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| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/0-introduction.html">Introduction</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/1-nparray.html">Step 1: Manipulate data with NP on MXNet</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/2-create-nn.html">Step 2: Create a neural network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/3-autograd.html">Step 3: Automatic differentiation with autograd</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/4-components.html">Step 4: Necessary components that are not in the network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html">Step 5: <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s and <code class="docutils literal notranslate"><span class="pre">DataLoader</span></code></a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#Using-own-data-with-included-Datasets">Using own data with included <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#Using-your-own-data-with-custom-Datasets">Using your own data with custom <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#New-in-MXNet-2.0:-faster-C++-backend-dataloaders">New in MXNet 2.0: faster C++ backend dataloaders</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/6-train-nn.html">Step 6: Train a Neural Network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/7-use-gpus.html">Step 7: Load and Run a NN using GPU</a></li> |
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| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/to-mxnet/index.html">Moving to MXNet from Other Frameworks</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/to-mxnet/pytorch.html">PyTorch vs Apache MXNet</a></li> |
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| <li class="toctree-l3"><a class="reference external" href="https://mxnet.apache.org/api/python/docs/tutorials/packages/gluon/image/mnist.html">MNIST</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../data/datasets.html#Appendix:-Upgrading-from-Module-DataIter-to-Gluon-DataLoader">Appendix: Upgrading from Module <code class="docutils literal notranslate"><span class="pre">DataIter</span></code> to Gluon <code class="docutils literal notranslate"><span class="pre">DataLoader</span></code></a></li> |
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| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_finder.html">Learning Rate Finder</a></li> |
| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_schedules.html">Learning Rate Schedules</a></li> |
| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_schedules_advanced.html">Advanced Learning Rate Schedules</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../training/normalization/index.html">Normalization Blocks</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../kvstore/kvstore.html">Distributed Key-Value Store</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/01-ndarray-intro.html">An Intro: Manipulate Data the MXNet Way with NDArray</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/gotchas_numpy_in_mxnet.html">Gotchas using NumPy in Apache MXNet</a></li> |
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| <li class="toctree-l3"><a class="reference internal" href="../../np/index.html">What is NP on MXNet</a><ul> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../performance/backend/dnnl/dnnl_quantization_inc.html">Improving accuracy with Intel® Neural Compressor</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.flipud.html">mxnet.np.flipud</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.io.html">Input and output</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.genfromtxt.html">mxnet.np.genfromtxt</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ndarray.tolist.html">mxnet.np.ndarray.tolist</a></li> |
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| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.linalg.html">Linear algebra (<code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy.linalg</span></code>)</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.dot.html">mxnet.np.dot</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.tensordot.html">mxnet.np.tensordot</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.einsum.html">mxnet.np.einsum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.multi_dot.html">mxnet.np.linalg.multi_dot</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.matmul.html">mxnet.np.matmul</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.matrix_power.html">mxnet.np.linalg.matrix_power</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.kron.html">mxnet.np.kron</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.svd.html">mxnet.np.linalg.svd</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.cholesky.html">mxnet.np.linalg.cholesky</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.qr.html">mxnet.np.linalg.qr</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eig.html">mxnet.np.linalg.eig</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eigh.html">mxnet.np.linalg.eigh</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eigvals.html">mxnet.np.linalg.eigvals</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eigvalsh.html">mxnet.np.linalg.eigvalsh</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.norm.html">mxnet.np.linalg.norm</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.trace.html">mxnet.np.trace</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.cond.html">mxnet.np.linalg.cond</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.det.html">mxnet.np.linalg.det</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.matrix_rank.html">mxnet.np.linalg.matrix_rank</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.slogdet.html">mxnet.np.linalg.slogdet</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.solve.html">mxnet.np.linalg.solve</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.tensorsolve.html">mxnet.np.linalg.tensorsolve</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.lstsq.html">mxnet.np.linalg.lstsq</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.inv.html">mxnet.np.linalg.inv</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.pinv.html">mxnet.np.linalg.pinv</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.math.html">Mathematical