In this paper, we take a very unusual approach to the design of a deep neural network accelerator in hardware: for us, the nodes in the neural network are Boolean lookup tables.
We were motivated initially by the fact that in very low precision FPGA neural network architectures, lookup tables are often used for arithmetic, but they are often used for very specific functions: while a -LUT is capable of implementing any nonlinear Boolean function with inputs, it ends up getting used for only a tiny fraction of these functions. A good example is binarised neural networks (BNNs) such as FINN, where LUTs end up being used to implement XNOR gates (multiplication over ) and popcount functions. Our research question is therefore: rather than restricting ourselves to these functions, can we make better use of the LUTs by embracing the nonlinearity and the -input support they give us?
We show that this is indeed possible. Our basic approach is to start with a weight-binarised neural network, add inputs to each node to bring them up to support, and then retrain the Boolean function implemented by that node. Retraining Boolean functions is a bit tricky, of course, because neural network training algorithms are not designed for this purpose. We generate a smooth interpolating function over the LUT entries, allowing us to use standard neural network training software (we use TensorFlow).
The end result is that the re-trained neural network is far more prunable than the original, because the extra inputs to the -LUTs compensate for the removal of other nodes. Thus we end up with a much sparser neural network for the same classification accuracy. The sparsity improves our area by a factor of two or more, yet the more complex inference functions at each node are effectively provided “for free” by the FPGA architecture.
I very much enjoyed the format of the meeting: select interesting speakers and allow them 30 minutes to talk about a topic of their choosing related to the theme of the meeting, with 10 full minutes for discussion and in-depth questions after each talk. Posters were also presented by a wide variety of researchers, with each poster presenter given a one-minute lightning-talk slot. Two of my PhD students, Erwei Wang and He Li, took this opportunity. Erwei presented a preview of our LUTNet paper appearing at FCCM very soon (separate blog post to follow), while He presented some of our work on arbitrary precision iterative compute.
Talks by others included:
David Keyes (KAUST) on the topic “Hierarchical algorithms on hierarchical architectures”. He discussed some very interesting hierarchical low-rank decompositions and also hierarchies of numerical precision.
Kathy Yelick (Berkeley) spoke on “Antisocial parallelism: avoiding, hiding and managing communication”, a very fruitful area of research in recent years. A few years ago, Abid Rafique, one of my former PhD students (joint with Nachiket Kapre) made use of this work, and it was good to catch up with the current state of research.
Anna Scaife (Manchester) gave a fascinating insight into the Square Kilometre Array. The sheer volumes of data are mind boggling (zettabytes annually) and pose unique algorithmic challenges.
Michela Taufer (UTK) discussed molecular dynamics workflows, and how we may be able to harness machine learning to reduce the human bottlenecks in such workflows in the future.
Rick Stevens (Argonne) gave a very engaging talk about the intersection of machine learning with computational science, exemplified by the Candle project, using deep learning in cancer research. He mentioned many of the emerging architectures for deep learning and their optimisation for low-precision compute. Interested readers may also like our recent survey article on the topic.
John Shalf (LBNL) spoke about alternative computational models beyond CMOS, new materials for switches, and the growth of hardware specialisation. He proposed the strategy of: hardware-driven algorithm design, algorithm-driven hardware design, and co-design of hardware and algorithm. Having worked in the FPGA community for decades where this has been our mantra, it is great to see hardware specialisation spreading the message of co-design into the HPC community.
Doug Kothe (ORNL) provided a very interesting insight into exascale computational motifs at the DOE.
Tony Hey (STFC) set out a compelling argument that academics in applied deep learning should focus on deep learning for scientific data, on the basis that (i) scientific data sets are huge and open and (ii) head-to-head competition with industrial giants on profit-oriented deep-learning applications, without access to their data sets, is a poor choice for academia. I think the same argument could be made for academic computer architecture too. His team are developing benchmarks for scientific machine learning, complementary to MLPerf.
