How to Compute and Prove
Lower and Upper Boundson the
Communication Costsof Your Algorithm
Part II: Geometric embedding
Oded Schwartz
CS294, Lecture #3 Fall, 2011Communication-Avoiding Algorithms
www.cs.berkeley.edu/~odedsc/CS294
Based on:
D. Irony, S. Toledo, and A. Tiskin:
Communication lower bounds for distributed-memory matrix multiplication.
G. Ballard, J. Demmel, O. Holtz, and O. Schwartz:
Minimizing communication in linear algebra.
2
Last time: the models
Two kinds of costs:Arithmetic (FLOPs)Communication: moving data between
• levels of a memory hierarchy (sequential case) • over a network connecting processors (parallel case)
CPURAM
CPURAM
CPURAM
CPURAM
Parallel
CPUCache
RAM
Sequential
M1
M2
M3
Mk =
Hierarchy
3
Last time: Communication Lower Bounds
Approaches:
1. Reduction [Ballard, Demmel, Holtz, S. 2009]
2. Geometric Embedding[Irony,Toledo,Tiskin 04],
[Ballard, Demmel, Holtz, S. 2011a]
3. Graph Analysis [Hong & Kung 81], [Ballard, Demmel, Holtz, S. 2011b]
Proving that your algorithm/implementation is as good as it gets.
4
Last time: Lower bounds for matrix multiplicationBandwidth:
[Hong & Kung 81]• Sequential
[Irony,Toledo,Tiskin 04]• Sequential and parallel
Latency: Divide by M.
M
M
n3
P
M
M
n3
5
Last time: Reduction (1st approach) [Ballard, Demmel, Holtz, S. 2009a]
Thm:Cholesky and LU decompositions are(communication-wise) as hard as matrix-multiplication
Proof:By a reduction (from matrix-multiplication) that preserves communication bandwidth, latency, and arithmetic.
Cor: Any classical O(n3) algorithm for Cholesky and LU decomposition requires:
Bandwidth: (n3 / M1/2)Latency: (n3 / M3/2)
(similar cor. for the parallel model).
6
Today: Communication Lower Bounds
Approaches:
1. Reduction [Ballard, Demmel, Holtz, S. 2009]
2. Geometric Embedding[Irony,Toledo,Tiskin 04],
[Ballard, Demmel, Holtz, S. 2011a]
3. Graph Analysis [Hong & Kung 81], [Ballard, Demmel, Holtz, S. 2011b]
Proving that your algorithm/implementation is as good as it gets.
7
Lower bounds: for matrix multiplicationusing geometric embedding
[Hong & Kung 81]• Sequential
[Irony,Toledo,Tiskin 04]• Sequential and parallel
Now: prove both, using the geometric embedding approach of [Irony,Toledo,Tiskin 04].
M
M
n3
P
M
M
n3
8
Geometric Embedding (2nd approach) [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
Matrix multiplication form:
(i,j) n x n, C(i,j) = k A(i,k) B(k,j),
Thm: If an algorithm agrees with this form (regardless of the order of computation) then
BW = (n3/ M1/2)
BW = (n3/ PM1/2) in P-parallel model.
9
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M
Example of a partition,M = 3
For a given run (algorithm, machine, input)1. Partition computations into segments
of M reads / writes
2. Any segment S has 3M inputs/outputs.
3. Show that #multiplications in S k
4. The total communication BW isBW = BW of one segment #segments
M #mults / k
...
Geometric Embedding (2nd approach) [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
10
Volume of boxV = x·y·z = ( xz · zy · yx)1/2
Thm: (Loomis & Whitney, 1949) Volume of 3D set V ≤ (area(A shadow) · area(B shadow) · area(C shadow) ) 1/2
x
z
z
y
x
y
A BC
“A shadow”
“B shadow”
“C shadow”
A BC
VV
Matrix multiplication form:
(i,j) n x n, C(i,j) = k A(i,k)B(k,j),
Geometric Embedding (2nd approach) [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
11
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M
Example of a partition,M = 3
For a given run (algorithm, machine, input)1. Partition computations into segments
of M reads / writes
2. Any segment S has 3M inputs/outputs.
3. Show that #multiplications in S k
4. The total communication BW isBW = BW of one segment #segments
M #mults / k = M n3 / k
5. By Loomis-Whitney:BW M n3 / (3M)3/2 ..
.
