avoiding communication in sparse matrix-vector multiply ( spmv )

Avoiding Communication in Sparse Matrix-Vector Multiply (SpMV)

• Sequential and shared-memory performance is dominated by off-chip communication

• Distributed-memory performance is dominated by network communication

The problem:SpMV has low arithmetic intensity

dimension: n = 5 number of nonzeros: nnz = 3n-2 (tridiagonal A)

overcounts flops by up to n (diagonal A)

SpMV

floating point operations 2⋅nnz

floating point words moved nnz + 2⋅n

Assumption: A is invertible⇒ nonzero in every row ⇒ nnz ≥ n

SpMV Arithmetic Intensity (1)

• Arithmetic intensity := Total flops / Total DRAM bytes• Upper bound: compulsory traffic

– further diminished by conflict or capacity misses

A r i t h m e t i c I n t e n s i t y

O( n )O( lg(n) )

O( 1 )

SpMV, BLAS1,2

Stencils (PDEs)

Lattice Methods

FFTsDense Linear Algebra

(BLAS3)Particle Methods

more flops per byte

SpMV

flops 2⋅nnzwords moved nnz + 2⋅narith. intensity 2


actual flop:byte ratio

atta

inab

le g

flop/

s

Opteron 2356(Barcelona)

0.5

1.0

1/8

2.0

4.0

8.0

16.0

32.0

64.0

128.0

256.0

1/41/2 1 2 4 8 16

strea

m bandwidth

peak double precision

floating-point rate

In practice, A requires at least nnz words:• indexing data, zero padding• depends on nonzero structure, eg,

banded or dense blocks• depends on data structure, eg,

• CSR/C, COO, SKY, DIA, JDS, ELL, DCS/C, …

• blocked generalizations• depends on optimizations, eg,

index compression or variable block splitting

• 2 flops per word of data• 8 bytes per double• flop:byte ratio ≤ ¼• Can’t beat 1/16 of peak!

How to do more flops per byte?

Reuse data (x, y, A) across multiple SpMVs


(1) used in:• Block Krylov methods• Krylov methods for multiple

systems (AX = B)

(1) k independent SpMVs

(2) used in: • s-step Krylov methods,• Communication-avoiding Krylov

methods…to compute k Krylov basis vectors

Def. Krylov space (given A, x, s):

What if we can amortize cost of reading A over k SpMVs ?• (k-fold reuse of A)

Combining multiple SpMVs

(2) k dependent SpMVs

(3) k dependent SpMVs, in-place variant

(3) used in: • multigrid smoothers, power method• Related to Streaming Matrix Powers

optimization for CA-Krylov methods

SpMM optimization:• Compute row-by-row• Stream A only once

=

1 SpMV k independent SpMVs k independent SpMVs (using SpMM)

flops 2⋅nnz 2k⋅nnz 2k⋅nnz

words moved nnz + 2n k⋅nnz + 2kn 1⋅nnz + 2kn

arith. intensity,nnz = ω(n)

2 2 2k

(1) k independent SpMVs (SpMM)

Akx (Akx) optimization: • Must satisfy data

dependencies while keeping working set in cache

Naïve algorithm (no reuse):

1 SpMV k dependent SpMVs k dependent SpMVs (using Akx)


words moved nnz + 2n k⋅nnz + 2kn 1⋅nnz + (k+1)n


2 2 2k

(2) k dependent SpMVs (Akx)

1 SpMV k dependent SpMVs k dependent SpMVs (using Akx)


words moved nnz + 2n k⋅nnz + 2kn 1⋅nnz + (k+1)n


2 2 2k

(2) k dependent SpMVs (Akx)

Akx algorithm (reuse nonzeros of A):

1 SpMV k dependent SpMVs, in-place

Akx, last-vector-only


words moved nnz + 2n k⋅nnz + 2kn 1⋅nnz + 2n

arith. intensity,nnz = anything

2 2 2k

Last-vector-only Akx optimization:• Reuses matrix and vector k times, instead of once.• Overwrites intermediates without memory traffic• Attains O(k) reuse, even when nnz < n

• eg, A is a stencil (implicit values and structure)

(3) k dependent SpMVs, in-place (Akx, last-vector-only)

Problem flops words moved optimization words

moved

relative bandwidth savings(n, nnz ⟶ ∞)

nnz = ω(n) nnz = c⋅n nnz = o(n)

