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<!DOCTYPE html>
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<li class="toctree-l2"><a class="reference internal" href="nn.html#parameters">Parameters</a></li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#containers">Containers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#module"><span class="hidden-section">Module</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#sequential"><span class="hidden-section">Sequential</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#modulelist"><span class="hidden-section">ModuleList</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#moduledict"><span class="hidden-section">ModuleDict</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#parameterlist"><span class="hidden-section">ParameterList</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#parameterdict"><span class="hidden-section">ParameterDict</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#convolution-layers">Convolution layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv1d"><span class="hidden-section">Conv1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv2d"><span class="hidden-section">Conv2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv3d"><span class="hidden-section">Conv3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#convtranspose1d"><span class="hidden-section">ConvTranspose1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#convtranspose2d"><span class="hidden-section">ConvTranspose2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#convtranspose3d"><span class="hidden-section">ConvTranspose3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#unfold"><span class="hidden-section">Unfold</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#fold"><span class="hidden-section">Fold</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#pooling-layers">Pooling layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxpool1d"><span class="hidden-section">MaxPool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxpool2d"><span class="hidden-section">MaxPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxpool3d"><span class="hidden-section">MaxPool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxunpool1d"><span class="hidden-section">MaxUnpool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxunpool2d"><span class="hidden-section">MaxUnpool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#maxunpool3d"><span class="hidden-section">MaxUnpool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avgpool1d"><span class="hidden-section">AvgPool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avgpool2d"><span class="hidden-section">AvgPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avgpool3d"><span class="hidden-section">AvgPool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#fractionalmaxpool2d"><span class="hidden-section">FractionalMaxPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lppool1d"><span class="hidden-section">LPPool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lppool2d"><span class="hidden-section">LPPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptivemaxpool1d"><span class="hidden-section">AdaptiveMaxPool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptivemaxpool2d"><span class="hidden-section">AdaptiveMaxPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptivemaxpool3d"><span class="hidden-section">AdaptiveMaxPool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptiveavgpool1d"><span class="hidden-section">AdaptiveAvgPool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptiveavgpool2d"><span class="hidden-section">AdaptiveAvgPool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptiveavgpool3d"><span class="hidden-section">AdaptiveAvgPool3d</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#padding-layers">Padding layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#reflectionpad1d"><span class="hidden-section">ReflectionPad1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#reflectionpad2d"><span class="hidden-section">ReflectionPad2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#replicationpad1d"><span class="hidden-section">ReplicationPad1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#replicationpad2d"><span class="hidden-section">ReplicationPad2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#replicationpad3d"><span class="hidden-section">ReplicationPad3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#zeropad2d"><span class="hidden-section">ZeroPad2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#constantpad1d"><span class="hidden-section">ConstantPad1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#constantpad2d"><span class="hidden-section">ConstantPad2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#constantpad3d"><span class="hidden-section">ConstantPad3d</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#non-linear-activations-weighted-sum-nonlinearity">Non-linear activations (weighted sum, nonlinearity)</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#elu"><span class="hidden-section">ELU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#hardshrink"><span class="hidden-section">Hardshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#hardtanh"><span class="hidden-section">Hardtanh</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#leakyrelu"><span class="hidden-section">LeakyReLU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#logsigmoid"><span class="hidden-section">LogSigmoid</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#prelu"><span class="hidden-section">PReLU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#relu"><span class="hidden-section">ReLU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#relu6"><span class="hidden-section">ReLU6</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#rrelu"><span class="hidden-section">RReLU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#selu"><span class="hidden-section">SELU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#sigmoid"><span class="hidden-section">Sigmoid</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softplus"><span class="hidden-section">Softplus</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softshrink"><span class="hidden-section">Softshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softsign"><span class="hidden-section">Softsign</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#tanh"><span