = g_0\)
\(v_0 = 0\)
\(v_1 = g_1 - g_0\)
\(n_0 = g_0^2\)
Finally, on each iteration, we update the variables as follows:
\[\begin{align*} m_t &\gets (1 - \beta_1) m_{t-1} + \beta_1 g_t \\ v_t &\gets (1 - \beta_2) v_{t-1} + \beta_2 (g_t - g_{t-1}) \\ n_t &\gets (1 - \beta_3) n_{t-1} + \beta_3 (g_t + (1 - \beta_2) (g_t - g_{t-1}))^2 \\ \eta_t &\gets \eta / ({\sqrt{n_t + \bar{\varepsilon}} + \varepsilon}) \\ u_t &\gets (\theta_t - \eta_t \circ (m_t + (1 - \beta_2) v_t)) / (1 + \lambda \eta) \\ S_t &\leftarrow (m_t, v_t, n_t). \end{align*}\]
learning_rate – this is a fixed global scaling factor.
b1 – Decay rate for the EWMA of gradients.
b2 – Decay rate for the EWMA of differences of gradients.
b3 – Decay rate for the EMWA of the algorithm’s squared term.
eps – Term added to the denominator to improve numerical stability.
eps_root – Term added to the denominator inside the square-root to improve numerical stability when backpropagating gradients through the rescaling.
weight_decay – Strength of the weight decay regularization.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the weight decay to, and False for those you want to skip.
the corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> f = lambda x: x @ x # simple quadratic function
>>> solver = optax.adan(learning_rate=1e-1)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.28E+01
Objective function: 1.17E+01
Objective function: 1.07E+01
Objective function: 9.68E+00
Objective function: 8.76E+00
References
Xie et al, Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models, 2022
The Adagrad optimizer.
AdaGrad is a sub-gradient algorithm for stochastic optimization that adapts the learning rate individually for each feature based on its gradient history.
The updated parameters adopt the form:
\[w_{t+1}^{(i)} = w_{t}^{(i)} - \eta \frac{g_{t}^{(i)}} {\sqrt{\sum_{\tau=1}^{t} (g_{\tau}^{(i)})^2 + \epsilon}}\]
\(w_t^{(i)}\) is the parameter \(i\) at time step \(t\),
\(\eta\) is the learning rate,
\(g_t^{(i)}\) is the gradient of parameter \(i\) at time step \(t\),
\(\epsilon\) is a small constant to ensure numerical stability.
Defining \(G = \sum_{t=1}^\tau g_t g_t^\top\), the update can be written as
\[w_{t+1} = w_{t} - \eta \cdot \text{diag}(G + \epsilon I)^{-1/2} \cdot g_t\]
where \(\text{diag} (G) = (G_{ii})_{i=1}^p\) is the vector of diagonal entries of \(G \in \mathbb{R}^p\) and \(I\) is the identity matrix in \(\mathbb{R}^p\).
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
initial_accumulator_value – Initial value for the accumulator.
eps – A small constant applied to denominator inside of the square root (as in RMSProp) to avoid dividing by zero when rescaling.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adagrad(learning_rate=1.0)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 5.01E+00
Objective function: 2.40E+00
Objective function: 1.25E+00
Objective function: 6.86E-01
Objective function: 3.85E-01
References
Duchi et al, Adaptive Subgradient Methods for Online Learning and Stochastic Optimization, 2011
Warning
Adagrad’s main limit is the monotonic accumulation of squared gradients in the denominator: since all terms are >0, the sum keeps growing during training and the learning rate eventually becomes vanishingly small.
The Adafactor optimizer.
Adafactor is an adaptive learning rate optimizer that focuses on fast training of large scale neural networks. It saves memory by using a factored estimate of the second order moments used to scale gradients.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate(). Note that the natural scale for Adafactor’s LR is markedly different from Adam, one doesn’t use the 1/sqrt(hidden) correction for this optim with attention-based models.
min_dim_size_to_factor – Only factor the statistics if two array dimensions have at least this size.
decay_rate – Controls second-moment exponential decay schedule.
decay_offset – For fine-tuning, one may set this to the starting step number of the fine-tuning phase.
multiply_by_parameter_scale – If True, then scale learning_rate by parameter norm. If False, provided learning_rate is absolute step size.
clipping_threshold – Optional clipping threshold. Must be >= 1. If None, clipping is disabled.
momentum – Optional value between 0 and 1, enables momentum and uses extra memory if non-None! None by default.
dtype_momentum – Data type of momentum buffers.
weight_decay_rate – Optional rate at which to decay weights.
eps – Regularization constant for root mean squared gradient.
factored – Whether to use factored second-moment estimates.
weight_decay_mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the transformation to, and False for those you want to skip.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adafactor(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.36E+01
References
Shazeer et al, Adafactor: Adaptive Learning Rates with Sublinear Memory Cost, 2018
The Adam optimizer.
Adam is an SGD variant with gradient scaling adaptation. The scaling used for each parameter is computed from estimates of first and second-order moments of the gradients (using suitable exponential moving averages).
