# coding: utf-8 # Licensed to the Apache Software Foundation (ASF) under one # or more contributor license agreements. See the NOTICE file # distributed with this work for additional information # regarding copyright ownership. The ASF licenses this file # to you under the Apache License, Version 2.0 (the # "License"); you may not use this file except in compliance # with the License. You may obtain a copy of the License at # # http://www.apache.org/licenses/LICENSE-2.0 # # Unless required by applicable law or agreed to in writing, # software distributed under the License is distributed on an # "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY # KIND, either express or implied. See the License for the # specific language governing permissions and limitations # under the License. # pylint: disable=too-many-lines """Weight updating functions.""" from __future__ import absolute_import import logging import math import pickle import warnings import os import numpy from ..base import py_str from ..ndarray import (NDArray, zeros, clip, sqrt, cast, maximum, abs as NDabs, array, multiply, multi_sum_sq, multi_lars, norm as NDnorm) from ..ndarray import (sgd_update, sgd_mom_update, adam_update, rmsprop_update, rmspropalex_update, mp_sgd_update, mp_sgd_mom_update, square, ftrl_update, ftml_update, signsgd_update, signum_update, nag_mom_update, mp_nag_mom_update, multi_sgd_update, multi_sgd_mom_update, multi_mp_sgd_update, multi_mp_sgd_mom_update, preloaded_multi_sgd_update, preloaded_multi_sgd_mom_update, preloaded_multi_mp_sgd_update, preloaded_multi_mp_sgd_mom_update, lamb_update_phase1, lamb_update_phase2, mp_lamb_update_phase1, mp_lamb_update_phase2) from ..ndarray import sparse from ..random import normal from ..util import is_np_array __all__ = [ 'AdaDelta', 'AdaGrad', 'Adam', 'Adamax', 'DCASGD', 'FTML', 'Ftrl', 'LARS', 'LBSGD', 'NAG', 'NDabs', 'Nadam', 'Optimizer', 'RMSProp', 'SGD', 'SGLD', 'Signum', 'LAMB', 'Test', 'Updater', 'ccSGD', 'create', 'get_updater', 'register' ] def _flatten_list(nested_list): return [item for sublist in nested_list for item in sublist] class Optimizer(object): """The base class inherited by all optimizers. Parameters ---------- rescale_grad : float, optional, default 1.0 Multiply the gradient with `rescale_grad` before updating. Often choose to be ``1.0/batch_size``. param_idx2name : dict from int to string, optional, default None A dictionary that maps int index to string name. clip_gradient : float, optional, default None Clip the gradient by projecting onto the box ``[-clip_gradient, clip_gradient]``. learning_rate : float, optional, default None The initial learning rate. If None, the optimization will use the learning rate from ``lr_scheduler``. If not None, it will overwrite the learning rate in ``lr_scheduler``. If None and ``lr_scheduler`` is also None, then it will be set to 0.01 by default. lr_scheduler : LRScheduler, optional, default None The learning rate scheduler. wd : float, optional, default 0.0 The weight decay (or L2 regularization) coefficient. Modifies objective by adding a penalty for having large weights. sym: Symbol, optional, default None The Symbol this optimizer is applying to. begin_num_update : int, optional, default 0 The initial number of updates. multi_precision : bool, optional, default False Flag to control the internal precision of the optimizer. False: results in using the same precision as the weights (default), True: makes internal 32-bit copy of the weights and applies gradients in 32-bit precision even if actual weights used in the model have lower precision. Turning this on can improve convergence and accuracy when training with float16. param_dict : dict of int -> gluon.Parameter, default None Dictionary of parameter index to gluon.Parameter, used to lookup parameter attributes such as lr_mult, wd_mult, etc. param_dict shall not be deep copied. Properties ---------- learning_rate : float The current learning rate of the optimizer. Given an Optimizer object optimizer, its learning rate can be accessed as optimizer.learning_rate. """ def __init__(self, rescale_grad=1., param_idx2name=None, wd=0., clip_gradient=None, learning_rate=None, lr_scheduler=None, sym=None, begin_num_update=0, multi_precision=False, param_dict=None): self.rescale_grad = rescale_grad self.lr_scheduler = lr_scheduler if self.lr_scheduler is None and learning_rate is None: learning_rate = 0.01 self.lr = learning_rate if self.lr_scheduler is not None and learning_rate is not None: if self.lr_scheduler.base_lr != learning_rate: print(UserWarning("learning rate from ``lr_scheduler`` has been " "overwritten by ``learning_rate`` in optimizer.")) self.lr_scheduler.base_lr = learning_rate self.wd = wd self.lr_mult = {} self.wd_mult = {} self.begin_num_update = begin_num_update self.num_update = begin_num_update self._all_index_update_counts = {0 : {}} self._index_update_count = self._all_index_update_counts[0] self.clip_gradient = clip_gradient self.multi_precision = multi_precision self.aggregate_num = 0 if param_idx2name is None: param_idx2name = {} assert isinstance(param_idx2name, dict), \ 'param_idx2name should be a dict of param indexes to names.' self.idx2name = param_idx2name.copy() self.sym_info = (sym.attr_dict(), sym.list_arguments()) if sym is not None else () self.param_dict = param_dict if param_dict else {} self.allow_np_array = is_np_array() self.set_lr_mult({}) self.set_wd_mult({}) opt_registry = {} @staticmethod def register(klass): """Registers a new optimizer. Once an optimizer is registered, we can create an instance of this optimizer with `create_optimizer` later. Examples -------- >>> @mx.optimizer.Optimizer.register ... class MyOptimizer(mx.optimizer.Optimizer): ... pass >>> optim = mx.optimizer.Optimizer.create_optimizer('MyOptimizer') >>> print(type(optim)) """ assert(isinstance(klass, type)) name = klass.__name__.lower() if name in Optimizer.opt_registry: warnings.warn('WARNING: New optimizer %s.%s is overriding ' 'existing optimizer %s.%s' % (klass.__module__, klass.__name__, Optimizer.opt_registry[name].__module__, Optimizer.opt_registry[name].__name__)) Optimizer.opt_registry[name] = klass return klass @staticmethod def create_optimizer(name, **kwargs): """Instantiates an optimizer with a given name and kwargs. .. note:: We can use the alias `create` for ``Optimizer.create_optimizer``. Parameters ---------- name: str Name of the optimizer. Should be the name of a subclass of Optimizer. Case insensitive. kwargs: dict Parameters for the optimizer. Returns ------- Optimizer An instantiated optimizer. Examples -------- >>> sgd = mx.optimizer.Optimizer.create_optimizer('sgd') >>> type(sgd) >>> adam = mx.optimizer.create('adam', learning_rate=.1) >>> type(adam) """ if name.lower() in Optimizer.opt_registry: return Optimizer.opt_registry[name.lower()](**kwargs) else: raise ValueError('Cannot find optimizer %s' % name) @property def learning_rate(self): if self.lr_scheduler is not None: return self.lr_scheduler(self.num_update) else: return self.lr def create_state(self, index, weight): """Creates auxiliary state for a given weight. Some optimizers require additional states, e.g. as momentum, in addition to gradients in order to update weights. This function creates state for a given weight which will be used in `update`. This function is called only once for each weight. Parameters ---------- index : int An unique index to identify the weight. weight : NDArray The weight. Returns ------- state : any obj The state associated with the weight. """ def create_state_multi_precision(self, index, weight): """Creates auxiliary state for a given weight, including FP32 high precision copy if original weight is FP16. This method is provided to perform automatic mixed precision training for optimizers that do not support it themselves. Parameters ---------- index : int An unique index to identify the weight. weight : NDArray The weight. Returns ------- state : any obj The state associated with the weight. """ weight_master_copy = None if self.multi_precision and weight.dtype == numpy.float16: weight_master_copy = weight.astype(numpy.float32) return (weight_master_copy,) + (self.create_state(index, weight_master_copy),) if