gradient

Evaluate gradient of function approximator object given observation and action input data

Since R2022a

Syntax

grad = gradient(fcnAppx,'output-input',inData)

grad = gradient(fcnAppx,'output-parameters',inData)

grad = gradient(fcnAppx,lossFcn,inData,fcnData)

Description

grad = gradient(fcnAppx,'output-input',inData) evaluates the gradient of the sum of the outputs of the function approximator object fcnAppx with respect to its inputs. It returns the value of the gradient grad when the input of fcnAppx is inData.

grad = gradient(fcnAppx,'output-parameters',inData) evaluates the gradient of the sum of the outputs of fcnAppx with respect to its parameters.

grad = gradient(fcnAppx,lossFcn,inData,fcnData) evaluates the gradient of a loss function associated to the function handle lossFcn, with respect to the parameters of fcnAppx. The last optional argument fcnData can contain additional inputs for the loss function.

Examples

Calculate Gradients for Continuous Gaussian Actor

Create observation and action specification objects (or alternatively use getObservationInfo and getActionInfo to extract the specification objects from an environment). For this example, define an observation space with of three channels. The first channel carries an observation from a continuous three-dimensional space, so that a single observation is a column vector containing three doubles. The second channel carries a discrete observation made of a two-dimensional row vector that can take one of five different values. The third channel carries a discrete scalar observation that can be either zero or one. Finally, the action space is a continuous four-dimensional space, so that a single action is a column vector containing four doubles, each between -10 and 10.

obsInfo = [rlNumericSpec([3 1]) 
           rlFiniteSetSpec({[1 2],[3 4],[5 6],[7 8],[9 10]})
           rlFiniteSetSpec([0 1])];

actInfo = rlNumericSpec([4 1], ...
                UpperLimit= 10*ones(4,1), ...
                LowerLimit=-10*ones(4,1) );

To approximate the policy within the actor, use a recurrent deep neural network. For a continuous Gaussian actor, the network must have two output layers (one for the mean values the other for the standard deviation values), each having as many elements as the dimension of the action space.

Create a the network, defining each path as an array of layer objects. Use sequenceInputLayer as the input layer and include an lstmLayer as one of the other network layers. Also use a softplus layer to enforce nonnegativity of the standard deviations and a ReLU layer to scale the mean values to the desired output range. Get the dimensions of the observation and action spaces from the environment specification objects, and specify a name for the input layers, so you can later explicitly associate them with the appropriate environment channel.

inPath1 = [ sequenceInputLayer( ...
                prod(obsInfo(1).Dimension), ...
                Name="netObsIn1")
            fullyConnectedLayer(prod(actInfo.Dimension), ...
                Name="infc1") ];

inPath2 = [ sequenceInputLayer( ...
                prod(obsInfo(2).Dimension), ...
                Name="netObsIn2")
            fullyConnectedLayer( ...
                prod(actInfo.Dimension), ...
                Name="infc2") ];

inPath3 = [ sequenceInputLayer( ...
                prod(obsInfo(3).Dimension), ...
                Name="netObsIn3")
            fullyConnectedLayer( ...
                prod(actInfo.Dimension), ...
                Name="infc3") ];

% Concatenate inputs along the first available dimension
jointPath = [ concatenationLayer(1,3,Name="cat")
              tanhLayer(Name="tanhJnt");
              lstmLayer(8,OutputMode="sequence",Name="lstm")
              fullyConnectedLayer( ...
                prod(actInfo.Dimension), ...
                Name="jntfc");
             ];

% Path layers for mean value 
% Using scalingLayer to scale range from (-1,1) to (-10,10)
meanPath = [ tanhLayer(Name="tanhMean");
             fullyConnectedLayer(prod(actInfo.Dimension));
             scalingLayer(Name="scale", ...
                Scale=actInfo.UpperLimit) ];

% Path layers for standard deviations
% Using softplus layer to make them nonnegative
sdevPath = [ tanhLayer(Name="tanhStdv");
             fullyConnectedLayer(prod(actInfo.Dimension));
             softplusLayer(Name="splus") ];