functions</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sin.html">mxnet.np.sin</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cos.html">mxnet.np.cos</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.tan.html">mxnet.np.tan</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.arcsin.html">mxnet.np.arcsin</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.arccos.html">mxnet.np.arccos</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.radians.html">mxnet.np.radians</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.unwrap.html">mxnet.np.unwrap</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cosh.html">mxnet.np.cosh</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cumprod.html">mxnet.np.cumprod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nancumprod.html">mxnet.np.nancumprod</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.subtract.html">mxnet.np.subtract</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.modf.html">mxnet.np.modf</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sqrt.html">mxnet.np.sqrt</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cbrt.html">mxnet.np.cbrt</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.square.html">mxnet.np.square</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.absolute.html">mxnet.np.absolute</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sign.html">mxnet.np.sign</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.maximum.html">mxnet.np.maximum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.minimum.html">mxnet.np.minimum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fabs.html">mxnet.np.fabs</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.heaviside.html">mxnet.np.heaviside</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fmin.html">mxnet.np.fmin</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nan_to_num.html">mxnet.np.nan_to_num</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.interp.html">mxnet.np.interp</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/random/index.html">np.random</a><ul> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.shuffle.html">mxnet.np.random.shuffle</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.gamma.html">mxnet.np.random.gamma</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.gumbel.html">mxnet.np.random.gumbel</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.logistic.html">mxnet.np.random.logistic</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.lognormal.html">mxnet.np.random.lognormal</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.multinomial.html">mxnet.np.random.multinomial</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.multivariate_normal.html">mxnet.np.random.multivariate_normal</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.pareto.html">mxnet.np.random.pareto</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.power.html">mxnet.np.random.power</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.rayleigh.html">mxnet.np.random.rayleigh</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.weibull.html">mxnet.np.random.weibull</a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.sort.html">Sorting, searching, and counting</a><ul> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.partition.html">mxnet.np.partition</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.argpartition.html">mxnet.np.argpartition</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanargmax.html">mxnet.np.nanargmax</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.statistics.html">Statistics</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.min.html">mxnet.np.min</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.mean.html">mxnet.np.mean</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.std.html">mxnet.np.std</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.average.html">mxnet.np.average</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanmedian.html">mxnet.np.nanmedian</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanstd.html">mxnet.np.nanstd</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.corrcoef.html">mxnet.np.corrcoef</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.correlate.html">mxnet.np.correlate</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cov.html">mxnet.np.cov</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.histogram.html">mxnet.np.histogram</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.digitize.html">mxnet.np.digitize</a></li> |
| </ul> |
| </li> |
| </ul> |
| </li> |
| </ul> |
| </li> |
| <li class="toctree-l2"><a class="reference internal" href="../../../../api/npx/index.html">NPX: NumPy Neural Network Extension</a><ul> |
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| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.cpu.html">mxnet.npx.cpu</a></li> |
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| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.activation.html">mxnet.npx.activation</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.batch_norm.html">mxnet.npx.batch_norm</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.convolution.html">mxnet.npx.convolution</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.dropout.html">mxnet.npx.dropout</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.embedding.html">mxnet.npx.embedding</a></li> |
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| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.layer_norm.html">mxnet.npx.layer_norm</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.pooling.html">mxnet.npx.pooling</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.rnn.html">mxnet.npx.rnn</a></li> |
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| <span class="mdl-layout-title toc">Table Of Contents</span> |
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| <ul class="current"> |
| <li class="toctree-l1 current"><a class="reference internal" href="../../../index.html">Python Tutorials</a><ul class="current"> |
| <li class="toctree-l2"><a class="reference internal" href="../../../getting-started/index.html">Getting Started</a><ul> |
| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/crash-course/index.html">Crash Course</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/0-introduction.html">Introduction</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/1-nparray.html">Step 1: Manipulate data with NP on MXNet</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/2-create-nn.html">Step 2: Create a neural network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/3-autograd.html">Step 3: Automatic differentiation with autograd</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/4-components.html">Step 4: Necessary components that are not in the network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html">Step 5: <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s and <code class="docutils literal notranslate"><span class="pre">DataLoader</span></code></a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#Using-own-data-with-included-Datasets">Using own data with included <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#Using-your-own-data-with-custom-Datasets">Using