from keras.src import initializers from keras.src import ops from keras.src.api_export import keras_export from keras.src.optimizers import optimizer @keras_export(["keras.optimizers.Ftrl"]) class Ftrl(optimizer.Optimizer): r"""Optimizer that implements the FTRL algorithm. "Follow The Regularized Leader" (FTRL) is an optimization algorithm developed at Google for click-through rate prediction in the early 2010s. It is most suitable for shallow models with large and sparse feature spaces. The algorithm is described by [McMahan et al., 2013](https://research.google.com/pubs/archive/41159.pdf). The Keras version has support for both online L2 regularization (the L2 regularization described in the paper above) and shrinkage-type L2 regularization (which is the addition of an L2 penalty to the loss function). Initialization: ```python n = 0 sigma = 0 z = 0 ``` Update rule for one variable `w`: ```python prev_n = n n = n + g ** 2 sigma = (n ** -lr_power - prev_n ** -lr_power) / lr z = z + g - sigma * w if abs(z) < lambda_1: w = 0 else: w = (sgn(z) * lambda_1 - z) / ((beta + sqrt(n)) / alpha + lambda_2) ``` Notation: - `lr` is the learning rate - `g` is the gradient for the variable - `lambda_1` is the L1 regularization strength - `lambda_2` is the L2 regularization strength - `lr_power` is the power to scale n. Check the documentation for the `l2_shrinkage_regularization_strength` parameter for more details when shrinkage is enabled, in which case gradient is replaced with a gradient with shrinkage. Args: learning_rate: A float, a `keras.optimizers.schedules.LearningRateSchedule` instance, or a callable that takes no arguments and returns the actual value to use. The learning rate. Defaults to `0.001`. learning_rate_power: A float value, must be less or equal to zero. Controls how the learning rate decreases during training. Use zero for a fixed learning rate. initial_accumulator_value: The starting value for accumulators. Only zero or positive values are allowed. l1_regularization_strength: A float value, must be greater than or equal to zero. Defaults to `0.0`. l2_regularization_strength: A float value, must be greater than or equal to zero. Defaults to `0.0`. l2_shrinkage_regularization_strength: A float value, must be greater than or equal to zero. This differs from L2 above in that the L2 above is a stabilization penalty, whereas this L2 shrinkage is a magnitude penalty. When input is sparse shrinkage will only happen on the active weights. beta: A float value, representing the beta value from the paper. Defaults to `0.0`. {{base_optimizer_keyword_args}} """ def __init__( self, learning_rate=0.001, learning_rate_power=-0.5, initial_accumulator_value=0.1, l1_regularization_strength=0.0, l2_regularization_strength=0.0, l2_shrinkage_regularization_strength=0.0, beta=0.0, weight_decay=None, clipnorm=None, clipvalue=None, global_clipnorm=None, use_ema=False, ema_momentum=0.99, ema_overwrite_frequency=None, loss_scale_factor=None, gradient_accumulation_steps=None, name="ftrl", **kwargs, ): super().__init__( learning_rate=learning_rate, name=name, weight_decay=weight_decay, clipnorm=clipnorm, clipvalue=clipvalue, global_clipnorm=global_clipnorm, use_ema=use_ema, ema_momentum=ema_momentum, ema_overwrite_frequency=ema_overwrite_frequency, loss_scale_factor=loss_scale_factor, gradient_accumulation_steps=gradient_accumulation_steps, **kwargs, ) if initial_accumulator_value < 0.0: raise ValueError( "`initial_accumulator_value` needs to be positive or zero. " "Received: initial_accumulator_value=" f"{initial_accumulator_value}." ) if learning_rate_power > 0.0: raise ValueError( "`learning_rate_power` needs to be negative or zero. Received: " f"learning_rate_power={learning_rate_power}." ) if l1_regularization_strength < 0.0: raise ValueError( "`l1_regularization_strength` needs to be positive or zero. " "Received: l1_regularization_strength=" f"{l1_regularization_strength}." ) if l2_regularization_strength < 0.0: raise ValueError( "`l2_regularization_strength` needs to be positive or zero. " "Received: l2_regularization_strength=" f"{l2_regularization_strength}." ) if l2_shrinkage_regularization_strength < 0.0: raise ValueError( "`l2_shrinkage_regularization_strength` needs to be positive " "or zero. Received: l2_shrinkage_regularization_strength" f"={l2_shrinkage_regularization_strength}." ) self.learning_rate_power = learning_rate_power self.initial_accumulator_value = initial_accumulator_value self.l1_regularization_strength = l1_regularization_strength self.l2_regularization_strength = l2_regularization_strength self.l2_shrinkage_regularization_strength = ( l2_shrinkage_regularization_strength ) self.beta = beta def build(self, var_list): """Initialize optimizer variables. Args: var_list: list of model variables to build Ftrl variables on. """ if self.built: return super().build(var_list) accumulator_initializer = initializers.Constant( self.initial_accumulator_value, ) self._accumulators, self._linears = self.add_optimizer_variables( var_list, ["accumulator", "linear"], initializer=[accumulator_initializer, "zeros"], ) def update_step(self, gradient, variable, learning_rate): """Update step given gradient and the associated model variable.""" lr = ops.cast(learning_rate, variable.dtype) gradient = ops.cast(gradient, variable.dtype) accum = self._accumulators[self._get_variable_index(variable)] linear = self._linears[self._get_variable_index(variable)] lr_power = self.learning_rate_power l2_reg = self.l2_regularization_strength l2_reg = l2_reg + self.beta / (2.0 * lr) grad_to_use = ops.add( gradient, ops.multiply( 2 * self.l2_shrinkage_regularization_strength, variable ), ) new_accum = ops.add(accum, ops.square(gradient)) self.assign_add( linear, ops.subtract( grad_to_use, ops.multiply( ops.divide( ops.subtract( ops.power(new_accum, -lr_power), ops.power(accum, -lr_power), ), lr, ), variable, ), ), ) quadratic = ops.add( ops.divide(ops.power(new_accum, (-lr_power)), lr), 2 * l2_reg ) linear_clipped = ops.clip( linear, -self.l1_regularization_strength, self.l1_regularization_strength, ) self.assign( variable, ops.divide(ops.subtract(linear_clipped, linear), quadratic), ) self.assign(accum, new_accum) def get_config(self): config = super().get_config() config.update( { "learning_rate_power": self.learning_rate_power, "initial_accumulator_value": self.initial_accumulator_value, "l1_regularization_strength": self.l1_regularization_strength, "l2_regularization_strength": self.l2_regularization_strength, "l2_shrinkage_regularization_strength": self.l2_shrinkage_regularization_strength, # noqa: E501 "beta": self.beta, } ) return config Ftrl.__doc__ = Ftrl.__doc__.replace( "{{base_optimizer_keyword_args}}", optimizer.base_optimizer_keyword_args )