# Iterating Exactly

I’m very excited to share that my PhD student, He Li, will tomorrow be presenting his paper ARCHITECT: Arbitrary-precision Constant-hardware Iterative Compute at the IEEE International Conference on Field-Programmable Technology 2017 (joint work also with James Davis and John Wickerson.)

Anyone who has done any numerical computation will sooner or later encounter a loop like this:

```while( P(x) )
x = f(x);
```

Where $P(x)$ denotes a predicate determining when the loop will exit, $f$ is a function transforming the state of the loop at each iteration, and $x$ is – critically – a vector of real numbers. Such examples crop up everywhere, for example the Jacobi method, conjugate gradient, etc.

How do people tend to implement such loops? They approximate them by using a finite precision number system like floating point instead of reals.

OK, let’s say you’ve done your implementation. You run for 1000 iterations and still the loop hasn’t quit. Is that because you need to run for a few more iterations? Or is it because you computed in single precision instead of double precision? (Or double instead of quad, etc.) Do you have to throw away all your computation, go back to the first iteration, and try again in a higher precision? Often we just don’t know.

He’s paper solves this problem. As time progresses, we increase both the iteration and the accuracy to which a given iterate is known, snaking through the two-dimensional iteration / precision space, linearising two countably infinite dimensions into the single countably infinite dimension of time (clock cycle) using a trick due to Cantor.

This is the essence of our contribution.

To make it work in practice, efficiently in hardware, requires some tricks. For a start, we need to be able to support arbitrary precision arithmetic on finite computational hardware (only memory space growing with precision, not compute hardware). Secondly, we need to compute from most-significant to least-significant digit, iteratively refining our computation as we proceed. This form of computation is not supported naturally by standard binary arithmetic, but is supported by redundant arithmetic. We make use of online arithmetic to enable this transformation.

So now you don’t need to worry – rounding error will not stop you getting your answer. There’s an FPGA design for that.

# Passing Data Structures to FPGAs

Next week, my former PhD student and postdoctoral researcher, Felix Winterstein, will present our paper Pass a Pointer: Exploring Shared Virtual Memory Abstractions in OpenCL Tools for FPGAs at the IEEE International Conference on Field-Programmable Technology in Melbourne, Australia.

Before launching his current startup, Xelera, Felix and I worked together on the problem of automating the production of custom memory systems for FPGA-based accelerators. I previously blogged about some highly novel work we’d done during his PhD on high-level synthesis for code manipulating complex data structures like trees and linked lists. Full detail can be found in the book version of his PhD thesis. All this work – as exciting as it is – was based on sequential C code description as the input format to a high-level synthesis tool.

Many readers of this blog will be aware that OpenCL is rapidly becoming viewed as an alternative way to write correctness-portable code for FPGA development, with both Intel and Xilinx offering OpenCL flows based around OpenCL 1.X. However, OpenCL 2.0 offers a number of interesting features around shared virtual memory which could radically simplify programming, at the cost of making the compiler significantly more complex for FPGA-based computation. It is this issue we address in the paper Felix will present next week.

There’s lots of exciting program analysis work that could be built on top of Felix’s framework, and I’m keen to explore this further – if a reader of this blog would like to collaborate in this direction or like to do a PhD in this field, feel free to get in touch.

Perhaps most importantly, Felix’s framework is open source – check it out at https://github.com/constantinides/FPGA-shared-mem and let us know if you use it!

# HLS and Power: Some FPL Contributions

This week sees the IEEE International Conference on Field-Programmable Logic and Applications, in Ghent, Belgium.  Two of my team are attending to present their research papers on High Level Synthesis and on Run-time Power Estimation. In this post, I briefly summarise the key contributions of these papers.