your own data with custom <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/5-datasets.html#New-in-MXNet-2.0:-faster-C++-backend-dataloaders">New in MXNet 2.0: faster C++ backend dataloaders</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/6-train-nn.html">Step 6: Train a Neural Network</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/crash-course/7-use-gpus.html">Step 7: Load and Run a NN using GPU</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/to-mxnet/index.html">Moving to MXNet from Other Frameworks</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../../getting-started/to-mxnet/pytorch.html">PyTorch vs Apache MXNet</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/gluon_from_experiment_to_deployment.html">Gluon: from experiment to deployment</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/gluon_migration_guide.html">Gluon2.0: Migration Guide</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../getting-started/logistic_regression_explained.html">Logistic regression explained</a></li> |
| <li class="toctree-l3"><a class="reference external" href="https://mxnet.apache.org/api/python/docs/tutorials/packages/gluon/image/mnist.html">MNIST</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l2 current"><a class="reference internal" href="../../index.html">Packages</a><ul class="current"> |
| <li class="toctree-l3"><a class="reference internal" href="../../autograd/index.html">Automatic Differentiation</a></li> |
| <li class="toctree-l3 current"><a class="reference internal" href="../index.html">Gluon</a><ul class="current"> |
| <li class="toctree-l4"><a class="reference internal" href="../blocks/index.html">Blocks</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/custom-layer.html">Custom Layers</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/hybridize.html">Hybridize</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/init.html">Initialization</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/naming.html">Parameter and Block Naming</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/nn.html">Layers and Blocks</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/parameters.html">Parameter Management</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/save_load_params.html">Saving and Loading Gluon Models</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../blocks/activations/activations.html">Activation Blocks</a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../data/index.html">Data Tutorials</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../data/data_augmentation.html">Image Augmentation</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../data/datasets.html">Gluon <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s and <code class="docutils literal notranslate"><span class="pre">DataLoader</span></code></a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../data/datasets.html#Using-own-data-with-included-Datasets">Using own data with included <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../data/datasets.html#Using-own-data-with-custom-Datasets">Using own data with custom <code class="docutils literal notranslate"><span class="pre">Dataset</span></code>s</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../data/datasets.html#Appendix:-Upgrading-from-Module-DataIter-to-Gluon-DataLoader">Appendix: Upgrading from Module <code class="docutils literal notranslate"><span class="pre">DataIter</span></code> to Gluon <code class="docutils literal notranslate"><span class="pre">DataLoader</span></code></a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../image/index.html">Image Tutorials</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../image/info_gan.html">Image similarity search with InfoGAN</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../image/mnist.html">Handwritten Digit Recognition</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l4 current"><a class="reference internal" href="index.html">Losses</a><ul class="current"> |
| <li class="toctree-l5"><a class="reference internal" href="custom-loss.html">Custom Loss Blocks</a></li> |
| <li class="toctree-l5 current"><a class="current reference internal" href="#">Kullback-Leibler (KL) Divergence</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="loss.html">Loss functions</a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../text/index.html">Text Tutorials</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../text/gnmt.html">Google Neural Machine Translation</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../text/transformer.html">Machine Translation with Transformer</a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../training/index.html">Training</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../training/fit_api_tutorial.html">MXNet Gluon Fit API</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../training/trainer.html">Trainer</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../training/learning_rates/index.html">Learning Rates</a><ul> |
| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_finder.html">Learning Rate Finder</a></li> |
| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_schedules.html">Learning Rate Schedules</a></li> |
| <li class="toctree-l6"><a class="reference internal" href="../training/learning_rates/learning_rate_schedules_advanced.html">Advanced Learning Rate Schedules</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l5"><a class="reference internal" href="../training/normalization/index.html">Normalization Blocks</a></li> |
| </ul> |
| </li> |
| </ul> |
| </li> |
| <li class="toctree-l3"><a class="reference internal" href="../../kvstore/index.html">KVStore</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../kvstore/kvstore.html">Distributed Key-Value Store</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l3"><a class="reference internal" href="../../legacy/index.html">Legacy</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../legacy/ndarray/index.html">NDArray</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/01-ndarray-intro.html">An Intro: Manipulate Data the MXNet Way with NDArray</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/02-ndarray-operations.html">NDArray Operations</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/03-ndarray-contexts.html">NDArray Contexts</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/gotchas_numpy_in_mxnet.html">Gotchas using NumPy in Apache MXNet</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../legacy/ndarray/sparse/index.html">Tutorials</a><ul> |
| <li class="toctree-l6"><a class="reference internal" href="../../legacy/ndarray/sparse/csr.html">CSRNDArray - NDArray in Compressed Sparse Row Storage Format</a></li> |