Let \(\alpha_t\) represent the learning rate and \(\beta_1, \beta_2\), \(\varepsilon\), \(\bar{\varepsilon}\) represent the arguments b1, b2, eps and eps_root respectively. The learning rate is indexed by \(t\) since the learning rate may also be provided by a schedule function.
The init function of this optimizer initializes an internal state \(S_0 := (m_0, v_0) = (0, 0)\), representing initial estimates for the first and second moments. In practice these values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\) and optimizer state \(S_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have,
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ v_t &\leftarrow \beta_2 \cdot v_{t-1} + (1-\beta_2) \cdot {g_t}^2 \\ \hat{m}_t &\leftarrow m_t / {(1-\beta_1^t)} \\ \hat{v}_t &\leftarrow v_t / {(1-\beta_2^t)} \\ u_t &\leftarrow -\alpha_t \cdot \hat{m}_t / \left({\sqrt{\hat{v}_t + \bar{\varepsilon}} + \varepsilon} \right)\\ S_t &\leftarrow (m_t, v_t). \end{align*}\]
With the keyword argument nesterov=True, the optimizer uses Nesterov momentum, replacing the above \(\hat{m}_t\) with
\[\hat{m}_t \leftarrow \beta_1 m_t / {(1-\beta_1^{t+1})} + (1 - \beta_1) g_t / {(1-\beta_1^t)}. \]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for example when computing (meta-)gradients through Adam.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
nesterov – Whether to use Nesterov momentum. The solver with nesterov=True is equivalent to the optax.nadam() optimizer, and described in [Dozat 2016].
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adam(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Kingma et al, Adam: A Method for Stochastic Optimization, 2014
Dozat, Incorporating Nesterov Momentum into Adam, 2016
Warning
PyTorch and optax’s implementation follow Algorithm 1 of [Kingma et al. 2014]. Note that TensorFlow used instead the formulation just before Section 2.1 of the paper. See deepmind/optax#571 for more detail.
A variant of the Adam optimizer that uses the infinity norm.
AdaMax is a variant of the optax.adam() optimizer. By generalizing Adam’s \(L^2\) norm to an \(L^p\) norm and taking the limit as \(p \rightarrow \infty\), we obtain a simple and stable update rule.
Let \(\alpha_t\) represent the learning rate and \(\beta_1, \beta_2\), \(\varepsilon\) represent the arguments b1, b2 and eps respectively. The learning rate is indexed by \(t\) since the learning rate may also be provided by a schedule function.
The init function of this optimizer initializes an internal state \(S_0 := (m_0, v_0) = (0, 0)\), representing initial estimates for the first and second moments. In practice these values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\) and optimizer state \(S_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have,
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ v_t &\leftarrow \max(\left| g_t \right| + \varepsilon, \beta_2 \cdot v_{t-1}) \\ \hat{m}_t &\leftarrow m_t / (1-\beta_1^t) \\ u_t &\leftarrow -\alpha_t \cdot \hat{m}_t / v_t \\ S_t &\leftarrow (m_t, v_t). \end{align*}\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the maximum of past gradients.
eps – A small constant applied to denominator to avoid dividing by zero when rescaling.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adamax(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Kingma et al, 2014: https://arxiv.org/abs/1412.6980
Adamax with weight decay regularization.
AdamaxW uses weight decay to regularize learning towards small weights, as this leads to better generalization. In SGD you can also use L2 regularization to implement this as an additive loss term, however L2 regularization does not behave as intended for adaptive gradient algorithms such as Adam.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the maximum of past gradients.
eps – A small constant applied to denominator to avoid dividing by zero when rescaling.
weight_decay – Strength of the weight decay regularization. Note that this weight decay is multiplied with the learning rate. This is consistent with other frameworks such as PyTorch, but different from (Loshchilov et al, 2019) where the weight decay is only multiplied with the “schedule multiplier”, but not the base learning rate.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the weight decay to, and False for those you want to skip. Note that the Adamax gradient transformations are applied to all parameters.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adamaxw(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Loshchilov et al, 2019: https://arxiv.org/abs/1711.05101
Warning
Sometimes you may want to skip weight decay for BatchNorm scale or for the bias parameters. You can use optax.masked to make your own AdamaxW variant where additive_weight_decay is applied only to a subset of params.
Adam with weight decay regularization.
AdamW uses weight decay to regularize learning towards small weights, as this leads to better generalization. In SGD you can also use L2 regularization to implement this as an additive loss term, however L2 regularization does not behave as intended for adaptive gradient algorithms such as Adam, see [Loshchilov et al, 2019].
Let \(\alpha_t\) represent the learning rate and \(\beta_1, \beta_2\), \(\varepsilon\), \(\bar{\varepsilon}\) represent the arguments b1, b2, eps and eps_root respectively. The learning rate is indexed by \(t\) since the learning rate may also be provided by a schedule function. Let \(\lambda\) be the weight decay and \(\theta_t\) the parameter vector at time \(t\).
