diff --git a/examples/MNIST.jl b/examples/MNIST.jl
index 88970f0f..e768e166 100644
--- a/examples/MNIST.jl
+++ b/examples/MNIST.jl
@@ -1,5 +1,5 @@
 using Flux, MNIST
-using Flux: accuracy
+using Flux: accuracy, onehot
 
 data = [(trainfeatures(i), onehot(trainlabel(i), 0:9)) for i = 1:60_000]
 train = data[1:50_000]
diff --git a/src/backend/tensorflow/graph.jl b/src/backend/tensorflow/graph.jl
index 63c4f4e2..b34e23a7 100644
--- a/src/backend/tensorflow/graph.jl
+++ b/src/backend/tensorflow/graph.jl
@@ -28,10 +28,13 @@ graph(::typeof(all), x, dim=nothing) = TensorFlow.reduce_all(x;axis=dim)
 graph(::typeof(any), x, dim=nothing) = TensorFlow.reduce_any(x;axis=dim)
 graph(::typeof(mean), x, dim=nothing) = TensorFlow.reduce_mean(x;axis=dim)
 graph(::typeof(svd), x) = svd(x)
+graph(::typeof(size), x, dim) = TensorFlow.size(x,convert(Tensor{Int32}, dim))
+graph(::typeof(size), x) = TensorFlow.size(x)
+graph(::typeof(chol), args...) = TensorFlow.transpose(TensorFlow.cholesky(args...))
 
 for op in (*, .*, .+, .^, log, exp, ceil, floor, sqrt, abs, cos,
            sin, tan, atan, asin, acos, tanh, lgamma, erf, erfc, real, imag, conj,
-           inv, det)
+           inv, det, transpose, permutedims, cat, length, diag, diagm)
   @eval graph(::typeof($op), args...) = $op(args...)
 end
 
diff --git a/src/core.jl b/src/core.jl
index d3953849..66e33440 100644
--- a/src/core.jl
+++ b/src/core.jl
@@ -6,11 +6,11 @@ module FluxCore
 """
     back!(model, ΔY, X...) => ΔX
 
-Backpropagate the gradient `ΔY` through the model `m`, accumulating the
+Backpropagate the gradient `ΔY` through the model `model`, accumulating the
 gradients of any parameters. Returns the gradient of the input `X`. Gradients
 may be arrays or tuples of arrays (for multiple inputs/outputs).
 """
-back!(model, Δ, xs...) = error("Backprop not implemented for $(typeof(m))")
+back!(model, Δ, xs...) = error("Backprop not implemented for $(typeof(model))")
 
 """
     update!(model, η) => m
diff --git a/src/layers/affine.jl b/src/layers/affine.jl
index 9608efcc..ca79c004 100644
--- a/src/layers/affine.jl
+++ b/src/layers/affine.jl
@@ -9,3 +9,16 @@ Affine(in::Integer, out::Integer; init = initn) =
 
 inferred(::Type{Affine}, in::Tuple{Dims{2}}, out::Integer) =
   Affine(in[1][2], out)
+
+function back!(m::Affine, Δ, x)
+  W, b = m.W, m.b
+  W.Δx[:] = x' * Δ
+  b.Δx[:] = sum(Δ, 1)
+  Δ * W.x'
+end
+
+function update!(m::Affine, η)
+  update!(m.W, η)
+  update!(m.b, η)
+  m
+end
diff --git a/src/layers/control.jl b/src/layers/control.jl
index 7851f902..d0c5e61b 100644
--- a/src/layers/control.jl
+++ b/src/layers/control.jl
@@ -7,9 +7,19 @@ end
 @forward Chain.layers Base.start, Base.next, Base.done
 
 (s::Chain)(x) = foldl((x, m) -> m(x), x, s.layers)
-back!(s::Chain, Δ) = foldr((m, Δ) -> back!(m, Δ), Δ, s.layers)
 update!(s::Chain, η) = foreach(l -> update!(l, η), s.layers)
 