| <li class="toctree-l6"><a class="reference internal" href="../../legacy/ndarray/sparse/row_sparse.html">RowSparseNDArray - NDArray for Sparse Gradient Updates</a></li> |
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| </li> |
| </ul> |
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| </ul> |
| </li> |
| <li class="toctree-l3"><a class="reference internal" href="../../np/index.html">What is NP on MXNet</a><ul> |
| <li class="toctree-l4"><a class="reference internal" href="../../np/cheat-sheet.html">The NP on MXNet cheat sheet</a></li> |
| <li class="toctree-l4"><a class="reference internal" href="../../np/np-vs-numpy.html">Differences between NP on MXNet and NumPy</a></li> |
| </ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.io.html">Input and output</a><ul> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.linalg.html">Linear algebra (<code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy.linalg</span></code>)</a><ul> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.svd.html">mxnet.np.linalg.svd</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eigvals.html">mxnet.np.linalg.eigvals</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.eigvalsh.html">mxnet.np.linalg.eigvalsh</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.norm.html">mxnet.np.linalg.norm</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.linalg.cond.html">mxnet.np.linalg.cond</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.math.html">Mathematical functions</a><ul> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cos.html">mxnet.np.cos</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.rint.html">mxnet.np.rint</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fix.html">mxnet.np.fix</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.floor.html">mxnet.np.floor</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ceil.html">mxnet.np.ceil</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.trunc.html">mxnet.np.trunc</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.around.html">mxnet.np.around</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.round_.html">mxnet.np.round_</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sum.html">mxnet.np.sum</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cumsum.html">mxnet.np.cumsum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanprod.html">mxnet.np.nanprod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nansum.html">mxnet.np.nansum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cumprod.html">mxnet.np.cumprod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nancumprod.html">mxnet.np.nancumprod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nancumsum.html">mxnet.np.nancumsum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.diff.html">mxnet.np.diff</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ediff1d.html">mxnet.np.ediff1d</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cross.html">mxnet.np.cross</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.trapz.html">mxnet.np.trapz</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.exp.html">mxnet.np.exp</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.expm1.html">mxnet.np.expm1</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.log.html">mxnet.np.log</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.log10.html">mxnet.np.log10</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.log2.html">mxnet.np.log2</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.log1p.html">mxnet.np.log1p</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.logaddexp.html">mxnet.np.logaddexp</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.i0.html">mxnet.np.i0</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ldexp.html">mxnet.np.ldexp</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.signbit.html">mxnet.np.signbit</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.copysign.html">mxnet.np.copysign</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.frexp.html">mxnet.np.frexp</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.spacing.html">mxnet.np.spacing</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.lcm.html">mxnet.np.lcm</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.gcd.html">mxnet.np.gcd</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.add.html">mxnet.np.add</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.divide.html">mxnet.np.divide</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.power.html">mxnet.np.power</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.subtract.html">mxnet.np.subtract</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.mod.html">mxnet.np.mod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.multiply.html">mxnet.np.multiply</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.true_divide.html">mxnet.np.true_divide</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.remainder.html">mxnet.np.remainder</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.positive.html">mxnet.np.positive</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.float_power.html">mxnet.np.float_power</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fmod.html">mxnet.np.fmod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.modf.html">mxnet.np.modf</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.divmod.html">mxnet.np.divmod</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.floor_divide.html">mxnet.np.floor_divide</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.clip.html">mxnet.np.clip</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sqrt.html">mxnet.np.sqrt</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cbrt.html">mxnet.np.cbrt</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.square.html">mxnet.np.square</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.absolute.html">mxnet.np.absolute</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.sign.html">mxnet.np.sign</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.maximum.html">mxnet.np.maximum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.minimum.html">mxnet.np.minimum</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fabs.html">mxnet.np.fabs</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.heaviside.html">mxnet.np.heaviside</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fmax.html">mxnet.np.fmax</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.fmin.html">mxnet.np.fmin</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nan_to_num.html">mxnet.np.nan_to_num</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.interp.html">mxnet.np.interp</a></li> |
