Source code for pymc.variational.updates

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"""
Functions to generate PyTensor update dictionaries for training.

The update functions implement different methods to control the learning
rate for use with stochastic gradient descent.

Update functions take a loss expression or a list of gradient expressions and
a list of parameters as input and return an ordered dictionary of updates:

.. autosummary::
    :nosignatures:

    sgd
    momentum
    nesterov_momentum
    adagrad
    rmsprop
    adadelta
    adam
    adamax

Two functions can be used to further modify the updates to include momentum:

.. autosummary::
    :nosignatures:

    apply_momentum
    apply_nesterov_momentum

Finally, we provide two helper functions to constrain the norm of tensors:

.. autosummary::
    :nosignatures:

    norm_constraint
    total_norm_constraint

:func:`norm_constraint()` can be used to constrain the norm of parameters
(as an alternative to weight decay), or for a form of gradient clipping.
:func:`total_norm_constraint()` constrain the total norm of a list of tensors.
This is often used when training recurrent neural networks.

Examples
--------
>>> import lasagne
>>> import pytensor
>>> from lasagne.nonlinearities import softmax
>>> from lasagne.layers import InputLayer, DenseLayer, get_output
>>> from lasagne.updates import sgd, apply_momentum
>>> l_in = InputLayer((100, 20))
>>> l1 = DenseLayer(l_in, num_units=3, nonlinearity=softmax)
>>> x = at.matrix('x')  # shp: num_batch x num_features
>>> y = at.ivector('y') # shp: num_batch
>>> l_out = get_output(l1, x)
>>> params = lasagne.layers.get_all_params(l1)
>>> loss = at.mean(at.nnet.categorical_crossentropy(l_out, y))
>>> updates_sgd = sgd(loss, params, learning_rate=0.0001)
>>> updates = apply_momentum(updates_sgd, params, momentum=0.9)
>>> train_function = pytensor.function([x, y], updates=updates)

Notes
-----
Taken from the Lasagne project: http://lasagne.readthedocs.io/en/latest/

"""
from collections import OrderedDict
from functools import partial

import numpy as np
import pytensor
import pytensor.tensor as at

import pymc as pm

__all__ = [
    "sgd",
    "apply_momentum",
    "momentum",
    "apply_nesterov_momentum",
    "nesterov_momentum",
    "adagrad",
    "adagrad_window",
    "rmsprop",
    "adadelta",
    "adam",
    "adamax",
    "norm_constraint",
    "total_norm_constraint",
]


def get_or_compute_grads(loss_or_grads, params):
    """Helper function returning a list of gradients

    Parameters
    ----------
    loss_or_grads: symbolic expression or list of expressions
        A scalar loss expression, or a list of gradient expressions
    params: list of shared variables
        The variables to return the gradients for

    Returns
    -------
    list of expressions
        If `loss_or_grads` is a list, it is assumed to be a list of
        gradients and returned as is, unless it does not match the length
        of `params`, in which case a `ValueError` is raised.
        Otherwise, `loss_or_grads` is assumed to be a cost expression and
        the function returns `pytensor.grad(loss_or_grads, params)`.

    Raises
    ------
    ValueError
        If `loss_or_grads` is a list of a different length than `params`, or if
        any element of `params` is not a shared variable (while we could still
        compute its gradient, we can never update it and want to fail early).
    """
    if any(not isinstance(p, pytensor.compile.SharedVariable) for p in params):
        raise ValueError(
            "params must contain shared variables only. If it "
            "contains arbitrary parameter expressions, then "
            "lasagne.utils.collect_shared_vars() may help you."
        )
    if isinstance(loss_or_grads, list):
        if not len(loss_or_grads) == len(params):
            raise ValueError(
                "Got %d gradient expressions for %d parameters" % (len(loss_or_grads), len(params))
            )
        return loss_or_grads
    else:
        return pytensor.grad(loss_or_grads, params)


def _get_call_kwargs(_locals_):
    _locals_ = _locals_.copy()
    _locals_.pop("loss_or_grads")
    _locals_.pop("params")
    return _locals_