The init function of this optimizer initializes an internal state \(S_0 := (m_0, v_0) = (0, 0)\), representing initial estimates for the first and second moments. In practice these values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\), the optimizer state \(S_t\) and the parameters \(\theta_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have,
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ v_t &\leftarrow \beta_2 \cdot v_{t-1} + (1-\beta_2) \cdot {g_t}^2 \\ \hat{m}_t &\leftarrow m_t / {(1-\beta_1^t)} \\ \hat{v}_t &\leftarrow v_t / {(1-\beta_2^t)} \\ u_t &\leftarrow -\alpha_t \cdot \left( \hat{m}_t / \left({\sqrt{\hat{v}_t + \bar{\varepsilon}} + \varepsilon} \right) + \lambda \theta_{t} \right)\\ S_t &\leftarrow (m_t, v_t). \end{align*}\]
This implementation can incorporate a momentum a la Nesterov introduced by [Dozat 2016]. The resulting optimizer is then often referred as NAdamW. With the keyword argument nesterov=True, the optimizer uses Nesterov momentum, replacing the above \(\hat{m}_t\) with
\[\hat{m}_t \leftarrow \beta_1 m_t / {(1-\beta_1^{t+1})} + (1 - \beta_1) g_t / {(1-\beta_1^t)}. \]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
weight_decay – Strength of the weight decay regularization. Note that this weight decay is multiplied with the learning rate. This is consistent with other frameworks such as PyTorch, but different from (Loshchilov et al, 2019) where the weight decay is only multiplied with the “schedule multiplier”, but not the base learning rate.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the weight decay to, and False for those you want to skip. Note that the Adam gradient transformations are applied to all parameters.
nesterov – Whether to use Nesterov momentum. The solver with nesterov=True is equivalent to the optax.nadamw() optimizer. This modification is described in [Dozat 2016].
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.adamw(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Loshchilov et al, Decoupled Weight Decay Regularization, 2019
Dozat, Incorporating Nesterov Momentum into Adam, 2016
The AMSGrad optimizer.
The original Adam can fail to converge to the optimal solution in some cases. AMSGrad guarantees convergence by using a long-term memory of past gradients.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.amsgrad(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Reddi et al, On the Convergence of Adam and Beyond, 2023
The Frobenius matched gradient descent (Fromage) optimizer.
Fromage is a learning algorithm that does not require learning rate tuning. The optimizer is based on modeling neural network gradients via deep relative trust (a distance function on deep neural networks). Fromage is similar to the LARS optimizer and can work on a range of standard neural network benchmarks, such as natural language Transformers and generative adversarial networks.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
min_norm – A minimum value that the norm of the gradient updates and the norm of the layer parameters can be clipped to to avoid dividing by zero when computing the trust ratio (as in the LARS paper).
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.fromage(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.37E+01
Objective function: 1.36E+01
References
Bernstein et al, On the distance between two neural networks and the stability of learning, 2020
The LAMB optimizer.
LAMB is a general purpose layer-wise adaptive large batch optimizer designed to provide consistent training performance across a wide range of tasks, including those that use attention-based models (such as Transformers) and ResNet-50. The optimizer is able to work with small and large batch sizes. LAMB was inspired by the LARS learning algorithm.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
weight_decay – Strength of the weight decay regularization.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the transformation to, and False for those you want to skip.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.lamb(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.36E+01
References
You et al, Large Batch Optimization for Deep Learning: Training BERT in 76 minutes, 2020
The LARS optimizer.
LARS is a layer-wise adaptive optimizer introduced to help scale SGD to larger batch sizes. LARS later inspired the LAMB optimizer.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
weight_decay – Strength of the weight decay regularization.
weight_decay_mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the transformation to, and False for those you want to skip.
trust_coefficient – A multiplier for the trust ratio.
eps – Optional additive constant in the trust ratio denominator.
trust_ratio_mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the transformation to, and False for those you want to skip.
momentum – Decay rate for momentum.
nesterov – Whether to use Nesterov momentum.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.lars(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
References
You et al, Large Batch Training of Convolutional Networks, 2017
L-BFGS optimizer.
L-BFGS is a quasi-Newton method that multiplies the update (gradient) with an approximation of the inverse Hessian. This algorithm does not need access to the Hessian, as this approximation is constructed from the gradient evaluations seen during optimization. L-BFGS is a limited-memory variant of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The BFGS algorithm requires storing a matrix of size \(p \times p\) with \(p\) the dimension of the parameters. The limited variant circuments this issue by computing the approximation of the inverse using only \(m\) (memory_size) past differences of parameters/gradients. Namely, the approximation of the Hessian inverse is denoted \(P_k = P_{k, k}\), where
\[\begin{align*} P_{k, j+1} & = V_j^\top P_{k, j} V_j + \rho_j \delta w_j \delta w_j^\top \quad \text{for} \ j \in \{k-m, \ldots, k-1\}\\ P_{k, k-m} & = \gamma_k I \\ V_k & = I - \rho_k \delta u_k \delta w_k^\top \\ \rho_k & = 1/(\delta u_k^\top \delta w_k) \\ \delta w_k & = w_{k+1} - w_k \\ \delta u_k & = u_{k+1} - u_k \\ \gamma_k & = \begin{cases} (\delta w_{k-1}^\top \delta u_{k-1}) / (\delta u_{k-1}^\top \delta u_{k-1}) & \text{if} \ \texttt{scale\_init\_hess} \\ 1 & \text{otherwise} \end{cases}, \end{align*}\]
for \(u_k\) the gradients/updates at iteration \(k\), \(w_k\) the parameters at iteration \(k\).