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| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/random/index.html">np.random</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.choice.html">mxnet.np.random.choice</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.shuffle.html">mxnet.np.random.shuffle</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.normal.html">mxnet.np.random.normal</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.uniform.html">mxnet.np.random.uniform</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.rand.html">mxnet.np.random.rand</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.randint.html">mxnet.np.random.randint</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.chisquare.html">mxnet.np.random.chisquare</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.exponential.html">mxnet.np.random.exponential</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.f.html">mxnet.np.random.f</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.gamma.html">mxnet.np.random.gamma</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.gumbel.html">mxnet.np.random.gumbel</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.laplace.html">mxnet.np.random.laplace</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.logistic.html">mxnet.np.random.logistic</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.lognormal.html">mxnet.np.random.lognormal</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.multinomial.html">mxnet.np.random.multinomial</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.multivariate_normal.html">mxnet.np.random.multivariate_normal</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.pareto.html">mxnet.np.random.pareto</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.power.html">mxnet.np.random.power</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.rayleigh.html">mxnet.np.random.rayleigh</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/random/generated/mxnet.np.random.weibull.html">mxnet.np.random.weibull</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.sort.html">Sorting, searching, and counting</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ndarray.sort.html">mxnet.np.ndarray.sort</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.lexsort.html">mxnet.np.lexsort</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.argsort.html">mxnet.np.argsort</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.flatnonzero.html">mxnet.np.flatnonzero</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.where.html">mxnet.np.where</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.extract.html">mxnet.np.extract</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.count_nonzero.html">mxnet.np.count_nonzero</a></li> |
| </ul> |
| </li> |
| <li class="toctree-l4"><a class="reference internal" href="../../../../api/np/routines.statistics.html">Statistics</a><ul> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.min.html">mxnet.np.min</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.max.html">mxnet.np.max</a></li> |
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| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanmax.html">mxnet.np.nanmax</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.ptp.html">mxnet.np.ptp</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.percentile.html">mxnet.np.percentile</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanpercentile.html">mxnet.np.nanpercentile</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.quantile.html">mxnet.np.quantile</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanquantile.html">mxnet.np.nanquantile</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.mean.html">mxnet.np.mean</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.std.html">mxnet.np.std</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.var.html">mxnet.np.var</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.median.html">mxnet.np.median</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.average.html">mxnet.np.average</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanmedian.html">mxnet.np.nanmedian</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanstd.html">mxnet.np.nanstd</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.nanvar.html">mxnet.np.nanvar</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.corrcoef.html">mxnet.np.corrcoef</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.correlate.html">mxnet.np.correlate</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.cov.html">mxnet.np.cov</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.histogram.html">mxnet.np.histogram</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.histogram2d.html">mxnet.np.histogram2d</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.histogramdd.html">mxnet.np.histogramdd</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.bincount.html">mxnet.np.bincount</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.histogram_bin_edges.html">mxnet.np.histogram_bin_edges</a></li> |
| <li class="toctree-l5"><a class="reference internal" href="../../../../api/np/generated/mxnet.np.digitize.html">mxnet.np.digitize</a></li> |
| </ul> |
| </li> |
| </ul> |
| </li> |
| </ul> |
| </li> |
| <li class="toctree-l2"><a class="reference internal" href="../../../../api/npx/index.html">NPX: NumPy Neural Network Extension</a><ul> |
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| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.reset_np.html">mxnet.npx.reset_np</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.cpu.html">mxnet.npx.cpu</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.cpu_pinned.html">mxnet.npx.cpu_pinned</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.gpu.html">mxnet.npx.gpu</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.gpu_memory_info.html">mxnet.npx.gpu_memory_info</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.current_device.html">mxnet.npx.current_device</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.num_gpus.html">mxnet.npx.num_gpus</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.activation.html">mxnet.npx.activation</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.batch_norm.html">mxnet.npx.batch_norm</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.convolution.html">mxnet.npx.convolution</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.dropout.html">mxnet.npx.dropout</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.embedding.html">mxnet.npx.embedding</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.fully_connected.html">mxnet.npx.fully_connected</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.layer_norm.html">mxnet.npx.layer_norm</a></li> |
| <li class="toctree-l3"><a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.pooling.html">mxnet.npx.pooling</a></li> |