weight.dtype == numpy.float16 and not self.multi_precision: warnings.warn("Accumulating with float16 in optimizer can lead to " "poor accuracy or slow convergence. " "Consider using multi_precision=True option of the " "optimizer") return self.create_state(index, weight) def update(self, index, weight, grad, state): """Updates the given parameter using the corresponding gradient and state. Parameters ---------- index : int The unique index of the parameter into the individual learning rates and weight decays. Learning rates and weight decay may be set via `set_lr_mult()` and `set_wd_mult()`, respectively. weight : NDArray The parameter to be updated. grad : NDArray The gradient of the objective with respect to this parameter. state : any obj The state returned by `create_state()`. """ raise NotImplementedError() def update_multi_precision(self, index, weight, grad, state): """Updates the given parameter using the corresponding gradient and state. Mixed precision version. Parameters ---------- index : int The unique index of the parameter into the individual learning rates and weight decays. Learning rates and weight decay may be set via `set_lr_mult()` and `set_wd_mult()`, respectively. weight : NDArray The parameter to be updated. grad : NDArray The gradient of the objective with respect to this parameter. state : any obj The state returned by `create_state()`. """ if self.multi_precision and weight.dtype == numpy.float16: # Wrapper for mixed precision weight_master_copy = state[0] original_state = state[1] grad32 = grad.astype(numpy.float32) self.update(index, weight_master_copy, grad32, original_state) cast(weight_master_copy, dtype=weight.dtype, out=weight) else: self.update(index, weight, grad, state) def set_learning_rate(self, lr): """Sets a new learning rate of the optimizer. Parameters ---------- lr : float The new learning rate of the optimizer. """ if self.lr_scheduler is not None: # pylint: disable=no-else-raise raise UserWarning("LRScheduler of the optimizer has already been " "defined. Note that set_learning_rate can mutate " "the value of the learning rate of the optimizer " "only when the LRScheduler of the optimizer is " "undefined.") else: self.lr = lr def set_lr_scale(self, args_lrscale): # pylint: disable=unused-argument """[DEPRECATED] Sets lr scale. Use set_lr_mult instead.""" raise DeprecationWarning def set_lr_mult(self, args_lr_mult): """Sets an individual learning rate multiplier for each parameter. If you specify a learning rate multiplier for a parameter, then the learning rate for the parameter will be set as the product of the global learning rate `self.lr` and its multiplier. .. note:: The default learning rate multiplier of a `Variable` can be set with `lr_mult` argument in the constructor. Parameters ---------- args_lr_mult : dict of str/int to float For each of its key-value entries, the learning rate multipler for the parameter specified in the key will be set as the given value. You can specify the parameter with either its name or its index. If you use the name, you should pass `sym` in the constructor, and the name you specified in the key of `args_lr_mult` should match the name of the parameter in `sym`. If you use the index, it should correspond to the index of the parameter used in the `update` method. Specifying a parameter by its index is only supported for backward compatibility, and we recommend to use the name instead. """ self.lr_mult = {} if self.sym_info: attr, arg_names = self.sym_info for name in arg_names: if name in attr and '__lr_mult__' in attr[name]: self.lr_mult[name] = float(attr[name]['__lr_mult__']) self.lr_mult.update(args_lr_mult) def set_wd_mult(self, args_wd_mult): """Sets an individual weight decay multiplier for each parameter. By default, if `param_idx2name` was provided in the constructor, the weight decay multipler is set as 0 for all parameters whose name don't end with ``_weight`` or ``_gamma``. .. note:: The default weight decay multiplier for a `Variable` can be set with its `wd_mult` argument in the constructor. Parameters ---------- args_wd_mult : dict of string/int to float For each of its key-value entries, the weight decay multipler for the parameter specified in the key will be set as the given value. You can specify the parameter with either its name or its index. If you use the name, you should pass `sym` in the constructor, and the name you specified in the key of `args_lr_mult` should match the name of the parameter in `sym`. If you use the index, it should correspond to the index of the parameter used in the `update` method. Specifying a parameter by its index is only supported for backward compatibility, and we recommend to use the name instead. """ self.wd_mult = {} for n in self.idx2name.values(): if not (n.endswith('_weight') or n.endswith('_gamma')): self.wd_mult[n] = 0.0 if self.sym_info: attr, arg_names = self.sym_info for name in arg_names: if name in attr and '__wd_mult__' in attr[name]: self.wd_mult[name] = float(attr[name]['__wd_mult__']) self.wd_mult.update(args_wd_mult) def _set_current_context(self, device_id): """Sets the number of the currently handled device. Parameters ---------- device_id : int The number of current device. """ if device_id not in self._all_index_update_counts: self._all_index_update_counts[device_id] = {} self._index_update_count = self._all_index_update_counts[device_id] def _update_count(self, index): """Updates num_update. Parameters ---------- index : int or list of int The index to be updated. """ if not isinstance(index, (list, tuple)): index = [index] for idx in index: if idx not in self._index_update_count: self._index_update_count[idx] = self.begin_num_update self._index_update_count[idx] += 1 self.num_update = max(self._index_update_count[idx], self.num_update) def _get_lrs(self, indices): """Gets the learning rates given the indices of the weights. Parameters ---------- indices : list of int Indices corresponding to weights. Returns ------- lrs : list of float Learning rates for those indices. """ if self.lr_scheduler is not None: lr = self.lr_scheduler(self.num_update) else: lr = self.lr lrs = [lr for _ in indices] for i, index in enumerate(indices): if index in self.param_dict: lrs[i] *= self.param_dict[index].lr_mult elif index in self.lr_mult: lrs[i] *= self.lr_mult[index] elif index in self.idx2name: lrs[i] *= self.lr_mult.get(self.idx2name[index], 1.0) return lrs def _get_lr(self, index): """Gets the learning rate given the index of the weight. Parameters ---------- index : int The index corresponding to the weight. Returns ------- lr : float Learning rate for this index. """ return self._get_lrs([index])[0] def _get_wds(self, indices): """Gets weight decays for indices. Returns 0 for non-weights if the name of weights are provided for `__init__`. Parameters ---------- indices : list of int Indices of weights. Returns ------- wds : list of float Weight decays for those indices. """ wds = [self.wd for _ in indices] for i, index in enumerate(indices): if index in self.param_dict: wds[i] *= self.param_dict[index].wd_mult elif index in self.wd_mult: wds[i] *= self.wd_mult[index] elif index in self.idx2name: wds[i] *= self.wd_mult.get(self.idx2name[index], 1.0) return wds def _get_wd(self, index): """Gets weight decay for index. Returns 0 for non-weights if the name of weights are provided for `__init__`. Parameters ---------- index : int The index of weight. Returns ------- wd : float Weight decay for this index. """ return self._get_wds([index])[0] def __getstate__(self): ret = self.__dict__.copy() # do not include param_dict in the state del ret['param_dict'] return ret def __setstate__(self, state): self.