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+# Design Doc: Computations as Graphs
+
+A primary goal of the refactorization of PaddlePaddle is a more flexible representation of deep learning computation, in particular, a graph of operators and variables, instead of sequences of layers as before.
+
+This document explains that the construction of a graph as three steps:
+
+- construct the forward part
+- construct the backward part
+- construct the optimization part
+
+Let us take the problem of image classification as a simple example.  The application program that trains the model looks like:
+
+```python
+x = layer.data("images")
+l = layer.data("label")
+y = layer.fc(x)
+cost = layer.mse(y, l)
+optimize(cost)
+train(cost, reader=mnist.train())
+```
+
+### Forward Part
+
+The first four lines of above program build the forward part of the graph.
+
+![](images/graph_construction_example_forward_only.png)
+
+In particular, the first line `x = layer.data("images")` creates variable x and a Feed operator that copies a column from the minibatch to x.  `y = layer.fc(x)` creates not only the FC operator and output variable y, but also two parameters, W and b.
+
+In this example, all operators are created as `OpDesc` protobuf messages, and all variables are `VarDesc`.  These protobuf messages are saved in a `BlockDesc` protobuf message.
+
+### Backward Part
+
+The fifth line `optimize(cost)` calls two functions, `ConstructBackwardGraph` and `ConstructOptimizationGraph`.
+
+`ConstructBackwardGraph` traverses the forward graph in the `BlockDesc` protobuf message and builds the backward part.
+
+![](images/graph_construction_example_forward_backward.png)
+
+According to the chain rule of gradient computation, `ConstructBackwardGraph` would
+
+1. create a gradient operator G for each operator F,
+1. make all inputs, outputs, and outputs' gradient of F as inputs of G,
+1. create gradients for all inputs of F, except for those who don't have gradients, like x and l, and
+1. make all these gradients as outputs of G.
+
+### Optimization Part
+
+For each parameter, like W and b created by `layer.fc`, marked as double circles in above graphs, `ConstructOptimizationGraph` creates an optimization operator to apply its gradient.  Here results in the complete graph:
+
+![](images/graph_construction_example_all.png)
diff --git a/doc/design/images/graph_construction_example.bash b/doc/design/images/graph_construction_example.bash
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+cat ./graph_construction_example.dot | \
+    sed 's/color=red/color=red, style=invis/g' | \
+    sed 's/color=green/color=green, style=invis/g' | \
+    dot -Tpng > graph_construction_example_forward_only.png
+
+cat ./graph_construction_example.dot | \
+    sed 's/color=green/color=green, style=invis/g' | \
+    dot -Tpng > graph_construction_example_forward_backward.png
+
+cat ./graph_construction_example.dot | \
+    dot -Tpng > graph_construction_example_all.png
diff --git a/doc/design/images/graph_construction_example.dot b/doc/design/images/graph_construction_example.dot
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+digraph ImageClassificationGraph {
+        ///////// The forward part /////////
+        FeedX [label="Feed", color=blue, shape=box];
+        FeedY [label="Feed", color=blue, shape=box];
+        FC [label="FC", color=blue, shape=box];
+        MSE [label="MSE", color=blue, shape=box];
+
+        x [label="x", color=blue, shape=oval];
+        l [label="l", color=blue, shape=oval];
+        y [label="y", color=blue, shape=oval];
+        W [label="W", color=blue, shape=doublecircle];
+        b [label="b", color=blue, shape=doublecircle];
+        cost [label="cost", color=blue, shape=oval];
+
+        FeedX -> x -> FC -> y -> MSE -> cost [color=blue];
+        FeedY -> l [color=blue];
+        W -> FC [color=blue];
+        b -> FC [color=blue];
+        l -> MSE [color=blue];
+
+        ////////// The backward part /////////
+        MSE_Grad [label="MSE_grad", color=red, shape=box];
+        FC_Grad [label="FC_grad", color=red, shape=box];
+
+        d_cost [label="d cost", color=red, shape=oval];
+        d_y [label="d y", color=red, shape=oval];
+        d_b [label="d b", color=red, shape=oval];
+        d_W [label="d W", color=red, shape=oval];
+
+        cost -> MSE_Grad [color=red];
+        d_cost -> MSE_Grad [color=red];
+        x -> MSE_Grad [color=red];
+        l -> MSE_Grad [color=red];
+        y -> MSE_Grad -> d_y [color=red];
+
+        x -> FC_Grad [color=red];
+        y -> FC_Grad [color=red];
+        d_y -> FC_Grad [color=red];
+        W -> FC_Grad -> d_W [color=red];
+        b -> FC_Grad -> d_b [color=red];
+
+        ////////// The optimizaiton part //////////
+
+        OPT_W [label="SGD", color=green, shape=box];
+        OPT_b [label="SGD", color=green, shape=box];
+
+        W -> OPT_W [color=green];
+        b -> OPT_b [color=green];
+        d_W -> OPT_W -> W [color=green];
+        d_b -> OPT_b -> b [color=green];
+
+        ////////// Groupings //////////
+
+        subgraph clusterMSE {
+                style=invis;
+                MSE;
+                MSE_Grad;
+        }
+
+        subgraph clusterFC {
+                style=invis;
+                FC;
+                FC_Grad;
+        }
+}
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