% Add layers to network object
net = layerGraph;
net = addLayers(net,inPath1);
net = addLayers(net,inPath2);
net = addLayers(net,inPath3);
net = addLayers(net,jointPath);
net = addLayers(net,meanPath);
net = addLayers(net,sdevPath);

% Connect layers
net = connectLayers(net,"infc1","cat/in1");
net = connectLayers(net,"infc2","cat/in2");
net = connectLayers(net,"infc3","cat/in3");
net = connectLayers(net,"jntfc","tanhMean/in");
net = connectLayers(net,"jntfc","tanhStdv/in");

% Plot network
plot(net)

% Convert to dlnetwork
net = dlnetwork(net);

% Display the number of weights
summary(net)

   Initialized: true

   Number of learnables: 784

   Inputs:
      1   'netObsIn1'   Sequence input with 3 dimensions
      2   'netObsIn2'   Sequence input with 2 dimensions
      3   'netObsIn3'   Sequence input with 1 dimensions

Create the actor with rlContinuousGaussianActor, using the network, the observations and action specification objects, as well as the names of the network input layer and the options object.

actor = rlContinuousGaussianActor(net, obsInfo, actInfo, ...
    ActionMeanOutputNames="scale",...
    ActionStandardDeviationOutputNames="splus",...
    ObservationInputNames=["netObsIn1","netObsIn2","netObsIn3"]);

To return mean value and standard deviations of the Gaussian distribution as a function of the current observation, use evaluate.

[prob,state] = evaluate(actor, {rand([obsInfo(1).Dimension 1 1]) , ...
                                rand([obsInfo(2).Dimension 1 1]) , ...
                                rand([obsInfo(3).Dimension 1 1]) });

The result is a cell array with two elements, the first one containing a vector of mean values, and the second containing a vector of standard deviations.

prob{1}

ans = 4x1 single column vector

   -1.5454
    0.4908
   -0.1697
    0.8081

prob{2}

ans = 4x1 single column vector

    0.6913
    0.6141
    0.7291
    0.6475

To return an action sampled from the distribution, use getAction.

act = getAction(actor, {rand(obsInfo(1).Dimension) , ...
                        rand(obsInfo(2).Dimension) , ...
                        rand(obsInfo(3).Dimension) });
act{1}

ans = 4x1 single column vector

   -3.2003
   -0.0534
   -1.0700
   -0.4032

Calculate the gradients of the sum of the outputs (all the mean values plus all the standard deviations) with respect to the inputs, given a random observation.

gro = gradient(actor,"output-input", ...
                        {rand(obsInfo(1).Dimension) , ...
                         rand(obsInfo(2).Dimension) , ...
                         rand(obsInfo(3).Dimension)} )

gro=3×1 cell array
    {3x1 single}
    {2x1 single}
    {[  0.1311]}

The result is a cell array with as many elements as the number of input channels. Each element contains the derivatives of the sum of the outputs with respect to each component of the input channel. Display the gradient with respect to the element of the second channel.

gro{2}

ans = 2x1 single column vector

   -1.3404
    0.6642

Obtain the gradient with respect of five independent sequences, each one made of nine sequential observations.

gro_batch = gradient(actor,"output-input", ...
                        {rand([obsInfo(1).Dimension 5 9]) , ...
                         rand([obsInfo(2).Dimension 5 9]) , ...
                         rand([obsInfo(3).Dimension 5 9])} )

gro_batch=3×1 cell array
    {3x5x9 single}
    {2x5x9 single}
    {1x5x9 single}

Display the derivative of the sum of the outputs with respect to the third observation element of the first input channel, after the seventh sequential observation in the fourth independent batch.

gro_batch{1}(3,4,7)

ans = single
    0.2020

Set the option to accelerate the gradient computations.

actor = accelerate(actor,true);

Calculate the gradients of the sum of the outputs with respect to the parameters, given a random observation.

