Stochastic computing
Introduction '''Stochastic computing''' is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams. Stochastic computing is distinct from the study of randomized algorithms.
Motivation and a simple example Suppose that  is given, and we wish to compute . Stochastic computing performs this operation using probability instead of arithmetic.Specifically, suppose that there are two random, independent bit streams called ''stochastic number''s (i.e. Bernoulli processes), where the probability of a one in the first stream is , and the probability in the second stream is . We can take the logical AND of the two streams. 1  0  1  1  0  1  ...  1  1  0  1  1  0  ...  1  0  0  1  0  0  ...The probability of a one in the output stream is . By observing enough output bits and measuring the frequency of ones, it is possible to estimate  to arbitrary accuracy.The operation above converts a fairly complicated computation (multiplication of  and ) into a series of very simple operations (evaluation of ) on random bits.To put in another perspective, assuming the truth table of an AND gate. Conventional interpretation is that the output is true if and only if input A and B are true. However, if the table is interpreted vertically, (0011) AND (0101) is (0001), i.e., 1/2 x 1/2 = 1/4, which is exactly an arithmetic multiplication. As the information is presented in problability distribution, probability multiplication is literally an AND operation.+ABOut000010100111More generally speaking, stochastic computing represents numbers as streams of random bits and reconstructs numbers by calculating frequencies. The computations are performed on the streams and translate complicated operations on  and  into simple operations on their stream representations. (Because of the method of reconstruction, devices that perform these operations are sometimes called stochastic averaging processors.) In modern terms, stochastic computing can be viewed as an interpretation of calculations in probabilistic terms, which are then evaluated with a Gibbs sampler. It can also be interpreted as a hybrid analog/digital computer.
History The RASCEL stochastic computer, circa 1969Stochastic computing was first introduced in a pioneering paper by John von Neumann in 1953.However, thetheory could not be fully developed until advances in computing of the 1960s,mostly through a series of simultaneous and parallel efforts in the USand the UK.By the late 1960s, attention turned to the design ofspecial-purpose hardware to perform stochastic computation. A hostof these machines were constructed between 1969 and 1974; RASCELis pictured in this article.Despite the intense interest in the 1960s and 1970s, stochasticcomputing ultimately failed to compete with more traditional digitallogic, for reasons outlined below. The first (and last)International Symposium on Stochastic Computingtook place in 1978; active research in the area dwindled over the nextfew years.Although stochastic computing declined as a general method ofcomputing, it has shown promise in several applications. Research hastraditionally focused on certain tasks in machine learning andcontrol.Somewhat recently, interest has turned towards stochasticdecoding, which applies stochastic computing to the decoding of errorcorrecting codes. More recently, stochastic circuits have been successfully used in image processing tasks such as edge detection and image thresholding. Recent advancement in stochastic circuits also shows promising speed and energy efficiency advantages in artificial intelligence (AI) hardware acceleration on edge computing.
Strengths and weaknesses Although stochastic computing was a historical failure, it may still remain relevant forsolving certain problems. To understand when it remains relevant, it is useful tocompare stochastic computing with more traditional methods of digital computing.=== Strengths ===Suppose we wish to multiplytwo numbers each with  bits of precision.Using the typical longmultiplication method, we need to perform operations. With stochastic computing, we canAND together any number of bits and the expected value will alwaysbe correct. (However, with a small number of samples the variance willrender the actual result highly inaccurate).Moreover, the underlying operations in a digital multiplier arefull adders, whereas a stochasticcomputer only requires an AND gate. Additionally,a digital multiplier would naively require  input wires,whereas a stochastic multiplier would only require 2 input wires. (If the digital multiplier serialized its output, however, it would alsorequire only 2 input wires. )Additionally, stochastic computing is robust against noise; if a fewbits in a stream are flipped, those errors will have no significant impacton the solution.Furthermore, stochastic computing elements can tolerate skew in the arrival time of the inputs.Circuits work properly even when the inputs are misaligned temporally. As a result, stochasticsystems can be designed to work with inexpensive locally generated clocks instead of using a global clock and an expensive clock distribution network.Finally, stochastic computing provides an estimate of the solutionthat grows more accurate as we extend the bit stream. In particular,it provides a rough estimate very rapidly. This property is usually referred to as ''progressive precision'', which suggests that the precisionof stochastic numbers (bit streams) increases as computation proceeds.It is as if the most significant bits of the numberarrive before its least significant bits; unlike theconventional arithmetic circuits where the mostsignificant bits usually arrive last. In someiterative systems the partial solutions obtained through progressive precision can provide faster feedbackthan through traditional computing methods, leading to fasterconvergence.