The document presents the design and performance optimization of a low-power, high-performance linear algebra core (LAC) tailored for matrix multiplication kernels, specifically focusing on the General Matrix Multiply (GEMM). It discusses the challenges posed by traditional architectures in handling linear algebra operations and proposes a specialized structure that leverages SIMD (Single Instruction, Multiple Data) capabilities and efficient data handling mechanisms. Finally, it provides performance metrics and emphasizes the potential for future enhancements, including the implementation of broader BLAS functionalities in hardware.

1. Op#mizing
for
the
Inner
Kernel:
A
Low
Power,
High
Performance
Core
for
Matrix
Mul#plica#on
Kernels
Ardavan
Pedram

2. The
GEMM
Micro-­‐kernel
• BLIS
exposes
outer
two
loops
of
the
inner
kernel
• Key
observa+on:
Virtually
all
of
the
differences
between
level-­‐3
inner
kernels
reside
in
the
outer
two
loops
9/8/13
2
for ( 0 to NC: NR )
for ( 0 to MC: MR )
for ( 0 to KC: 1 )
// block-dot product
endfor
endfor
endfor
inner
kernel
/
macro-­‐kernel
micro-­‐kernel
©
A.
Pedram

3. 9/8/13
Era of Heterogeneous Computing
• Physical limits of technology scaling
– Power wall and dark silicon
• Only a fraction of a chip may be active
• Efficiency/optimality vs. flexibility/generality
• GFLOPS/W (energy per operation)
Ø Opportunity for specialization
Ø Heterogeneous multi-core /
Asynchronous CMP
Ø On-chip accelerators
Nvidia
Tegra
2
System
on
Chip
© A. Pedram 3

4. Implementa#on
Spectrum
9/8/13
4
E F F I CI E N C Y
FLEXIBILITY
Source: T. Noll, RWTH Aachen, via R. Leupers, “From ASIP to MPSoC”,
Computer Engineering Colloquium, TU Delft, 2006
Linear
Algebra
Processor
©
A.
Pedram

5. The
GEMM
Micro-­‐kernel
9/8/13
5
+=
KC
NR
MR
NR
C
A
B
for ( 0 to NC: NR )
for ( 0 to MC: MR )
for ( 0 to KC: 1 )
// block-dot product
endfor
endfor
endfor
©
A.
Pedram

6. C
The
GEMM
Micro-­‐kernel
9/8/13
6
+=
KC
NR
MR
NR
α1
α2
α3
α0
β1
β0
β2
β3
γ00
γ10
γ20
γ30
γ01
γ11
γ21
γ31
γ02
γ12
γ22
γ32
γ03
γ13
γ23
γ33
+=
A
B
for ( 0 to NC: NR )
for ( 0 to MC: MR )
for ( 0 to KC: 1 )
// block-dot product
endfor
endfor
endfor
• Typical
micro-­‐kernel
loop
itera#on
(“block-­‐dot
product”)
– Load
column
of
packed
A
– Load
row
of
packed
B
– Compute
outer
product
– Update
C
(kept
in
registers)
©
A.
Pedram

7. 9/8/13
Linear Algebra Core (LAC)
• Scalable 2-D array of nr×nr processing elements (PEs)
– Specialized floating-point units w/ 1 MAC/cycle throughput
– Broadcast busses (no need to pipeline up to nr=16)
– Distributed memory architecture
– Distributed, PE-local control
PE(0,0) PE(0,1) PE(0,2) PE(0,3)
PE(1,0) PE(1,1) PE(1,2) PE(1,3)
PE(2,0) PE(2,1) PE(2,2) PE(2,3)
PE(3,0) PE(3,1) PE(3,2) PE(3,3)
`
MEM B
Addr1
Row Bus Write (RBW)
Column Bus Write (CBW)
A B
Controller
Column Bus
Read (CBR)
Row Bus
Read (RBR)
MAC
ACC_in
Accumulator
Cin
Memory Interface
Addr2
RF
MEM A
© A. Pedram 7

8. 9/8/13
The “Rank-1” Machine
BpCi
Ai,p
+= x
+=
+
+
nr
nr
+ …
x
x
x
BpCi
Ai,p
+= x
+=
+
+
nr
nr
+ …
x
x
x
C += A x B
© A. Pedram 8