class="hidden-section">Tanh</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#tanhshrink"><span class="hidden-section">Tanhshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#threshold"><span class="hidden-section">Threshold</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#non-linear-activations-other">Non-linear activations (other)</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softmin"><span class="hidden-section">Softmin</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softmax"><span class="hidden-section">Softmax</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softmax2d"><span class="hidden-section">Softmax2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#logsoftmax"><span class="hidden-section">LogSoftmax</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptivelogsoftmaxwithloss"><span class="hidden-section">AdaptiveLogSoftmaxWithLoss</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#normalization-layers">Normalization layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#batchnorm1d"><span class="hidden-section">BatchNorm1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#batchnorm2d"><span class="hidden-section">BatchNorm2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#batchnorm3d"><span class="hidden-section">BatchNorm3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#groupnorm"><span class="hidden-section">GroupNorm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#instancenorm1d"><span class="hidden-section">InstanceNorm1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#instancenorm2d"><span class="hidden-section">InstanceNorm2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#instancenorm3d"><span class="hidden-section">InstanceNorm3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#layernorm"><span class="hidden-section">LayerNorm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#localresponsenorm"><span class="hidden-section">LocalResponseNorm</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#recurrent-layers">Recurrent layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#rnn"><span class="hidden-section">RNN</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lstm"><span class="hidden-section">LSTM</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#gru"><span class="hidden-section">GRU</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#rnncell"><span class="hidden-section">RNNCell</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lstmcell"><span class="hidden-section">LSTMCell</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#grucell"><span class="hidden-section">GRUCell</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#linear-layers">Linear layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#linear"><span class="hidden-section">Linear</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#bilinear"><span class="hidden-section">Bilinear</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#dropout-layers">Dropout layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#dropout"><span class="hidden-section">Dropout</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#dropout2d"><span class="hidden-section">Dropout2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#dropout3d"><span class="hidden-section">Dropout3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#alphadropout"><span class="hidden-section">AlphaDropout</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#sparse-layers">Sparse layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#embedding"><span class="hidden-section">Embedding</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#embeddingbag"><span class="hidden-section">EmbeddingBag</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#distance-functions">Distance functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#cosinesimilarity"><span class="hidden-section">CosineSimilarity</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pairwisedistance"><span class="hidden-section">PairwiseDistance</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#loss-functions">Loss functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#l1loss"><span class="hidden-section">L1Loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#mseloss"><span class="hidden-section">MSELoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#crossentropyloss"><span class="hidden-section">CrossEntropyLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#nllloss"><span class="hidden-section">NLLLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#poissonnllloss"><span class="hidden-section">PoissonNLLLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#kldivloss"><span class="hidden-section">KLDivLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#bceloss"><span class="hidden-section">BCELoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#bcewithlogitsloss"><span class="hidden-section">BCEWithLogitsLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#marginrankingloss"><span class="hidden-section">MarginRankingLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#hingeembeddingloss"><span class="hidden-section">HingeEmbeddingLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multilabelmarginloss"><span class="hidden-section">MultiLabelMarginLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#smoothl1loss"><span class="hidden-section">SmoothL1Loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#softmarginloss"><span class="hidden-section">SoftMarginLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multilabelsoftmarginloss"><span class="hidden-section">MultiLabelSoftMarginLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#cosineembeddingloss"><span class="hidden-section">CosineEmbeddingLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multimarginloss"><span class="hidden-section">MultiMarginLoss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#tripletmarginloss"><span class="hidden-section">TripletMarginLoss</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#vision-layers">Vision layers</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pixelshuffle"><span class="hidden-section">PixelShuffle</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#upsample"><span class="hidden-section">Upsample</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#upsamplingnearest2d"><span class="hidden-section">UpsamplingNearest2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#upsamplingbilinear2d"><span class="hidden-section">UpsamplingBilinear2d</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#dataparallel-layers-multi-gpu-distributed">DataParallel