Optimization Algorithms

---

Adam

Scala:

val optim = new Adam(learningRate=1e-3, learningRateDecay=0.0, beta1=0.9, beta2=0.999, Epsilon=1e-8)

Python:

optim = Adam(learningrate=1e-3, learningrate_decay=0.0, beta1=0.9, beta2=0.999, epsilon=1e-8, bigdl_type="float")

An implementation of Adam optimization, first-order gradient-based optimization of stochastic objective functions. http://arxiv.org/pdf/1412.6980.pdf

learningRate learning rate. Default value is 1e-3.

learningRateDecay learning rate decay. Default value is 0.0.

beta1 first moment coefficient. Default value is 0.9.

beta2 second moment coefficient. Default value is 0.999.

Epsilon for numerical stability. Default value is 1e-8.

Scala example:

import com.intel.analytics.bigdl.optim._
import com.intel.analytics.bigdl.tensor.Tensor
import com.intel.analytics.bigdl.tensor.TensorNumericMath.TensorNumeric.NumericFloat
import com.intel.analytics.bigdl.utils.T

val optm = new Adam(learningRate=0.002)
def rosenBrock(x: Tensor[Float]): (Float, Tensor[Float]) = {
    // (1) compute f(x)
    val d = x.size(1)

    // x1 = x(i)
    val x1 = Tensor[Float](d - 1).copy(x.narrow(1, 1, d - 1))
    // x(i + 1) - x(i)^2
    x1.cmul(x1).mul(-1).add(x.narrow(1, 2, d - 1))
    // 100 * (x(i + 1) - x(i)^2)^2
    x1.cmul(x1).mul(100)

    // x0 = x(i)
    val x0 = Tensor[Float](d - 1).copy(x.narrow(1, 1, d - 1))
    // 1-x(i)
    x0.mul(-1).add(1)
    x0.cmul(x0)
    // 100*(x(i+1) - x(i)^2)^2 + (1-x(i))^2
    x1.add(x0)

    val fout = x1.sum()

    // (2) compute f(x)/dx
    val dxout = Tensor[Float]().resizeAs(x).zero()
    // df(1:D-1) = - 400*x(1:D-1).*(x(2:D)-x(1:D-1).^2) - 2*(1-x(1:D-1));
    x1.copy(x.narrow(1, 1, d - 1))
    x1.cmul(x1).mul(-1).add(x.narrow(1, 2, d - 1)).cmul(x.narrow(1, 1, d - 1)).mul(-400)
    x0.copy(x.narrow(1, 1, d - 1)).mul(-1).add(1).mul(-2)
    x1.add(x0)
    dxout.narrow(1, 1, d - 1).copy(x1)

    // df(2:D) = df(2:D) + 200*(x(2:D)-x(1:D-1).^2);
    x0.copy(x.narrow(1, 1, d - 1))
    x0.cmul(x0).mul(-1).add(x.narrow(1, 2, d - 1)).mul(200)
    dxout.narrow(1, 2, d - 1).add(x0)

    (fout, dxout)
  }  
val x = Tensor(2).fill(0)
> print(optm.optimize(rosenBrock, x))
(0.0019999996
0.0
[com.intel.analytics.bigdl.tensor.DenseTensor$mcD$sp of size 2],[D@302d88d8)

Python example:

optim_method = Adam(learningrate=0.002)

optimizer = Optimizer(
    model=mlp_model,
    training_rdd=train_data,
    criterion=ClassNLLCriterion(),
    optim_method=optim_method,
    end_trigger=MaxEpoch(20),
    batch_size=32)

SGD

Scala:

val optimMethod = new SGD(learningRate= 1e-3,learningRateDecay=0.0,
                      weightDecay=0.0,momentum=0.0,dampening=Double.MaxValue,
                      nesterov=false,learningRateSchedule=Default(),
                      learningRates=null,weightDecays=null)

Python:

optim_method = SGD(learningrate=1e-3,learningrate_decay=0.0,weightdecay=0.0,
                   momentum=0.0,dampening=DOUBLEMAX,nesterov=False,
                   leaningrate_schedule=None,learningrates=None,
                   weightdecays=None,bigdl_type="float")

A plain implementation of SGD which provides optimize method. After setting optimization method when create Optimize, Optimize will call optimization method at the end of each iteration.