grp = gradient(actor,"output-parameters", ...
                {rand(obsInfo(1).Dimension) , ...
                 rand(obsInfo(2).Dimension) , ...
                 rand(obsInfo(3).Dimension)} )

grp=15×1 cell array
    { 4x3  single}
    { 4x1  single}
    { 4x2  single}
    { 4x1  single}
    { 4x1  single}
    { 4x1  single}
    {32x12 single}
    {32x8  single}
    {32x1  single}
    { 4x8  single}
    { 4x1  single}
    { 4x4  single}
    { 4x1  single}
    { 4x4  single}
    { 4x1  single}

Each array within a cell contains the gradient of the sum of the outputs with respect to a group of parameters.

grp_batch = gradient(actor,"output-parameters", ...
                        {rand([obsInfo(1).Dimension 5 9]) , ...
                         rand([obsInfo(2).Dimension 5 9]) , ...
                         rand([obsInfo(3).Dimension 5 9])} )

grp_batch=15×1 cell array
    { 4x3  single}
    { 4x1  single}
    { 4x2  single}
    { 4x1  single}
    { 4x1  single}
    { 4x1  single}
    {32x12 single}
    {32x8  single}
    {32x1  single}
    { 4x8  single}
    { 4x1  single}
    { 4x4  single}
    { 4x1  single}
    { 4x4  single}
    { 4x1  single}

If you use a batch of inputs, gradient uses the whole input sequence (in this case nine steps), and all the gradients with respect to the independent batch dimensions (in this case five) are added together. Therefore, the returned gradient has always the same size as the output from getLearnableParameters.

Calculate Gradients for Vector Q-Value Function

Create observation and action specification objects (or alternatively use getObservationInfo and getActionInfo to extract the specification objects from an environment). For this example, define an observation space made of two channels. The first channel carries an observation from a continuous four-dimensional space. The second carries a discrete scalar observation that can be either zero or one. Finally, the action space consist of a scalar that can be -1, 0, or 1.

obsInfo = [rlNumericSpec([4 1]) 
           rlFiniteSetSpec([0 1])];

actInfo =  rlFiniteSetSpec([-1 0 1]);

To approximate the vector Q-value function within the critic, use a recurrent deep neural network. The output layer must have three elements, each one expressing the value of executing the corresponding action, given the observation.

Create the neural network, defining each network path as an array of layer objects. Get the dimensions of the observation and action spaces from the environment specification objects, use sequenceInputLayer as the input layer, and include an lstmLayer as one of the other network layers.

inPath1 = [ sequenceInputLayer( ...
                prod(obsInfo(1).Dimension), ...
                Name="netObsIn1")
            fullyConnectedLayer( ...
                prod(actInfo.Dimension), ...
                Name="infc1") ];

inPath2 = [ sequenceInputLayer( ...
                prod(obsInfo(2).Dimension), ...
                Name="netObsIn2")
            fullyConnectedLayer( ...
                prod(actInfo.Dimension), ...
                Name="infc2") ];

% Concatenate inputs along first available dimension
jointPath = [ concatenationLayer(1,2,Name="cct")
              tanhLayer(Name="tanhJnt")
              lstmLayer(8,OutputMode="sequence")
              fullyConnectedLayer(prod(numel(actInfo.Elements))) ];

% Add layers to network object
net = layerGraph;
net = addLayers(net,inPath1);
net = addLayers(net,inPath2);
net = addLayers(net,jointPath);

% Connect layers
net = connectLayers(net,"infc1","cct/in1");
net = connectLayers(net,"infc2","cct/in2");

% Plot network
plot(net)

% Convert to dlnetwork
net = dlnetwork(net);

% Display the number of weights
summary(net)

   Initialized: true

   Number of learnables: 386

   Inputs:
      1   'netObsIn1'   Sequence input with 4 dimensions
      2   'netObsIn2'   Sequence input with 1 dimensions

Create the critic with rlVectorQValueFunction, using the network and the observation and action specification objects.