=== Weaknesses ===Stochastic computing is, by its very nature, random. When we examinea random bit stream and try to reconstruct the underlying value, the effective precisioncan be measured by the variance of our sample. In the example above, the digital multipliercomputes a number to  bits of accuracy, so theprecision is . If we are using a random bitstream to estimate a number and want the standard deviation of ourestimate of the solution to be at least , wewould need  samples. This represents anexponential increase in work. In certain applications, however, theprogressive precision property of stochastic computing can be exploitedto compensate this exponential loss.Second, stochastic computing requires a method of generating randombiased bit streams. In practice, these streams are generated withpseudo-random number generators. Unfortunately, generating(pseudo-)random bits is fairly costly (compared to the expense of,e.g., a full adder). Therefore, the gate-level advantage ofstochastic computing is typically lost.Third, the analysis of stochastic computing assumes that the bitstreams are independent (uncorrelated). If this assumption does nothold, stochastic computing can fail dramatically. For instance, if wetry to compute  by multiplying a bit stream for by itself, the process fails: since , the stochastic computation would yield , which is not generally true (unless 0 or 1).In systems with feedback, the problem of decorrelation can manifest inmore complicated ways. Systems of stochastic processors are prone to''latching'', where feedback between different components can achievea deadlocked state.A great deal of effort must be spent decorrelating the system toattempt to remediate latching.Fourth, although some digital functions have very simple stochasticcounterparts (such as the translation between multiplication and theAND gate), many do not. Trying to express these functions stochasticallymay cause various pathologies. For instance, stochastic decoding requiresthe computation of the function .There is no single bit operation that can compute this function; the usual solutioninvolves producing correlated output bits, which, as we have seen above, can causea host of problems.Other functions (such as the averaging operator  require either stream decimation or inflation. Tradeoffs between precision and memorycan be challenging.
Stochastic decoding Although stochastic computing has a number of defects when consideredas a method of general computation, there are certain applicationsthat highlight its strengths. One notable case occurs in thedecoding of certain error correcting codes.In developments unrelated to stochastic computing, highly effectivemethods of decoding LDPC codes usingthe belief propagation algorithm weredeveloped. Belief propagation in this context involves iterativelyreestimating certain parameters using two basic operations(essentially, a probabilistic XOR operation and an averagingoperation).In 2003, researchers realized that these two operations could bemodeled very simply with stochastic computing.Moreover, since thebelief propagation algorithm is iterative, stochastic computing provides partialsolutions that may lead to faster convergence.Hardware implementations of stochastic decoders have been built on FPGAs.The proponents of these methods argue that the performance of stochastic decoding iscompetitive with digital alternatives.
Deterministic Methods to Stochastic Computing Deterministic methods of SC has been developed to perform completely accurate computation with SC circuits. The essential principle of these methods is that every bit of one bit-streams interacts with every bit of the other bit-streams exactly once. To produce completely accurate result with these methods, the operation must run for the product of the length of input bit-streams. Deterministic methods are developed based on unary bit-streams, pseudo-random bit-streams, and low-discrepancy bit-streams.
Variants of stochastic computing There are a number of variants of the basic stochastic computingparadigm. Further information can be found in the referenced book byMars and Poppelbaum. ''Bundle Processing'' involves sending a fixed number ofbits instead of a stream. One of the advantages of this approach isthat the precision is improved. To see why, suppose we transmit bits. In regular stochastic computing, we canrepresent a precision of roughly  differentvalues, because of the variance of the estimate. In bundleprocessing, we can represent a precision of .However, bundle processing retains the same robustness to error ofregular stochastic processing. ''Ergodic Processing'' involves sending a stream of bundles, whichcaptures the benefits of regular stochastic and bundle processing. ''Burst Processing'' encodes a number by a higher base increasingstream. For instance, we would encode 4.3 with ten decimal digits as::: 4444444555since the average value of the preceding stream is 4.3.Thisrepresentation offers various advantages: there is no randomizationsince the numbers appear in increasing order,so the PRNG issues are avoided, but many of the advantages ofstochastic computing are retained (such as partial estimates of thesolution). Additionally, it retains the linear precision of bundleand ergodic processing.
See also * Unconventional computing
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