__dict__ = state # param_dict needs to be explicitly set by the trainer self.param_dict = {} # convenience wrapper for Optimizer.Register register = Optimizer.register # pylint: disable=invalid-name # pylint: disable=line-too-long @register class SGD(Optimizer): """The SGD optimizer with momentum and weight decay. If the storage types of grad is ``row_sparse`` and ``lazy_update`` is True, \ **lazy updates** are applied by:: for row in grad.indices: rescaled_grad[row] = lr * (rescale_grad * clip(grad[row], clip_gradient) + wd * weight[row]) state[row] = momentum[row] * state[row] + rescaled_grad[row] weight[row] = weight[row] - state[row] The sparse update only updates the momentum for the weights whose row_sparse gradient indices appear in the current batch, rather than updating it for all indices. Compared with the original update, it can provide large improvements in model training throughput for some applications. However, it provides slightly different semantics than the original update, and may lead to different empirical results. In the case when ``update_on_kvstore`` is set to False (either globally via MXNET_UPDATE_ON_KVSTORE=0 environment variable or as a parameter in :class:`~mxnet.gluon.Trainer`) SGD optimizer can perform aggregated update of parameters, which may lead to improved performance. The aggregation size is controlled by MXNET_OPTIMIZER_AGGREGATION_SIZE environment variable and defaults to 4. Otherwise, **standard updates** are applied by:: rescaled_grad = lr * (rescale_grad * clip(grad, clip_gradient) + wd * weight) state = momentum * state + rescaled_grad weight = weight - state For details of the update algorithm see :class:`~mxnet.ndarray.sgd_update` and :class:`~mxnet.ndarray.sgd_mom_update`. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- momentum : float, optional The momentum value. lazy_update : bool, optional Default is True. If True, lazy updates are applied \ if the storage types of weight and grad are both ``row_sparse``. multi_precision: bool, optional Flag to control the internal precision of the optimizer. False: results in using the same precision as the weights (default), True: makes internal 32-bit copy of the weights and applies gradients in 32-bit precision even if actual weights used in the model have lower precision. Turning this on can improve convergence and accuracy when training with float16. """ def __init__(self, momentum=0.0, lazy_update=True, **kwargs): super(SGD, self).__init__(**kwargs) self.momentum = momentum self.lazy_update = lazy_update self.aggregate_num = int(os.getenv('MXNET_OPTIMIZER_AGGREGATION_SIZE', "4")) def create_state_multi_precision(self, index, weight): weight_master_copy = None if self.multi_precision and weight.dtype == numpy.float16: weight_master_copy = weight.astype(numpy.float32) return (self.create_state(index, weight_master_copy), weight_master_copy) if weight.dtype == numpy.float16 and not self.multi_precision: warnings.warn("Accumulating with float16 in optimizer can lead to " "poor accuracy or slow convergence. " "Consider using multi_precision=True option of the " "SGD optimizer") return self.create_state(index, weight) def create_state(self, index, weight): momentum = None if self.momentum != 0.0: stype = weight.stype if self.lazy_update else 'default' momentum = zeros(weight.shape, weight.context, dtype=weight.dtype, stype=stype) return momentum def _update_impl(self, indices, weights, grads, states, multi_precision=False): aggregate = True if not isinstance(indices, (tuple, list)): indices = [indices] weights = [weights] grads = [grads] states = [states] for weight, grad in zip(weights, grads): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) aggregate = (aggregate and weight.stype == 'default' and grad.stype == 'default') self._update_count(indices) lrs = self._get_lrs(indices) wds = self._get_wds(indices) kwargs = {'rescale_grad': self.rescale_grad} if self.momentum > 0: kwargs['momentum'] = self.momentum if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if aggregate: if not multi_precision: if self.momentum > 0: multi_sgd_mom_update(*_flatten_list(zip(weights, grads, states)), out=weights, num_weights=len(weights), lrs=lrs, wds=wds, **kwargs) else: multi_sgd_update(*_flatten_list(zip(weights, grads)), out=weights, num_weights=len(weights), lrs=lrs, wds=wds, **kwargs) else: if self.momentum > 0: multi_mp_sgd_mom_update(*_flatten_list(zip(weights, grads, *zip(*states))), out=weights, num_weights=len(weights), lrs=lrs, wds=wds, **kwargs) else: multi_mp_sgd_update(*_flatten_list(zip(weights, grads, list(zip(*states))[1])), out=weights, num_weights=len(weights), lrs=lrs, wds=wds, **kwargs) else: for weight, grad, state, lr, wd in zip(weights, grads, states, lrs, wds): if not multi_precision: if state is not None: sgd_mom_update(weight, grad, state, out=weight, lazy_update=self.lazy_update, lr=lr, wd=wd, **kwargs) else: sgd_update(weight, grad, out=weight, lazy_update=self.lazy_update, lr=lr, wd=wd, **kwargs) else: if state[0] is not None: mp_sgd_mom_update(weight, grad, state[0], state[1], out=weight, lr=lr, wd=wd, **kwargs) else: mp_sgd_update(weight, grad, state[1], out=weight, lr=lr, wd=wd, **kwargs) def update(self, index, weight, grad, state): self._update_impl(index, weight, grad, state, multi_precision=False) def update_multi_precision(self, index, weight, grad, state): if not isinstance(index, (tuple, list)): use_multi_precision = self.multi_precision and weight.dtype == numpy.float16 else: use_multi_precision = self.multi_precision and weight[0].dtype == numpy.float16 self._update_impl(index, weight, grad, state, multi_precision=use_multi_precision) @register class Signum(Optimizer): r"""The Signum optimizer that takes the sign of gradient or momentum. The optimizer updates the weight by:: rescaled_grad = rescale_grad * clip(grad, clip_gradient) + wd * weight state = momentum * state + (1-momentum)*rescaled_grad weight = (1 - lr * wd_lh) * weight - lr * sign(state) References ---------- Jeremy Bernstein, Yu-Xiang Wang, Kamyar Azizzadenesheli & Anima Anandkumar. (2018). signSGD: Compressed Optimisation for Non-Convex Problems. In ICML'18. See: https://arxiv.org/abs/1802.04434 For details of the update algorithm see :class:`~mxnet.ndarray.signsgd_update` and :class:`~mxnet.ndarray.signum_update`. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- momentum : float, optional The momentum value. wd_lh : float, optional The amount of decoupled weight decay regularization, see details in the original paper at:\ https://arxiv.org/abs/1711.05101 """ def __init__(self, learning_rate=0.01, momentum=0.9, wd_lh=0.0, **kwargs): super(Signum, self).__init__(learning_rate=learning_rate, **kwargs) self.momentum = momentum self.wd_lh = wd_lh def create_state(self, index, weight): momentum = None if self.momentum != 0.0: momentum = zeros(weight.shape, weight.context, dtype=weight.dtype, stype=weight.stype) return momentum def _update_impl(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) kwargs = {'rescale_grad': self.rescale_grad} if self.momentum > 0: kwargs['momentum'] = self.momentum if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if self.wd_lh: kwargs['wd_lh'] = self.wd_lh if state is not None: signum_update(weight, grad, state, out=weight, lr=lr, wd=wd, **kwargs) else: signsgd_update(weight, grad, out=weight, lr=lr, wd=wd, **kwargs) def update(self, index, weight, grad, state): self._update_impl(index, weight, grad, state) @register class FTML(Optimizer): """The FTML optimizer. This class implements the optimizer described in *FTML - Follow the Moving Leader in Deep Learning*, available at http://proceedings.mlr.press/v70/zheng17a/zheng17a.pdf. Denote time step by t. The optimizer updates the weight by:: rescaled_grad = clip(grad * rescale_grad + wd * weight, clip_gradient) v = beta2 * v + (1 - beta2) * square(rescaled_grad) d_t = (1 - power(beta1, t)) / lr * square_root(v / (1 - power(beta2, t))) + epsilon) z = beta1 * z + (1 - beta1) * rescaled_grad - (d_t - beta1 * d_(t-1)) * weight weight = - z / d_t For details of the update algorithm, see :class:`~mxnet.ndarray.ftml_update`. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- beta1 : float, optional 0 < beta1 < 1. Generally close to 0.5. beta2 : float, optional 0 < beta2 < 1. Generally close to 1. epsilon : float, optional Small value to avoid division by 0. """ def __init__(self, beta1=0.6, beta2=0.999, epsilon=1e-8, **kwargs): super(FTML, self).