Geometric Embedding (2nd approach) [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
12
From Sequential Lower bound to Parallel Lower Bound
We showed:Any classical O(n3) algorithm for matrix multiplication on sequential model requires:Bandwidth: (n3 / M1/2)Latency: (n3 / M3/2)
Cor:Any classical O(n3) algorithm for matrix multiplication on P-processors machine (with balanced workload) requires:
2D-layout: M=O(n2/P)Bandwidth: (n3 /PM1/2) (n2/P1/2) Latency: (n3 / PM3/2) (P1/2)
13
From Sequential Lower bound to Parallel Lower Bound
Proof:Observe one processor.
Is it always true?“A shadow”
“B shadow”
“C shadow”
A BC
Let Alg be an algorithm with communication lower bound B = B(n,M).
Then any parallel implementation of Alg has a communication lower bound B’(n, M, p) = B(n, M)/p
?
Proof of Loomis-Whitney inequality
• T = 3D set of 1x1x1 cubes on lattice • N = |T| = #cubes
• Tx = projection of T onto x=0 plane
• Nx = |Tx| = #squares in Tx, same for Ty, Ny, etc
• Goal: N ≤ (Nx · Ny · Nz)1/2
14
• T(x=i) = subset of T with x=i• T(x=i | y ) = projection of T(x=i) onto y=0 plane• N(x=i) = |T(x=i)| etc
• N = i N(x=i) = i (N(x=i))1/2 · (N(x=i))1/2
≤ i (Nx)1/2 · (N(x=i))1/2
≤ (Nx)1/2 · i (N(x=i | y ) · N(x=i | z ) )1/2
= (Nx)1/2 · i (N(x=i | y ) )1/2 · (N(x=i | z ) )1/2
≤ (Nx)1/2 · (i N(x=i | y ) )1/2 · (i N(x=i | z ) )1/2
= (Nx)1/2 · (Ny)1/2 · (Nz)1/2
z
y
x
T(x=i)
T(x=i | y)
T
x=i
N(x=i|y)
N(x=i) N(x=i|y) ·N(x=i|z)
N(x
=i|z
)
T(x=i)
15
Communication Lower Bounds
Approaches:
1. Reduction [Ballard, Demmel, Holtz, S. 2009]
2. Geometric Embedding[Irony,Toledo,Tiskin 04],
[Ballard, Demmel, Holtz, S. 2011a]
3. Graph Analysis [Hong & Kung 81], [Ballard, Demmel, Holtz, S. 2011b]
Proving that your algorithm/implementation is as good as it gets.
How to generalize this lower bound
16
Matrix multiplication form:
(i,j) n x n, C(i,j) = k A(i,k)B(k,j),
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
• C(i,j) any unique memory location.Same for A(i,k) and B(k,j). A,B and C may overlap.
• Lower bound for all reorderings. Incorrect ones too.• It does assume each operand generate load/store.
• Turns out QR, eig, SVD all may do this• Need a different analysis. Not today…
• fij and gijk are “nontrivial” functions
17
Geometric Embedding (2nd approach)
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
Thm: [Ballard, Demmel, Holtz, S. 2011a]If an algorithm agrees with the generalized form then
BW = (G/ M1/2) where G = |{g(i,j,k) | (i,j) S, k Sij }
BW = (G/ PM1/2) in P-parallel model.
18
Example: Application to Cholesky decomposition
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
jikjAjiAjiAjjL
jiLik
,,,,,
1,
1
1
2,,,ik
kiAiiAiiL
19
From Sequential Lower bound to Parallel Lower Bound
We showed:Any algorithm that agrees with Form (1) on sequential model requires:Bandwidth: (G / M1/2)Latency: (G / M3/2)where G is the gijk count.
Cor:Any algorithm that agrees with Form (1), on a P-processors machine, where at least two processors perform (1/P) of G each requires:
Bandwidth: (G /PM1/2)Latency: (G / PM3/2)
20
Geometric Embedding (2nd approach) [Ballard, Demmel, Holtz, S. 2011a]Follows [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
Lower bounds: for algorithms with “flavor” of 3 nested loopsBLAS, LU, Cholesky, LDLT, and QR factorizations, eigenvalues and SVD, i.e., essentially all direct methods of linear algebra.