9. 9/8/13
Core-Level Implementation
• 4x4 LAC
• 256 kB of local SRAM
• @ 1GHz
• Running GEMM @45nm technology
W
mm2
GFLOPS
mm2
GFLOPS
W
Utilization
Cell SPE (SP) 0.4 6.4 16 83%
NVidia GTX480 SM (SP) 0.5 4.5 8.4 70%
NVidia GTX480 SM (DP) 0.5 2.05 4.1 70%
Intel Core (DP) 0.5 0.4 0.9 95%
ClearSpeed (DP) 0.02 0.3 13 80%
LAC (SP) 0.2 20 104 95+%
LAC (DP) 0.3 16 47 95+%
© A. Pedram 9

10. Design
Trade-­‐off
Challenge
• Some
Linear
algebra
data
handling
requirements
– Matrix
Transpose
(SYRK)
– Vector
Transpose
(Matrix-­‐vector
mul#ply)
(LU)
– Broadcast
(GEMM)
• Common
Sources
of
Inefficiencies
in
CPUs
&
GPUs
– Instruc#on
handling
– Mul#-­‐ported
register
file
– Cache
overheads:
tags
and
coherency
• What
is
the
best
solu#on
for
those
data
handling
requirements?
9/8/13
10
©
A.
Pedram

11. Studied
Register
File
Organiza#ons
9/8/13
11
16 Wide SIMD
SIMD Register File
/ Bypass network
Memory Interface
FPU
+LS
FPU
+LS
FPU
+LS
FPU
+LS
FPU
+LS
FPU
+LS
FPU
+LS
FPU
+LS
FPU FPU FPU FPU
Load/
Store
Unit
Load/
Store
Unit
RF/Bypass
4 Wide SIMD
RF/Bypass
4 Wide SIMD
RF/Bypass
4 Wide SIMD
RF/Bypass
4 Wide SIMD
FPU FPU FPU .. FPU
Multi-Port Flat Register File
/ Bypass network Load/
Store
Unit
..
..
..
(c)(b)(a)
• Organiza#ons
for
16
FPUs
– (a)
Flat
mul#ple-­‐ported
register
file
– (b)
Wide
SIMD
with
wide
ports
– (c)
Array
of
short
SIMD
units
• No
limits
– load-­‐store
unit
data
movement
and
bandwidth
– Scalar
pipeline
(#
instruc#ons
that
are
executed)
©
A.
Pedram

12. Data
Handling
Trade-­‐off
in
Core
• GEMM[ASAP2011]
– Regular
data
movement
– Regular
access
paoern
– Easily
mapped
on
most
SIMD
Cores
• SYRK[ASAP2012],[SBAC-­‐PAD2012]
– Matrix
Transpose
needed
– Data
manipula#on
for
Transpose
is
a
challenge
9/8/13
12
©
A.
Pedram

13. Memory
Hierarchy
of
GEMM
• Blocking
in
higher
levels
of
memory
is
almost
the
same
9/8/13
13
+=
xAi,p
C A B
C
x
x
+=
x+=
Accumulator
(Register Level)
Local Store Level
(LAC)
On-chip Memory
n
Bp
Ap
Ci
Bp,j
Main Memory
LAP
1 x w SIMD
h x w SIMD
L2 Cache
(CPUs) L1 Cache
(CPUs)
L3 Cache
(CPUs)
©
A.
Pedram

14. GEMM
Register-­‐level
Opera#ons
• Accumulator
delay
mul#plies
the
number
of
registers
needed
in
core
9/8/13
14
Ai+dM-1,k Bk,j
cij
Ci+1,j
Blocks of Ci,j
1-Stream Out
2-Current
3-Prefetch
+=
mc
nr
Ci+1,j
Ci+2,j
Ci+d-1,j
dA
dM
dA
©
A.
Pedram
C A Bx+=
+=
+=
+=
x
x
+= xnr
nr nr
w
dA
w x dA
w
h
w x dA
LAP
1 x w SIMD
h x w SIMD
Ai,p in Local Store of PEs Bp,j Bp,j+1
Blocks of Ci,j
1-Stream Out
2-Current
3-Prefetch
+=
x
kc
mc kc

15. 9/8/13
Non-Blocked nr x nr SYRK on LAC
• LAC takes advantage of the
broadcast buses
• Overlap computation with
communication with transposition
• Least operation count in part of
bigger SYRK problem
• Simple control
– 3-stage state machine in PEs
PE(0,0) PE(0,1) PE(0,2) PE(0,3)
PE(1,0) PE(1,1) PE(1,2) PE(1,3)
PE(2,0) PE(2,1) PE(2,2) PE(2,3)
PE(3,0) PE(3,1) PE(3,2) PE(3,3)
Broadcast a0(S1)
© A. Pedram 15