+function back!(s::Chain, Δ, x)
+  crumbs = foldl([x], s.layers[1:end-1]) do crumbs, layer
+    push!(crumbs, layer(crumbs[end]))
+  end
+
+  foldr(Δ, collect(zip(crumbs, s.layers))) do pack, Δ
+    x, layer = pack
+    back!(layer, Δ, x)
+  end
+end
+
 graph(s::Chain) =
   foldl((v, m) -> vertex(m, v), constant(inputnode(1)), s.layers)
 
diff --git a/test/backend/tensorflow.jl b/test/backend/tensorflow.jl
index 178de761..1dcfdf53 100644
--- a/test/backend/tensorflow.jl
+++ b/test/backend/tensorflow.jl
@@ -29,6 +29,24 @@ end
   @test A ≈ u*diagm(s)*transpose(v)
   @test tf(@net x -> inv(x))(A) ≈ inv(A)
   @test tf(@net x -> det(x))(A) ≈ det(A)
+  A = randn(Float32,(6,3))
+  @test tf(@net x -> transpose(x))(A) ≈ transpose(A)
+  A = randn(Float32,(6,3,2))
+  @test tf(@net (x,y) -> permutedims(x,y))(A,[3,2,1]) ≈ permutedims(A,[3,2,1])
+  A1 = randn(Float32,(4,1))
+  A2 = randn(Float32,(4,1))
+  @test tf(@net (x,y) -> cat(2,x,y))(A1,A2) ≈ cat(2,A1,A2)
+  @test tf(@net x -> length(x))(A1) == length(A1)
+  A = randn(Float32,(5,5))
+  @test tf(@net x -> diag(x))(A) ≈ diag(A)
+  A = randn(Float32,(5,))
+  @test tf(@net x -> diagm(x))(A) ≈ diagm(A)
+  A = randn(4,5)
+  @test tf(@net x -> size(x))(A) == [4,5]
+  @test tf(@net (x,y) -> size(x,y))(A,1) == 4
+  A = randn(6,5)
+  A = A'*A
+  @test tf(@net x -> chol(x))(A) ≈ chol(A)
 end
 
 end
diff --git a/test/optimizer.jl b/test/optimizer.jl
new file mode 100644
index 00000000..57f1d011
--- /dev/null
+++ b/test/optimizer.jl
@@ -0,0 +1,38 @@
+@testset "training julia models" begin
+
+    @testset "linear regression" begin
+        srand(0)
+
+        model = Affine(10, 1)
+
+        truth = Float32[0, 4, 2, 2, -3, 6, -1, 3, 2, -5]'
+
+        data = map(1:256) do i
+            x = rand(Float32, 10)
+            x, truth * x + 3rand(Float32)
+        end
+
+        Flux.train!(model, data, epoch=5)
+
+        @test cor(reshape.((model.W.x, truth), 10)...) > .99
+    end
+
+    @testset "logistic regression" begin
+        srand(0)
+
+        model = Chain(Affine(10, 1), σ)
+
+        truth = Float32[0, 4, 2, 2, -3, 6, -1, 3, 2, -5]'
+
+        data = map(1:256) do i
+            x = rand(Float32, 10)
+            x, truth * x + 2rand(Float32) > 5f0
+        end
+
+        Flux.train!(model, data, epoch=10)
+
+        @test cor(reshape.((model.layers[1].W.x, truth), 10)...) > .99
+    end
+
+end
+
diff --git a/test/runtests.jl b/test/runtests.jl
index 8dd1dd8e..1e4981f7 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -15,5 +15,7 @@ include("backend/common.jl")
 
 include("basic.jl")
 include("recurrent.jl")
+include("optimizer.jl")
+
 @tfonly include("backend/tensorflow.jl")
 @mxonly include("backend/mxnet.jl")