__init__(**kwargs) self.beta1 = beta1 self.beta2 = beta2 self.epsilon = epsilon def create_state(self, index, weight): return (zeros(weight.shape, weight.context, dtype=weight.dtype), # d_0 zeros(weight.shape, weight.context, dtype=weight.dtype), # v_0 zeros(weight.shape, weight.context, dtype=weight.dtype)) # z_0 def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) t = self._index_update_count[index] kwargs = {'beta1': self.beta1, 'beta2': self.beta2, 'epsilon': self.epsilon, 'rescale_grad': self.rescale_grad, 't': t} if self.clip_gradient: kwargs['clip_grad'] = self.clip_gradient prev_d, prev_v, prev_z = state ftml_update(weight, grad, prev_d, prev_v, prev_z, out=weight, lr=lr, wd=wd, **kwargs) @register class LARS(Optimizer): """the LARS optimizer from 'Large Batch Training of Convolution Networks' \ (https://arxiv.org/abs/1708.03888) Behave mostly like SGD with momentum and weight decay but is scaling \ adaptively the learning for each layer (except bias and batch norm parameters): w_norm = L2norm(weights) g_norm = L2norm(gradients) if w_norm > 0 and g_norm > 0: lr_layer = lr * lr_mult * eta * w_norm / (g_norm + weight_decay * w_norm + eps) else: lr_layer = lr * lr_mult Parameters ---------- momentum : float, optional The momentum value. lazy_update : bool, optional Default is True. If True, lazy updates are applied \ if the storage types of weight and grad are both ``row_sparse``. lars_eta : float, optional LARS coefficient used to scale the learning rate. Default set to 0.001. lars_epsilon : float, optional Optional epsilon in case of very small gradients. Default set to 0. momentum_correction : bool, optional If True scale momentum w.r.t global learning rate change (with an lr_scheduler) \ as indicated in 'Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour` \ (https://arxiv.org/pdf/1706.02677.pdf) Default set to True. """ def __init__(self, momentum=0.0, lazy_update=True, eta=0.001, eps=0, momentum_correction=True, **kwargs): super(LARS, self).__init__(**kwargs) self.momentum = momentum self.momentum_correction = momentum_correction self.lazy_update = lazy_update self.aggregate_num = int(os.getenv('MXNET_OPTIMIZER_AGGREGATION_SIZE', "4")) self.eta = eta self.eps = eps self.skip = 0 self.last_lr = None self.cur_lr = None def _get_lrs(self, indices): """Gets the learning rates given the indices of the weights. Parameters ---------- indices : list of int Indices corresponding to weights. Returns ------- lrs : list of float Learning rates for those indices. """ if self.cur_lr is not None: self.last_lr = self.cur_lr if self.lr_scheduler is not None: lr = self.lr_scheduler(self.num_update) else: lr = self.lr if self.cur_lr is None: self.last_lr = lr self.cur_lr = lr lrs = [lr for _ in indices] for i, index in enumerate(indices): if index in self.param_dict: lrs[i] *= self.param_dict[index].lr_mult elif index in self.lr_mult: lrs[i] *= self.lr_mult[index] elif index in self.idx2name: lrs[i] *= self.lr_mult.get(self.idx2name[index], 1.0) return lrs def set_wd_mult(self, args_wd_mult): self.wd_mult = {} for n in self.idx2name.values(): is_weight = n.endswith('_weight') if not is_weight: self.wd_mult[n] = 0.0 if self.sym_info: attr, arg_names = self.sym_info for name in arg_names: if name in attr and '__wd_mult__' in attr[name]: self.wd_mult[name] = float(attr[name]['__wd_mult__']) self.wd_mult.update(args_wd_mult) def create_state_multi_precision(self, index, weight): weight_master_copy = None if self.multi_precision and weight.dtype == numpy.float16: weight_master_copy = weight.astype(numpy.float32) return (self.create_state(index, weight_master_copy), weight_master_copy) if weight.dtype == numpy.float16 and not self.multi_precision: warnings.warn("Accumulating with float16 in optimizer can lead to " "poor accuracy or slow convergence. " "Consider using multi_precision=True option of the " "SGD optimizer") return self.create_state(index, weight) def create_state(self, index, weight): momentum = None if self.momentum != 0.0: stype = weight.stype if self.lazy_update else 'default' momentum = zeros(weight.shape, weight.context, dtype=weight.dtype, stype=stype) return momentum def _l2norm(self, v, rescale=False): """L2 Norm implementation""" v = v.astype('float32') if rescale: v *= self.rescale_grad norm = NDnorm(v).asnumpy()[0] return norm def _get_lars(self, i, weight, g, lr, wd): """Returns a scaling factor for the learning rate for this layer""" name = self.idx2name[i] if i in self.idx2name else str(i) if name.endswith('gamma') or name.endswith('beta') or name.endswith('bias'): return lr w_norm = self._l2norm(weight) g_norm = self._l2norm(g, rescale=True) if w_norm > 0.0 and g_norm > 0.0: lars = self.eta * w_norm/(g_norm + wd * w_norm + self.eps) else: lars = 1.0 return lars * lr def _update_impl(self, indices, weights, grads, states, multi_precision=False): aggregate = True if not isinstance(indices, (tuple, list)): indices = [indices] weights = [weights] grads = [grads] states = [states] for weight, grad in zip(weights, grads): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) aggregate = (aggregate and weight.stype == 'default' and grad.stype == 'default') self._update_count(indices) lrs = self._get_lrs(indices) wds = self._get_wds(indices) kwargs = {'rescale_grad': self.rescale_grad} if self.momentum > 0: kwargs['momentum'] = (self.momentum * (self.cur_lr / self.last_lr)) \ if (self.momentum_correction and self.last_lr != 0) else \ self.momentum if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if aggregate: nb_params = len(indices) names = [self.idx2name[i] if i in self.idx2name else str(i) for i in indices] lars_idx = [i for i in range(nb_params) if not(names[i].endswith('gamma') or names[i].endswith('beta') or names[i].endswith('bias'))] nb_lars = len(lars_idx) no_lars_idx = [i for i in range(nb_params) if (names[i].endswith('gamma') or names[i].endswith('beta') or names[i].endswith('bias'))] cur_ctx = weights[0].context full_idx = lars_idx + no_lars_idx new_lrs = array([lrs[i] for i in full_idx], ctx=cur_ctx, dtype='float32') new_wds = array([wds[i] for i in full_idx], ctx=cur_ctx, dtype='float32') new_weights = [weights[i] for i in full_idx] new_grads = [grads[i] for i in full_idx] new_states = [states[i] for i in full_idx] if nb_lars > 0: w_sum_sq = multi_sum_sq(*new_weights[:nb_lars], num_arrays=nb_lars) g_sum_sq = multi_sum_sq(*new_grads[:nb_lars], num_arrays=nb_lars) multi_lars(new_lrs[:nb_lars], w_sum_sq, g_sum_sq, new_wds[:nb_lars], eta=self.eta, eps=self.eps, rescale_grad=self.rescale_grad, out=new_lrs[:nb_lars]) # Same than usual using preloaded sgd functions sidx = 0 while sidx < len(indices): eidx = sidx + len(new_weights[sidx:sidx+self.aggregate_num]) if not multi_precision: if self.momentum > 0: preloaded_multi_sgd_mom_update( *(_flatten_list(zip(new_weights[sidx:eidx], new_grads[sidx:eidx], new_states[sidx:eidx])) + [new_lrs[sidx:eidx], new_wds[sidx:eidx]]), out=new_weights[sidx:eidx], num_weights=len(new_weights[sidx:eidx]), **kwargs) else: preloaded_multi_sgd_update( *(_flatten_list(zip(new_weights[sidx:eidx], new_grads[sidx:eidx])) + [new_lrs[sidx:eidx], new_wds[sidx:eidx]]), out=new_weights[sidx:eidx], num_weights=len(new_weights[sidx:eidx]), **kwargs) else: if self.momentum > 0: preloaded_multi_mp_sgd_mom_update( *(_flatten_list(zip(new_weights[sidx:eidx], new_grads[sidx:eidx], *zip(*new_states[sidx:eidx]))) + [new_lrs[sidx:eidx], new_wds[sidx:eidx]]), out=new_weights[sidx:eidx], num_weights=len(new_weights[sidx:eidx]), **kwargs) else: preloaded_multi_mp_sgd_update( *(_flatten_list(zip(new_weights[sidx:eidx], new_grads[sidx:eidx], list(zip(*new_states[sidx:eidx]))[1])) + [new_lrs[sidx:eidx], new_wds[sidx:eidx]]), out=new_weights[sidx:eidx], num_weights=len(new_weights[sidx:eidx]), **kwargs) sidx += self.aggregate_num else: lrs = [self._get_lars(i, w, g, lr, wd) for (i, w, g, lr, wd) in zip(indices, weights, grads, lrs, wds)] for weight, grad, state, lr, wd in zip(weights, grads, states, lrs, wds): if not multi_precision: if state is not None: sgd_mom_update(weight, grad, state, out=weight, lazy_update=self.lazy_update, lr=lr, wd=wd, **kwargs) else: sgd_update(weight, grad, out=weight, lazy_update=self.lazy_update, lr=lr, wd=wd, **kwargs) else: if state[0] is not None: mp_sgd_mom_update(weight, grad, state[0], state[1], out=weight, lr=lr, wd=wd, **kwargs) else: mp_sgd_update(weight, grad, state[1], out=weight, lr=lr, wd=wd, **kwargs) def update(self, index, weight, grad, state): self._update_impl(index, weight, grad, state, multi_precision=False) def update_multi_precision(self, index, weight, grad, state): if not isinstance(index, (tuple, list)): use_multi_precision = self.multi_precision and weight.dtype == numpy.float16 else: use_multi_precision = self.multi_precision and weight[0].dtype == numpy.float16 self._update_impl(index, weight, grad, state, multi_precision=use_multi_precision) # @register class LBSGD(Optimizer): """The Large Batch SGD optimizer with momentum and weight decay. The optimizer updates the weight by:: state = momentum * state + lr * rescale_grad * clip(grad, clip_gradient) + wd * weight weight = weight - state For details of the update algorithm see :class:`~mxnet.ndarray.sgd_update` and :class:`~mxnet.ndarray.sgd_mom_update`. In addition to the SGD updates the LBSGD optimizer uses the LARS, Layer-wise Adaptive Rate Scaling, algorithm to have a separate learning rate for each layer of the network, which leads to better stability over large batch sizes. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- momentum : float, optional The momentum value. multi_precision: bool, optional Flag to control the internal precision of the optimizer. False: results in using the same precision as the weights (default), True: makes internal 32-bit copy of the weights and applies gradients in 32-bit precision even if actual weights used in the model have lower precision. Turning this on can improve convergence and accuracy when training with float16. warmup_strategy: string ('linear', 'power2', 'sqrt'. , 'lars' default : 'linear') warmup_epochs: unsigned, default: 5 batch_scale: unsigned, default: 1 (same as batch size * numworkers) updates_per_epoch: updates_per_epoch (default: 32, Default might not reflect true number batches per epoch. Used for warmup.) begin_epoch: unsigned, default 0, starting epoch. """ def __init__(self, momentum=0.0, multi_precision=False, warmup_strategy='linear', warmup_epochs=5, batch_scale=1, updates_per_epoch=32, begin_epoch=0, num_epochs=60, **kwargs): super(LBSGD, self).__init__(**kwargs) logging.info('Running Large-Batch SGD Algorithm') logging.info('(Batch_scale=%f, warmup_epochs=%d, warmup_strategy=%s, updates_per_epoch=%d)', batch_scale, warmup_epochs, warmup_strategy, updates_per_epoch) self.momentum = momentum self.multi_precision = multi_precision # new user parameters for large batch self.warmup_strategy = warmup_strategy self.warmup_epochs = warmup_epochs self.batch_scale = batch_scale self.updates_per_epoch = updates_per_epoch self.init_updates = begin_epoch * updates_per_epoch self.num_epochs = num_epochs # addl internal usage parameters and storage self.lbmult = 1 self.cumgrads = {} # for adaptive lr self.adaptive = False self.admult = 1 # adaptation constant def create_state(self, index, weight): momentum = None weight_master_copy = None if self.multi_precision and weight.dtype == numpy.float16: weight_master_copy = array(weight, ctx=weight.context, dtype=numpy.float32) if self.momentum != 0.0: momentum = zeros(weight.shape, weight.context, dtype=numpy.float32, stype=weight.stype) return (momentum, weight_master_copy) if weight.dtype == numpy.float16 and not self.multi_precision: warnings.warn("Accumulating with float16 in optimizer can lead to " "poor accuracy or slow convergence. " "Consider using multi_precision=True option of the " "SGD optimizer") if self.momentum != 0.0: momentum = zeros(weight.shape, weight.context, dtype=weight.dtype, stype=weight.stype) return momentum def _get_lbmult(self, nup): """Returns lr scaling factor for large batch according to warmup schedule (to be implemented) """ nwup = self.warmup_epochs * self.updates_per_epoch strategy = self.warmup_strategy maxmult = float(self.batch_scale) if nup >= nwup: mult = maxmult elif nwup <= 1: mult = 1.0 else: if (strategy == 'linear'): mult = 1.0 + (maxmult - 1) * nup / nwup elif (strategy == 'power2'): mult = 1.0 + (maxmult-1) * (nup*nup)/(nwup*nwup) elif (strategy == 'sqrt'): mult = 1.0 + (maxmult - 1) * math.sqrt(float(nup) / nwup) else: mult = 1.0 return mult def _get_lars(self, weight, g, wd): """Returns a scaling factor for the learning rate for this layer default is 1 """ weight2 = self._l2norm(weight) grad2 = self._l2norm(g) lars = math.sqrt(weight2 / (grad2 + wd * weight2 + 1e-18)) if lars < 0.01: lars = 0.01 elif lars > 100: lars = 100 return lars def _l2norm(self, v): "inner product implementation" norm = multiply(v, v).asnumpy().sum() return norm def _reset_cum_gradient(self, index): "called every macro-batch to reset cumulated gradients to 0 for a given index" self.cumgrads[index]['cum_grad'] = 0 def _get_cum_gradient(self, index): "get the cumulated gradient for index" if index in self.cumgrads: return self.cumgrads[index] else: return {} def _put_cum_gradient(self, index, cgrad): "store cumulated gradient for index" self.cumgrads[index] = cgrad def _cumulate_gradient(self, grad, index): "Cumulate gradients for large-batch emulation. Cumulated by index (layer)" cgrad = self._get_cum_gradient(index) if cgrad: num_cums = cgrad['num_cums'] if num_cums > 0: cum_grad = cgrad['cum_grad'] + grad num_cums += 1 else: cum_grad = grad num_cums = self.init_updates + 1 else: cum_grad = grad num_cums = self.init_updates + 1 cgrad = {'cum_grad': cum_grad, 'num_cums': num_cums} self._put_cum_gradient(index, cgrad) return cgrad def update(self, index, weight, grad, state): assert (isinstance(weight, NDArray)) assert (isinstance(grad, NDArray)) lr = self._get_lr(index) wd = self._get_wd(index) self._update_count(index) # new stuff for large batch cgrad = self._cumulate_gradient(grad, index) if (cgrad['num_cums'] % self.batch_scale) == 0: grad = cgrad['cum_grad'] / self.batch_scale if self.warmup_strategy == 'lars': lbmult = self._get_lars(weight, grad, wd) else: lbmult = self._get_lbmult(cgrad['num_cums']) lr = lr * lbmult # do the regular sgd update flow kwargs = {'rescale_grad': self.rescale_grad} if self.momentum > 0: kwargs['momentum'] = self.momentum if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient use_multi_precision = isinstance(state, (list, tuple)) if not use_multi_precision: if state is not None: sgd_mom_update(weight, grad, state, out=weight, lr=lr, wd=wd, **kwargs) else: sgd_update(weight, grad, out=weight, lr=lr, wd=wd, **kwargs) else: if state[0] is not None: mp_sgd_mom_update(weight, grad, state[0], state[1], out=weight, lr=lr, wd=wd, **kwargs) else: mp_sgd_update(weight, grad, state[1], out=weight, lr=lr, wd=wd, **kwargs) # reset update count and cumulated gradient per large batch self._reset_cum_gradient(index) else: lr = 0.0 kwargs = {} sgd_update(weight, grad, out=weight, lr=lr, wd=wd, **kwargs) @register class LAMB(Optimizer): """LAMB Optimizer. """ def __init__(self, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-6, lower_bound=None, upper_bound=None, bias_correction=True, **kwargs): super(LAMB, self).__init__(learning_rate=learning_rate, **kwargs) self.beta1 = beta1 self.beta2 = beta2 self.epsilon = epsilon self.lower_bound = lower_bound self.upper_bound = upper_bound self.bias_correction = bias_correction def create_state(self, index, weight): stype = weight.stype return (zeros(weight.shape, weight.context, dtype=numpy.float32, stype=stype), zeros(weight.shape, weight.context, dtype=numpy.float32, stype=stype)) def _update_impl(self, index, weight, grad, state, multi_precision=False): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) t = self._index_update_count[index] kwargs = {'beta1': self.beta1, 'beta2': self.beta2, 'epsilon': self.epsilon, 'bias_correction': self.bias_correction, 't': t, 'rescale_grad': self.rescale_grad} if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if multi_precision: mean, var = state[1] weight32 = state[0] g = mp_lamb_update_phase1(weight, grad, mean, var, weight32, wd=wd, **kwargs) kwargs = {} if self.lower_bound: kwargs['lower_bound'] = self.lower_bound if self.upper_bound: kwargs['upper_bound'] = self.upper_bound r_1 = weight32.norm() r_2 = g.norm() mp_lamb_update_phase2(weight, g, r_1, r_2, weight32, lr=lr, out=weight, **kwargs) else: mean, var = state g = lamb_update_phase1(weight, grad, mean, var, wd=wd, **kwargs) kwargs = {} if self.lower_bound: kwargs['lower_bound'] = self.lower_bound if self.upper_bound: kwargs['upper_bound'] = self.upper_bound r_1 = weight.norm() r_2 = g.norm() lamb_update_phase2(weight, g, r_1, r_2, lr=lr, out=weight, **kwargs) def update(self, index, weight, grad, state): self._update_impl(index, weight, grad, state, multi_precision=False) def update_multi_precision(self, index, weight, grad, state): if not isinstance(index, (tuple, list)): use_multi_precision = self.multi_precision and weight.dtype == numpy.float16 else: use_multi_precision = self.multi_precision and weight[0].dtype == numpy.float16 self._update_impl(index, weight, grad, state, multi_precision=use_multi_precision) # pylint: enable=line-too-long @register class DCASGD(Optimizer): """The DCASGD optimizer. This class implements the optimizer described in *Asynchronous Stochastic Gradient Descent with Delay Compensation for Distributed Deep Learning*, available at https://arxiv.org/abs/1609.08326. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- momentum : float, optional The momentum value. lamda : float, optional Scale DC value. """ def __init__(self, momentum=0.0, lamda=0.04, **kwargs): super(DCASGD, self).__init__(**kwargs) self.momentum = momentum self.weight_previous = {} self.lamda = lamda def create_state(self, index, weight): if self.momentum == 0.0: return (None, weight.copy()) # previous weight else: return (zeros(weight.shape, weight.context, dtype=weight.dtype), # momentum weight.copy()) # previous weight def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) grad = grad * self.rescale_grad if self.clip_gradient is not None: grad = clip(grad, -self.clip_gradient, self.clip_gradient) mom, previous_weight = state if mom: mom[:] *= self.momentum mom[:] += -lr * (grad + wd * weight + self.lamda \ * grad * grad * (weight - previous_weight)) else: assert(self.momentum == 0.0) mom = -lr * (grad + wd * weight + self.lamda \ * grad * grad * (weight - previous_weight)) previous_weight[:] = weight weight[:] += mom @register class NAG(Optimizer): """Nesterov accelerated gradient. This optimizer updates each weight by:: state = momentum * state + grad + wd * weight weight = weight - (lr * (grad + momentum * state)) Parameters ---------- momentum : float, optional The momentum value. multi_precision: bool, optional Flag to control the internal precision of the optimizer. False: results in using the same precision as the weights (default), True: makes internal 32-bit copy of the weights and applies gradients in 32-bit precision even if actual weights used in the model have lower precision. Turning this on can improve convergence and accuracy when training with float16. """ def __init__(self, momentum=0.0, **kwargs): super(NAG, self).__init__(**kwargs) self.momentum = momentum def create_state_multi_precision(self, index, weight): weight_master_copy = None if self.multi_precision and weight.dtype == numpy.float16: weight_master_copy = weight.astype(numpy.float32) return (self.create_state(index, weight_master_copy), weight_master_copy) if weight.dtype == numpy.float16 and not self.multi_precision: warnings.warn("Accumulating with float16 in optimizer can lead to " "poor accuracy or slow convergence. " "Consider using multi_precision=True option of the " "NAG optimizer") return self.create_state(index, weight) def create_state(self, index, weight): momentum = None if self.momentum != 0.0: momentum = zeros(weight.shape, weight.context, dtype=weight.dtype) return momentum def _update_impl(self, index, weight, grad, state, multi_precision=False): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) kwargs = {'rescale_grad': self.rescale_grad} if self.momentum > 0: kwargs['momentum'] = self.momentum if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if not multi_precision: if state is not None: nag_mom_update(weight, grad, state, out=weight, lr=lr, wd=wd, **kwargs) else: sgd_update(weight, grad, out=weight, lr=lr, wd=wd, **kwargs) else: if state[0] is not None: mp_nag_mom_update(weight, grad, state[0], state[1], out=weight, lr=lr, wd=wd, **kwargs) else: mp_sgd_update(weight, grad, state[1], out=weight, lr=lr, wd=wd, **kwargs) def update(self, index, weight, grad, state): self._update_impl(index, weight, grad, state, multi_precision=False) def update_multi_precision(self, index, weight, grad, state): use_multi_precision = self.multi_precision and weight.dtype == numpy.float16 \ and isinstance(state, (tuple, list)) self._update_impl(index, weight, grad, state, multi_precision=use_multi_precision) @register class SGLD(Optimizer): """Stochastic Gradient Riemannian Langevin Dynamics. This class implements the optimizer described in the paper *Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex*, available at https://papers.nips.cc/paper/4883-stochastic-gradient-riemannian-langevin-dynamics-on-the-probability-simplex.pdf. """ def __init__(self, **kwargs): super(SGLD, self).__init__(**kwargs) def create_state(self, index, weight): return None def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) grad = grad * self.rescale_grad if self.clip_gradient is not None: grad = clip(grad, -self.clip_gradient, self.clip_gradient) weight[:] += - lr/2 * (grad + wd * weight) weight[:] += normal(0, math.sqrt(lr), shape=weight.shape, dtype=weight.dtype, ctx=weight.context) @register # pylint: disable=invalid-name class ccSGD(SGD): """[DEPRECATED] Same as `SGD`. Left here for backward compatibility.""" def __init__(self, *args, **kwargs): super(ccSGD, self).__init__(*args, **kwargs) @register class Adam(Optimizer): """The Adam optimizer. This class implements the optimizer described in *Adam: A Method for Stochastic Optimization*, available at http://arxiv.org/abs/1412.6980. If the storage types of grad is ``row_sparse``, and ``lazy_update`` is True, \ **lazy updates** at step t are applied by:: for row in grad.indices: rescaled_grad[row] = clip(grad[row] * rescale_grad + wd * weight[row], clip_gradient) m[row] = beta1 * m[row] + (1 - beta1) * rescaled_grad[row] v[row] = beta2 * v[row] + (1 - beta2) * (rescaled_grad[row]**2) lr = learning_rate * sqrt(1 - beta1**t) / (1 - beta2**t) w[row] = w[row] - lr * m[row] / (sqrt(v[row]) + epsilon) The lazy update only updates the mean and var for the weights whose row_sparse gradient indices appear in the current batch, rather than updating it for all indices. Compared with the original update, it can provide large improvements in model training throughput for some applications. However, it provides slightly different semantics than the original update, and may lead to different empirical results. Otherwise, **standard updates** at step t are applied by:: rescaled_grad = clip(grad * rescale_grad + wd * weight, clip_gradient) m = beta1 * m + (1 - beta1) * rescaled_grad v = beta2 * v + (1 - beta2) * (rescaled_grad**2) lr = learning_rate * sqrt(1 - beta1**t) / (1 - beta2**t) w = w - lr * m / (sqrt(v) + epsilon) This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. For details of the update algorithm, see :class:`~mxnet.ndarray.adam_update`. Parameters ---------- beta1 : float, optional Exponential decay rate for the first moment estimates. beta2 : float, optional Exponential decay rate for the second moment estimates. epsilon : float, optional Small value to avoid division by 0. lazy_update : bool, optional Default is True. If True, lazy updates are applied \ if the storage types of weight and grad are both ``row_sparse``. """ def __init__(self, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8, lazy_update=True, **kwargs): super(Adam, self).