• Dense or sparse matricesIn sparse cases: bandwidth is a function NNZ.
• Bandwidth and latency.• Sequential, hierarchical, and
parallel – distributed and shared memory models.• Compositions of linear algebra operations.• Certain graph optimization problems
[Demmel, Pearson, Poloni, Van Loan, 11], [Ballard, Demmel, S. 11]• Tensor contraction
M
M
n3
P
M
M
n3
For dense:
21
Do conventional dense algorithms as implemented in LAPACK and ScaLAPACK attain these bounds?
Mostly not.
Are there other algorithms that do?Mostly yes.
22
Dense Linear Algebra: Sequential Model
Lower bound Attaining algorithm
Algorithm Bandwidth Latency Bandwidth Latency
Matrix-Multiplication
[Ballard, Demmel, Holtz, S. 11]
[Ballard, Demmel, Holtz, S. 11]
[Frigo, Leiserson, Prokop, Ramachandran 99]
Cholesky [Ahmad, Pingali 00][Ballard, Demmel, Holtz, S. 09]
LU [Toledo97] [DGX08]
QR [EG98] [DGHL08a]
Symmetric Eigenvalues
[Ballard,Demmel,Dumitriu 10]
SVD [Ballard,Demmel,Dumitriu 10]
(Generalized) Nonsymetric Eigenvalues
[Ballard,Demmel,Dumitriu 10]
M
M
n3
3
M
n
Dense 2D parallel algorithms • Assume nxn matrices on P processors, memory per processor = O(n2 / P)
• ScaLAPACK assumes best block size b chosen
• Many references (see reports), Blue are new
• Recall lower bounds:
#words_moved = ( n2 / P1/2 ) and #messages = ( P1/2 )Algorithm Reference Factor exceeding
lower bound for #words_moved
Factor exceeding lower bound for
#messages
Matrix multiply ]Cannon, 69[ 1 1
Cholesky ScaLAPACK log P log P
LU ]GDX08[ScaLAPACK
log P log P
log P( N / P1/2 ) · log P
QR ]DGHL08 [ScaLAPACK
log Plog P
log3 P( N / P1/2 ) · log P
Sym Eig, SVD ]BDD10[ScaLAPACK
log Plog P
log3 PN / P1/2
Nonsym Eig ]BDD10[ScaLAPACK
log PP1/2 · log P
log3 PN · log P
Relax: 2.5D AlgorithmsSolomonik & Demmel ‘11
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M
Example of a partition,M = 3
For a given run (algorithm, machine, input)1. Partition computations into segments
of M reads / writes
2. Any segment S has 3M inputs/outputs.
3. Show that S performs k FLOPs gijk
4. The total communication BW isBW = BW of one segment #segments
M G / k, where G is #gi,j,k
...
Geometric Embedding (2nd approach)
25
Geometric Embedding (2nd approach) [Ballard, Demmel, Holtz, S. 2011a]Follows [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
Volume of boxV = x·y·z = ( xz · zy · yx)1/2
Thm: (Loomis & Whitney, 1949) Volume of 3D set V ≤ (area(A shadow) · area(B shadow) · area(C shadow) ) 1/2
x
z
z
y
x
y
A BC
“A shadow”
“B shadow”
“C shadow”
A BC
VV
26
S1
S2
S3
Read
Read
Read
Read
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Read
Write
Write
Write
FLOP
FLOP
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Tim
e
...
M
Example of a partition,M = 3
For a given run (algorithm, machine, input)1. Partition computations into segments
of M reads / writes
2. Any segment S has 3M inputs/outputs.
3. Show that S performs k FLOPs gijk
4. The total communication BW isBW = BW of one segment #segments
M G / kwhere G is #gi,j,k
5. By Loomis-Whitney:BW M G / (3M)3/2
...
Geometric Embedding (2nd approach)
27
Applications
27
BW = (G/ M1/2) where G = |{g(i,j,k) | (i,j) S, k Sij }BW = (G/ PM1/2) in P-parallel model.