The formula for updating \(P_k\) is obtained by computing the optimal preconditioning matrix subject to some secant condition, see references for more details. Computing \(P_k u_k\) can be done by a sequence of vector operations using past differences of parameters and gradients stored in a memory bufffer.
The present function just outputs the LBFGS direction \(P_k u_k\). It can be chained with a linesearch ensuring sufficient decrease and low curvature, such as a zoom linesearch. The linesearch computes a stepsize \(\eta_k\), such that the updated parameters (using optax.apply_updates()) take the form \(w_{k+1} = w_k - \eta_k P_k u_k\).
learning_rate – optional global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate(). By default the learning rate is handled by a linesearch.
memory_size – number of past updates to keep in memory to approximate the Hessian inverse.
scale_init_precond – whether to use a scaled identity as the initial preconditioner, see formula of \(\gamma_k\) above.
linesearch – an instance of optax.GradientTransformationExtraArgs such as optax.scale_by_zoom_linesearch() that computes a learning rate, a.k.a. stepsize, to satisfy some criterion such as a sufficient decrease of the objective by additional calls to the objective.
A optax.GradientTransformationExtraArgs object.
Example
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2)
>>> solver = optax.lbfgs()
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> value_and_grad = optax.value_and_grad_from_state(f)
>>> for _ in range(2):
... value, grad = value_and_grad(params, state=opt_state)
... updates, opt_state = solver.update(
... grad, opt_state, params, value=value, grad=grad, value_fn=f
... )
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 7.52E+00
Objective function: 7.46E-14
References
Algorithms 7.4, 7.5 (page 199) of Nocedal et al, Numerical Optimization , 1999
Liu et al., On the limited memory BFGS method for large scale optimization , 1989.
Warning
This optimizer is memory intensive and best used for small to medium scale problems.
Warning
This optimizer works best with a linesearch (current default is a zoom linesearch). See example above for best use in a non-stochastic setting, where we can recycle gradients computed by the linesearch using optax.value_and_grad_from_state().
Note
We initialize the scaling of the identity as a capped reciprocal of the gradient norm. This avoids wasting linesearch iterations for the first step by taking into account the magnitude of the gradients. In other words, we constrain the trust-region of the first step to an Euclidean ball of radius 1 at the first iteration. The choice of \(\gamma_0\) is not detailed in the references above, so this is a heuristic choice.
Note
The algorithm can support complex inputs.
The Lion optimizer.
Lion is discovered by symbolic program search. Unlike most adaptive optimizers such as AdamW, Lion only tracks momentum, making it more memory-efficient. The update of Lion is produced through the sign operation, resulting in a larger norm compared to updates produced by other optimizers such as SGD and AdamW. A suitable learning rate for Lion is typically 3-10x smaller than that for AdamW, the weight decay for Lion should be in turn 3-10x larger than that for AdamW to maintain a similar strength (lr * wd).
Let \(\alpha_t\) represent the learning rate and \(\beta_1, \beta_2\), represent the arguments b1 and b2 respectively. The learning rate is indexed by \(t\) since the learning rate may also be provided by a schedule function. Let \(\lambda\) be the weight decay and \(\theta_t\) the parameter vector at time \(t\).
The init function of this optimizer initializes an internal state \(S_0 := (m_0) = (0)\), representing the intial estimate for the first moment. In practice these values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\), the optimizer state \(S_t\) and the parameters \(\theta_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have,
\[\begin{align*} c_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ u_t &\leftarrow -\alpha_t \cdot \left( sign \left( c_t \right) + \lambda \theta_{t} \right)\\ m_t &\leftarrow \beta_2 \cdot m_{t-1} + (1-\beta_2) \cdot g_t \\ S_t &\leftarrow (m_t). \end{align*}\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Rate to combine the momentum and the current gradient.
b2 – Exponential decay rate to track the momentum of past gradients.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
weight_decay – Strength of the weight decay regularization. Note that this weight decay is multiplied with the learning rate. This is consistent with other frameworks such as PyTorch, but different from (Loshchilov et al, 2019) where the weight decay is only multiplied with the “schedule multiplier”, but not the base learning rate.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the weight decay to, and False for those you want to skip. Note that the Adam gradient transformations are applied to all parameters.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.lion(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Chen et al, Symbolic Discovery of Optimization Algorithms, 2023
The NAdam optimizer.
Nadam is a variant of optax.adam() with Nesterov’s momentum. The update rule of this solver is as follows:
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t-1} + (1-\beta_1) \cdot g_t \\ v_t &\leftarrow \beta_2 \cdot v_{t-1} + (1-\beta_2) \cdot {g_t}^2 \\ \hat{m}_t &\leftarrow \beta_1 m_t / {(1-\beta_1^{t+1})} + (1 - \beta_1) g_t / {(1-\beta_1^t)}\\ \hat{v}_t &\leftarrow v_t / {(1-\beta_2^t)} \\ u_t &\leftarrow -\alpha_t \cdot \hat{m}_t / \left({\sqrt{\hat{v}_t + \bar{\varepsilon}} + \varepsilon} \right)\\ S_t &\leftarrow (m_t, v_t). \end{align*}\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for example when computing (meta-)gradients through Adam.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.nadam(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.38E+01
References
Dozat, Incorporating Nesterov Momentum into Adam, 2016
Added in version 0.1.9.