__init__(learning_rate=learning_rate, **kwargs) self.beta1 = beta1 self.beta2 = beta2 self.epsilon = epsilon self.lazy_update = lazy_update def create_state(self, index, weight): stype = weight.stype if self.lazy_update else 'default' return (zeros(weight.shape, weight.context, dtype=weight.dtype, stype=stype), # mean zeros(weight.shape, weight.context, dtype=weight.dtype, stype=stype)) # variance def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) t = self._index_update_count[index] coef1 = 1. - self.beta1**t coef2 = 1. - self.beta2**t lr *= math.sqrt(coef2)/coef1 kwargs = {'beta1': self.beta1, 'beta2': self.beta2, 'epsilon': self.epsilon, 'rescale_grad': self.rescale_grad} if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient mean, var = state adam_update(weight, grad, mean, var, out=weight, lazy_update=self.lazy_update, lr=lr, wd=wd, **kwargs) @register class AdaGrad(Optimizer): """AdaGrad optimizer. This class implements the AdaGrad optimizer described in *Adaptive Subgradient Methods for Online Learning and Stochastic Optimization*, and available at http://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf. This optimizer updates each weight by:: grad = clip(grad * rescale_grad, clip_gradient) history += square(grad) div = grad / sqrt(history + float_stable_eps) weight += (div + weight * wd) * -lr This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. See Also ---------- :meth:`mxnet.ndarray.sparse.adagrad_update`. Parameters ---------- eps: float, optional Initial value of the history accumulator. Avoids division by 0. """ def __init__(self, eps=1e-7, **kwargs): super(AdaGrad, self).__init__(**kwargs) self.float_stable_eps = eps def create_state(self, index, weight): return zeros(weight.shape, weight.context, stype=weight.stype) # history def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) is_sparse = grad.stype == 'row_sparse' history = state if is_sparse: kwargs = {'epsilon': self.float_stable_eps, 'rescale_grad': self.rescale_grad} if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient sparse.adagrad_update(weight, grad, history, out=weight, lr=lr, wd=wd, **kwargs) else: grad = grad * self.rescale_grad if self.clip_gradient is not None: grad = clip(grad, -self.clip_gradient, self.clip_gradient) history[:] += square(grad) div = grad / sqrt(history + self.float_stable_eps) weight[:] += (div + weight * wd) * -lr @register class RMSProp(Optimizer): """The RMSProp optimizer. Two versions of RMSProp are implemented: If ``centered=False``, we follow http://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf by Tieleman & Hinton, 2012. For details of the update algorithm see :class:`~mxnet.ndarray.rmsprop_update`. If ``centered=True``, we follow http://arxiv.org/pdf/1308.0850v5.pdf (38)-(45) by Alex Graves, 2013. For details of the update algorithm see :class:`~mxnet.ndarray.rmspropalex_update`. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- gamma1: float, optional A decay factor of moving average over past squared gradient. gamma2: float, optional A "momentum" factor. Only used if `centered`=``True``. epsilon : float, optional Small value to avoid division by 0. centered : bool, optional Flag to control which version of RMSProp to use.:: True: will use Graves's version of `RMSProp`, False: will use Tieleman & Hinton's version of `RMSProp`. clip_weights : float, optional Clips weights into range ``[-clip_weights, clip_weights]``. """ def __init__(self, learning_rate=0.001, gamma1=0.9, gamma2=0.9, epsilon=1e-8, centered=False, clip_weights=None, **kwargs): super(RMSProp, self).__init__(learning_rate=learning_rate, **kwargs) self.gamma1 = gamma1 self.gamma2 = gamma2 self.centered = centered self.epsilon = epsilon self.clip_weights = clip_weights def create_state(self, index, weight): if self.centered: return ( zeros(weight.shape, weight.context, stype=weight.stype), # n zeros(weight.shape, weight.context, stype=weight.stype), # g zeros(weight.shape, weight.context, stype=weight.stype)) # delta else: return (zeros(weight.shape, weight.context, stype=weight.stype),) # n def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) kwargs = {'gamma1': self.gamma1, 'epsilon': self.epsilon, 'rescale_grad': self.rescale_grad} if self.centered: kwargs['gamma2'] = self.gamma2 if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient if self.clip_weights: kwargs['clip_weights'] = self.clip_weights if not self.centered: (n, ) = state rmsprop_update( weight, grad, n, out=weight, lr=lr, wd=wd, **kwargs) else: n, g, delta = state rmspropalex_update(weight, grad, n, g, delta, out=weight, lr=lr, wd=wd, **kwargs) @register class AdaDelta(Optimizer): """The AdaDelta optimizer. This class implements AdaDelta, an optimizer described in *ADADELTA: An adaptive learning rate method*, available at https://arxiv.org/abs/1212.5701. This optimizer updates each weight by:: grad = clip(grad * rescale_grad + wd * weight, clip_gradient) acc_grad = rho * acc_grad + (1. - rho) * grad * grad delta = sqrt(acc_delta + epsilon) / sqrt(acc_grad + epsilon) * grad acc_delta = rho * acc_delta + (1. - rho) * delta * delta weight -= (delta + wd * weight) This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- rho: float Decay rate for both squared gradients and delta. epsilon : float Small value to avoid division by 0. """ def __init__(self, rho=0.90, epsilon=1e-5, **kwargs): super(AdaDelta, self).__init__(**kwargs) self.rho = rho self.epsilon = epsilon def create_state(self, index, weight): return (zeros(weight.shape, weight.context), # accumulated g zeros(weight.shape, weight.context)) # accumulated delta def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) wd = self._get_wd(index) self._update_count(index) # preprocess grad grad *= self.rescale_grad if self.clip_gradient is not None: grad = clip(grad, - self.clip_gradient, self.clip_gradient) # accumulated g and delta initlization acc_g, acc_delta = state # update g, delta acc_g[:] *= self.rho acc_g[:] += (1. - self.rho) * grad * grad current_delta = sqrt(acc_delta + self.epsilon) / sqrt(acc_g + self.epsilon) * grad acc_delta[:] *= self.rho acc_delta[:] += (1. - self.rho) * current_delta * current_delta # update weight weight[:] -= current_delta + wd * weight #pylint: disable=invalid-name #pylint: disable=line-too-long @register class Ftrl(Optimizer): """The Ftrl optimizer. Referenced from *Ad Click Prediction: a View from the Trenches*, available at http://dl.acm.org/citation.cfm?id=2488200. eta : .. math:: \\eta_{t,i} = \\frac{learningrate}{\\beta+\\sqrt{\\sum_{s=1}^tg_{s,i}^2}} The optimizer updates the weight by:: rescaled_grad = clip(grad * rescale_grad, clip_gradient) z += rescaled_grad - (sqrt(n + rescaled_grad**2) - sqrt(n)) * weight / learning_rate n += rescaled_grad**2 w = (sign(z) * lamda1 - z) / ((beta + sqrt(n)) / learning_rate + wd) * (abs(z) > lamda1) If the storage types of weight, state and grad are all ``row_sparse``, \ **sparse updates** are applied by:: for row in grad.indices: rescaled_grad[row] = clip(grad[row] * rescale_grad, clip_gradient) z[row] += rescaled_grad[row] - (sqrt(n[row] + rescaled_grad[row]**2) - sqrt(n[row])) * weight[row] / learning_rate n[row] += rescaled_grad[row]**2 w[row] = (sign(z[row]) * lamda1 - z[row]) / ((beta + sqrt(n[row])) / learning_rate + wd) * (abs(z[row]) > lamda1) The sparse update only updates the z and n for the weights whose row_sparse gradient indices appear in the current batch, rather than updating it for all indices. Compared with the original update, it can provide large improvements in model training throughput for some applications. However, it provides slightly different semantics than the original update, and may lead to different empirical results. For details of the update algorithm, see :class:`~mxnet.ndarray.ftrl_update`. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- lamda1 : float, optional L1 regularization coefficient. learning_rate : float, optional The initial learning rate. beta : float, optional Per-coordinate learning rate correlation parameter. """ def __init__(self, lamda1=0.01, learning_rate=0.1, beta=1, **kwargs): super(Ftrl, self).