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
28
Geometric Embedding (2nd approach) [Ballard, Demmel, Holtz, S. 2011a]Follows [Irony,Toledo,Tiskin 04], based on [Loomis & Whitney 49]
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
But many algorithms just don’t fit the generalized form!
For example: Strassen’s fast matrix multiplication
29
Beyond 3-nested loops
How about the communication costs of algorithmsthat have a more complex structure?
30
Communication Lower Bounds – to be continued…
Approaches:
1. Reduction [Ballard, Demmel, Holtz, S. 2009]
2. Geometric Embedding[Irony,Toledo,Tiskin 04],
[Ballard, Demmel, Holtz, S. 2011a]
3. Graph Analysis [Hong & Kung 81], [Ballard, Demmel, Holtz, S. 2011b]
Proving that your algorithm/implementation is as good as it gets.
31
Further reduction techniques: Imposing reads and writes
Example: Computing ||A∙B|| where each matrix element is a formulas, computed only once.
Problem: Input/output do not agree with Form (1).Solution: •Impose writes/reads of (computed) entries of A and B.•Impose writes of the entries of C.•The new algorithm has lower bound
For the original algorithm
i.e., for(which we assume anyway).
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
M
M
nBW
3
2
3
cnMM
nBW
M
M
nBW
3
223 ''/ ncMncMn
32
Further reduction techniques: Imposing reads and writes
The previous example can be generalized to other “black-box” uses of algorithms that fit Form (1).
Consider a more general class of algorithms:•Some arguments of the generalized form may be computed “on the fly” and discarded immediately after used. …
33
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Example of a partition,M = 3
For a given run (Algorithm, Machine, Input)1. Partition computations into segments
of 3M reads / writes
2. Any segment S has M inputs/outputs.
3. Show that S performs G(3M) FLOPs gijk
4. The total communication BW isBW = BW of one segment #segments
M G / G(3M)
But now some operands inside a segmentmay be computed on-the fly and discarded.So no read/write performed.
...
Recall…
How to generalize this lower bound:How to deal with on-the-fly generated operands
34
• Need to distinguish Sources, Destinations of each operand in fast memory during a segment:
• Possible Sources:R1: Already in fast memory at start of segment,
or read; at most 2MR2: Created during segment;
no bound without more information• Possible Destinations:
D1: Left in fast memory at end of segment, or written; at most 2M
D2: Discarded; no bound without more information
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How to generalize this lower bound:How to deal with on-the-fly generated operands
35
There are at most 4M of types: R1/D1, R1/D2, R2/D1.
Need to assume/prove: not too many R2/D2 arguments;
Then we can use LW, and obtain the lower bound of Form (1).
Bounding R2/D2 is sometimes quite subtle.
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“A shadow”“B shadow”
“C shadow”
A BC
V
36
Composition of algorithms
Many algorithms and applications use composition of other (linear algebra) algorithms.
How to compute lower and upper bounds for such cases?
Example - Dense matrix poweringCompute An by (log n times) repeated squaring:
A A2 A4 … An
Each squaring step agrees with Form (1).Do we get
or is there a way to reorder (interleave) computations to reduce communication?
M
M
nnBW
3
log
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Communication hiding vs. Communication avoiding
Q.The Model assumes that computation and communication do not overlap.Is this a realistic assumption? Can we not gain time by such overlapping?
A.Right. This is called communication hiding. It is done in practice, and ignored in our model. It may save up to a factor of 2 in the running time. Note that the speedup gained by avoiding (minimizing) communication is typically larger than a constant factor.
38
Two-nested loops: when the input/output size dominates
Q. Do two-nested-loops algorithms fall into the paradigm of Form (1)?For example, what lower bound do we obtain for computing Matrix-vector multiplication?
A. Yes, but the lower bound we obtain is
Where just reading the input costsMore generally, the communication cost lower bound for algorithms that agree with Form (1) is
where LW is the one we obtain from the geometric embedding, and #inputs+#outputs is the size of the inputs and outputs.