[docs]def sgd(loss_or_grads=None, params=None, learning_rate=1e-3): """Stochastic Gradient Descent (SGD) updates Generates update expressions of the form: * ``param := param - learning_rate * gradient`` Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float or symbolic scalar The learning rate controlling the size of update steps Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- Optimizer can be called without both loss_or_grads and params in that case partial function is returned Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = sgd(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = sgd(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(sgd, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") grads = get_or_compute_grads(loss_or_grads, params) updates = OrderedDict() for param, grad in zip(params, grads): updates[param] = param - learning_rate * grad return updates
[docs]def apply_momentum(updates, params=None, momentum=0.9): """Returns a modified update dictionary including momentum Generates update expressions of the form: * ``velocity := momentum * velocity + updates[param] - param`` * ``param := param + velocity`` Parameters ---------- updates: OrderedDict A dictionary mapping parameters to update expressions params: iterable of shared variables, optional The variables to apply momentum to. If omitted, will apply momentum to all `updates.keys()`. momentum: float or symbolic scalar, optional The amount of momentum to apply. Higher momentum results in smoothing over more update steps. Defaults to 0.9. Returns ------- OrderedDict A copy of `updates` with momentum updates for all `params`. Notes ----- Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by `1 - momentum`. See Also -------- momentum: Shortcut applying momentum to SGD updates """ if params is None: params = updates.keys() updates = OrderedDict(updates) for param in params: value = param.get_value(borrow=True) velocity = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) x = momentum * velocity + updates[param] updates[velocity] = x - param updates[param] = x return updates
[docs]def momentum(loss_or_grads=None, params=None, learning_rate=1e-3, momentum=0.9): """Stochastic Gradient Descent (SGD) updates with momentum Generates update expressions of the form: * ``velocity := momentum * velocity - learning_rate * gradient`` * ``param := param + velocity`` Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float or symbolic scalar The learning rate controlling the size of update steps momentum: float or symbolic scalar, optional The amount of momentum to apply. Higher momentum results in smoothing over more update steps. Defaults to 0.9. Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by `1 - momentum`. Optimizer can be called without both loss_or_grads and params in that case partial function is returned See Also -------- apply_momentum: Generic function applying momentum to updates nesterov_momentum: Nesterov's variant of SGD with momentum Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = momentum(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = momentum(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(pm.updates.momentum, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") updates = sgd(loss_or_grads, params, learning_rate) return apply_momentum(updates, momentum=momentum)
[docs]def apply_nesterov_momentum(updates, params=None, momentum=0.9): """Returns a modified update dictionary including Nesterov momentum Generates update expressions of the form: * ``velocity := momentum * velocity + updates[param] - param`` * ``param := param + momentum * velocity + updates[param] - param`` Parameters ---------- updates: OrderedDict A dictionary mapping parameters to update expressions params: iterable of shared variables, optional The variables to apply momentum to. If omitted, will apply momentum to all `updates.keys()`. momentum: float or symbolic scalar, optional The amount of momentum to apply. Higher momentum results in smoothing over more update steps. Defaults to 0.9. Returns ------- OrderedDict A copy of `updates` with momentum updates for all `params`. Notes ----- Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by `1 - momentum`. The classic formulation of Nesterov momentum (or Nesterov accelerated gradient) requires the gradient to be evaluated at the predicted next position in parameter space. Here, we use the formulation described at https://github.com/lisa-lab/pylearn2/pull/136#issuecomment-10381617, which allows the gradient to be evaluated at the current parameters. See Also -------- nesterov_momentum: Shortcut applying Nesterov momentum to SGD updates """ if params is None: params = updates.keys() updates = OrderedDict(updates) for param in params: value = param.get_value(borrow=True) velocity = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) x = momentum * velocity + updates[param] - param updates[velocity] = x updates[param] = momentum * x + updates[param] return updates
