confirmation of exponential speedup - using memcomputing ... · solve complex optimization...

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Confirmation of Exponential Speed-up Using MemComputing to Solve

Hard Optimization Problems

White Paper

White PaperConfirmation of Exponential Speed-up Using MemComputing to Solve Hard Optimization Problems

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Introduction

Self-Organizing Logic Gates Nonlocal Collective State Computation Scalability in Linear Time

MemComputing Advantage

• The Challenge• The Procedure• The Result

Demonstrated Performance

Conclusion

5 References

Optimization is a core component of every data-based scientific discipline and industry.However, the explosive growth of Big Data combined with classical computing architecturecan cause significant issues in solving optimization problems. The difficulty of computation isespecially prominent in a set of problems classified as NP-hard. Currently, no algorithmexists for NP-hard problems that is able to find a solution in polynomial time.

Furthermore, even reaching approximations to an optimal solution is challenging to discover for the most difficult optimization problems. This is because as the number of inputs and constraints increase linearly, the computational time needed to find an approximation to the optimal solution can grow exponentially. It is common in industry for a single problem to take days, even weeks, to run.

EXECUTIVE SUMMARY

© Copyright MemComputing, Inc. 2019

INTRODUCTION

For industrial applications involving autonomousvehicles, delivery routing, traffic control, circuitdesign, drug discovery, and targeted clinical therapy(to name a few), improving the accuracy ofapproximations to recurring optimization problems ata fraction of the current computation time wouldresult in significant improvements in operationalefficiencies and related costs. It is not unreasonableto expect millions, if not hundreds of millions ofdollars in cost savings or improved revenuesannually as well as improved operationalperformance.

Currently, strategies for improving accuracy andovercoming this computational challenge can beseparated into four distinct categories. Each,however, presents shortcomings in its approach:

● Heuristic algorithm research anddevelopment have historically beenresponsible for reducing the time required tosolve complex optimization problems. Whilethe implementation of heuristic algorithms isthe most widely accepted solution acrossacademia and commercial industries, thechallenges facing heuristics (namelyexponentially increasing computation time anddifficulty refining candidate solutions) can limittheir performance.

• High Performance Computing (HPC) can beapplied to heuristics that can be broken downinto a set of smaller problems that can besolved simultaneously and then merged into acomposite solution. Exponential growth of runtimes is not eliminated but the speed at whichthe approximations can be calculated can beimproved significantly. A downside is that theaccuracy of the approximation can be reducedsince the problem isn’t solved as a whole.

● Cloud computing allows users to tap intoshared system resources provided via theinternet. This allows organizations tooutsource the burden of maintaining acomputer infrastructure and potentially spreadthe computational burden across manyprocessors. Even though cloud computing cansupplement an organization’s processingpower, it does not resolve the reliance onheuristic algorithms nor overcome certainexponential requirements to solve problems.

● Quantum computing in theory presents theability to store and process huge amounts ofinformation through quantum bits that canexist in a superimposition of 0s and 1s.However, there are significant physics,engineering, and scaling challenges hinderingthe realization of fully functional quantumcomputers. In fact, using quantum computersto solve optimization problems with more thana few dozen variables has shown to be verydifficult. While it isn’t possible to predictexactly when and if these breakthroughs willoccur, many estimates assume availability isat least 5-10 years away.

MemComputing is not a heuristic approach. Itintroduces a new paradigm where these problemsare converted from a combinatorial problem into aphysics problem. Solving the physical problemavoids the exponential growth associated withrelated heuristic approaches. This approach allowsMemComputing to provide performanceenhancements that are dramatically faster thanHPC, Cloud Computing and Quantum Computing.

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MEMCOMPUTING ADVANTAGE (1/2)

The unprecedented performance of MemComputingtechnology is based on the pioneering theoreticalconcept of universal memcomputing machines(UMMs). UMMs are a class of scalablememcomputing machines built with interconnectedmemory units (memprocessors) capable ofperforming computation.

UMMs are practically realized in the form of digitalmemcomputing machines (DMMs). DMMsharness the power of self-organizing logic circuits(SOLCs) which are differentiated from traditionalcircuits through the unique properties of the self-organizing logic gates (SOLGs) used in theirconstruction. These SOLGs, in turn, can be realizedin hardware with available technology (non-quantum) or simulated efficiently in software.