16. 9/8/13
Non-Blocked nr x nr SYRK on LAC
• LAC takes advantage of the
broadcast buses
• Overlap computation with
communication with transposition
• Least operation count in part of
bigger SYRK problem
• Simple control
– 3-stage state machine in PEs
(0,0) PE(0,1) PE(0,2) PE(0,3)
(1,0) PE(1,1) PE(1,2) PE(1,3)
(2,0) PE(2,1) PE(2,2) PE(2,3)
(3,0) PE(3,1) PE(3,2) PE(3,3)
Broadcast a0(S1)
PE(0,0) PE(0,1) PE(0,2) PE(0,3)
PE(1,0) PE(1,1) PE(1,2) PE(1,3)
PE(2,0) PE(2,1) PE(2,2) PE(2,3)
PE(3,0) PE(3,1) PE(3,2) PE(3,3)
Broadcast a1
Broadcast a0
T
Rank-1 update C+=a0a0
T
(S1)
(S1)
(S1)
© A. Pedram 16

17. Power
and
Performance
Analysis
• Analy#cal
formulae
– U#liza#on
– Bandwidth
– Size
of
local
stores
– Size
of
register
file
– #ports
of
memory
levels
• Component
selec#ons
– Storage
model
[CACTI
6.5]
• Pure
SRAM
Model
for
Register
file
• Cache
Models
for
L1
&
L2
• Configura#on
– 1
GHz
– 16
FPUs
– LAC
• 256+32
KB
aggregate
Local
store
– Register
file
organiza#ons
• 256
KB
L2,
64B
line,
8
way
• 32
KB
L1,
64B
line,
8way
• Register
file
Size
variable
– Aimed
for
peak
performance
9/8/13
17
©
A.
Pedram

18. GEMM
Power
and
Area
• SIMD
architectures
– Effec#vely
reduce
the
power
consump#on
• LAC
is
the
best
not
using
caches
• The
flexibility
of
flat
register
files
is
of
no
use
for
GEMM.
• 4×4-­‐wide
array
of
SIMD
requires
less
area.
• Most
of
the
area
is
used
by
Caches
9/8/13
18
0
2
4
6
8
10
12
14
FLAT SIMD A SIMD X h LAC
mm^2
SRAM
L2 Cache
L1 Cache
RF
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
FLAT SIMD A SIMD X h LAC
W/GFLOP
SRAM
L2 Cache
L1 Cache
RF
©
A.
Pedram

19. SYRK
Power
and
Area
• Wide
SIMD
takes
many
cycles
to
transpose
• 4×4-­‐wide
SIMD
has
best
efficiency
– Less
power
than
FLAT
– Higher
u#liza#on
then
Wide
SIMD
• 4×4-­‐wide
array
of
SIMD
requires
less
area.
• Most
of
the
area
is
used
by
Caches
9/8/13
19
0
2
4
6
8
10
12
14
16
18
FLAT SIMD A SIMD X h LAC
mm^2
SRAM
L2 Cache
L1 Cache
RF
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
FLAT SIMD A SIMD X h LAC
W/GFLOP
SRAM
L2 Cache
L1 Cache
RF
©
A.
Pedram

20. Conclusion
• Architectures
Studied
• Linear
Algebra
Core
(LAC)
• FLAT
register
file
• 1D
wide
SIMD
• 4×4-­‐wide
array
of
SIMD
• Algorithm/Architecture
co-­‐design
• GEMM
(emphasize
on
data
bandwidth
to-­‐from
core)
• SYRK
(
emphasize
on
data
manipula#on
in
core)
• Analy#cal
formulae
for
different
layers
of
memory
hierarchy
• Power
and
efficiency
es#ma#on
9/8/13
20
©
A.
Pedram

21. Linear
Algebra
Core
• Results @ 1GHz
– DP: 32 GFLOPS, 47 GFLOPS/W
– 0.6 Watts
– 2.8 mm2 in 45nm
– 4 GB/s external BW
– Orders of magnitude improvement
Ø On-going and future work
– Generalization: full Level-3 BLAS in hardware
– Implementation: synthesis and tape-out
9/8/13
21
©
A.
Pedram

22. Ques#ons
?
9/8/13
22
©
A.
Pedram