High-Level Synthesis (HLS) is an important technology, which aims to automatically generate hardware designs from high-level (typically software) descriptions of their behaviour. In a previous blog post, I described some work from my PhD student Junyi Liu (joint with Sam Bayliss) on extending a common paradigm for analysis memory dependences – the polyhedral model – to a parametric version, for efficient pipelining in HLS. This week, Junyi presents an alternative use for the same parametric polyhedral HLS framework: automatic loop tiling (joint work with John Wickerson). Loop tiling is a very common compiler transformation – for example it is often used in matrix-matrix multiplication. The key advantage is to make sure that you only have a small set of data you’re working with at any given moment in time (traditionally for cache, in the FPGA context for embedded scratch-pad memories). The size of this working set can be traded off against the amount of off-chip memory traffic by selection of tile sizes. In a multi-dimensional loop, there are many possible options, and navigating this space is non-trivial. Junyi’s work provides a way to produce an explicit formula for both the memory requirement and the amount of off-chip data traffic required for any given tile size. He can then use nonlinear optimisation techniques to explicitly optimise the traffic subject to any given constraint on buffer size. This work is available as an open-source tool at https://github.com/Junyi-Liu/PolyTSS.

Back in 2016, some work I did with Eddie Hung, James Davis, Josh Levine, Ed Stott and Peter Cheung won the best paper prize at FCCM 2016. We showed that it is possible to use an online (recursive least squares) algorithm to learn the instantaneous power consumption of individual components in an FPGA design, with a view to some kind of run-time manager using this information. The solution worked by monitoring certain signal activity at run-time, but the missing part of the puzzle was which signals to monitor. James’s latest paper, STRIPE, with the same co-authors, answers this question. It turns out that the answer to this problem – as with so many in engineering (and life?) – lies in linear algebra. Golub and Van Loan describe in their classic textbook how QR factorisation can be used to heuristically select a subset of “nearly linearly independent” vectors from a larger set, and it’s this approach that tends to win out when given enough data to work with.

# Overclocking For Fun and Profit

This week at the Design, Automation and Test in Europe (DATE) conference, Kevin Murray is presenting some exciting work I’ve done in collaboration with Kevin, his supervisor Vaughn Betz at the University of Toronto, and Andrea Suardi at Imperial College.

I’ve been working for a while on the idea that one form of approximate computing derives from circuit overclocking. The idea is that if you overclock a circuit then this may induce some error. However the error may be small or rare, despite a very significant performance enhancement. We’ve shown, for example, that such tradeoffs make sense for image processing hardware and – excitingly – that the tradeoffs themselves can be improved by adopting “overclocking-friendly” number representations.

In the work I’ve done on this topic to date, the intuition that a given circuit is “overclocking friendly” for a certain set of input data has been a human one. In this latest paper we move to an automated approach.

Once we accept the possibility of overclocking, our circuit timing analysis has to totally change – we can’t any longer be content with bounding the worst-case delay of a circuit, because we’re aiming to violate this worst case by design. What we’re really after is a histogram of timing critical paths – allowing us to answer questions like “what’s the chance that we’ll see a critical path longer than this in any given clock period?” Different input values and different switching activities give rise to the sensitization of different paths, leading to different timing on each clock cycle.

This paper’s fundamental contribution is to show that the #SAT probem can be efficiently used to quantify these probabilities, giving rise to the first method for determining at synthesis time the affinity of a given circuit to approximation-by-overclocking.

# Concurrent Programming in High-Level Synthesis

This week, my student Nadesh Ramanathan presents our FPGA 2017 paper “Hardware Synthesis of Weakly Consistent C Concurrency”, a piece of work jointly done with John Wickerson and Shane Fleming.

High-Level Synthesis, the automatic mapping of programs – typically C programs – into hardware, has had a lot of recent success. The basic idea is straightforward to understand, but difficult to do: automatically convert a vanilla C program into hardware, extracting parallelism, making memory decisions, etc., as you go. As these tools gain industry adoption, people will begin using them not only for code originally specified as sequential C, but for code specified as concurrent C.

There are a few tricky issues to deal with when mapping concurrent C programs into hardware. One approach, which seems modular and therefore scalable, has been adopted by LegUp: schedule threads independently and then build a multithreaded piece of hardware out of multiple hardware threads. This all works fine, indeed there is an existing pthreads library for LegUp. The challenge comes when there’s complex interactions between these threads. What if they talk to each other? Do we need to use locks to ensure synchronisation?