[docs]def nesterov_momentum(loss_or_grads=None, params=None, learning_rate=1e-3, momentum=0.9): """Stochastic Gradient Descent (SGD) updates with Nesterov momentum Generates update expressions of the form: * ``velocity := momentum * velocity - learning_rate * gradient`` * ``param := param + momentum * velocity - learning_rate * gradient`` Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float or symbolic scalar The learning rate controlling the size of update steps momentum: float or symbolic scalar, optional The amount of momentum to apply. Higher momentum results in smoothing over more update steps. Defaults to 0.9. Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by `1 - momentum`. The classic formulation of Nesterov momentum (or Nesterov accelerated gradient) requires the gradient to be evaluated at the predicted next position in parameter space. Here, we use the formulation described at https://github.com/lisa-lab/pylearn2/pull/136#issuecomment-10381617, which allows the gradient to be evaluated at the current parameters. Optimizer can be called without both loss_or_grads and params in that case partial function is returned See Also -------- apply_nesterov_momentum: Function applying momentum to updates Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = nesterov_momentum(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = nesterov_momentum(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(nesterov_momentum, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") updates = sgd(loss_or_grads, params, learning_rate) return apply_nesterov_momentum(updates, momentum=momentum)
[docs]def adagrad(loss_or_grads=None, params=None, learning_rate=1.0, epsilon=1e-6): """Adagrad updates Scale learning rates by dividing with the square root of accumulated squared gradients. See [1]_ for further description. Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float or symbolic scalar The learning rate controlling the size of update steps epsilon: float or symbolic scalar Small value added for numerical stability Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- Using step size eta Adagrad calculates the learning rate for feature i at time step t as: .. math:: \\eta_{t,i} = \\frac{\\eta} {\\sqrt{\\sum^t_{t^\\prime} g^2_{t^\\prime,i}+\\epsilon}} g_{t,i} as such the learning rate is monotonically decreasing. Epsilon is not included in the typical formula, see [2]_. Optimizer can be called without both loss_or_grads and params in that case partial function is returned References ---------- .. [1] Duchi, J., Hazan, E., & Singer, Y. (2011): Adaptive subgradient methods for online learning and stochastic optimization. JMLR, 12:2121-2159. .. [2] Chris Dyer: Notes on AdaGrad. http://www.ark.cs.cmu.edu/cdyer/adagrad.pdf Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = adagrad(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = adagrad(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(adagrad, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") grads = get_or_compute_grads(loss_or_grads, params) updates = OrderedDict() for param, grad in zip(params, grads): value = param.get_value(borrow=True) accu = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) accu_new = accu + grad**2 updates[accu] = accu_new updates[param] = param - (learning_rate * grad / at.sqrt(accu_new + epsilon)) return updates
[docs]def adagrad_window(loss_or_grads=None, params=None, learning_rate=0.001, epsilon=0.1, n_win=10): """Returns a function that returns parameter updates. Instead of accumulated estimate, uses running window Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float Learning rate. epsilon: float Offset to avoid zero-division in the normalizer of adagrad. n_win: int Number of past steps to calculate scales of parameter gradients. Returns ------- OrderedDict A dictionary mapping each parameter to its update expression """ if loss_or_grads is None and params is None: return partial(adagrad_window, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") grads = get_or_compute_grads(loss_or_grads, params) updates = OrderedDict() for param, grad in zip(params, grads): i = pytensor.shared(pm.floatX(0)) i_int = i.astype("int32") value = param.get_value(borrow=True) accu = pytensor.shared(np.zeros(value.shape + (n_win,), dtype=value.dtype)) # Append squared gradient vector to accu_new accu_new = at.set_subtensor(accu[..., i_int], grad**2) i_new = at.switch((i + 1) < n_win, i + 1, 0) updates[accu] = accu_new updates[i] = i_new accu_sum = accu_new.sum(axis=-1) updates[param] = param - (learning_rate * grad / at.sqrt(accu_sum + epsilon)) return updates