Erin Carson (Charles University) presented an enchanting talk on iterative linear algebra in low and multiple precision compute. I’m a fan of her earlier work with Nick, and it was great to hear her current thinking and discussion of least-squares iterative refinement.
Steve Furber (Manchester) spoke about arithmetic in the context of the SpiNNaker machine, and a particular approach they have taken to numerical solution of neural ODEs in fixed-point arithmetic, demonstrating that stochastic rounding can radically improve the quality of their results.
Tim Palmer (Oxford) argued for low precision compute in weather and climate models, allowing the recouped computational cost to be recycled into better resolution weather models, resulting in higher overall accuracy. This reminded me of the argument I made with my PhD student Antonio Roldão Lopes and collaborator Eric Kerrigan in our paper More FLOPS or More Precision?
Guillaume Aupy (INRIA) discussed memory-efficient approaches for automatic differentiation and back-propagation.
Satoshi Matsuoka (RIKEN Centre) took us through the work being done on Post-K, a new Japanese supercomputer being designed to provide compute infrastructure for future workloads at the intersection of big data and AI/ML.
Mike Heroux (Sandia) spoke about his work developing programming infrastructure for future HPC, in particular for performance portability and for system reliability.
My own talk was entitled “Rethinking Deep Learning: Architectures and Algorithms” – I will save summarising the content for a future blog post. Slides for all these talks will appear on the Manchester Numerical Linear Algebra group website. In addition, each speaker has received an invitation to author an article for a special issue of Philosophical Transactions A – this should be a very interesting read.
I was impressed by the great attendance at the meeting and by the quality of the technical interaction; I met several new and interesting people at the intersection of numerical analysis and scientific computing.
Readers can find a summary of some of the talks I found most interesting below.
On Tuesday, my colleague Martin Trefzer chaired a session on Computational and Resource-efficiency in Quantum and Approximate Computing. The work by Sekanina was interesting, using information on data distribution to drive the construction of approximate circuits. Circuits are constructed from the baseline using a technique called Cartesian Genetic Programming. I have recently been collaborating with Ilaria Scarabottolo and others from Laura Pozzi‘s group on a related problem – see our DAC 2019 paper (to appear) for details – so this was of particular interest.
On Wednesday, I chaired a session When Approximation Meets Dependability, together with my colleague Rishad Shafik. Ioannis Tsiokanos from Queen’s University Belfast presented an interesting approach that dynamically truncates precision in order to avoid timing violations. This is interestingly complementary to the approach I developed with my former PhD student Kan Shi where we first simply allowed timing violations [FCCM 2013], and secondly redesigned data representation based on Ercegovac‘s online arithmetic, so that timing violations caused low-magnitude error [DAC 2014].
On Thursday morning, I attended the session Architectures for Emerging Machine Learning Techniques. Interestingly, there was a paper there making use of Gustafson’s posits within hardware-accelerated deep learning, a technique they dub Deep Positron.
The highlight talk for me was Ed Lee‘s Thursday keynote, A Fundamental Look at Models and Intelligence. Although I’ve been aware of Lee’s work, especially on Ptolemy, since I did my own PhD, I don’t think I’ve ever had the pleasure of hearing him lecture before. It was insightful and entertaining. A central theme of the talk was that models mean two different things for scientists and engineers: a scientist builds a model to correspond closely to a ‘thing’; an engineer builds a ‘thing’ to correspond closely to a model. He used dichotomy to illuminate some of the differences we see between neuroscience-inspired artificial intelligence and the kind of AI we see as very popular at the moment, such as deep learning. Lee’s general-readership book Plato and the Nerd – which has been on my “to read” list since my colleague and friend Steve Neuendorffer mentioned it to me a few years ago – has just climbed several notches up that list!
On Thursday afternoon, I attended the session on The Art of Synthesizing Logic. My favourite talk in this session was from Heinz Reiner, who presented a collaboration between EPFL and UC Berkeley on Boolean rewriting for logic synthesis, in which exact synthesis methods are used to replace circuit cuts, rather than resorting to a pre-computed database of optimal function implementations. During the talk, Reiner also pointed the audience to an impressive-looking GitHub repo, featuring what looks like some very useful tools.