Scala example:

val optimMethod = new SGD[Float](learningRate= 1e-3,learningRateDecay=0.0,
                               weightDecay=0.0,momentum=0.0,dampening=Double.MaxValue,
                               nesterov=false,learningRateSchedule=Default(),
                               learningRates=null,weightDecays=null)
optimizer.setOptimMethod(optimMethod)

Python example:

optim_method = SGD(learningrate=1e-3,learningrate_decay=0.0,weightdecay=0.0,
                  momentum=0.0,dampening=DOUBLEMAX,nesterov=False,
                  leaningrate_schedule=None,learningrates=None,
                  weightdecays=None,bigdl_type="float")

optimizer = Optimizer(
    model=mlp_model,
    training_rdd=train_data,
    criterion=ClassNLLCriterion(),
    optim_method=optim_method,
    end_trigger=MaxEpoch(20),
    batch_size=32)

Adadelta

AdaDelta implementation for SGD It has been proposed in ADADELTA: An Adaptive Learning Rate Method. http://arxiv.org/abs/1212.5701.

Scala:

val optimMethod = new Adadelta(decayRate = 0.9, Epsilon = 1e-10)

Python:

optim_method = AdaDelta(decayrate = 0.9, epsilon = 1e-10)

Scala example:

optimizer.setOptimMethod(new Adadelta(0.9, 1e-10))

Python example:

optimizer = Optimizer(
    model=mlp_model,
    training_rdd=train_data,
    criterion=ClassNLLCriterion(),
    optim_method=Adadelta(0.9, 0.00001),
    end_trigger=MaxEpoch(20),
    batch_size=32)

RMSprop

An implementation of RMSprop (Reference: http://arxiv.org/pdf/1308.0850v5.pdf, Sec 4.2)

• learningRate : learning rate
• learningRateDecay : learning rate decay
• decayRate : decayRate, also called rho
• Epsilon : for numerical stability

Adamax

An implementation of Adamax http://arxiv.org/pdf/1412.6980.pdf

Arguments:

• learningRate : learning rate
• beta1 : first moment coefficient
• beta2 : second moment coefficient
• Epsilon : for numerical stability

Returns:

the new x vector and the function list {fx}, evaluated before the update

Adagrad

Scala:

val adagrad = new Adagrad(learningRate = 1e-3,
                          learningRateDecay = 0.0,
                          weightDecay = 0.0)

An implementation of Adagrad. See the original paper: http://jmlr.org/papers/volume12/duchi11a/duchi11a.pdf

Scala example:

import com.intel.analytics.bigdl.tensor.TensorNumericMath.TensorNumeric.NumericFloat
import com.intel.analytics.bigdl.optim._
import com.intel.analytics.bigdl.tensor._
import com.intel.analytics.bigdl.utils.T

val adagrad = new Adagrad(0.01, 0.0, 0.0)
    def feval(x: Tensor[Float]): (Float, Tensor[Float]) = {
      // (1) compute f(x)
      val d = x.size(1)
      // x1 = x(i)
      val x1 = Tensor[Float](d - 1).copy(x.narrow(1, 1, d - 1))
      // x(i + 1) - x(i)^2
      x1.cmul(x1).mul(-1).add(x.narrow(1, 2, d - 1))
      // 100 * (x(i + 1) - x(i)^2)^2
      x1.cmul(x1).mul(100)
      // x0 = x(i)
      val x0 = Tensor[Float](d - 1).copy(x.narrow(1, 1, d - 1))
      // 1-x(i)
      x0.mul(-1).add(1)
      x0.cmul(x0)
      // 100*(x(i+1) - x(i)^2)^2 + (1-x(i))^2
      x1.add(x0)
      val fout = x1.sum()
      // (2) compute f(x)/dx
      val dxout = Tensor[Float]().resizeAs(x).zero()
      // df(1:D-1) = - 400*x(1:D-1).*(x(2:D)-x(1:D-1).^2) - 2*(1-x(1:D-1));
      x1.copy(x.narrow(1, 1, d - 1))
      x1.cmul(x1).mul(-1).add(x.narrow(1, 2, d - 1)).cmul(x.narrow(1, 1, d - 1)).mul(-400)
      x0.copy(x.narrow(1, 1, d - 1)).mul(-1).add(1).mul(-2)
      x1.add(x0)
      dxout.narrow(1, 1, d - 1).copy(x1)
      // df(2:D) = df(2:D) + 200*(x(2:D)-x(1:D-1).^2);
      x0.copy(x.narrow(1, 1, d - 1))
      x0.cmul(x0).mul(-1).add(x.narrow(1, 2, d - 1)).mul(200)
      dxout.narrow(1, 2, d - 1).add(x0)
      (fout, dxout)
    }
val x = Tensor(2).fill(0)
val config = T("learningRate" -> 1e-1)
for (i <- 1 to 10) {
  adagrad.optimize(feval, x, config, config)
}
x after optimize: 0.27779138
0.07226955
[com.intel.analytics.bigdl.tensor.DenseTensor$mcF$sp of size 2]