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| <!--- Licensed to the Apache Software Foundation (ASF) under one --><!--- or more contributor license agreements. See the NOTICE file --><!--- distributed with this work for additional information --><!--- regarding copyright ownership. The ASF licenses this file --><!--- to you under the Apache License, Version 2.0 (the --><!--- "License"); you may not use this file except in compliance --><!--- with the License. You may obtain a copy of the License at --><!--- http://www.apache.org/licenses/LICENSE-2.0 --><!--- Unless required by applicable law or agreed to in writing, --><!--- software distributed under the License is distributed on an --><!--- "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY --><!--- KIND, either express or implied. See the License for the --><!--- specific language governing permissions and limitations --><!--- under the License. --><div class="section" id="Kullback-Leibler-(KL)-Divergence"> |
| <h1>Kullback-Leibler (KL) Divergence<a class="headerlink" href="#Kullback-Leibler-(KL)-Divergence" title="Permalink to this headline">¶</a></h1> |
| <p>Kullback-Leibler (KL) Divergence is a measure of how one probability distribution is different from a second, reference probability distribution. Smaller KL Divergence values indicate more similar distributions and, since this loss function is differentiable, we can use gradient descent to minimize the KL divergence between network outputs and some target distribution. As an example, this can be used in Variational Autoencoders (VAEs), and reinforcement learning policy networks such as <a class="reference external" href="https://arxiv.org/abs/1502.05477">Trust |
| Region Policy Optimization (TRPO)</a>.</p> |
| <p>In MXNet Gluon, we can use <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> to compare categorical distributions. One important thing to note is that the KL Divergence is an asymmetric measure (i.e. <code class="docutils literal notranslate"><span class="pre">KL(P,Q)</span> <span class="pre">!=</span> <span class="pre">KL(Q,P)</span></code>): order matters and we should compare our predicted distribution with our target distribution in that order. Another thing to note is that there are two ways to use <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> |
| that depend on how we set <code class="docutils literal notranslate"><span class="pre">from_logits</span></code> (which has a default value of true).</p> |
| <p>As an example, let’s compare a few categorical distributions (<code class="docutils literal notranslate"><span class="pre">dist_1</span></code>, <code class="docutils literal notranslate"><span class="pre">dist_2</span></code> and <code class="docutils literal notranslate"><span class="pre">dist_3</span></code>), each with 4 categories.</p> |
| <div class="nbinput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span> |
| <span class="kn">import</span> <span class="nn">mxnet</span> <span class="k">as</span> <span class="nn">mx</span> |
| <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> |
| |
| <span class="n">idx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span> |
| <span class="n">dist_1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span> |
| <span class="n">dist_2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">])</span> |
| <span class="n">dist_3</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">])</span> |
| |
| <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">top</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">dist_1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'black'</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">dist_2</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'aqua'</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Distributions 1 & 2'</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="n">top</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">dist_1</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'black'</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">idx</span><span class="p">,</span> <span class="n">dist_3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'aqua'</span><span class="p">)</span> |
| <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">'Distributions 1 & 3'</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nboutput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]: |
| </pre></div> |
| </div> |
| <div class="output_area highlight-none notranslate"><div class="highlight"><pre> |
| <span></span>Text(0.5, 1.0, 'Distributions 1 & 3') |
| </pre></div> |
| </div> |
| </div> |
| <div class="nboutput nblast docutils container"> |
| <div class="prompt empty docutils container"> |
| </div> |
| <div class="output_area docutils container"> |
| <img alt="../../../../_images/tutorials_packages_gluon_loss_kl_divergence_1_1.png" src="../../../../_images/tutorials_packages_gluon_loss_kl_divergence_1_1.png" /> |
| </div> |
| </div> |
| <p>We can see visually that distributions 1 and 2 are more similar than distributions 1 and 3. We’ll confirm this result using <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a>. When using <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> with the default <code class="docutils literal notranslate"><span class="pre">from_logits=True</span></code> we need:</p> |
| <ol class="arabic simple"> |
| <li><p>our predictions to be parameters of a logged probability distribution.</p></li> |
| <li><p>our targets to be parameters of a probability distribution (i.e. not logged).</p></li> |
| </ol> |
| <p>We often apply a <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.softmax.html"><span class="doc">softmax</span></a> operation to the output of our network to get a distribution, but this can have a numerically unstable gradient calculation. As as stable alternative, we use <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.log_softmax.html"><span class="doc">log_softmax</span></a> and so this is what is expected by <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> when <code class="docutils literal notranslate"><span class="pre">from_logits=True</span></code>. We also usually work with batches of |
| predictions, so the predictions and targets need to have a batch dimension (the first axis by default).</p> |
| <p>Since we’re already working with distributions in this example, we don’t need to apply the softmax and only need to apply <a class="reference internal" href="../../../../api/np/generated/mxnet.np.log.html"><span class="doc">log</span></a>. And we’ll create batch dimensions even though we’re working with single distributions.</p> |