layers (multi-GPU, distributed)</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#dataparallel"><span class="hidden-section">DataParallel</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#distributeddataparallel"><span class="hidden-section">DistributedDataParallel</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#utilities">Utilities</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#clip-grad-norm"><span class="hidden-section">clip_grad_norm_</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#clip-grad-value"><span class="hidden-section">clip_grad_value_</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#parameters-to-vector"><span class="hidden-section">parameters_to_vector</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#vector-to-parameters"><span class="hidden-section">vector_to_parameters</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#weight-norm"><span class="hidden-section">weight_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#remove-weight-norm"><span class="hidden-section">remove_weight_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#spectral-norm"><span class="hidden-section">spectral_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#remove-spectral-norm"><span class="hidden-section">remove_spectral_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#packedsequence"><span class="hidden-section">PackedSequence</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pack-padded-sequence"><span class="hidden-section">pack_padded_sequence</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pad-packed-sequence"><span class="hidden-section">pad_packed_sequence</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pad-sequence"><span class="hidden-section">pad_sequence</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pack-sequence"><span class="hidden-section">pack_sequence</span></a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="nn.html#torch-nn-functional">torch.nn.functional</a><ul>
<li class="toctree-l2"><a class="reference internal" href="nn.html#convolution-functions">Convolution functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id20"><span class="hidden-section">conv1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id21"><span class="hidden-section">conv2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id22"><span class="hidden-section">conv3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv-transpose1d"><span class="hidden-section">conv_transpose1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv-transpose2d"><span class="hidden-section">conv_transpose2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#conv-transpose3d"><span class="hidden-section">conv_transpose3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id23"><span class="hidden-section">unfold</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id24"><span class="hidden-section">fold</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#pooling-functions">Pooling functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avg-pool1d"><span class="hidden-section">avg_pool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avg-pool2d"><span class="hidden-section">avg_pool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#avg-pool3d"><span class="hidden-section">avg_pool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-pool1d"><span class="hidden-section">max_pool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-pool2d"><span class="hidden-section">max_pool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-pool3d"><span class="hidden-section">max_pool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-unpool1d"><span class="hidden-section">max_unpool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-unpool2d"><span class="hidden-section">max_unpool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#max-unpool3d"><span class="hidden-section">max_unpool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lp-pool1d"><span class="hidden-section">lp_pool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#lp-pool2d"><span class="hidden-section">lp_pool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-max-pool1d"><span class="hidden-section">adaptive_max_pool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-max-pool2d"><span class="hidden-section">adaptive_max_pool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-max-pool3d"><span class="hidden-section">adaptive_max_pool3d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-avg-pool1d"><span class="hidden-section">adaptive_avg_pool1d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-avg-pool2d"><span class="hidden-section">adaptive_avg_pool2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#adaptive-avg-pool3d"><span class="hidden-section">adaptive_avg_pool3d</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#non-linear-activation-functions">Non-linear activation functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id25"><span class="hidden-section">threshold</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id26"><span class="hidden-section">relu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id27"><span class="hidden-section">hardtanh</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id28"><span class="hidden-section">relu6</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id29"><span class="hidden-section">elu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id30"><span class="hidden-section">selu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#leaky-relu"><span class="hidden-section">leaky_relu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id31"><span class="hidden-section">prelu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id32"><span class="hidden-section">rrelu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#glu"><span class="hidden-section">glu</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id33"><span class="hidden-section">logsigmoid</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id34"><span class="hidden-section">hardshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id35"><span class="hidden-section">tanhshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id36"><span class="hidden-section">softsign</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id37"><span class="hidden-section">softplus</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id38"><span class="hidden-section">softmin</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id39"><span