Friday is always workshop day at DATE, featuring a number of satellite workshops. I attended the workshop entitled Quo Vadis, Logic Synthesis?, organised by Tiziano Villa and Luca Carloni. This was a one-off workshop in celebration of the 35th anniversary of the publication of the influential Espresso book on two-level logic minimisation.
My favourite talk in this workshop was from Jordi Cortadella, who spoke about a method for synthesising Boolean relations. This is the problem of synthesising a the cheapest implementation of a function , which one is free to choose from amongst the given relation , i.e. viewing as a relation , one is free to chose – and its implementation – subject to the requirement that for some given relation (not necessarily a function). This is a strict generalisation of the well-known problem of Boolean ‘don’t care’ conditions, a.k.a. incompletely specified functions. Cortadella presented a method leveraging the known approaches to this latter problem, by exploring the semi-lattice of relations between these sets generated by in a structured way, using a form of branch-and-bound.
Soeken also presented a very interesting summary of three uses of SAT within logic synthesis, namely Schmitt’s ASP-DAC paper on SAT-based LUT mapping, Eén’s paper on using logic synthesis for efficient SAT (rather than SAT for efficient logic synthesis) and Haaswijk‘s recent PhD on making exact logic synthesis more scalable by providing partial topological information – a topic that interestingly has some echoes in work I’m soon to present at the Royal Society.
The workshop was an enjoyable way to end DATE, and I was disappointed to have to leave half-way through – there may well have been other interesting talks presented in the afternoon.
This was my first SIAM conference. I was kindly invited to speak on the topic of floating-point error analysis by Pierre Blanchard, Nick Higham and Theo Mary. I very much enjoyed the sessions they organised and indeed the CSE conference, which I hope to be able to attend more regularly from now on.
My own talk was entitled Approximate Arithmetic – A Hardware perspective. I spoke about the rise of architecture specialisation as driving the need for closer collaboration between computer architects and numerical analysts, about some of our work on automatic error bounds Boland and Constantinides (2011) and Magron, Constantinides and Donaldson (2017), on code refactoring Gao and Constantinides (2015), as well as some of our most recent work on machine learning (I will blog separately about this latter topic over the next couple of months.)
The CSE conference is very large – with 30-40 small parallel sessions happening at any given moment – so I cannot begin to summarise the conference. However, I include some notes below on other talks I found particularly interesting.
I very much enjoyed the plenary presentation by Rachel Ward on Stochastic Gradient Descent (SGD) in Theory and Practice. She introduced the SGD method very nicely, and looked at various assumptions for convergence. She took a particularly illuminating approach, by looking at applying SGD to the simple special case of solving a system of linear equations by minimising in the case where . She showed that if the system is under-determined, then SGD converges to the solution of minimum 2-norm, and therefore has an inherent regularising effect. I was surprised by some of the results on overparameterised neural networks, showing that SGD finds global minimisers and that there really doesn’t tend to be much overfitting despite the huge number of parameters, pointing to the implicit regularisation caused by the SGD algorithm itself. I learnt a lot from this talk, and have several papers on my “to read” list as a result, in particular:
There was also an interesting plenary from Anima Anandkumar on the role of tensors in machine learning. The mathematical structure of tensors and multi-linear algebra are topics I’ve not explored before – mainly because I’ve not seen the need to spend time on them. Anandkumar certainly provided me with motivation to do that!