| <div class="nbinput nblast docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[2]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="k">def</span> <span class="nf">kl_divergence</span><span class="p">(</span><span class="n">dist_a</span><span class="p">,</span> <span class="n">dist_b</span><span class="p">):</span> |
| <span class="c1"># add batch dimension</span> |
| <span class="n">pred_batch</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">dist_a</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> |
| <span class="n">target_batch</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">dist_b</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> |
| <span class="c1"># log the distribution</span> |
| <span class="n">pred_batch</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">pred_batch</span><span class="p">)</span> |
| <span class="c1"># create loss (assuming we have a logged prediction distribution)</span> |
| <span class="n">loss_fn</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">gluon</span><span class="o">.</span><span class="n">loss</span><span class="o">.</span><span class="n">KLDivLoss</span><span class="p">(</span><span class="n">from_logits</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> |
| <span class="n">divergence</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">pred_batch</span><span class="p">,</span> <span class="n">target_batch</span><span class="p">)</span> |
| <span class="k">return</span> <span class="n">divergence</span><span class="o">.</span><span class="n">item</span><span class="p">()</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nbinput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[3]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="nb">print</span><span class="p">(</span><span class="s2">"Distribution 1 compared with Distribution 2: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> |
| <span class="n">kl_divergence</span><span class="p">(</span><span class="n">dist_1</span><span class="p">,</span> <span class="n">dist_2</span><span class="p">)))</span> |
| <span class="nb">print</span><span class="p">(</span><span class="s2">"Distribution 1 compared with Distribution 3: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> |
| <span class="n">kl_divergence</span><span class="p">(</span><span class="n">dist_1</span><span class="p">,</span> <span class="n">dist_3</span><span class="p">)))</span> |
| <span class="nb">print</span><span class="p">(</span><span class="s2">"Distribution 1 compared with Distribution 1: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> |
| <span class="n">kl_divergence</span><span class="p">(</span><span class="n">dist_1</span><span class="p">,</span> <span class="n">dist_1</span><span class="p">)))</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nboutput docutils container"> |
| <div class="prompt empty docutils container"> |
| </div> |
| <div class="output_area docutils container"> |
| <div class="highlight"><pre> |
| Distribution 1 compared with Distribution 2: 0.025424212217330933 |
| Distribution 1 compared with Distribution 3: 0.2656409740447998 |
| Distribution 1 compared with Distribution 1: 0.0 |
| </pre></div></div> |
| </div> |
| <div class="nboutput nblast docutils container"> |
| <div class="prompt empty docutils container"> |
| </div> |
| <div class="output_area stderr docutils container"> |
| <div class="highlight"><pre> |
| [04:48:32] /work/mxnet/src/storage/storage.cc:202: Using Pooled (Naive) StorageManager for CPU |
| </pre></div></div> |
| </div> |
| <p>As expected we see a smaller KL Divergence for distributions 1 & 2 than 1 & 3. And we also see the KL Divergence of a distribution with itself is 0.</p> |
| <div class="section" id="from_logits=False"> |
| <h2><code class="docutils literal notranslate"><span class="pre">from_logits=False</span></code><a class="headerlink" href="#from_logits=False" title="Permalink to this headline">¶</a></h2> |
| <p>Alternatively, instead of manually applying the <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.log_softmax.html"><span class="doc">log_softmax</span></a> to our network outputs, we can leave that to the loss function. When setting <code class="docutils literal notranslate"><span class="pre">from_logits=False</span></code> on <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a>, the <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.log_softmax.html"><span class="doc">log_softmax</span></a> is applied to the first argument passed to <code class="docutils literal notranslate"><span class="pre">loss_fn</span></code>. As an example, let’s assume our network outputs us the values |
| below (favorably chosen so that when we <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.softmax.html"><span class="doc">softmax</span></a> these values we get the same distribution parameters as <code class="docutils literal notranslate"><span class="pre">dist_1</span></code>).</p> |
| <div class="nbinput nblast docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[4]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="n">output</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.39056206</span><span class="p">,</span> <span class="mf">1.3068528</span><span class="p">,</span> <span class="mf">0.39056206</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.30258512</span><span class="p">])</span> |
| </pre></div> |
| </div> |
| </div> |
| <p>We can pass this to our <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> loss function (with <code class="docutils literal notranslate"><span class="pre">from_logits=False</span></code>) and get the same KL Divergence between <code class="docutils literal notranslate"><span class="pre">dist_1</span></code> and <code class="docutils literal notranslate"><span class="pre">dist_2</span></code> as before, because the <a class="reference internal" href="../../../../api/npx/generated/mxnet.npx.log_softmax.html"><span class="doc">log_softmax</span></a> is applied within the loss function.</p> |
| <div class="nbinput nblast docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[5]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="k">def</span> <span class="nf">kl_divergence_not_from_logits</span><span class="p">(</span><span class="n">dist_a</span><span class="p">,</span> <span class="n">dist_b</span><span class="p">):</span> |
| <span class="c1"># add batch dimension</span> |
| <span class="n">pred_batch</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">dist_a</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> |
| <span class="n">target_batch</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">dist_b</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> |
| <span class="c1"># create loss (assuming we have a logged prediction distribution)</span> |