class="hidden-section">softmax</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id40"><span class="hidden-section">softshrink</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#gumbel-softmax"><span class="hidden-section">gumbel_softmax</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#log-softmax"><span class="hidden-section">log_softmax</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id41"><span class="hidden-section">tanh</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id42"><span class="hidden-section">sigmoid</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#normalization-functions">Normalization functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#batch-norm"><span class="hidden-section">batch_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#instance-norm"><span class="hidden-section">instance_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#layer-norm"><span class="hidden-section">layer_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#local-response-norm"><span class="hidden-section">local_response_norm</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#normalize"><span class="hidden-section">normalize</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#linear-functions">Linear functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id43"><span class="hidden-section">linear</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id44"><span class="hidden-section">bilinear</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#dropout-functions">Dropout functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id45"><span class="hidden-section">dropout</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#alpha-dropout"><span class="hidden-section">alpha_dropout</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id46"><span class="hidden-section">dropout2d</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id47"><span class="hidden-section">dropout3d</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#sparse-functions">Sparse functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id48"><span class="hidden-section">embedding</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#embedding-bag"><span class="hidden-section">embedding_bag</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#id49">Distance functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pairwise-distance"><span class="hidden-section">pairwise_distance</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#cosine-similarity"><span class="hidden-section">cosine_similarity</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#id50">Loss functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#binary-cross-entropy"><span class="hidden-section">binary_cross_entropy</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#poisson-nll-loss"><span class="hidden-section">poisson_nll_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#cosine-embedding-loss"><span class="hidden-section">cosine_embedding_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#cross-entropy"><span class="hidden-section">cross_entropy</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#hinge-embedding-loss"><span class="hidden-section">hinge_embedding_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#kl-div"><span class="hidden-section">kl_div</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#l1-loss"><span class="hidden-section">l1_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#mse-loss"><span class="hidden-section">mse_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#margin-ranking-loss"><span class="hidden-section">margin_ranking_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multilabel-margin-loss"><span class="hidden-section">multilabel_margin_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multilabel-soft-margin-loss"><span class="hidden-section">multilabel_soft_margin_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#multi-margin-loss"><span class="hidden-section">multi_margin_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#nll-loss"><span class="hidden-section">nll_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#binary-cross-entropy-with-logits"><span class="hidden-section">binary_cross_entropy_with_logits</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#smooth-l1-loss"><span class="hidden-section">smooth_l1_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#soft-margin-loss"><span class="hidden-section">soft_margin_loss</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#triplet-margin-loss"><span class="hidden-section">triplet_margin_loss</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#vision-functions">Vision functions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pixel-shuffle"><span class="hidden-section">pixel_shuffle</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#pad"><span class="hidden-section">pad</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#interpolate"><span class="hidden-section">interpolate</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#id51"><span class="hidden-section">upsample</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#upsample-nearest"><span class="hidden-section">upsample_nearest</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#upsample-bilinear"><span class="hidden-section">upsample_bilinear</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#grid-sample"><span class="hidden-section">grid_sample</span></a></li>
<li class="toctree-l3"><a class="reference internal" href="nn.html#affine-grid"><span class="hidden-section">affine_grid</span></a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="nn.html#dataparallel-functions-multi-gpu-distributed">DataParallel functions (multi-GPU, distributed)</a><ul>
<li class="toctree-l3"><a class="reference internal" href="nn.html#data-parallel"><span class="hidden-section">data_parallel</span></a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="nn.html#torch-nn-init">torch.nn.init</a></li>
<li class="toctree-l1"><a class="reference internal" href="optim.html">torch.optim</a><ul>
<li class="toctree-l2"><a class="reference internal" href="optim.html#how-to-use-an-optimizer">How to use an optimizer</a><ul>
<li class="toctree-l3"><a class="reference internal" href="optim.html#constructing-it">Constructing it</a></li>
<li class="toctree-l3"><a class="reference internal" href="optim.html#per-parameter-options">Per-parameter options</a></li>
