1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
|
Layer Normalization {#dev_guide_layer_normalization}
====================================================
>
> [API Reference](@ref dnnl_api_layer_normalization)
>
## General
The layer normalization primitive performs a forward or backward layer
normalization operation on a 2-5D data tensor.
### Forward
The layer normalization operation performs normalization over the last logical
axis of the data tensor and is defined by the following formulas. We show
formulas only for 3D data, which are straightforward to generalize to
cases of higher dimensions. Variable names follow the standard
@ref dev_guide_conventions.
\f[
\dst(t, n, c) =
\gamma(c) \cdot
\frac{\src(t, n, c) - \mu(t, n)} {\sqrt{\sigma^2(t, n) + \varepsilon}}
+ \beta(c),
\f]
where
- \f$\gamma(c), \beta(c)\f$ are optional scale and shift for a channel
(see #dnnl_use_scaleshift flag),
- \f$\mu(t, n), \sigma^2(t, n)\f$ are mean and variance (see
#dnnl_use_global_stats flag), and
- \f$\varepsilon\f$ is a constant to improve numerical stability.
Mean and variance are computed at runtime or provided by a user. When mean and variance are computed at runtime, the following formulas are used:
- \f$\mu(t, n) = \frac{1}{C} \sum\limits_{c} \src(t, n, c)_{}\f$,
- \f$\sigma^2(t, n) = \frac{1}{C} \sum\limits_{c} {}_{} (\src(t, n, c) - \mu(t, n))^2\f$.
The \f$\gamma(c)\f$ and \f$\beta(c)\f$ tensors are considered learnable.
#### Difference Between Forward Training and Forward Inference
* If mean and variance are computed at runtime (i.e., #dnnl_use_global_stats
is not set), they become outputs for the propagation kind
#dnnl_forward_training (because they would be required during the backward
propagation). Data layout for mean and variance must be specified during
initialization of the layer normalization descriptor by passing the memory
descriptor for statistics (e.g., by passing stat_desc in
dnnl::layer_normalization_forward::desc::desc()). Mean and variance are
not exposed for the propagation kind #dnnl_forward_inference.
### Backward
The backward propagation computes
\f$\diffsrc(t, n, c)\f$,
\f$\diffgamma(c)^*\f$, and \f$\diffbeta(c)^*\f$
based on
\f$\diffdst(t, n, c)\f$, \f$src(t, n, c)\f$, \f$\mu(t, n)\f$,
\f$\sigma^2(t, n)\f$, \f$\gamma(c) ^*\f$, and \f$\beta(c) ^*\f$.
The tensors marked with an asterisk are used only when the primitive is
configured to use \f$\gamma(c)\f$, and \f$\beta(c)\f$
(i.e., #dnnl_use_scaleshift is set).
## Execution Arguments
Depending on the [flags](@ref dnnl_normalization_flags_t) and
[propagation kind](@ref dnnl_prop_kind_t), the layer normalization primitive
requires different inputs and outputs. For clarity, a summary is shown below.
| | #dnnl_forward_inference | #dnnl_forward_training | #dnnl_backward | #dnnl_backward_data |
| :-- | :-- | :-- | :-- | :-- |
| #dnnl_normalization_flags_none | *Inputs*: \src <br><br> *Outputs*: \dst | *Inputs*: \src <br><br> *Outputs*: \dst, \f$\mu\f$, \f$\sigma^2\f$ | *Inputs*: \diffdst, \src, \f$\mu\f$, \f$\sigma^2\f$ <br><br> *Outputs*: \diffsrc | Same as for #dnnl_backward |
| #dnnl_use_global_stats | *Inputs*: \src, \f$\mu\f$, \f$\sigma^2\f$ <br><br> *Outputs*: \dst | *Inputs*: \src, \f$\mu\f$, \f$\sigma^2\f$ <br><br> *Outputs*: \dst | *Inputs*: \diffdst, \src, \f$\mu\f$, \f$\sigma^2\f$ <br><br> *Outputs*: \diffsrc | Same as for #dnnl_backward |
| #dnnl_use_scaleshift | *Inputs*: \src, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \dst | *Inputs*: \src, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \dst, \f$\mu\f$, \f$\sigma^2\f$ | *Inputs*: \diffdst, \src, \f$\mu\f$, \f$\sigma^2\f$, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \diffsrc, \diffgamma, \diffbeta | Not supported |
| #dnnl_use_global_stats \| #dnnl_use_scaleshift | *Inputs*: \src, \f$\mu\f$, \f$\sigma^2\f$, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \dst | *Inputs*: \src, \f$\mu\f$, \f$\sigma^2\f$, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \dst | *Inputs*: \diffdst, \src, \f$\mu\f$, \f$\sigma^2\f$, \f$\gamma\f$, \f$\beta\f$ <br><br> *Outputs*: \diffsrc, \diffgamma, \diffbeta | Not supported |
When executed, the inputs and outputs should be mapped to an execution
argument index as specified by the following table.