Floating-Point Error Analysis
Theo Mary from the University of Manchester gave a very good presentation of his work with Nick Higham on probabilistic rounding error analysis, treating numerical roundoff errors as zero-mean independent random variables of arbitrary distribution, making use of Hoeffding’s inequality to a produce a backward error analysis. Their work is described in more detail on their own blog post and – in more depth – in their their very interesting paper. It’s a really exciting and useful direction, I think, given the greater emphasis on average-case performance from modern applications, together with both very large data sets and very low precision computation, the combination of which renders many worst-case analyses meaningless. In a similar vein, Ilse Ipsen also presented a very interesting approach: a forward error analysis, more specialised in that she only looked at inner products, but also without the assumption of independence, making use of Azuma’s inequality. The paper on this topic has not yet been finished, but I certainly look forward to reading it in due course!
Reducing Communication Costs
There were a number of interesting talks on mitigating communication costs. Lawrence Livermore National Labs presented several papers relating to the ZFP format they’ve recently proposed for (lossily) compressed floating-point vectors, at a mini-symposium organised by Alyson Fox, Jeffrey Hittinger, and James Diffenderfer. Diffenderfer’s talk developed a bound on the norm-wise relative error of vectors reconstructed from ZFP; Alyson Fox’s talk then extended this to the setting of iterative methods, noting as future work their interest in probabilistic analyses. In the same session, Nick Higham gave a crystal clear and well-motivated talk on his recent work with Srikara Pranesh and Mawussi Zunon – slides and paper are available. This work extends the applicability of Nick’s earlier work with Erin Carson to cases that would have over- or under-flowed, or led to subnormal numbers, without the scaling technique developed and analysed here. They use matrix equilibration – this reminded me of some work I did with my former PhD student Juan Jerez and colleague Eric Kerrigan, but in our case for a different algorithm kernel and targeting fixed-point arithmetic, where making use of the full dynamic range is particularly important. The Higham, Pranesh and Zunon results are both interesting and practically very useful.
In a different session, Hartwig Anzt spoke about the work he and others have been doing to explicitly decouple storage precision from compute precision in sparse linear algebra. The idea is simple but effective: take the high-order bits of the mantissa (and the sign / exponent) and store them in one chunk of data and – separately – store the low-order bits in another chunk. Perform all arithmetic in high precision (because it’s not the computation that’s the bottleneck), but convert low-precision stored data to high precision on the fly at data load (e.g. by packing low-order bits with zeros.) Then, at run-time, decide whether to load the full-precision data or only the low-precision data, based on current estimates of convergence. This approach could also make a good case study application for the run-time adaptation methodology we developed with U. Southampton in the PRiME project.
Beyond the technical talks, there were two things that stood out for me since I’m new to the conference. Firstly, there were many more women than in the typical engineering conferences I attend. I don’t know whether the statistics on maths versus engineering are in line with this observation, but clearly maths is doing something right from which we could learn. Secondly, there were clear sessions devoted to community building: mentoring sessions, tutorials for new research students, SIAM student chapter presentations, early career panels, presentations on funding programmes, diversity and inclusion sessions, a session on helping people improve their CV, an explicit careers fair, etc. Partly this may simply reflect the size of the conference, but even so, this seems to be something SIAM does particularly well.
This week, I attended the ACM FPGA 2019 conference in Seaside (nr. Monterey), California, the annual premier ACM event on FPGAs and associated technology. I’ve been involved in this conference for many years, as author, TPC member, TPC and general chair, and now steering committee member. Fashions have come and gone over this time, including in the applications of FPGA technology, but the programme at FPGA is always interesting and high quality. This year particular thanks should go to Steve Neuendorffer for organising the conference programme and to Kia Bazargan in his role as General Chair.
Below, I summarise my personal highlights of the conference. These are by no means my view of the “best” papers – they are all good – but rather those that interested me the most.
Efficient and Effective Sparse LSTM on FPGA with Bank-Balanced Sparsity, a collaboration between Tsinghua, Beihang, Harbin Institute of Technology, and Microsoft Research, tackled the problem of ensuring that an inference implementation, when sparsified, gets sparsified in a way that leads to balanced load across the various memory banks. The idea is simple but effective, and leads to an interesting tradeoff between the quality of LSTM output and performance. I think it would be interesting to try to design a training method / regulariser that encourages this kind of structured sparsity in the first place.