[docs]def rmsprop(loss_or_grads=None, params=None, learning_rate=1.0, rho=0.9, epsilon=1e-6): """RMSProp updates Scale learning rates by dividing with the moving average of the root mean squared (RMS) gradients. See [1]_ for further description. Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float or symbolic scalar The learning rate controlling the size of update steps rho: float or symbolic scalar Gradient moving average decay factor epsilon: float or symbolic scalar Small value added for numerical stability Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- `rho` should be between 0 and 1. A value of `rho` close to 1 will decay the moving average slowly and a value close to 0 will decay the moving average fast. Using the step size :math:`\\eta` and a decay factor :math:`\\rho` the learning rate :math:`\\eta_t` is calculated as: .. math:: r_t &= \\rho r_{t-1} + (1-\\rho)*g^2\\\\ \\eta_t &= \\frac{\\eta}{\\sqrt{r_t + \\epsilon}} Optimizer can be called without both loss_or_grads and params in that case partial function is returned References ---------- .. [1] Tieleman, at. and Hinton, G. (2012): Neural Networks for Machine Learning, Lecture 6.5 - rmsprop. Coursera. http://www.youtube.com/watch?v=O3sxAc4hxZU (formula @5:20) Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = rmsprop(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = rmsprop(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(rmsprop, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") grads = get_or_compute_grads(loss_or_grads, params) updates = OrderedDict() # Using pytensor constant to prevent upcasting of float32 one = at.constant(1) for param, grad in zip(params, grads): value = param.get_value(borrow=True) accu = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) accu_new = rho * accu + (one - rho) * grad**2 updates[accu] = accu_new updates[param] = param - (learning_rate * grad / at.sqrt(accu_new + epsilon)) return updates
[docs]def adadelta(loss_or_grads=None, params=None, learning_rate=1.0, rho=0.95, epsilon=1e-6): r""" Adadelta updates Scale learning rates by the ratio of accumulated gradients to accumulated updates, see [1]_ and notes for further description. Parameters ---------- loss_or_grads : symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params : list of shared variables The variables to generate update expressions for learning_rate : float or symbolic scalar The learning rate controlling the size of update steps rho : float or symbolic scalar Squared gradient moving average decay factor epsilon : float or symbolic scalar Small value added for numerical stability Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- rho should be between 0 and 1. A value of rho close to 1 will decay the moving average slowly and a value close to 0 will decay the moving average fast. rho = 0.95 and epsilon=1e-6 are suggested in the paper and reported to work for multiple datasets (MNIST, speech). In the paper, no learning rate is considered (so learning_rate=1.0). Probably best to keep it at this value. epsilon is important for the very first update (so the numerator does not become 0). Using the step size eta and a decay factor rho the learning rate is calculated as: .. math:: r_t &= \\rho r_{t-1} + (1-\\rho)*g^2\\\\ \\eta_t &= \\eta \\frac{\\sqrt{s_{t-1} + \\epsilon}} {\sqrt{r_t + \epsilon}}\\\\ s_t &= \\rho s_{t-1} + (1-\\rho)*(\\eta_t*g)^2 Optimizer can be called without both loss_or_grads and params in that case partial function is returned References ---------- .. [1] Zeiler, M. D. (2012): ADADELTA: An Adaptive Learning Rate Method. arXiv Preprint arXiv:1212.5701. Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = adadelta(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = adadelta(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(adadelta, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") grads = get_or_compute_grads(loss_or_grads, params) updates = OrderedDict() # Using pytensor constant to prevent upcasting of float32 one = at.constant(1) for param, grad in zip(params, grads): value = param.get_value(borrow=True) # accu: accumulate gradient magnitudes accu = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) # delta_accu: accumulate update magnitudes (recursively!) delta_accu = pytensor.shared( np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape ) # update accu (as in rmsprop) accu_new = rho * accu + (one - rho) * grad**2 updates[accu] = accu_new # compute parameter update, using the 'old' delta_accu update = grad * at.sqrt(delta_accu + epsilon) / at.sqrt(accu_new + epsilon) updates[param] = param - learning_rate * update # update delta_accu (as accu, but accumulating updates) delta_accu_new = rho * delta_accu + (one - rho) * update**2 updates[delta_accu] = delta_accu_new return updates