NAdamW optimizer, implemented as part of the AdamW optimizer.
NadamW is variant of optax.adamw() with Nesterov’s momentum. Compared to AdamW, this optimizer replaces the assignment
\[\hat{m}_t \leftarrow m_t / {(1-\beta_1^t)}\]
with
\[\hat{m}_t \leftarrow \beta_1 m_t / {(1-\beta_1^{t+1})} + (1 - \beta_1) g_t / {(1-\beta_1^t)}.\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
weight_decay – Strength of the weight decay regularization. Note that this weight decay is multiplied with the learning rate. This is consistent with other frameworks such as PyTorch, but different from (Loshchilov et al, 2019) where the weight decay is only multiplied with the “schedule multiplier”, but not the base learning rate.
mask – A tree with same structure as (or a prefix of) the params PyTree, or a Callable that returns such a pytree given the params/updates. The leaves should be booleans, True for leaves/subtrees you want to apply the weight decay to, and False for those you want to skip. Note that the Adam gradient transformations are applied to all parameters.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.nadamw(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.38E+01
References
Loshchilov et al, Decoupled Weight Decay Regularization, 2019
Dozat, Incorporating Nesterov Momentum into Adam, 2016
Added in version 0.1.9.
A variant of SGD with added noise.
Noisy SGD is a variant of optax.sgd() that incorporates Gaussian noise into the updates. It has been found that adding noise to the gradients can improve both the training error and the generalization error in very deep networks.
The update \(u_t\) is modified to include this noise as follows:
\[u_t \leftarrow -\alpha_t (g_t + N(0, \sigma_t^2)), \]
where \(N(0, \sigma_t^2)\) represents Gaussian noise with zero mean and a variance of \(\sigma_t^2\).
The variance of this noise decays over time according to the formula
\[\sigma_t^2 = \frac{\eta}{(1+t)^\gamma}, \]
where \(\gamma\) is the decay rate parameter gamma and \(\eta\) represents the initial variance eta.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
eta – Initial variance for the Gaussian noise added to gradients.
gamma – A parameter controlling the annealing of noise over time t, the variance decays according to (1+t)**(-gamma).
seed – Seed for the pseudo-random generation process.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.noisy_sgd(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.35E+01
Objective function: 1.33E+01
Objective function: 1.32E+01
References
Neelakantan et al, Adding Gradient Noise Improves Learning for Very Deep Networks, 2015
NovoGrad optimizer.
NovoGrad is more robust to the initial learning rate and weight initialization than other methods. For example, NovoGrad works well without LR warm-up, while other methods require it. NovoGrad performs exceptionally well for large batch training, e.g. it outperforms other methods for ResNet-50 for all batches up to 32K. In addition, NovoGrad requires half the memory compared to Adam. It was introduced together with Jasper ASR model.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – An exponential decay rate to track the first moment of past gradients.
b2 – An exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
weight_decay – Strength of the weight decay regularization.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.novograd(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.37E+01
References
Ginsburg et al, Stochastic Gradient Methods with Layer-wise Adaptive Moments for Training of Deep Networks, 2019
Li et al, Jasper: An End-to-End Convolutional Neural Acoustic Model, 2019
An Optimistic Gradient Descent optimizer.
Optimistic gradient descent is an approximation of extra-gradient methods which require multiple gradient calls to compute the next update. It has strong formal guarantees for last-iterate convergence in min-max games, for which standard gradient descent can oscillate or even diverge.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
alpha – Coefficient for generalized OGD.
beta – Coefficient for generalized OGD negative momentum.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.optimistic_gradient_descent(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.35E+01
Objective function: 1.33E+01
Objective function: 1.32E+01
References
Mokhtari et al, A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach, 2019
The Optimistic Adam optimizer.
This is an optimistic version of the Adam optimizer. It addresses the issue of limit cycling behavior in training Generative Adversarial Networks and other saddle-point min-max problems.
The algorithm is as follows. First, we define the following parameters:
\(\alpha\): the learning rate.
\(o\) the optimism rate.
\(\beta_1\) the exponential decay rate for the first moment estimate.
\(\beta_2\) the exponential decay rate for the second moment estimate.
Second, we define the following variables:
\(g_t\): the incoming gradient.
\(m_t\): the biased first moment estimate.
\(v_t\): the biased second raw moment estimate.
\(\hat{m}_t\): the bias-corrected first moment estimate.
\(\hat{v}_t\): the bias-corrected second raw moment estimate.
\(r_t\): the signal-to-noise ratio (SNR) vector.
\(u_t\): the outgoing update vector.
\(S_t\): the state of the optimizer.