<li class="toctree-l3"><a class="reference internal" href="optim.html#taking-an-optimization-step">Taking an optimization step</a><ul>
<li class="toctree-l4"><a class="reference internal" href="optim.html#optimizer-step"><code class="docutils literal notranslate"><span class="pre">optimizer.step()</span></code></a></li>
<li class="toctree-l4"><a class="reference internal" href="optim.html#optimizer-step-closure"><code class="docutils literal notranslate"><span class="pre">optimizer.step(closure)</span></code></a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="optim.html#algorithms">Algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="optim.html#how-to-adjust-learning-rate">How to adjust Learning Rate</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="autograd.html">torch.autograd</a><ul>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#locally-disabling-gradient-computation">Locally disabling gradient computation</a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#in-place-operations-on-tensors">In-place operations on Tensors</a><ul>
<li class="toctree-l3"><a class="reference internal" href="autograd.html#in-place-correctness-checks">In-place correctness checks</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#variable-deprecated">Variable (deprecated)</a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#tensor-autograd-functions">Tensor autograd functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#function"><span class="hidden-section">Function</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#numerical-gradient-checking">Numerical gradient checking</a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#profiler">Profiler</a></li>
<li class="toctree-l2"><a class="reference internal" href="autograd.html#anomaly-detection">Anomaly detection</a></li>
</ul>
</li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">torch.distributions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#score-function">Score function</a></li>
<li class="toctree-l2"><a class="reference internal" href="#pathwise-derivative">Pathwise derivative</a></li>
<li class="toctree-l2"><a class="reference internal" href="#distribution"><span class="hidden-section">Distribution</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#exponentialfamily"><span class="hidden-section">ExponentialFamily</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#bernoulli"><span class="hidden-section">Bernoulli</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#beta"><span class="hidden-section">Beta</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#binomial"><span class="hidden-section">Binomial</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#categorical"><span class="hidden-section">Categorical</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#cauchy"><span class="hidden-section">Cauchy</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#chi2"><span class="hidden-section">Chi2</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#dirichlet"><span class="hidden-section">Dirichlet</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#exponential"><span class="hidden-section">Exponential</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#fishersnedecor"><span class="hidden-section">FisherSnedecor</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#gamma"><span class="hidden-section">Gamma</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#geometric"><span class="hidden-section">Geometric</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#gumbel"><span class="hidden-section">Gumbel</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#halfcauchy"><span class="hidden-section">HalfCauchy</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#halfnormal"><span class="hidden-section">HalfNormal</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#independent"><span class="hidden-section">Independent</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#laplace"><span class="hidden-section">Laplace</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#lognormal"><span class="hidden-section">LogNormal</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#multinomial"><span class="hidden-section">Multinomial</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#multivariatenormal"><span class="hidden-section">MultivariateNormal</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#normal"><span class="hidden-section">Normal</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#onehotcategorical"><span class="hidden-section">OneHotCategorical</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#pareto"><span class="hidden-section">Pareto</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#poisson"><span class="hidden-section">Poisson</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#relaxedbernoulli"><span class="hidden-section">RelaxedBernoulli</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#relaxedonehotcategorical"><span class="hidden-section">RelaxedOneHotCategorical</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#studentt"><span class="hidden-section">StudentT</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#transformeddistribution"><span class="hidden-section">TransformedDistribution</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#uniform"><span class="hidden-section">Uniform</span></a></li>
<li class="toctree-l2"><a class="reference internal" href="#module-torch.distributions.kl"><cite>KL Divergence</cite></a></li>
<li class="toctree-l2"><a class="reference internal" href="#module-torch.distributions.transforms"><cite>Transforms</cite></a></li>
<li class="toctree-l2"><a class="reference internal" href="#module-torch.distributions.constraints"><cite>Constraints</cite></a></li>
<li class="toctree-l2"><a class="reference internal" href="#module-torch.distributions.constraint_registry"><cite>Constraint Registry</cite></a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="multiprocessing.html">torch.multiprocessing</a><ul>
<li class="toctree-l2"><a class="reference internal" href="multiprocessing.html#strategy-management">Strategy management</a></li>
<li class="toctree-l2"><a class="reference internal" href="multiprocessing.html#sharing-cuda-tensors">Sharing CUDA tensors</a></li>
<li class="toctree-l2"><a class="reference internal" href="multiprocessing.html#sharing-strategies">Sharing strategies</a><ul>