[docs]def adam( loss_or_grads=None, params=None, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8 ): """Adam updates Adam updates implemented as in [1]_. Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float Learning rate beta1: float Exponential decay rate for the first moment estimates. beta2: float Exponential decay rate for the second moment estimates. epsilon: float Constant for numerical stability. Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- The paper [1]_ includes an additional hyperparameter lambda. This is only needed to prove convergence of the algorithm and has no practical use (personal communication with the authors), it is therefore omitted here. Optimizer can be called without both loss_or_grads and params in that case partial function is returned References ---------- .. [1] Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980. Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = adam(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = adam(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(adam, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") all_grads = get_or_compute_grads(loss_or_grads, params) t_prev = pytensor.shared(pm.pytensorf.floatX(0.0)) updates = OrderedDict() # Using pytensor constant to prevent upcasting of float32 one = at.constant(1) t = t_prev + 1 a_t = learning_rate * at.sqrt(one - beta2**t) / (one - beta1**t) for param, g_t in zip(params, all_grads): value = param.get_value(borrow=True) m_prev = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) v_prev = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) m_t = beta1 * m_prev + (one - beta1) * g_t v_t = beta2 * v_prev + (one - beta2) * g_t**2 step = a_t * m_t / (at.sqrt(v_t) + epsilon) updates[m_prev] = m_t updates[v_prev] = v_t updates[param] = param - step updates[t_prev] = t return updates
[docs]def adamax( loss_or_grads=None, params=None, learning_rate=0.002, beta1=0.9, beta2=0.999, epsilon=1e-8 ): """Adamax updates Adamax updates implemented as in [1]_. This is a variant of the Adam algorithm based on the infinity norm. Parameters ---------- loss_or_grads: symbolic expression or list of expressions A scalar loss expression, or a list of gradient expressions params: list of shared variables The variables to generate update expressions for learning_rate: float Learning rate beta1: float Exponential decay rate for the first moment estimates. beta2: float Exponential decay rate for the weighted infinity norm estimates. epsilon: float Constant for numerical stability. Returns ------- OrderedDict A dictionary mapping each parameter to its update expression Notes ----- Optimizer can be called without both loss_or_grads and params in that case partial function is returned References ---------- .. [1] Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980. Examples -------- >>> a = pytensor.shared(1.) >>> b = a*2 >>> updates = adamax(b, [a], learning_rate=.01) >>> isinstance(updates, dict) True >>> optimizer = adamax(learning_rate=.01) >>> callable(optimizer) True >>> updates = optimizer(b, [a]) >>> isinstance(updates, dict) True """ if loss_or_grads is None and params is None: return partial(adamax, **_get_call_kwargs(locals())) elif loss_or_grads is None or params is None: raise ValueError("Please provide both `loss_or_grads` and `params` to get updates") all_grads = get_or_compute_grads(loss_or_grads, params) t_prev = pytensor.shared(pm.pytensorf.floatX(0.0)) updates = OrderedDict() # Using pytensor constant to prevent upcasting of float32 one = at.constant(1) t = t_prev + 1 a_t = learning_rate / (one - beta1**t) for param, g_t in zip(params, all_grads): value = param.get_value(borrow=True) m_prev = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) u_prev = pytensor.shared(np.zeros(value.shape, dtype=value.dtype), shape=param.type.shape) m_t = beta1 * m_prev + (one - beta1) * g_t u_t = at.maximum(beta2 * u_prev, abs(g_t)) step = a_t * m_t / (u_t + epsilon) updates[m_prev] = m_t updates[u_prev] = u_t updates[param] = param - step updates[t_prev] = t return updates