For some algorithms LW dominates, for others #inputs+#outputs dominate.
outputsinputsLWMaxBW ##,
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nBW
2
2nBW
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Composition of algorithms
Claim: any implementation of An by (log n times) repeated squaring requires
Therefore we cannot reduce communication by more than a constant factor (compared to log n separate calls to matrix multiplications)by reordering computations.
M
M
nnBW
3
log
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
40
Composition of algorithms
Proof: by imposing reads/writes on each entry of every intermediate matrix.The total number of gi,j,k is (n3log n).
The total number of imposed reads/writes is (n2log n).
The lower bound for the original algorithm is
(1) Generalized form: (i,j) S, C(i,j) = fij( gi,j,k1 (A(i,k1), B(k1,j)),
gi,j,k2 (A(i,k2), B(k2,j)), …, k1,k2,… Sij
other arguments)
nnM
M
nnBW loglog 2
3
M
M
nn
3
log
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Composition of algorithms: when interleaving does matter
Example 1: Input: A,v1,v2,…,vn
Output: Av1,Av2,…,Avn
The phased solution costs
But we already know that we can save a M1/2 factor:Set B = (v1,v2,…,vn), and compute AB, then the cost is
Other examples?
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nBW
3
3nBW
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Composition of algorithms: when interleaving does matter
Example 2: Input: A,B, t
Output: C(k) = A B(k) for k = 1,2,…,t
where Bi,j(k) = Bi,j
1/k
Phased solution:
Upper bound:
(by adding up the BW cost of t matrix multiplication calls).
Lower bound:
(by imposing writes/reads between phases).
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ntBW
3
M
M
ntOBW
3
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Composition of algorithms: when interleaving does matter
Example 2: Input: A,B, t
Output: C(k) = A B(k) for k = 1,2,…,t
where Bi,j(k) = Bi,j
1/k
Can we do better than ?
M
M
ntBW
3
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Composition of algorithms: when interleaving does matter
Example 2: Input: A,B, t
Output: C(k) = A B(k) for k = 1,2,…,t
where Bi,j(k) = Bi,j
1/k
Can we do better than ?
Yes.Claim:There exists an implementation for the above algorithm, with communication cost (tight lower and upper bounds):
M
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ntBW
3
M
M
ntBW
3
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Composition of algorithms: when interleaving does matter
Example 2: Input: A,B, t
Output: C(k) = A B(k) for k = 1,2,…,t
where Bi,j(k) = Bi,j
1/k
Proofs idea:
•Upper bound: Having both Ai,k and Bk,j in fast memory lets us do up to t evaluations of gijk.
•Lower bound: The union of all these tn3 operations does not match Form (1), since the inputs Bk,j cannot be indexed in a one-to-one fashion. We need a more careful argument regarding the numbers of gijk. Operations in a segment as a function of the number of accessed elements of A, B and C(k).
46
Composition of algorithms: when interleaving does matter
Can you think of natural examples where reordering / interleaving of known algorithms may improve the communication costs, compared to the phased implementation?
47
Summary
How to compute an upper bound on the communication costs of your algorithm?
Typically straightforward. Not always.
How to compute and prove a lower bound on the communication costs of your algorithm?
Reductions: from another algorithm/problem from another model of computing By using the generalized form (“flavor” of 3 nested loops) and imposing reads/writes – black-box-wise or bounding the number of R2/D2 operands By carefully composing the lower bounds of the building blocks.
Next time: by graph analysis
48
Open Problems
Find algorithms that attain the lower bounds:• Sparse matrix algorithms• that auto-tune or are cache oblivious• cache oblivious for parallel (distributed memory) • Cache oblivious parallel matrix multiplication? (Cilk++ ?)
Address complex heterogeneous hardware:• Lower bounds and algorithms
[Demmel, Volkov 08],[Ballard, Demmel, Gearhart 11]
How to Compute and Prove
Lower Boundson the
Communication Costsof your Algorithm
Oded Schwartz
CS294, Lecture #2 Fall, 2011Communication-Avoiding Algorithms
Based on:
D. Irony, S. Toledo, and A. Tiskin:
Communication lower bounds for distributed-memory matrix multiplication.
G. Ballard, J. Demmel, O. Holtz, and O. Schwartz:
Minimizing communication in linear algebra.
Thank you!
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