<li class="toctree-l3"><a class="reference internal" href="multiprocessing.html#file-descriptor-file-descriptor">File descriptor - <code class="docutils literal notranslate"><span class="pre">file_descriptor</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="multiprocessing.html#file-system-file-system">File system - <code class="docutils literal notranslate"><span class="pre">file_system</span></code></a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="distributed.html">torch.distributed</a><ul>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#basics">Basics</a></li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#initialization">Initialization</a><ul>
<li class="toctree-l3"><a class="reference internal" href="distributed.html#tcp-initialization">TCP initialization</a></li>
<li class="toctree-l3"><a class="reference internal" href="distributed.html#shared-file-system-initialization">Shared file-system initialization</a></li>
<li class="toctree-l3"><a class="reference internal" href="distributed.html#environment-variable-initialization">Environment variable initialization</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#groups">Groups</a></li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#point-to-point-communication">Point-to-point communication</a></li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#collective-functions">Collective functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#multi-gpu-collective-functions">Multi-GPU collective functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="distributed.html#launch-utility">Launch utility</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="bottleneck.html">torch.utils.bottleneck</a></li>
<li class="toctree-l1"><a class="reference internal" href="checkpoint.html">torch.utils.checkpoint</a></li>
<li class="toctree-l1"><a class="reference internal" href="cpp_extension.html">torch.utils.cpp_extension</a></li>
<li class="toctree-l1"><a class="reference internal" href="data.html">torch.utils.data</a></li>
<li class="toctree-l1"><a class="reference internal" href="dlpack.html">torch.utils.dlpack</a></li>
<li class="toctree-l1"><a class="reference internal" href="ffi.html">torch.utils.ffi</a></li>
<li class="toctree-l1"><a class="reference internal" href="model_zoo.html">torch.utils.model_zoo</a></li>
<li class="toctree-l1"><a class="reference internal" href="onnx.html">torch.onnx</a><ul>
<li class="toctree-l2"><a class="reference internal" href="onnx.html#example-end-to-end-alexnet-from-pytorch-to-caffe2">Example: End-to-end AlexNet from PyTorch to Caffe2</a></li>
<li class="toctree-l2"><a class="reference internal" href="onnx.html#limitations">Limitations</a></li>
<li class="toctree-l2"><a class="reference internal" href="onnx.html#supported-operators">Supported operators</a></li>
<li class="toctree-l2"><a class="reference internal" href="onnx.html#functions">Functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="legacy.html">torch.legacy</a></li>
</ul>
<p class="caption"><span class="caption-text">torchvision Reference</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="torchvision/index.html">torchvision</a><ul>
<li class="toctree-l2"><a class="reference internal" href="torchvision/datasets.html">torchvision.datasets</a><ul>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#mnist">MNIST</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#fashion-mnist">Fashion-MNIST</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#emnist">EMNIST</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#coco">COCO</a><ul>
<li class="toctree-l4"><a class="reference internal" href="torchvision/datasets.html#captions">Captions</a></li>
<li class="toctree-l4"><a class="reference internal" href="torchvision/datasets.html#detection">Detection</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#lsun">LSUN</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#imagefolder">ImageFolder</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#datasetfolder">DatasetFolder</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#imagenet-12">Imagenet-12</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#cifar">CIFAR</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#stl10">STL10</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#svhn">SVHN</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/datasets.html#phototour">PhotoTour</a></li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="torchvision/models.html">torchvision.models</a><ul>
<li class="toctree-l3"><a class="reference internal" href="torchvision/models.html#id1">Alexnet</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/models.html#id2">VGG</a></li>
<li class="toctree-l3"><a class="reference internal" href="torchvision/models.html#id3">ResNet</a></li>
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<div class="section" id="module-torch.distributions">
<span id="probability-distributions-torch-distributions"></span><h1>Probability distributions - torch.distributions<a class="headerlink" href="#module-torch.distributions" title="Permalink to this headline">¶</a></h1>
<p>The <code class="docutils literal notranslate"><span class="pre">distributions</span></code> package contains parameterizable probability distributions
and sampling functions. This allows the construction of stochastic computation
graphs and stochastic gradient estimators for optimization. This package
generally follows the design of the <a class="reference external" href="https://arxiv.org/abs/1711.10604">TensorFlow Distributions</a> package.</p>
<p>It is not possible to directly backpropagate through random samples. However,
there are two main methods for creating surrogate functions that can be
backpropagated through. These are the score function estimator/likelihood ratio
estimator/REINFORCE and the pathwise derivative estimator. REINFORCE is commonly
seen as the basis for policy gradient methods in reinforcement learning, and the
pathwise derivative estimator is commonly seen in the reparameterization trick
in variational autoencoders. Whilst the score function only requires the value
of samples <span class="math notranslate nohighlight">\(f(x)\)</span>, the pathwise derivative requires the derivative
<span class="math notranslate nohighlight">\(f'(x)\)</span>. The next sections discuss these two in a reinforcement learning
example. For more details see
<a class="reference external" href="https://arxiv.org/abs/1506.05254">Gradient Estimation Using Stochastic Computation Graphs</a> .</p>
<div class="section" id="score-function">
<h2>Score function<a class="headerlink" href="#score-function" title="Permalink to this headline">¶</a></h2>