Kees Vissers from Xilinx presented a keynote talk summarising their new Versal architecture, which the Imperial team had previously had the pleasure of hearing about from our alumnus Sam Bayliss. This is a really very different architecture to standard FPGA fare, and readers might well be interested in taking a look at Kees’s slides to learn more.
Vaughn Betz presented a paper from the University of Toronto, Math Doesn’t Have to be Hard: Logic Block Architectures to Enhance Low Precision Multiply-Accumulate on FPGAs. This work proposed a number of relatively minor tweaks to Intel FPGA architectures which might have a signifiant impact on low-precision MAC performance. Vaughn began by pointing out that in this application, very general LUTs often get wasted by being used as very simple gates – he gave the example of AND gates in partial product generation, and even as buffers. A number of architectural proposals were made to avoid this issue. I find this particularly interesting at the moment, because together with my PhD student Erwei Wang and others, I have proposed a new neural network architecture called LUTNet, motivated by exactly the same concern. However, our approach is the dual of that presented by Vaughn – we keep the FPGA architecture constant but modify the basic computations performed by the neural network to be more well-tuned to the underlying architecture. Expect a future blog post on our approach!
Lana Josipović presented the most recent work on the dynamically scheduled HLS tool from Paolo Ienne‘s group at EPFL, which they first presented at last year’s conference – see my blog post from last year. This time they have added speculative execution to their armoury. This is a very interesting line of work as HLS moves to encompass more and more complex algorithns, and Lana did a great job illustrating how it works.
The panel discussion – chaired by Deming Chen – was on the topic of whether FPGAs have a role to play in Supercomputing. As I pointed out in the discussion, to answer this question scientifically we need to have a working definition of “FPGA” and of “Supercomputing” – both seem to be on shifting sands at the moment, and we need to resist reducing a question like this to “does LINPACK run well on a Virtex or Stratix device.”
We also had the pleasure of congratulating Deming Chen and Paul Chow on their recently awarded fellowships, awarding a best paper prize, recognising several historical FPGA papers of significance, and last but by no means least welcoming the new baby of two of the stalwarts of the FPGA community – baby complete with “I am into FPGA” T-shirt! All this led to an excellent community feeling, which we should continue to nurture.
Multi-threaded programming is now a fairly mainstream activity, and has found its way into high-level synthesis tools, both through OpenCL and also LegUppthreads support. We focus here on the latter.
At FPL 2017, Joy and Jason had a paper that automatically decided how to partition shared arrays for multi-threaded code, aiming to reduce the amount of arbitration required between hardware units and chunks of memory. Their approach used a simulation trace to identify candidate partitions, and designed the arbiters so that, for example, if accesses to partition P were only observed in that trace to come from thread T, then there is very low latency access to P from T at execution time. In this way, they were able to significantly speed up synthesised multi-threaded code making use of shared memories.
However, the arbiters were still there. They were necessary because while no access by some other thread T’ was observed during simulation, there was no guarantee that such an access might not occur at run-time. So the arbiters sat there, taking up FPGA area and – for large enough numbers of ports – hitting the critical path of the design.
Enter our work.
In our paper, we show – building on the excellent PhD thesis by Nathan Chong that I examined a few years back – how the original multi-threaded code can be translated into single-threaded code in a verification language developed by Microsoft Research called Boogie. We then show how to automatically construct assertions in Boogie that, if passed, correspond to a formal proof that a particular thread can never access a particular partition. This lets us strip out the arbiters, gaining back the area and significantly boosting the clock frequency.
I think it’s a really neat approach. Please come and hear Jianyi give his talk and/or read the paper!
We can think of a DNN as a graph , where nodes perform computations and edges carry data. This graph can be interpreted (executed) as a function mapping input data to output data. The quality of this DNN is typically judged by a loss function . Let’s think about the supervised learning case: we typically evaluate the DNN on a set of test input data points and their corresponding desired output , and compute the mean loss:
Now let’s think about approximation. We can define the approximation problem as – starting with – coming up with a new graph , such that can be somehow much more efficiently implemented than , and yet is not significantly greater than – if at all. All the main methods for approximating NNs such as quantisation of activations and weights and sparsity – structured and unstructured – can be viewed in this way.