Finally, on each iteration, the variables are updated as follows:
\[\begin{align*} m_t &\leftarrow \beta_1 \cdot m_{t - 1} + (1 - \beta_1) \cdot g_t \\ v_t &\leftarrow \beta_2 \cdot v_{t - 1} + (1 - \beta_2) \cdot g_t^2 \\ \hat{m}_t &\leftarrow m_t / {(1 - \beta_1^t)} \\ \hat{v}_t &\leftarrow v_t / {(1 - \beta_2^t)} \\ r_t &\leftarrow \hat{m}_t / \left({\sqrt{\hat{v}_t + \bar{\varepsilon}} + \varepsilon} \right) \\ u_t &\leftarrow -\alpha r_t - o (r_t - r_{t - 1}) \\ S_t &\leftarrow (m_t, v_t, r_t). \end{align*}\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
optimism – The amount of optimism to be applied. If None, defaults to learning_rate, as in the paper.
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – Term added to the denominator to improve numerical stability.
eps_root – Term added to the second moment of the prediction error to improve numerical stability. If backpropagating gradients through the gradient transformation (e.g. for meta-learning), this must be non-zero.
mu_dtype – Optional dtype to be used for the first order accumulator; if None then the dtype is inferred from params and updates.
nesterov – Whether to use Nesterov momentum.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> from jax import numpy as jnp, lax
>>> def f(x, y):
... return x * y # simple bilinear function
>>> opt = optax.optimistic_adam(1e-2, 1.0)
>>> def step(state, _):
... params, opt_state = state
... distance = jnp.hypot(*params)
... grads = jax.grad(f, argnums=(0, 1))(*params)
... grads = grads[0], -grads[1]
... updates, opt_state = opt.update(grads, opt_state, params)
... params = optax.apply_updates(params, updates)
... return (params, opt_state), distance
>>> params = 1.0, 2.0
>>> opt_state = opt.init(params)
>>> _, distances = lax.scan(step, (params, opt_state), length=1025)
>>> for i in range(6):
... print(f"{distances[4**i]:.3f}")
2.243
2.195
2.161
2.055
0.796
0.001
References
Daskalakis et al, Training GANs with Optimism, 2017
SGD with Polyak step-size.
This solver implements the SGD with Polyak step size of (Loizou et al. 2021). It sets the step-size as
\[s \min\left\{\frac{f(x) - f^\star}{\|\nabla f(x)\|^2 + \epsilon}, \gamma_{\max}\right\}\,, \]
where \(f\) is the function from which a gradient is computed, \(\gamma_{\max}\) is a maximal acceptable learning rate set by max_learning_rate, \(\epsilon\) is a constant preventing division by zero set with eps, \(s\) scales the formula by scaling, and \(f^\star\) is a guess of the minimum value of the function set with f_min.
Setting variant="sps+" (Garrigos et al. 2023) uses only the non-negative part of the suboptimality gap. That is, it replaces \(f(x) - f^\star\) with \((f(x) - f^\star)_+\), where \(a_+ = \max \{x, 0\}\).
max_learning_rate – a maximum step size to use (defaults to 1).
scaling – A global scaling factor, either fixed or evolving along iterations with a scheduler (defaults to 1).
f_min – a lower bound on the objective function (defaults to 0). Corresponds to \(f^\star\) in the formula above.
eps – a value to add in the denominator of the update (defaults to 0).
variant – either 'sps' or 'sps+' (defaults to 'sps').
A optax.GradientTransformationExtraArgs, where the update functiontakes an additional keyword argument value containing the current value of the objective function.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.polyak_sgd()
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... value, grad = jax.value_and_grad(f)(params)
... params, opt_state = solver.update(grad, opt_state, params, value=value)
... print('Objective function: ', f(params))
Objective function: 3.5
Objective function: 0.875
Objective function: 0.21875
Objective function: 0.0546875
Objective function: 0.013671875
References
Loizou et al. Stochastic polyak step-size for SGD: An adaptive learning rate for fast convergence, 2021
Berrada et al., Training neural networks for and by interpolation, 2020
Garrigos et al., Function value learning: Adaptive learning rates based on the Polyak stepsize and function splitting in ERM, 2023
Warning
This method requires knowledge of an approximate value of the of the objective function minimum, passed through the f_min argument. For models that interpolate the data, this can be set to 0 (default value). Failing to set an appropriate value for f_min can lead to divergence or convergence to a suboptimal solution.
The Rectified Adam optimizer.
The adaptive learning rate in Adam has undesirably large variance in early stages of training, due to the limited number of training samples used to estimate the optimizer’s statistics. Rectified Adam addresses this issue by analytically reducing the large variance.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
eps_root – A small constant applied to denominator inside the square root (as in RMSProp), to avoid dividing by zero when rescaling. This is needed for instance when computing (meta-)gradients through Adam.
threshold – Threshold for variance tractability.
nesterov – Whether to use Nesterov momentum.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.radam(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.35E+01
Objective function: 1.33E+01
Objective function: 1.32E+01
References
Liu et al, 2020: On the Variance of the Adaptive Learning Rate and Beyond, 2020
A flexible RMSProp optimizer.