In the software world, this problem has been well studied. The approach proposed by Lamport was to provide the programmer with a view of memory known as “sequentially consistent” (SC). This is basically the intuitive way you would expect programs to execute. Consider the two threads below, one on the left and one on the right, each synthesised by an HLS tool. The shared variables `x` and `y` are both initialised to zero. The assertion is not an unreasonable expectation from a programmer: if `r0 = 0`, it follows that Line 2.3 has been executed (as otherwise `r0 = -1`). We can therefore conclude that Line 1.2 executed before Line 2.2. It’s reasonable for the programmer to assume, therefore that Line 1.1 also executed before Line 2.3, but then x` = 1` when it is read on Line 2.3, not zero! Within a single thread, dependence analysis implemented as standard in high-level synthesis would be enough to ensure consistency with the sequential order of the original code, by enforcing appropriate dependences. But not so in the multi-threaded case! Indeed, putting this code into an HLS tool does indeed result in cases where the assertion fails.

My PhD student’s paper shows that we can fix this issue neatly and simply within the modular framework of scheduling threads independently, by judicious additional dependences before scheduling. He also shows that you can improve the performance considerably by supporting the modern (C11) standard for atomic memory operations, which come in a variety of flavours from full sequential consistency to the relaxed approach natively supported by LegUp pthreads already. In particular, he shows for the first time that on an example piece of code chaining circular buffers together that you can get essentially near-zero performance overhead by using the so-called acquire / release atomics defined in the C11 standard as part of a HLS flow, opening the door to efficient synthesis of lock-free concurrent algorithms on FPGAs.

As FPGAs come of age in computing, it’s important to be able to synthesise a broad range range of software, including those making use of standard concurrent programming idioms. We hope this paper is a step in that direction.

# Time is Precision

Most modern FPGA arithmetic designs use bit-parallel binary arithmetic – typically two’s complement for signed computation. This generally makes for fast arithmetic, but has the distinct disadvantage that silicon area scales with precision of computation. Occasionally – much more often in the distant past when FPGA area was a precious resource – people compute with bit serial binary arithmetic. In this case, time scales with precision.

One problem with digit serial binary arithmetic for multiplication and division is that you need to know, in advance, the location of your least significant digit: while precision unfolds as time, precision is fixed a priori.

Milos Ercegovac‘s online arithmetic forms an interesting counterpoint to this: in this arithmetic, digits are produced most-significant-digit-first. This suggests that the longer we compute, the more precise our answer will be – and we can terminate whenever we’ve got a good enough answer. Or run out of time. The problem with this approach is that the hardware arithmetic units generally implemented for digit-serial multiplication or division still scale with precision requirements, placing an unwanted a priori bound on computational precision.

Enter my former BEng project student Aaron Zhao, who presented our paper “An Efficient Implementation of Online Arithmetic” at the IEEE International Conference on Field-Programmable Technology in Xi’an, China. Aaron’s contribution – for his BEng final year project – was to design a library of online arithmetic units whose logic area is constant with precision (of course, they still need more RAM for more precision.) This opens up a lot of practical possibilities for our research.

Precision (and energy) can scale elastically with time in FPGA-based compute.

# Memory Consistency Models, and how to compare them automatically

John Wickerson explains our POPL 2017 paper on memory consistency

My POPL 2017 paper with Mark Batty, Tyler Sorensen, and George Constantinides is all about memory consistency models, and how we can use an automatic constraint solver to compare them. In this blog post, I will discuss:

• What are memory consistency models, and why do we want to compare them?
• Why is comparing memory consistency models difficult, and how do we get around those difficulties in our work?
• What can we achieve, now that we can compare memory consistency models automatically?

# What is a memory consistency model?

Modern computers achieve blistering performance by having their computational tasks divided between many ‘mini-computers’ that run simultaneously. These mini-computers all have access to a collection of shared memory locations, through which they can communicate with one another.

It is tempting to assume that when one of the mini-computers stores some data into one of these shared memory locations, that data will immediately become visible to any other mini-computer that subsequently loads from that location…

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