SpMV 2⋅nnznnz + 2n

- - - - -

k independent

SpMVs2k⋅nnz

k⋅nnz +

2knSpMM

nnz +

2knk ≤ min(c, k) 1

k dependent SpMVs 2k⋅nnz

k⋅nnz +

2knAkx

nnz +

(k+1)nk ≤ min(c, k) 2

k dependent SpMVs, in-place

2k⋅nnzk⋅nnz

+ 2kn

Akx, last-vector-only

nnz + 2n

k k k

Combining multiple SpMVs(summary of sequential results)

Avoiding Serial Communication

• Reduce compulsory misses by reusing data:– more efficient use of memory– decreased bandwidth cost (Akx, asymptotic)

• Must also consider latency cost – How many cachelines?– depends on contiguous accesses

• When k = 16 ⇒ compute-bound? – Fully utilize memory system– Avoid additional memory traffic like capacity

and conflict misses– Fully utilize in-core parallelism – (Note: still assumes no indexing data)

• In practice, complex performance tradeoffs.– Autotune to find best k

actual flop:byte ratio

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1.0

1/8

2.0

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1/41/2 1 2 4 8 16

peak DP

Stream

Bandwidth

?

On being memory bound• Assume that off-chip communication (cache to memory) is bottleneck,

– eg, that we express sufficient ILP to hide hits in L3• When your multicore performance is bound by memory operations, is it because of

latency or bandwidth?– Latency-bound: expressed concurrency times the memory access rate does not fully utilize the

memory bandwidth• Traversing a linked list, pointer-chasing benchmarks

– Bandwidth-bound: expressed concurrency times the memory access rate exceeds the memory bandwidth• SpMV, stream benchmarks

– Either way, manifests as pipeline stalls on loads/stores (suboptimal throughput)

• Caches can improve memory bottlenecks – exploit them whenever possible– Avoid memory traffic when you have temporal or spatial locality – Increase memory traffic when cache line entries are unused (no locality)

• Prefetchers can allow you to express more concurrency– Hide memory traffic when your access pattern has sequential locality (clustered or regularly

strided access patterns)

Distributed-memory parallel SpMV• Harder to make general statements about performance:

– Many ways to partition x, y, and A to P processors– Communication, computation, and load-balance are partition-dependent– What fits in cache? (What is “cache”?!)

• A parallel SpMV involves 1 or 2 rounds of messages – (Sparse) collective communication, costly synchronization– Latency-bound (hard to saturate network bandwidth)– Scatter entries of x and/or gather entries of y across network– k SpMVs cost O(k) rounds of messages

• Can we do k SpMVs in one round of messages?– k independent vectors? SpMM generalizes

• Distribute all source vectors in one round of messages• Avoid further synchronization

– k dependent vectors? Akx generalizes• Distribute source vector plus additional ghost zone entries in one round of messages• Avoid further synchronization

– Last-vector-only Akx ≈ standard Akx in parallel• No savings discarding intermediates

Example: tridiagonal matrix, k = 3, n = 40, p = 4

Naïve algorithm:k messages per neighbor

Akx optimization:1 message per neighbor

Distributed-memory parallel Akx

Polynomial Basis for Akx

• Today we considered the special case of the monomials:

• Stability problems - tends to lose linear independence– Converges to principal eigenvector

• Given A, x, k > 0, compute

where pj(A) is a degree-j polynomial in A.– Choose p for stability.

Tuning space for Akx• DLP optimizations:

– vectorization• ILP optimizations:

– Software pipelining– Loop unrolling– Eliminate branches, inline functions

• TLP optimizations:– Explicit SMT

• Memory system optimizations:– NUMA-aware affinity– Software prefetching– TLB blocking

• Memory traffic optimizations:– Streaming stores (cache bypass)– Array padding– Cache blocking– Index compression– Blocked sparse formats– Stanza encoding

• Distributed memory optimizations:– Topology-aware sparse collectives

– Hypergraph partitioning– Dynamic load balancing– Overlapped communication and computation

• Algorithmic variants:– Compositions of distributed-memory parallel,

shared memory parallel, sequential algorithms– Streaming or explicitly buffered workspace– Explicit or implicit cache blocks– Avoiding redundant computation/storage/traffic– Last-vector-only optimization– Remove low-rank components (blocking covers)– Different polynomial bases pj(A)