<p>When the probability density function is differentiable with respect to its
parameters, we only need <code class="xref py py-meth docutils literal notranslate"><span class="pre">sample()</span></code> and
<code class="xref py py-meth docutils literal notranslate"><span class="pre">log_prob()</span></code> to implement REINFORCE:</p>
<div class="math notranslate nohighlight">
\[\Delta\theta = \alpha r \frac{\partial\log p(a|\pi^\theta(s))}{\partial\theta}\]</div>
<p>where <span class="math notranslate nohighlight">\(\theta\)</span> are the parameters, <span class="math notranslate nohighlight">\(\alpha\)</span> is the learning rate,
<span class="math notranslate nohighlight">\(r\)</span> is the reward and <span class="math notranslate nohighlight">\(p(a|\pi^\theta(s))\)</span> is the probability of
taking action <span class="math notranslate nohighlight">\(a\)</span> in state <span class="math notranslate nohighlight">\(s\)</span> given policy <span class="math notranslate nohighlight">\(\pi^\theta\)</span>.</p>
<p>In practice we would sample an action from the output of a network, apply this
action in an environment, and then use <code class="docutils literal notranslate"><span class="pre">log_prob</span></code> to construct an equivalent
loss function. Note that we use a negative because optimizers use gradient
descent, whilst the rule above assumes gradient ascent. With a categorical
policy, the code for implementing REINFORCE would be as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">probs</span> <span class="o">=</span> <span class="n">policy_network</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
<span class="c1"># Note that this is equivalent to what used to be called multinomial</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="p">)</span>
<span class="n">action</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>
<span class="n">next_state</span><span class="p">,</span> <span class="n">reward</span> <span class="o">=</span> <span class="n">env</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">action</span><span class="p">)</span>
<span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">m</span><span class="o">.</span><span class="n">log_prob</span><span class="p">(</span><span class="n">action</span><span class="p">)</span> <span class="o">*</span> <span class="n">reward</span>
<span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="pathwise-derivative">
<h2>Pathwise derivative<a class="headerlink" href="#pathwise-derivative" title="Permalink to this headline">¶</a></h2>
<p>The other way to implement these stochastic/policy gradients would be to use the
reparameterization trick from the
<code class="xref py py-meth docutils literal notranslate"><span class="pre">rsample()</span></code> method, where the
parameterized random variable can be constructed via a parameterized
deterministic function of a parameter-free random variable. The reparameterized
sample therefore becomes differentiable. The code for implementing the pathwise
derivative would be as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">params</span> <span class="o">=</span> <span class="n">policy_network</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="o">*</span><span class="n">params</span><span class="p">)</span>
<span class="c1"># Any distribution with .has_rsample == True could work based on the application</span>
<span class="n">action</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">rsample</span><span class="p">()</span>
<span class="n">next_state</span><span class="p">,</span> <span class="n">reward</span> <span class="o">=</span> <span class="n">env</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">action</span><span class="p">)</span> <span class="c1"># Assuming that reward is differentiable</span>
<span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">reward</span>
<span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="distribution">
<h2><span class="hidden-section">Distribution</span><a class="headerlink" href="#distribution" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.distribution.Distribution">
<em class="property">class </em><code class="descclassname">torch.distributions.distribution.</code><code class="descname">Distribution</code><span class="sig-paren">(</span><em>batch_shape=torch.Size([])</em>, <em>event_shape=torch.Size([])</em>, <em>validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference external" href="https://docs.python.org/3/library/functions.html#object" title="(in Python v3.7)"><code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></a></p>
<p>Distribution is the abstract base class for probability distributions.</p>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.arg_constraints">
<code class="descname">arg_constraints</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a dictionary from argument names to
<a class="reference internal" href="#torch.distributions.constraints.Constraint" title="torch.distributions.constraints.Constraint"><code class="xref py py-class docutils literal notranslate"><span class="pre">Constraint</span></code></a> objects that
should be satisfied by each argument of this distribution. Args that
are not tensors need not appear in this dict.</p>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.batch_shape">
<code class="descname">batch_shape</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.batch_shape" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the shape over which parameters are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.cdf">
<code class="descname">cdf</code><span class="sig-paren">(</span><em>value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.cdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.cdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the cumulative density/mass function evaluated at
<cite>value</cite>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.entropy">
<code class="descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.entropy" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns entropy of distribution, batched over batch_shape.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">Tensor of shape batch_shape.</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.enumerate_support">
<code class="descname">enumerate_support</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.enumerate_support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns tensor containing all values supported by a discrete