critic = rlVectorQValueFunction(net,obsInfo,actInfo);

To return the value of the actions as a function of the current observation, use getValue or evaluate.

val = evaluate(critic, ...
                {rand(obsInfo(1).Dimension), ...
                 rand(obsInfo(2).Dimension)})

val = 1x1 cell array
    {3x1 single}

When you use evaluate, the result is a single-element cell array, containing a vector with the values of all the possible actions, given the observation.

val{1}

ans = 3x1 single column vector

   -0.0054
   -0.0943
    0.0177

Calculate the gradients of the sum of the outputs with respect to the inputs, given a random observation.

gro = gradient(critic,"output-input", ...
                {rand(obsInfo(1).Dimension) , ...
                 rand(obsInfo(2).Dimension) } )

gro=2×1 cell array
    {4x1 single}
    {[ -0.0396]}

The result is a cell array with as many elements as the number of input channels. Each element contains the derivative of the sum of the outputs with respect to each component of the input channel. Display the gradient with respect to the element of the second channel.

gro{2}

ans = single
    -0.0396

Obtain the gradient with respect of five independent sequences each one made of nine sequential observations.

gro_batch = gradient(critic,"output-input", ...
                        {rand([obsInfo(1).Dimension 5 9]) , ...
                         rand([obsInfo(2).Dimension 5 9]) } )

gro_batch=2×1 cell array
    {4x5x9 single}
    {1x5x9 single}

Display the derivative of the sum of the outputs with respect to the third observation element of the first input channel, after the seventh sequential observation in the fourth independent batch.

gro_batch{1}(3,4,7)

ans = single
    0.0443

Set the option to accelerate the gradient computations.

critic = accelerate(critic,true);

Calculate the gradients of the sum of the outputs with respect to the parameters, given a random observation.

grp = gradient(critic,"output-parameters", ...
                {rand(obsInfo(1).Dimension) , ...
                 rand(obsInfo(2).Dimension) } )

grp=9×1 cell array
    {[-0.0101 -0.0097 -0.0039 -0.0065]}
    {[                        -0.0122]}
    {[                        -0.0078]}
    {[                        -0.0863]}
    {32x2 single                      }
    {32x8 single                      }
    {32x1 single                      }
    { 3x8 single                      }
    { 3x1 single                      }

Each array within a cell contains the gradient of the sum of the outputs with respect to a group of parameters.

grp_batch = gradient(critic,"output-parameters", ...
                        {rand([obsInfo(1).Dimension 5 9]) , ...
                         rand([obsInfo(2).Dimension 5 9]) } )

grp_batch=9×1 cell array
    {[-1.5801 -1.2618 -2.2168 -1.3463]}
    {[                        -3.7742]}
    {[                        -6.2158]}
    {[                       -12.6841]}
    {32x2 single                      }
    {32x8 single                      }
    {32x1 single                      }
    { 3x8 single                      }
    { 3x1 single                      }

If you use a batch of inputs, gradient uses the whole input sequence (in this case nine steps), and all the gradients with respect to the independent batch dimensions (in this case five) are added together. Therefore, the returned gradient always has the same size as the output from getLearnableParameters.

Input Arguments

fcnAppx — Function approximator object

function approximator object

Function approximator object, specified as one of the following:

• rlValueFunction object — Value function critic

• rlQValueFunction object — Q-value function critic

• rlVectorQValueFunction object — Multi-output Q-value function critic with a discrete action space

• rlContinuousDeterministicActor object — Deterministic policy actor with a continuous action space

• rlDiscreteCategoricalActor — Stochastic policy actor with a discrete action space

• rlContinuousGaussianActor object — Stochastic policy actor with a continuous action space

• rlContinuousDeterministicTransitionFunction object — Continuous deterministic transition function for a model based agent

• rlContinuousGaussianTransitionFunction object — Continuous Gaussian transition function for a model based agent

• rlContinuousDeterministicRewardFunction object — Continuous deterministic reward function for a model based agent

• rlContinuousGaussianRewardFunction object — Continuous Gaussian reward function for a model based agent.