There are a couple of interesting differences here to the different problem – often studied in approximate computing, or lossy synthesis – of approximating the original function . In this latter setting, we can define a distance between and (perhaps worst case or average case difference over the input data set), and our goal is to find a that keeps this distance bounded while improving the performance, power consumption, or area of the implementation. But in the deep learning setting, even the original network is imperfect, i.e. . In fact, we’re not really interested in keeping the distance between and bounded – we’re actually interested bounding the distance between and some oracle function defining the perfect classification behaviour. This means that there is a lot more room for approximation techniques. It also means that may even improve compared to , as sometimes seen – for example – through the implicit regularisation behaviour of rounding error in quantised networks. Secondly, we don’t even have access to the oracle function, only to a sample (the training set.) These features combine to make the DNN setting an ideal playground for novel approximation techniques, and I expect to see many such ideas emerging over the next few years, driven by the push to embed deep learning into edge devices.
I hope that the paper we’ve just published in ACM CSUR serves as a useful reference point for where we are at the moment with techniques that simultaneously affect classification performance (accuracy / loss) and computational performance (energy, throughput, area). These are currently mainly based around quantisation of the datatypes in (fixed point, binarisation, ternarisation, block floating point, etc.) topological changes to the network (pruning) and re-parametrisation of the network (weight sharing, low-rank factorisation, circulant matrices) as well as approximation of nonlinear activation functions. My view is that this is scratching the surface of the problem – expect to see many more developments in this area and consequent rapid changes in hardware architectures for neural networks!
Approximate Computing has been a buzzphrase for a while. The idea, generally, is to trade off quality of result / solution, for something else – performance, power consumption, silicon area. This is not a new topic, of course, because in numerical computation people have generally always worked with finite precision number representations. In my early work in 2001, before the phrase “Approximate Computing” was in circulation, I introduced this as “Lossy Synthesis” – the idea that circuit synthesis can be broadened to incorporate the automated control of loss of numerical quality in exchange for reduction in area and increase in performance.
Most approximate computing frameworks focus on domains where numerical error is tolerable. Perhaps we don’t care if our answer is 1% wrong, for example, or perhaps we don’t even care if it’s out by 100%, so long as that happens very infrequently.
However, there is another interesting class of computation. Consider a function producing a Boolean output , where . An interesting challenge is to produce another function with a ternary output bearing a close resemblance to . We can make the idea of bearing a close resemblance precise in the following way: if declares a value true (false), then so must . We can think of this as relation between fibres:
We can then think of the function as approximating if the fibre of the ‘don’t know’ element, , is small in some sense, e.g. if is small.
In the context of approximate computing, we can pose the following optimisation problem:
subject to and (1),
where represents the cost (energy, area, latency) of implementing a function. One application area for this kind of investigation is in computer graphics. It is often the case that, when rendering a scene, an algorithm first needs to decide which components of the scene will definitely not be visible, and therefore need not be considered further. Should this part of the graphics pipeline make a mistake by deciding a component may be visible when it is actually invisible, little harm is done – more computation is required downstream in the graphics pipelining, costing energy and time, but not a reduced quality rendering. On the other hand, if it makes a mistake by deciding that a component is invisible when it is actually visible, this may cause a significant visual artefact in the rendered scene.
Last year, I had a bright Masters student, Georgios Chatzianastasiou, who decided to explore this problem in the context of being the Slab Method in computer graphics and being one of a family of approximations , each produced by using interval arithmetic approximations to computed in floating-point with precision . In this way we get a family of approximate computing hardware IP blocks, all of which guarantee that, when given a ray and a bounding box, if the IP reports no intersection between the two, then there is provably no intersection. Yet each family member operates at a different precision, requiring different circuit area, trading off against the rate of `false positives’. Georgios wrote a paper on the implementation, which was accepted by FPL 2018 – he presents it next Wednesday.