• Other:– Preprocessing optimizations– Extended precision arithmetic– Scalable data structures (sparse representations)– Dynamic value and/or pattern updates

Krylov subspace methods (1)Want to solve Ax = b (still assume A is invertible)

How accurately can you hope to compute x? • Depends on condition number of A and the accuracy of your inputs A and b

• condition number with respect to matrix inversion• cond(A) – how much A distorts the unit sphere (in some norm)• 1/cond(A) – how close A is to a singular matrix• expect to lose log10(cond(A)) decimal digits relative to (relative) input accuracy

• Idea: Make successive approximations, terminate when accuracy is sufficient• How good is an approximation x0 to x?• Error: e0 = x0 - x

• If you know e0, then compute x = x0 - e0 (and you’re done.)• Finding e0 is as hard as finding x; assume you never have e0

• Residual: r0 = b – Ax0

• r0 = 0 ⇔ e0 = 0, but they do not necessarily vanish simultaneously cond(A) small (⇒ r0 small ⇒ e0 small)

Krylov subspace methods (2)1. Given approximation xold, refine by adding a correction xnew = xold + v

• Pick v as the ‘best possible choice’ from search space V• Krylov subspace methods: V :=

2. Expand V by one dimension3. xold = xnew. Repeat.

• Once dim(V) = dim(A) = n, xnew should be exact

Why Krylov subspaces?• Cheap to compute (via SpMV)• Search spaces V coincide with the residual spaces -

• makes it cheaper to avoid repeating search directions• K(A,z) = K(c1A - c2I, c3z) invariant under scaling, translation⇒

• Without loss, assume |λ(A)| ≤ 1• As s increases, Ks gets closer to the dominant eigenvectors of A• Intuitively, corrections v should target ‘largest-magnitude’ residual components

Convergence of Krylov methods• Convergence = process by which residual goes to zero

– If A isn’t too poorly conditioned, error should be small. • Convergence only governed by the angles θm between

spaces Km and AKm

– How fast does sin(θm) go to zero?– Not eigenvalues! You can construct a unitary system that

results in the same sequence of residuals r0, r1, …– If A is normal, λ(A) provides bounds on convergence.

• Preconditioning– Transforming A with hopes of ‘improving’ λ(A) or cond(A)

Conjugate Gradient (CG) MethodGiven starting approximation x0 to Ax = b, let p0 := r0 := b - Ax0.

For m = 0, 1, 2, … until convergence, do:

Vector iterates:• xm = candidate solution• rm = residual• pm = search direction

Correct candidate solution along search direction

Update residual according to new candidate solution

Expand search space

Communication-bound:• 1 SpMV operation per iteration• 2 dot products per iteration

1. Reformulate to use Akx2. Do something about the

dot products

Applying Akx to CG (1)1. Ignore x, α, and β, for now2. Unroll the CG loop s times (in your head)3. Observe that:

ie, two Akx calls4. This means we can represent rm+j and pm+j

symbolically as linear combinations:

5. And perform SpMV operations symbolically: (same holds for Rj-1)

vectors of length n vectors of

length 2j+1 CG loop:For m = 0,1,…, Do

SpMV performed symbolically by shifting coordinates:

Applying Akx to CG (2)6. Now substitute coefficient vectors for vector iterates (eg, for r)

CG loop:For m = 0,1,…, Do

7. Let’s also compute the 2j+1-by-2j+1 Gram matrices:

Now we can perform all dot products symbolically:

Blocking CG dot products


For m = 0, s, 2s, …, until convergence, Do{

For j = 0 to s - 1, Do {

} End For

} End For

Given approximation x0 to Ax = b, let

Represent SpMV operation as a change of basis (here, a shift):

Expand Krylov basis, using SpMM and Akx optimizations:

Represent the 2s+1 inner products of length n with a 2s+1-by-2s+1 Gram matrix

Represent vector iterates of length n with vectors of length 2s+1 and 2s+2:

Recover vector iterates:

Take s steps of CG without communication

Communication (sequential only)

Communication (sequential and parallel)


CA-CG

Kernel Computation costs Communication costs

s dependent SpMVs

• 2s⋅nnz flops (1 source vector)