LBFGS

Scala:

val optimMethod = new LBFGS(maxIter=20, maxEval=Double.MaxValue,
                            tolFun=1e-5, tolX=1e-9, nCorrection=100,
                            learningRate=1.0, lineSearch=None, lineSearchOptions=None)

Python:

optim_method = LBFGS(max_iter=20, max_eval=Double.MaxValue, \
                 tol_fun=1e-5, tol_x=1e-9, n_correction=100, \
                 learning_rate=1.0, line_search=None, line_search_options=None)

This implementation of L-BFGS relies on a user-provided line search function (state.lineSearch). If this function is not provided, then a simple learningRate is used to produce fixed size steps. Fixed size steps are much less costly than line searches, and can be useful for stochastic problems.

The learning rate is used even when a line search is provided.This is also useful for large-scale stochastic problems, where opfunc is a noisy approximation of f(x). In that case, the learning rate allows a reduction of confidence in the step size.

Parameters:

• maxIter - Maximum number of iterations allowed. Default: 20
• maxEval - Maximum number of function evaluations. Default: Double.MaxValue
• tolFun - Termination tolerance on the first-order optimality. Default: 1e-5
• tolX - Termination tol on progress in terms of func/param changes. Default: 1e-9
• learningRate - the learning rate. Default: 1.0
• lineSearch - A line search function. Default: None
• lineSearchOptions - If no line search provided, then a fixed step size is used. Default: None

Scala example:

val optimMethod = new LBFGS(maxIter=20, maxEval=Double.MaxValue,
                            tolFun=1e-5, tolX=1e-9, nCorrection=100,
                            learningRate=1.0, lineSearch=None, lineSearchOptions=None)
optimizer.setOptimMethod(optimMethod)

Python example:

optim_method = LBFGS(max_iter=20, max_eval=DOUBLEMAX, \
                 tol_fun=1e-5, tol_x=1e-9, n_correction=100, \
                 learning_rate=1.0, line_search=None, line_search_options=None)

optimizer = Optimizer(
    model=mlp_model,
    training_rdd=train_data,
    criterion=ClassNLLCriterion(),
    optim_method=optim_method,
    end_trigger=MaxEpoch(20),
    batch_size=32)

Ftrl

Scala:

val optimMethod = new Ftrl(
  learningRate = 1e-3, learningRatePower = -0.5,
  initialAccumulatorValue = 0.1, l1RegularizationStrength = 0.0,
  l2RegularizationStrength = 0.0, l2ShrinkageRegularizationStrength = 0.0)

Python:

optim_method = Ftrl(learningrate = 1e-3, learningrate_power = -0.5, \
                 initial_accumulator_value = 0.1, l1_regularization_strength = 0.0, \
                 l2_regularization_strength = 0.0, l2_shrinkage_regularization_strength = 0.0)

An implementation of (Ftrl)[https://www.eecs.tufts.edu/~dsculley/papers/ad-click-prediction.pdf.] Support L1 penalty, L2 penalty and shrinkage-type L2 penalty.

Parameters:

• learningRate: learning rate
• learningRatePower: double, must be less or equal to zero. Default is -0.5.
• initialAccumulatorValue: double, the starting value for accumulators, require zero or positive values. Default is 0.1.
• l1RegularizationStrength: double, must be greater or equal to zero. Default is zero.
• l2RegularizationStrength: double, must be greater or equal to zero. Default is zero.
• l2ShrinkageRegularizationStrength: double, must be greater or equal to zero. Default is zero. This differs from l2RegularizationStrength above. L2 above is a stabilization penalty, whereas this one is a magnitude penalty.

Scala example:

val optimMethod = new Ftrl(learningRate = 5e-3, learningRatePower = -0.5,
  initialAccumulatorValue = 0.01)
optimizer.setOptimMethod(optimMethod)

Python example:

optim_method = Ftrl(learningrate = 5e-3, \
    learningrate_power = -0.5, \
    initial_accumulator_value = 0.01)

optimizer = Optimizer(
    model=mlp_model,
    training_rdd=train_data,
    criterion=ClassNLLCriterion(),
    optim_method=optim_method,
    end_trigger=MaxEpoch(20),
    batch_size=32)

ParallelAdam

Multi-Thread version of Adam.

Scala:

val optim = new ParallelAdam(learningRate=1e-3, learningRateDecay=0.0, beta1=0.9, beta2=0.999, Epsilon=1e-8, parallelNum=Engine.coreNumber())

Python:

optim = ParallelAdam(learningrate=1e-3, learningrate_decay=0.0, beta1=0.9, beta2=0.999, epsilon=1e-8, parallel_num=get_node_and_core_number()[1], bigdl_type="float")