How would you use a benchmark like this to predict the performance of a well-designed asynchronous parallel conjugate gradient solver, like most modern deep learning neural networks that run on Internet HPC machines?

CG isn't truly asynchronous due to its reductions. It can be pipelined in various ways (we have several implementations in PETSc), but performance requires a quality implementation of asynchronous reduction (e.g., MPI_Iallreduce) which the vendors have been slow about developing (I've been working with some on fixing this and Cray has made recent progress).

With respect to deep learning and other applications using CG or related algorithms, the bottlenecks depend on the scale, and ability to expose locality, and operator/preconditioner representation. If there is no locality, then matrix-vector products require all-to-all communication which tend to dwarf the cost of the reductions in CG. Even with locality in the matrix-vector product, preconditioners often need to communicate globally in a scalable way similar to HPGMG. Operators need not be represented as a table of numbers or a sparse matrix format, but could use a tensor product, fast transform, or other information to compute the action using less storage. If they are represented explicitly (sparse or dense), then matrix-vector product performance (thus CG as a whole) is dominated by memory bandwidth for problem sizes that do not fit in cache. HPGMG tries to strike a balance between memory bandwidth demands and compute using a matrix-free representation. HPGMG also reports dynamic range expressed as Performance versus Time-to-solution as the problem size is varied, which allows applications to see performance barriers that might be relevant to them (e.g., see how Titan cannot do a solve in less than 200 ms while Edison can do 50 ms, and how that relates to climate simulation performance targets; see slide 7 of https://jedbrown.org/files/20150624-Versatility.pdf).

Is it possible to calculate the theoretical performance of a cluster under HPGMG and then do a practical run and come with an efficiency number like in HPL ?

One of the biggest reasons for use of HPL is that many sizing considerations can be based off of the theoretical calculations.

But anyway this is very interesting. I definitely need to check this out.

HPL has an abundance of flops at all scales (N^{1.5} flops on N data), so one can expect a decent fraction of peak flop/s on any architecture with enough memory and adequate cache performance. This is a problem because architectural tricks like doubling the vector registers without commensurate improvements in bandwidth, cache sizes, load/store/gather/scatter produce huge (nearly 2x) benefit for HPL and little or no benefit to a large fraction of real applications.

HPGMG is representative of most structure-exploiting algorithms in that it does not have this abundance of flops, thus theoretical performance is actively constrained by both memory bandwidth and flop/s. We see many active constraints in practice; e.g., improving any of peak flop/s, memory bandwidth, network latency, or network bandwidth produces a tangible improvement in HPGMG performance. Depending on the fidelity of the performance model, these dimensions can be a fairly accurate predictor of performance, but ILP, compiler quality, on-node synchronization latency, cache sizes, and similar factors also matter (more for HPGMG-FE than HPGMG-FV).

I think it is actually quite undesirable for benchmark performance to be trivially computed from one parameter in machine provisioning. No computing center has a mission statement asking for a place on a benchmark ranking list (like Top500). Instead, they have a scientific or engineering mandate. Press releases tend to overemphasize the ranking and I think it is harmful to the science any time the benchmark takes precedence over the expected scientific workload. HPGMG is intended to be representative in the sense that if you build an "HPGMG Machine", you'll get a balanced, versatile machine that scientists and engineers in most disciplines will be happy with. I'd still rather the centers focus on their workload instead of HPGMG.

The information I've seen about Sibyl (the Google ML system, not the genomics package (http://sybil.sourceforge.net/documentation.html)) says it is basically doing logistic regression using a parallel algorithm (Collins, Schapire, Singer) with a transpose on each iteration. Without knowing more about the problem sizes and data sparsity/irregularity, I expect the transpose to be a significant expense. I'd be happy to read more if you have access to further technical information, but it's not clear how this comment relates to your previous question about CG and deep learning. As it relates to HPGMG, I think my previous response covers the important performance dimensions. I'd be happy to discuss further over email.

Both logistic regression and deep learning are basically just big conjugate gradient minimizers.

What I meant by asynchronous is that not all terms in a gradient are required to be summed in the same step.

The transpose step in Sibyl is implemented in the Shuffle and Reduce phases. The filesystem is used to hold the temporary data. Nevertheless even for large systems, very few steps are required, and step times are reasonable, even compared to modern supercomputers. This is a tribute primarily to the design of sibyl and the implementation of MapReduce at Google.

This is all explained in online versions of the Sibyl presentation. I really wish more people from DOE who write modern solvers would pay attention to this stuff.

Interesting! I'll have to check this out, thanks!