If you’re at the FPL conference, please go and say hello to Georgios. If you’re interested in working with me to deepen and broaden the scope of this work, please get in touch!
Readers of this blog may remember that we previously came up with a neat way of computing arbitrarily precise values of arbitrarily deep iterations of an iterative real-number computation, while only using constant-area compute hardware. This latest paper extends our previous work in the following way.
In our previous work, we computed every digit of every iteration of the computation. While for any computable real function this will give a correct result, it tends to be wasteful in practice. There are two reasons it’s wasteful. Firstly, often the reason we’re computing an iteration is because that iteration converges. Convergence can be seen as agreement in most-significant digits – after a while they don’t change. So why do we recompute them? We see this again and again in standard numerical computing – each iteration might add just a couple of new correct digits, but we still end up wasting time and energy computing all of the digits in each iteration, even the stable ones. Secondly, not all iterations may contribute equally to the overall error resulting from early termination. This paper addresses these two issues.
The first, and more general, issue is the wastefulness of computing stabilised digits. But just because they look stable, are they really stable? Maybe we’ve stabilised to 0.9, 0.99, 0.999, 0.999, and then one more iteration might kick us over to 1.0001. So can we really afford not to recompute most-significant digits? Ercegovac‘s Online Arithmetic comes to our rescue again! If we compute in an appropriate redundant number representation, then we can prove that stability of digits means we don’t need to consider them any more. This is our first contribution – to recognise this and utilise it within an appropriately modified computational architecture.
The second, and more specific, issue is that some digits are effectively ‘don’t care’. In this paper, we only analyse the specific case of stationary iterative methods (Jacobi, SOR, etc.) for this kind of digit. We show that, in these cases, for a fixed digit budget (e.g. “compute at most D digits across all iterations”), you should allocate these digits by computing a constant more digits each iteration. This constant can be estimated from the infinity norm of a certain matrix involved in the computation. Again, we modify our hardware architecture to take advantage of this pattern.
The end result is that we end up tracing out a corridor of digits, shown in the figure below, where the vertical axis is iteration and the horizontal axis is precision / digit number. Some digits have provably stabilised and no longer need computation (marked “), some are irrelevant don’t cares (marked X). This corridor radically improves the storage requirements of the original ARCHITECT scheme.
In the 1970s, Miloš invented a rather nice method called the E-method for evaluating rational functions, i.e. ratios of two polynomials. The basic idea of his method is as follows. We may solve a system of linear equations where is a matrix of a special structure formed from constants together with variable :
If we further choose the vector , then it turns out that the first element of the solution vector is the rational function .
So we can use this to evaluate such rational functions. On the face of it, that doesn’t seem very interesting: why would we go to the bother of solving a system of linear equations to evaluate a rational function?
The answer lies in the combination of this idea with another one of Miloš’s key contributions, the idea of online arithmetic – computing results most-significant-digit first. In fact, if the matrix is sufficiently well conditioned then we may use a stationary iterative method to solve the system of equations in such a way that it produces one new correct digit of the solution for each iteration of the method, leading to very efficient evaluation.
Our paper at ARITH makes two novel contributions. Firstly, we show how to find such a matrix that is sufficiently well conditioned and for which the solution is close to a given function we’re trying to approximate, improving on the previous technique of Brisebarreet al. Secondly, we show how this method can be efficiently implemented in modern FPGA hardware, when aiming for high throughput.
The main domain of interest will be functions where rational approximation provides a much better fit than polynomials, as the computation required essentially provides rational computation for the price of polynomial computation. A buy-one-get-one-free offer, if you will.
I’m pleased to say that both the rational approximation generator and the hardware IP core generator will soon be open-sourced. Watch this space! Edit: I’m pleased to say this is now available at https://github.com/sfilip/emethod.