• rlIsDoneFunction object — Is-done function for a model based agent.

inData — Input data for the function approximator

cell array

Input data for the function approximator, specified as a cell array with as many elements as the number of input channels of fcnAppx. In the following section, the number of observation channels is indicated by N_O.

• If fcnAppx is an rlQValueFunction, an rlContinuousDeterministicTransitionFunction or an rlContinuousGaussianTransitionFunction object, then each of the first N_O elements of inData must be a matrix representing the current observation from the corresponding observation channel. They must be followed by a final matrix representing the action.

• If fcnAppx is a function approximator object representing an actor or critic (but not an rlQValueFunction object), inData must contain N_O elements, each one being a matrix representing the current observation from the corresponding observation channel.

• If fcnAppx is an rlContinuousDeterministicRewardFunction, an rlContinuousGaussianRewardFunction, or an rlIsDoneFunction object, then each of the first N_O elements of inData must be a matrix representing the current observation from the corresponding observation channel. They must be followed by a matrix representing the action, and finally by N_O elements, each one being a matrix representing the next observation from the corresponding observation channel.

Each element of inData must be a matrix of dimension M_C-by-L_B-by-L_S, where:

• M_C corresponds to the dimensions of the associated input channel.

• L_B is the batch size. To specify a single observation, set L_B = 1. To specify a batch of (independent) observations, specify L_B > 1. If inData has multiple elements, then L_B must be the same for all elements of inData.

• L_S specifies the sequence length (along the time dimension) for recurrent neural network. If fcnAppx does not use a recurrent neural network, (which is the case of environment function approximators, as they do not support recurrent neural networks) then L_S = 1. If inData has multiple elements, then L_S must be the same for all elements of inData.

For more information on input and output formats for recurrent neural networks, see the Algorithms section of lstmLayer.

Example: {rand(8,3,64,1),rand(4,1,64,1),rand(2,1,64,1)}

lossFcn — Loss function

function handle

Loss function, specified as a function handle to a user-defined function. The user defined function can either be an anonymous function or a function on the MATLAB path. The function first input parameter must be a cell array like the one returned from the evaluation of fcnAppx. For more information, see the description of outData in evaluate. The second, optional, input argument of lossFcn contains additional data that might be needed for the gradient calculation, as described below in fcnData. For an example of the signature that this function must have, see Train Reinforcement Learning Policy Using Custom Training Loop.

fcnData — Additional input data for loss function

any MATLAB^® data type

Additional data for the loss function, specified as any MATLAB data type, typically a structure or cell array. For an example see Train Reinforcement Learning Policy Using Custom Training Loop.

Output Arguments

grad — Value of the gradient

cell array

Value of the gradient, returned as a cell array.

When the type of gradient is from the sum of the outputs with respect to the inputs of fcnAppx, then grad is a cell array in which each element contains the gradient of the sum of all the outputs with respect to the corresponding input channel.

The numerical array in each cell has dimensions D-by-L_B-by-L_S, where:

• D corresponds to the dimensions of the input channel of fcnAppx.

• L_B is the batch size (length of a batch of independent inputs).

• L_S is the sequence length (length of the sequence of inputs along the time dimension) for a recurrent neural network. If fcnAppx does not use a recurrent neural network (which is the case of environment function approximators, as they do not support recurrent neural networks), then L_S = 1.

When the type of gradient is from the output with respect to the parameters of fcnAppx, then grad is a cell array in which each element contains the gradient of the sum of outputs belonging to an output channel with respect to the corresponding group of parameters. The gradient is calculated using the whole history of L_S inputs, and all the L_B gradients with respect to the independent input sequences are added together in grad. Therefore, grad has always the same size as the result from getLearnableParameters.

For more information on input and output formats for recurrent neural networks, see the Algorithms section of lstmLayer.

Version History

Introduced in R2022a