distribution. The result will enumerate over dimension 0, so the shape
of the result will be <cite>(cardinality,) + batch_shape + event_shape</cite>
(where <cite>event_shape = ()</cite> for univariate distributions).</p>
<p>Note that this enumerates over all batched tensors in lock-step
<cite>[[0, 0], [1, 1], …]</cite>. To iterate over the full Cartesian product
use <cite>itertools.product(m.enumerate_support())</cite>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">Tensor iterating over dimension 0.</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.event_shape">
<code class="descname">event_shape</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.event_shape" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the shape of a single sample (without batching).</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.icdf">
<code class="descname">icdf</code><span class="sig-paren">(</span><em>value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.icdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.icdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the inverse cumulative density/mass function evaluated at
<cite>value</cite>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.log_prob">
<code class="descname">log_prob</code><span class="sig-paren">(</span><em>value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.log_prob" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the log of the probability density/mass function evaluated at
<cite>value</cite>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.mean">
<code class="descname">mean</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.mean" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the mean of the distribution.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.perplexity">
<code class="descname">perplexity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.perplexity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.perplexity" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns perplexity of distribution, batched over batch_shape.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Returns:</th><td class="field-body">Tensor of shape batch_shape.</td>
</tr>
</tbody>
</table>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.rsample">
<code class="descname">rsample</code><span class="sig-paren">(</span><em>sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.rsample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.rsample" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates a sample_shape shaped reparameterized sample or sample_shape
shaped batch of reparameterized samples if the distribution parameters
are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.sample">
<code class="descname">sample</code><span class="sig-paren">(</span><em>sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.sample" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates a sample_shape shaped sample or sample_shape shaped batch of
samples if the distribution parameters are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.sample_n">
<code class="descname">sample_n</code><span class="sig-paren">(</span><em>n</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.sample_n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.sample_n" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates n samples or n batches of samples if the distribution
parameters are batched.</p>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.stddev">
<code class="descname">stddev</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.stddev" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the standard deviation of the distribution.</p>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.support">
<code class="descname">support</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.support" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a <a class="reference internal" href="#torch.distributions.constraints.Constraint" title="torch.distributions.constraints.Constraint"><code class="xref py py-class docutils literal notranslate"><span class="pre">Constraint</span></code></a> object
representing this distribution’s support.</p>
</dd></dl>
<dl class="attribute">
<dt id="torch.distributions.distribution.Distribution.variance">
<code class="descname">variance</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.variance" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the variance of the distribution.</p>
</dd></dl>
</dd></dl>
</div>
<div class="section" id="exponentialfamily">
<h2><span class="hidden-section">ExponentialFamily</span><a class="headerlink" href="#exponentialfamily" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.exp_family.ExponentialFamily">
<em class="property">class </em><code class="descclassname">torch.distributions.exp_family.</code><code class="descname">ExponentialFamily</code><span class="sig-paren">(</span><em>batch_shape=torch.Size([])</em>, <em>event_shape=torch.Size([])</em>, <em>validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/exp_family.html#ExponentialFamily"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.exp_family.ExponentialFamily" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.distribution.Distribution" title="torch.distributions.distribution.Distribution"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.distribution.Distribution</span></code></a></p>
<p>ExponentialFamily is the abstract base class for probability distributions belonging to an
exponential family, whose probability mass/density function has the form is defined below</p>
<div class="math notranslate nohighlight">
\[p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle) - F(\theta) + k(x))\]</div>
<p>where <span class="math notranslate nohighlight">\(\theta\)</span> denotes the natural parameters, <span class="math notranslate nohighlight">\(t(x)\)</span> denotes the sufficient statistic,
<span class="math notranslate nohighlight">\(F(\theta)\)</span> is the log normalizer function for a given family and <span class="math notranslate nohighlight">\(k(x)\)</span> is the carrier
measure.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">This class is an intermediary between the <cite>Distribution</cite> class and distributions which belong
to an exponential family mainly to check the correctness of the <cite>.entropy()</cite> and analytic KL
divergence methods. We use this class to compute the entropy and KL divergence using the AD frame-
work and Bregman divergences (courtesy of: Frank Nielsen and Richard Nock, Entropies and