Sequential:• Read s vectors of length n• Write s vectors of length n• Read A s times• bandwidth cost ≈ s⋅nnz + 2snParallel:• Distribute 1 source vector s times

Akx • 4s⋅nnz flops (2 source vectors)

Sequential:• Read 2 vectors of length n, • Write 2s-1 vectors of length n,• Read A once (both Akx and SpMM optimizations)• bandwidth cost ≈ nnz + (2s+1)nParallel:• Distribute 2 source vectors once • Communication volume and number of messages

increase with s (ghost zones)

CA-CG complexity (1)

Kernel Computation costs Communication costs

2s+1 dot products

Sequential:• 2(2s+1)n flops• ≈ 4nsParallel:• (2s+1)(2n+(p-1))/p• ≈ 4ns/p

Sequential:• Read a vector of length n 2s+1 timesParallel:• 2s+1 all-reduce collectives, each with lg(p) rounds

of messages: latency cost ≈ 2slg(p) • 1 word to/from each proc.: bandwidth cost ≈ 2s

Gram matrix

Sequential:• (2s+1)2n flops• ≈ 4ns2

Parallel:• (2s+1)2(n/p + (p-

1)/(2p))• ≈ 4ns2/pSymbolic dot products cost an additional (2s+1)2(2s+3) flops• ≈ 8s3

Sequential:• Read a matrix of size (2s+1)×n once.Parallel:• 1 all-reduce collective, with lg(p) rounds of

messages: latency cost ≈ lg(p)• (2s+1)2/2 words to/from each proc.: bandwidth

cost ≈ 4s2


Using Gram matrix and coefficient vectors have additional costs for CA-CG:• Dense work (besides dot products/Gram matrix) does not increase with s:

CG ≈ 6snCA-CG ≈ 3(2s+1)(2s+n) ≈ (6s+3)n

• Sequential memory traffic decreases (factor of 4.5)CG ≈ 6sn reads, 3sn writesCA-CG ≈ (2s+1)n reads, 3n writes

Method Sequential flops

Sequential bandwidth

Sequential latency

Parallel flops

Parallel bandwidth

Parallel latency

CG, s steps

2s⋅nnz +

10sn

s⋅nnz +

(13s+1)n

s⋅nnz/b+

(13s+1)n/b

2s⋅nnz/p+

10sn/p

s Expand(⋅ A) + 2s

s Expand(⋅ A) +

(2s+1)lg(p)

CA-CG(s), 1 step

4s⋅nnz +

(4s+10)sn

nnz +

(6s+6)n

nnz/b+

(6s+6)n/b

4s⋅nnz/p +

(4s+10)sn/p

Expand(|A|s)+ Expand(|A|s-1)

+ 4s2

Expand(|A|s)+ Expand(|A|s-1)

+ + lg(p)


b = cacheline size, p = number of processors

Performance optimizations:• 3-term recurrence CA-CG formulation:

• Avoid the auxiliary vector p.• 2x decrease in Akx flops, bandwidth cost and

serial latency cost (vectors only - A already optimal)

• 25% decrease in Gram matrix flops, bandwidth cost, and serial latency cost

• Roughly equivalent dense flops• Streaming Akx formulation:

• Interleave Akx and Gram matrix construction, then interleave Akx and vector reconstruction

• 2x increase in Akx flops• Factor of s decrease in Akx sequential

bandwidth and latency costs (vectors only - A already optimal)

• 2x increase in Akx parallel bandwidth, latency• Eliminate Gram matrix sequential bandwidth

and latency costs• Eliminate dense bandwidth costs other than 3n

writes• Decreases overall sequential bandwidth and

latency by O(s), regardless of nnz.

Stability optimizations:• Use scaled/shifted 2-term recurrence for Akx:

• Increase Akx flops by (2s+1)n• Increase dense flops by an O(s2) term

• Use scaled/shifted 3-term recurrence for Akx:• Increase Akx flops by 4sn• Increase dense flops by an O(s2) term

• Extended precision:• Constant factor cost increases

• Restarting:• Constant factor increase in Akx cost

• Preconditioning:• Structure-dependent costs

• Rank-revealing, reorthogonalization, etc…

Can also interleave Akx and Gram matrix construction without the extra work - • decreases sequential bandwidth and

latency by 33% rather than a factor of s

CA-CG tuning

avoiding communication in sparse matrix-vector multiply ( spmv )

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