oded schwartz

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.

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

Parallel

CPUCache

Sequential

Hierarchy

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.

Last time: Lower bounds for matrix multiplicationBandwidth:

[Hong & Kung 81]• Sequential

[Irony,Toledo,Tiskin 04]• Sequential and parallel

Latency: Divide by M.

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).

Today: Communication Lower Bounds

Approaches:

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].

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.

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

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

“A shadow”

“B shadow”

“C shadow”

(i,j) n x n, C(i,j) = k A(i,k)B(k,j),

of M reads / writes

3. Show that #multiplications in S k

M #mults / k = M n3 / k

5. By Loomis-Whitney:BW M n3 / (3M)3/2 ..

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)

Proof:Observe one processor.

Is it always true?“A shadow”

“B shadow”

“C shadow”

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

• 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

T(x=i)

T(x=i | y)

N(x=i|y)

N(x=i) N(x=i|y) ·N(x=i|z)

T(x=i)

Communication Lower Bounds

Approaches:

How to generalize this lower bound

(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

Geometric Embedding (2nd approach)

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.

Example: Application to Cholesky decomposition

other arguments)

jikjAjiAjiAjjL

2,,,ik

kiAiiAiiL

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)

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

For dense:

Do conventional dense algorithms as implemented in LAPACK and ScaLAPACK attain these bounds?

Mostly not.

Are there other algorithms that do?Mostly yes.

Dense Linear Algebra: Sequential Model

Lower bound Attaining algorithm

Algorithm Bandwidth Latency Bandwidth Latency

Matrix-Multiplication

[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]

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

of M reads / writes

3. Show that S performs k FLOPs gijk

M G / k, where G is #gi,j,k

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

“A shadow”

“B shadow”

“C shadow”

of M reads / writes

3. Show that S performs k FLOPs gijk

M G / kwhere G is #gi,j,k

5. By Loomis-Whitney:BW M G / (3M)3/2

Applications

BW = (G/ M1/2) where G = |{g(i,j,k) | (i,j) S, k Sij }BW = (G/ PM1/2) in P-parallel model.

other arguments)

But many algorithms just don’t fit the generalized form!

For example: Strassen’s fast matrix multiplication

Beyond 3-nested loops

How about the communication costs of algorithmsthat have a more complex structure?

Communication Lower Bounds – to be continued…

Approaches:

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).

other arguments)

223 ''/ ncMncMn

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. …

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

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

• 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

How to generalize this lower bound:How to deal with on-the-fly generated operands

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.

“A shadow”“B shadow”

“C shadow”

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?

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.

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 ##,

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.

other arguments)

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

other arguments)

nnBW loglog 2

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?

Example 2: Input: A,B, t

Output: C(k) = A B(k) for k = 1,2,…,t

where Bi,j(k) = Bi,j

Phased solution:

Upper bound:

(by adding up the BW cost of t matrix multiplication calls).

Lower bound:

(by imposing writes/reads between phases).

Can we do better than ?

Yes.Claim:There exists an implementation for the above algorithm, with communication cost (tight lower and upper bounds):

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).

Can you think of natural examples where reordering / interleaving of known algorithms may improve the communication costs, compared to the phased implementation?

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

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!

oded schwartz

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