| Primitive input/output | Execution argument index |
| --- | --- |
| \src | DNNL_ARG_SRC |
| \f$\gamma, \beta\f$ | DNNL_ARG_SCALE_SHIFT |
| mean (\f$\mu\f$) | DNNL_ARG_MEAN |
| variance (\f$\sigma\f$) | DNNL_ARG_VARIANCE |
| \dst | DNNL_ARG_DST |
| \diffdst | DNNL_ARG_DIFF_DST |
| \diffsrc | DNNL_ARG_DIFF_SRC |
| \diffgamma, \diffbeta | DNNL_ARG_DIFF_SCALE_SHIFT |
## Implementation Details
### General Notes
1. The different flavors of the primitive are partially controlled by the @p
flags parameter that is passed to the operation descriptor initialization
function (e.g., dnnl::layer_normalization_forward::desc::desc()). Multiple
flags can be set using the bitwise OR operator (`|`).
2. For forward propagation, the mean and variance might be either computed at
runtime (in which case they are outputs of the primitive) or provided by
a user (in which case they are inputs). In the latter case, a user must set
the #dnnl_use_global_stats flag. For the backward propagation, the mean and
variance are always input parameters.
3. The memory format and data type for `src` and `dst` are assumed to be the
same, and in the API they are typically referred to as `data` (e.g., see
`data_desc` in dnnl::layer_normalization_forward::desc::desc()). The same is
true for `diff_src` and `diff_dst`. The corresponding memory descriptors are
referred to as `diff_data_desc`.
4. Both forward and backward propagation support in-place operations, meaning
that \src can be used as input and output for forward propagation, and
\diffdst can be used as input and output for backward propagation. In case of
an in-place operation, the original data will be overwritten. Note, however,
that backward propagation requires original \src, hence the corresponding
forward propagation should not be performed in-place.
### Data Type Support
The operation supports the following combinations of data types:
| Propagation | Source / Destination | Mean / Variance / ScaleShift
| :-- | :-- | :--
| forward / backward | f32 | f32
| forward | f16 | f32
### Data Representation
#### Mean and Variance
The mean (\f$\mu\f$) and variance (\f$\sigma^2\f$) are separate tensors with
number of dimensions equal to (\f$data\_ndims - 1\f$) and size
\f$(data\_dim[0], data\_dim[1], ..., data\_dim[ndims - 2])\f$.
The corresponding memory object can have an arbitrary memory format. Unless mean
and variance are computed at runtime and not exposed (i.e., propagation kind is
#dnnl_forward_inference and #dnnl_use_global_stats is not set), the user should
provide a memory descriptor for statistics when initializing the layer
normalization descriptor. For best performance, it is advised to use the memory
format that follows the data memory format; i.e., if the data format is
#dnnl_tnc, the best performance can be expected for statistics with the
#dnnl_tn format and suboptimal for statistics with the #dnnl_nt format.
#### Scale and Shift
If used, the scale (\f$\gamma\f$) and shift (\f$\beta\f$) are
combined in a single 2D tensor of shape \f$2 \times C\f$.
The format of the corresponding memory object must be #dnnl_nc (#dnnl_ab).
#### Source, Destination, and Their Gradients
The layer normalization primitive works with an arbitrary data tensor; however,
it was designed for RNN data tensors (i.e., #dnnl_nc, #dnnl_tnc, #dnnl_ldnc).
Unlike CNN data tensors, RNN data tensors have a single feature dimension.
Layer normalization performs normalization over the last logical dimension
(feature dimension for RNN tensors) across non-feature dimensions.
The layer normalization primitive is optimized for the following memory formats:
| Logical tensor | Implementations optimized for memory formats
| :-- | :--
| NC | #dnnl_nc (#dnnl_ab)
| TNC | #dnnl_tnc (#dnnl_abc), #dnnl_ntc (#dnnl_bac)
| LDNC | #dnnl_ldnc (#dnnl_abcd)
## Performance Tips
1. For data tensors (`src`, `dst`, `diff_src`, `diff_dst`), use memory formats
for which the last logical axis is the last in the physical memory layout.
2. For `mean`/`variance`, use the memory format that follows the data memory
format; i.e., if the data format is #dnnl_tnc, the best performance can be
expected for statistics with #dnnl_tn and suboptimal for statistics with the
#dnnl_nt format.
3. For backward propagation, use the same memory format for `src`, `diff_dst`,
and `diff_src` (the format of `diff_dst` and `diff_src` are always the same
because of the API). Different formats are functionally supported but lead to
highly suboptimal performance.
4. Use in-place operations whenever possible (see caveats in General Notes).
## Examples
| Engine | Name | Comments
| :-- | :-- | :--
| CPU/GPU | @ref layer_normalization_example_cpp | @copydetails layer_normalization_example_cpp_short
|