[docs]def norm_constraint(tensor_var, max_norm, norm_axes=None, epsilon=1e-7): """Max weight norm constraints and gradient clipping This takes a TensorVariable and rescales it so that incoming weight norms are below a specified constraint value. Vectors violating the constraint are rescaled so that they are within the allowed range. Parameters ---------- tensor_var: TensorVariable PyTensor expression for update, gradient, or other quantity. max_norm: scalar This value sets the maximum allowed value of any norm in `tensor_var`. norm_axes: sequence (list or tuple) The axes over which to compute the norm. This overrides the default norm axes defined for the number of dimensions in `tensor_var`. When this is not specified and `tensor_var` is a matrix (2D), this is set to `(0,)`. If `tensor_var` is a 3D, 4D or 5D tensor, it is set to a tuple listing all axes but axis 0. The former default is useful for working with dense layers, the latter is useful for 1D, 2D and 3D convolutional layers. (Optional) epsilon: scalar, optional Value used to prevent numerical instability when dividing by very small or zero norms. Returns ------- TensorVariable Input `tensor_var` with rescaling applied to weight vectors that violate the specified constraints. Examples -------- >>> param = pytensor.shared( ... np.random.randn(100, 200).astype(pytensor.config.floatX)) >>> update = param + 100 >>> update = norm_constraint(update, 10) >>> func = pytensor.function([], [], updates=[(param, update)]) >>> # Apply constrained update >>> _ = func() >>> from lasagne.utils import compute_norms >>> norms = compute_norms(param.get_value()) >>> np.isclose(np.max(norms), 10) True Notes ----- When `norm_axes` is not specified, the axes over which the norm is computed depend on the dimensionality of the input variable. If it is 2D, it is assumed to come from a dense layer, and the norm is computed over axis 0. If it is 3D, 4D or 5D, it is assumed to come from a convolutional layer and the norm is computed over all trailing axes beyond axis 0. For other uses, you should explicitly specify the axes over which to compute the norm using `norm_axes`. """ ndim = tensor_var.ndim if norm_axes is not None: sum_over = tuple(norm_axes) elif ndim == 2: # DenseLayer sum_over = (0,) elif ndim in [3, 4, 5]: # Conv{1,2,3}DLayer sum_over = tuple(range(1, ndim)) else: raise ValueError( "Unsupported tensor dimensionality {}." "Must specify `norm_axes`".format(ndim) ) dtype = np.dtype(pytensor.config.floatX).type norms = at.sqrt(at.sum(at.sqr(tensor_var), axis=sum_over, keepdims=True)) target_norms = at.clip(norms, 0, dtype(max_norm)) constrained_output = tensor_var * (target_norms / (dtype(epsilon) + norms)) return constrained_output
[docs]def total_norm_constraint(tensor_vars, max_norm, epsilon=1e-7, return_norm=False): """Rescales a list of tensors based on their combined norm If the combined norm of the input tensors exceeds the threshold then all tensors are rescaled such that the combined norm is equal to the threshold. Scaling the norms of the gradients is often used when training recurrent neural networks [1]_. Parameters ---------- tensor_vars: List of TensorVariables. Tensors to be rescaled. max_norm: float Threshold value for total norm. epsilon: scalar, optional Value used to prevent numerical instability when dividing by very small or zero norms. return_norm: bool If true the total norm is also returned. Returns ------- tensor_vars_scaled: list of TensorVariables The scaled tensor variables. norm: PyTensor scalar The combined norms of the input variables prior to rescaling, only returned if ``return_norms=True``. Examples -------- >>> from lasagne.layers import InputLayer, DenseLayer >>> import lasagne >>> from lasagne.updates import sgd, total_norm_constraint >>> x = at.matrix() >>> y = at.ivector() >>> l_in = InputLayer((5, 10)) >>> l1 = DenseLayer(l_in, num_units=7, nonlinearity=at.special.softmax) >>> output = lasagne.layers.get_output(l1, x) >>> cost = at.mean(at.nnet.categorical_crossentropy(output, y)) >>> all_params = lasagne.layers.get_all_params(l1) >>> all_grads = at.grad(cost, all_params) >>> scaled_grads = total_norm_constraint(all_grads, 5) >>> updates = sgd(scaled_grads, all_params, learning_rate=0.1) Notes ----- The total norm can be used to monitor training. References ---------- .. [1] Sutskever, I., Vinyals, O., & Le, Q. V. (2014): Sequence to sequence learning with neural networks. In Advances in Neural Information Processing Systems (pp. 3104-3112). """ norm = at.sqrt(sum(at.sum(tensor**2) for tensor in tensor_vars)) dtype = np.dtype(pytensor.config.floatX).type target_norm = at.clip(norm, 0, dtype(max_norm)) multiplier = target_norm / (dtype(epsilon) + norm) tensor_vars_scaled = [step * multiplier for step in tensor_vars] if return_norm: return tensor_vars_scaled, norm else: return tensor_vars_scaled