RMSProp is an SGD variant with learning rate adaptation. The learning_rate used for each weight is scaled by a suitable estimate of the magnitude of the gradients on previous steps. Several variants of RMSProp can be found in the literature. This alias provides an easy to configure RMSProp optimizer that can be used to switch between several of these variants.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
decay – Decay used to track the magnitude of previous gradients.
eps – A small numerical constant to avoid dividing by zero when rescaling.
initial_scale – Initial value of accumulators tracking the magnitude of previous updates. PyTorch uses 0, TF1 uses 1. When reproducing results from a paper, verify the value used by the authors.
eps_in_sqrt – Whether to add eps in the square root of the denominator or outside the square root.
centered – Whether the second moment or the variance of the past gradients is used to rescale the latest gradients.
momentum – Decay rate used by the momentum term, when it is set to None, then momentum is not used at all.
nesterov – Whether Nesterov momentum is used.
bias_correction – Whether to apply bias correction to the estimates of the second moments (and first moment if centered=True).
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.rmsprop(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.39E+01
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.37E+01
Objective function: 1.36E+01
References
Hinton, Overview of mini-batch gradient descent <www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf>`_, 2012
Graves, Generating Sequences With Recurrent Neural Networks, 2014
Ziyin, LaProp: Separating Momentum and Adaptivity in Adam <https://arxiv.org/pdf/2002.04839>`_, 2021
Warning
Default behavior of optax’s RMSprop (eps_in_sqrt=True) differs from Pytorch’s implementation and could impact performance. If eps_in_sqrt=True, in the denominator, optax uses \(\sqrt{v + \epsilon}\) in the denominator whereas PyTorch uses \(\sqrt{v} + \epsilon\). Using eps_in_sqrt=False in optax will match PyTorch’s behavior. See google-deepmind/optax#532 for more detail.
The Rprop optimizer.
Rprop, short for resillient backpropogation, is a first order variant of gradient descent. It responds only to the sign of the gradient by increasing or decreasing the step size selected per parameter exponentially to speed up convergence and avoid oscillations.
learning_rate – The initial step size.
eta_minus – Multiplicative factor for decreasing step size. This is applied when the gradient changes sign from one step to the next.
eta_plus – Multiplicative factor for increasing step size. This is applied when the gradient has the same sign from one step to the next.
min_step_size – Minimum allowed step size. Smaller steps will be clipped to this value.
max_step_size – Maximum allowed step size. Larger steps will be clipped to this value.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.rprop(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Riedmiller et al. A direct adaptive method for faster backpropagation learning: the RPROP algorithm, 1993
Igel et al. Empirical evaluation of the improved Rprop learning algorithms, 2003
A canonical Stochastic Gradient Descent optimizer.
This implements stochastic gradient descent. It also includes support for momentum, and Nesterov acceleration, as these are standard practice when using stochastic gradient descent to train deep neural networks.
The canonical stochastic gradient descent returns an update \(u_t\) of the form
\[u_t \leftarrow -\alpha_t g_t, \]
where \(g_t\) is the gradient of the objective (potentially preprocessed by other transformations) and \(\alpha_t\) is the learning_rate at time \(t\) (constant or selected by an optax.Schedule).
Stochastic gradient descent with momentum takes two possible forms.
\[\begin{align*} m_t &\leftarrow g_t + \mu m_{t-1} \\ u_t &\leftarrow \begin{cases} -\alpha_t m_t & \text{ if } \texttt{nesterov = False} \\ -\alpha_t (g_t + \mu m_t) & \text{ if } \texttt{nesterov = True} \end{cases} \\ S_t &\leftarrow m_t, \end{align*}\]
where \(\mu\) is the momentum parameter and \(S_t\) is the state of the optimizer.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
momentum – Decay rate used by the momentum term, when it is set to None, then momentum is not used at all.
nesterov – Whether Nesterov momentum is used.
accumulator_dtype – Optional dtype to be used for the accumulator; if None then the dtype is inferred from params and updates.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.sgd(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.38E+01
Objective function: 1.37E+01
Objective function: 1.35E+01
Objective function: 1.33E+01
Objective function: 1.32E+01
References
Sutskever et al, On the importance of initialization and momentum in deep learning, 2013
A variant of SGD using only the signs of the gradient components.
SignSGD is a variant of SGD that uses the signs of the gradient components in the update, not their actual values. The update \(u_t\) is modified as follows:
\[u_t \leftarrow -\alpha_t\, \text{sign}\,(g_t), \]
for \(\alpha_t\) a given learning rate at iteration \(t\), and \(\text{sign}\,(g_t)\) the sign of each component of the gradient \(g_t\).
SGD variants that use only the signs of the gradient update have historically been used since RProp, with modern forms including RMSProp, Adam, and Lion. SignSGD uses only the signs of the gradient update. SignSGD enables significant gradient compression, substantially reducing the bottleneck imposed by communicating gradients when distributing learning across multiple workers.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.sign_sgd(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.38E+01
References
Bernstein et al., signSGD: Compressed optimization for Non-Convex Problems, 2018
Balles et al., The Geometry of Sign Gradient Descent, 2020
The SM3 optimizer.