| <span class="n">loss_fn</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">gluon</span><span class="o">.</span><span class="n">loss</span><span class="o">.</span><span class="n">KLDivLoss</span><span class="p">(</span><span class="n">from_logits</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span> |
| <span class="n">divergence</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">pred_batch</span><span class="p">,</span> <span class="n">target_batch</span><span class="p">)</span> |
| <span class="k">return</span> <span class="n">divergence</span><span class="o">.</span><span class="n">item</span><span class="p">()</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nbinput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[6]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="nb">print</span><span class="p">(</span><span class="s2">"Distribution 1 compared with Distribution 2: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> |
| <span class="n">kl_divergence_not_from_logits</span><span class="p">(</span><span class="n">output</span><span class="p">,</span> <span class="n">dist_2</span><span class="p">)))</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nboutput nblast docutils container"> |
| <div class="prompt empty docutils container"> |
| </div> |
| <div class="output_area docutils container"> |
| <div class="highlight"><pre> |
| Distribution 1 compared with Distribution 2: 0.025424206629395485 |
| </pre></div></div> |
| </div> |
| <div class="section" id="Advanced:-Common-Support"> |
| <h3>Advanced: Common Support<a class="headerlink" href="#Advanced:-Common-Support" title="Permalink to this headline">¶</a></h3> |
| <p>Occasionally, you might have issues with <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a>. One common issue arises when the support of the distributions being compared are not the same. ‘Support’ here is referring to the values of the distribution which have a non-zero probability. Conveniently, all our examples above had the same support, but we might have a case where some categories have a probability of 0.</p> |
| <div class="nbinput nblast docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[7]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="n">dist_4</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nbinput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[8]: |
| </pre></div> |
| </div> |
| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="nb">print</span><span class="p">(</span><span class="s2">"Distribution 4 compared with Distribution 1: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span> |
| <span class="n">kl_divergence</span><span class="p">(</span><span class="n">dist_4</span><span class="p">,</span> <span class="n">dist_1</span><span class="p">)))</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="nboutput nblast docutils container"> |
| <div class="prompt empty docutils container"> |
| </div> |
| <div class="output_area docutils container"> |
| <div class="highlight"><pre> |
| Distribution 4 compared with Distribution 1: nan |
| </pre></div></div> |
| </div> |
| <p>We can see that the result is <code class="docutils literal notranslate"><span class="pre">nan</span></code>, which will obviously cause issues when calculating the gradient. One option is to add a small value <code class="docutils literal notranslate"><span class="pre">epsilon</span></code> to all of the probabilities, and this is already done for the target distribution (using the value of 1e-12).</p> |
| </div> |
| <div class="section" id="Advanced:-Aggregation"> |
| <h3>Advanced: Aggregation<a class="headerlink" href="#Advanced:-Aggregation" title="Permalink to this headline">¶</a></h3> |
| <p>One minor difference between the true definition of KL Divergence and the result from <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> is how the aggregation of category contributions is performed. Although the true definition sums up these contributions, the default behaviour in MXNet Gluon is to average terms along the batch dimension. As a result, the <a class="reference internal" href="../../../../api/gluon/loss/index.html#mxnet.gluon.loss.KLDivLoss"><span class="std std-ref">KLDivLoss</span></a> output will be smaller than the |
| true definition by a factor of the number of categories.</p> |
| <div class="nbinput docutils container"> |
| <div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[9]: |
| </pre></div> |
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| <div class="input_area highlight-python notranslate"><div class="highlight"><pre> |
| <span></span><span class="n">true_divergence</span> <span class="o">=</span> <span class="p">(</span><span class="n">dist_2</span><span class="o">*</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">dist_2</span><span class="p">)</span><span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">dist_1</span><span class="p">)))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> |
| <span class="nb">print</span><span class="p">(</span><span class="s1">'true_divergence: </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">true_divergence</span><span class="p">))</span> |
| </pre></div> |
| </div> |
| </div> |
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| <div class="highlight"><pre> |
| true_divergence: 0.10169682996275987 |
| </pre></div></div> |
| </div> |
| <div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">num_categories</span> <span class="o">=</span> <span class="n">dist_1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> |
| <span class="n">divergence</span> <span class="o">=</span> <span class="n">kl_divergence</span><span class="p">(</span><span class="n">dist_1</span><span class="p">,</span> <span class="n">dist_2</span><span class="p">)</span> |
| <span class="nb">print</span><span class="p">(</span><span class="s1">'divergence: </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">divergence</span><span class="p">))</span> |
| <span class="nb">print</span><span class="p">(</span><span class="s1">'divergence * num_categories: </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">divergence</span> <span class="o">*</span> <span class="n">num_categories</span><span class="p">))</span> |
| </pre></div> |
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| <ul> |
| <li><a class="reference internal" href="#">Kullback-Leibler (KL) Divergence</a><ul> |
| <li><a class="reference internal" href="#from_logits=False"><code class="docutils literal notranslate"><span class="pre">from_logits=False</span></code></a><ul> |
| <li><a class="reference internal" href="#Advanced:-Common-Support">Advanced: Common Support</a></li> |
| <li><a class="reference internal" href="#Advanced:-Aggregation">Advanced: Aggregation</a></li> |
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