__init__(**kwargs) self.lamda1 = lamda1 self.beta = beta self.lr = learning_rate def create_state(self, index, weight): return (zeros(weight.shape, weight.context, stype=weight.stype), # z zeros(weight.shape, weight.context, stype=weight.stype)) # n def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) wd = self._get_wd(index) lr = self._get_lr(index) kwargs = {'lamda1': self.lamda1, 'beta': self.beta, 'rescale_grad': self.rescale_grad} if self.clip_gradient: kwargs['clip_gradient'] = self.clip_gradient # accumulated g and delta initialization z, n = state ftrl_update(weight, grad, z, n, out=weight, lr=lr, wd=wd, **kwargs) # pylint: enable=line-too-long @register class Adamax(Optimizer): """The AdaMax optimizer. It is a variant of Adam based on the infinity norm available at http://arxiv.org/abs/1412.6980 Section 7. The optimizer updates the weight by:: grad = clip(grad * rescale_grad + wd * weight, clip_gradient) m = beta1 * m_t + (1 - beta1) * grad u = maximum(beta2 * u, abs(grad)) weight -= lr / (1 - beta1**t) * m / u This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- beta1 : float, optional Exponential decay rate for the first moment estimates. beta2 : float, optional Exponential decay rate for the second moment estimates. """ def __init__(self, learning_rate=0.002, beta1=0.9, beta2=0.999, **kwargs): super(Adamax, self).__init__(learning_rate=learning_rate, **kwargs) self.beta1 = beta1 self.beta2 = beta2 def create_state(self, index, weight): return (zeros(weight.shape, weight.context, dtype=weight.dtype), # mean zeros(weight.shape, weight.context, dtype=weight.dtype)) # variance def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) t = self._index_update_count[index] lr /= (1. - self.beta1**t) # preprocess grad grad = grad * self.rescale_grad + wd * weight if self.clip_gradient is not None: grad = clip(grad, -self.clip_gradient, self.clip_gradient) # update m_t and u_t m_t, u_t = state m_t[:] *= self.beta1 m_t[:] += (1. - self.beta1) * grad u_t[:] = maximum(self.beta2 * u_t, NDabs(grad)) # update weight weight[:] -= lr * m_t / u_t @register class Nadam(Optimizer): """The Nesterov Adam optimizer. Much like Adam is essentially RMSprop with momentum, Nadam is Adam RMSprop with Nesterov momentum available at http://cs229.stanford.edu/proj2015/054_report.pdf. This optimizer accepts the following parameters in addition to those accepted by :class:`.Optimizer`. Parameters ---------- beta1 : float, optional Exponential decay rate for the first moment estimates. beta2 : float, optional Exponential decay rate for the second moment estimates. epsilon : float, optional Small value to avoid division by 0. schedule_decay : float, optional Exponential decay rate for the momentum schedule """ def __init__(self, learning_rate=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8, schedule_decay=0.004, **kwargs): super(Nadam, self).__init__(learning_rate=learning_rate, **kwargs) self.beta1 = beta1 self.beta2 = beta2 self.epsilon = epsilon self.schedule_decay = schedule_decay self.m_schedule = 1. def create_state(self, index, weight): return (zeros(weight.shape, weight.context, dtype=weight.dtype), # mean zeros(weight.shape, weight.context, dtype=weight.dtype)) # variance def update(self, index, weight, grad, state): assert(isinstance(weight, NDArray)) assert(isinstance(grad, NDArray)) self._update_count(index) lr = self._get_lr(index) wd = self._get_wd(index) t = self._index_update_count[index] # preprocess grad grad = grad * self.rescale_grad + wd * weight if self.clip_gradient is not None: grad = clip(grad, -self.clip_gradient, self.clip_gradient) # warming momentum schedule momentum_t = self.beta1 * (1. - 0.5 * (pow(0.96, t * self.schedule_decay))) momentum_t_1 = self.beta1 * (1. - 0.5 * (pow(0.96, (t + 1) * self.schedule_decay))) self.m_schedule = self.m_schedule * momentum_t m_schedule_next = self.m_schedule * momentum_t_1 # update m_t and v_t m_t, v_t = state m_t[:] *= self.beta1 m_t[:] += (1. - self.beta1) * grad v_t[:] *= self.beta2 v_t[:] += (1. - self.beta2) * grad * grad grad_prime = grad / (1. - self.m_schedule) m_t_prime = m_t / (1. - m_schedule_next) v_t_prime = v_t / (1. - pow(self.beta2, t)) m_t_bar = (1. - momentum_t) * grad_prime + momentum_t_1 * m_t_prime # update weight weight[:] -= lr * m_t_bar / (sqrt(v_t_prime) + self.epsilon) @register class Test(Optimizer): """The Test optimizer""" def __init__(self, **kwargs): super(Test, self).__init__(**kwargs) def create_state(self, index, weight): """Creates a state to duplicate weight.""" return zeros(weight.shape, weight.context) def update(self, index, weight, grad, state): """Performs w += rescale_grad * grad.""" weight[:] += grad * self.rescale_grad state[:] = weight # backward compatibility wrapper for Optimizer.CreateOptimizer create = Optimizer.create_optimizer # pylint: disable=invalid-name def _as_classic(a, allow_np): # TODO(junwu): This is a temp solution for allowing converting # np.ndarray to mx.nd.NDArray to be fed into the optimizer since # users may have custom optimizers implemented using mx.nd.NDArray ops. from ..numpy import ndarray as np_ndarray if isinstance(a, (tuple, list)): if any(isinstance(x, np_ndarray) for x in a): if allow_np: return [x.as_nd_ndarray() for x in a] else: raise ValueError('Converting np.ndarray to mx.nd.NDArray is not allowed') else: if isinstance(a, np_ndarray): if allow_np: return a.as_nd_ndarray() else: raise ValueError('Converting np.ndarray to mx.nd.NDArray is not allowed') return a class Updater(object): """Updater for kvstore.""" def __init__(self, optimizer): self.optimizer = optimizer self.states = {} self.states_synced = {} self.aggregate_updates = optimizer.aggregate_num > 0 def __call__(self, index, grad, weight): """Updates weight given gradient and index.""" allow_np = self.optimizer.allow_np_array if hasattr(self.optimizer, "allow_np_array") else is_np_array() if not isinstance(index, (list, tuple)): indices = [index] grads = [_as_classic(grad, allow_np)] weights = [_as_classic(weight, allow_np)] else: indices = index grads = _as_classic(grad, allow_np) weights = _as_classic(weight, allow_np) if weights: self.optimizer._set_current_context(weights[0].context.device_id) for i, idx in enumerate(indices): # convert ctypes.char_p.value back to python str if needed if isinstance(idx, bytes): indices[i] = py_str(idx) idx = indices[i] if idx not in self.states: self.states[idx] = self.optimizer.create_state_multi_precision(idx, weights[i]) self.states_synced[idx] = True elif not self.states_synced[idx]: self.states[idx] = \ self.sync_state_context(self.states[idx], weights[i].context) self.states_synced[idx] = True if self.aggregate_updates: # segregate values based on type type_map = {} for i, w, g in zip(indices, weights, grads): if w.dtype in type_map: type_map[w.dtype].append((i, w, g)) else: type_map[w.dtype] = [(i, w, g)] for idx in type_map: current_index = 0 indices, weights, grads = zip(*type_map[idx]) while current_index < len(indices): states = [] step = min(self.optimizer.aggregate_num, len(indices) - current_index) for j in range(step): states.append(self.states[indices[current_index + j]]) self.optimizer.update_multi_precision( indices[current_index:current_index + self.optimizer.aggregate_num], weights[current_index:current_index + self.optimizer.aggregate_num], grads[current_index:current_index + self.optimizer.aggregate_num], states) current_index += self.optimizer.aggregate_num else: for i, w, g in zip(indices, weights, grads): self.optimizer.update_multi_precision(i, w, g, self.states[i]) def sync_state_context(self, state, context): """sync state context.""" if isinstance(state, NDArray): return state.as_in_context(context) elif isinstance(state, (tuple, list)): synced_state = (self.sync_state_context(i, context) for i in state) if isinstance(state, tuple): return tuple(synced_state) else: return list(synced_state) else: return state def set_states(self, states): """Sets updater states.""" states = pickle.loads(states) if isinstance(states, tuple) and len(states) == 2: self.states, self.optimizer = states else: self.states = states self.states_synced = dict.fromkeys(self.states.keys(), False) def get_states(self, dump_optimizer=False): """Gets updater states. Parameters ---------- dump_optimizer : bool, default False Whether to also save the optimizer itself. This would also save optimizer information such as learning rate and weight decay schedules. """ return pickle.dumps((self.states, self.optimizer) if dump_optimizer else self.states) def get_updater(optimizer): """Returns a closure of the updater needed for kvstore. Parameters ---------- optimizer: Optimizer The optimizer. Returns ------- updater: function The closure of the updater. """ return Updater(optimizer)