SM3 (Square-root of Minima of Sums of Maxima of Squared-gradients Method) is a memory-efficient adaptive optimizer designed to decrease memory overhead when training very large models, such as the Transformer for machine translation, BERT for language modeling, and AmoebaNet-D for image classification. SM3: 1) applies to tensors of arbitrary dimensions and any predefined cover of the parameters; 2) adapts the learning rates in an adaptive and data-driven manner (like Adagrad and unlike Adafactor); and 3) comes with rigorous convergence guarantees in stochastic convex optimization settings.
The init function of this optimizer initializes an internal state \(S_0 := \{\mu_0, w_1\} = \{0, 0\}\), representing initial estimates for the cumulative squared gradients and the weights. These values are stored as pytrees containing all zeros, with the same shape as the model updates. At step \(t\), the update function of this optimizer takes as arguments the incoming gradients \(g_t\) and optimizer state \(S_t\) and computes updates \(u_t\) and new state \(S_{t+1}\). Thus, for \(t > 0\), we have:
SM3-I Algorithm
\[\begin{array}{l} \text{parameters: learning rate } \eta \\ \text{initialize } w_1 = 0; \forall r \in [k]: \mu_0(r) = 0 \\ \text{for } t = 1, \ldots, T \text{ do} \\ \quad \text{receive gradient } g_t = \nabla \ell_t(w_t) \\ \quad \text{for } r = 1, \ldots, k \text{ do} \\ \quad \quad \mu_t(r) \leftarrow \mu_{t-1}(r) + \max_{j \in S_r} g_t^2(j) \\ \quad \text{for } i = 1, \ldots, d \text{ do} \\ \quad \quad \nu_t(i) \leftarrow \min_{r:S_r \ni i} \mu_t(r) \\ \quad \quad w_{t+1}(i) \leftarrow w_t(i) - \eta \frac{g_t(i)}{\sqrt{\nu_t(i)}} \\ \quad \quad \text{with the convention that } 0/0 = 0 \end{array}\]
SM3-II Algorithm
The SM3-II optimizer initializes with parameters like the learning rate :math:eta and weight :math:w_1. It updates weights iteratively using gradients :math:g_t, adjusting each component with minimum accumulated values :math:nu’_t(i) and maintaining cumulative maximums :math:mu’_t(r) for subsets :math:S_r. SM3-II starts with an initial state :math:S_0 := (m_0, s_0) set to zero, storing estimates for first and second moments as pytrees matching model updates’ shape
\[\begin{array}{l} \text{parameters: learning rate } \eta \\ \text{initialize } w_1 = 0; \forall r \in [k]: \mu'_0(r) = 0 \\ \text{for } t = 1, \ldots, T \text{ do} \\ \quad \text{receive gradient } g_t = \nabla \ell_t(w_t) \\ \quad \text{initialize } \mu'_t(r) = 0 \text{ for all } r \in [k] \\ \quad \text{for } i = 1, \ldots, d \text{ do} \\ \quad \quad \nu'_t(i) \leftarrow \min_{r:S_r \ni i} \mu'_{t-1}(r) + g_t^2(i) \\ \quad \quad w_{t+1}(i) \leftarrow w_t(i) - \eta \frac{g_t(i)}{\sqrt{\nu'_t(i)}} \\ \quad \quad \text{with the convention that } 0/0 = 0 \\ \quad \text{for all } r : S_r \ni i \text{ do} \\ \quad \quad \mu'_t(r) \leftarrow \max\{\mu'_t(r), \nu'_t(i)\} \end{array}\]
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
momentum – Decay rate used by the momentum term (when it is not set to None, then momentum is not used at all).
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.sm3(learning_rate=0.003)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.40E+01
References
Anil et al, Memory-Efficient Adaptive Optimization, 2019
The Yogi optimizer.
Yogi is an adaptive optimizer, which provides control in tuning the effective learning rate to prevent it from increasing. By doing so, it focuses on addressing the issues of convergence and generalization in exponential moving average-based adaptive methods (such as Adam and RMSprop). Yogi is a modification of Adam and uses the same parameters.
learning_rate – A global scaling factor, either fixed or evolving along iterations with a scheduler, see optax.scale_by_learning_rate().
b1 – Exponential decay rate to track the first moment of past gradients.
b2 – Exponential decay rate to track the second moment of past gradients.
eps – A small constant applied to denominator outside of the square root (as in the Adam paper) to avoid dividing by zero when rescaling.
The corresponding optax.GradientTransformationExtraArgs.
Examples
>>> import optax
>>> import jax
>>> import jax.numpy as jnp
>>> def f(x): return jnp.sum(x ** 2) # simple quadratic function
>>> solver = optax.yogi(learning_rate=0.002)
>>> params = jnp.array([1., 2., 3.])
>>> print('Objective function: ', f(params))
Objective function: 14.0
>>> opt_state = solver.init(params)
>>> for _ in range(5):
... grad = jax.grad(f)(params)
... updates, opt_state = solver.update(grad, opt_state, params)
... params = optax.apply_updates(params, updates)
... print('Objective function: {:.2E}'.format(f(params)))
Objective function: 1.40E+01
Objective function: 1.40E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
Objective function: 1.39E+01
References
Zaheer et al, Adaptive Methods for Nonconvex Optimization, 2018
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