The proprietary components of the MemComputingsystem enable a non-combinatorial approach tosolve combinatorial optimization problems. Thus, itremoves the wide-spread dependency on heuristicalgorithms and stochastic local searches. Thefeatures of SOLGs, the foundational components ofMemComputing capabilities, provide the ultimatecomputational technology that is cost-effective,reduces computation time by orders of magnitude,and increases the quality of the solution to complexoptimization problems.

Self-Organizing Logic Gates

Logic gates are the building blocks for digital circuits.Traditional logic gates are essentially miniatureelectric circuits that receive incoming electricalcurrents (inputs) and send an outgoing electricalcurrent (output) based on what came in. The role oflogic gates is to perform a logical operation (e.g.,AND, OR, NOT) on its one or more binary inputs to

produce its singular binary output. Unlikeconventional logic gates, SOLGs can receive signalsfrom both the traditional input and output terminals.In other words, SOLGs are “terminal-agnostic” andmulti-directional. They accomplish this by self-organizing into their own logical proposition as wellas in logical relations to another gate. For example,if a SOLG represents the logical operation OR it hasan internal mechanism that propels its terminalstates to fulfill the relationship defined byx₀= x₁V x₂ (x₀ being the state of the traditionaloutput terminal and x₁, x₂ being the states of thetraditional input terminals). Then, setting x₀ willinitiate the logic gate’s self-organization to producex₁ and x₂ states consistent with x₀ and the truth tableof an OR gate. The ability to set the state of theoutput terminal is not available in traditional logicgates.

As previously mentioned, SOLGs can be realizedthrough standard, non-quantum electroniccomponents. This capability is due to its dynamiccorrection module which is designed to correctinconsistent logic gate configurations. The errorcorrection module reads voltages at the logic gate’sterminals and injects current when the gate is in aninconsistent configuration.

When SOLGs replace traditional logic gates, a self-organizing logic circuit (SOLC) is created. Thedynamic, collective self-organization of all theSOLGs in the circuit allows SOLCs to solve complexoptimization problems from any state selected atrandom by evolving to converge into equilibriumpoints. The equilibria represent approximations thatcome closer to the global optimum than current bestsolutions.

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Nonlocal Collective State Computation

The most significant feature of SOLGs are theirmanifested long-range order. Long-range orderdescribes physical systems which demonstratecorrelated behavior across remote particles. In otherwords, systems with long-range order containcomponents that correspond to the states of othercomponents regardless of distance. Thissimultaneous collective responsiveness of individualparts describes the temporal and spatial non-localityof the system.

The capability of SOLGs to realize long-range orderis due to the existence of instantons. Instantonsconnect topologically inequivalent critical points inthe phase space. They are the classical analogue ofquantum tunneling. Instantons create non-locality inthe system which generates the collective, dynamicbehavior of SOLGs to correlate at an arbitrarydistance. In effect, this collective behavior allowsSOLGs to efficiently adapt their truth value to satisfythe logical proposition of another gate withoutviolating their own internal logical proposition. Thenonlocality of SOLGs allows for simultaneousvariable flips which is a necessary task thatcombinatorial approaches cannot accomplish oncethey reach a certain number of satisfied constraints.

It is precisely the long-range order of SOLCs thatproduces computation acceleration by orders ofmagnitude. As discussed in greater detail in the nextsection on the demonstrated performance ofFalcon©, the system converges quickly to theequilibrium points which represent current closestapproximations to the global optimum for complexoptimization problems.

Scalability in Linear Time

Another key feature of the MemComputing solutionis the ability of DMMs to scale polynomially andparticularly linearly as demonstrated for the problemsets that map efficiently to a Max SAT problemspace. It is important to note that the linear scaling isindependent of the input size because the number oflogic gates grows linearly with each step requiringonly a linear number of floating-point operations andlinearly growing memory. In other words, the numberof variables can be increased without an exponentialgrowth in computation time which resolves theprimary issue with conventional computationsolutions.

The configuration of DMMs outlined here is able tosupport infinite-range correlations in the infinite sizelimit. This support allows for an ideal scale-freebehavior of the SOLC in which the correlations donot decay. This was derived analytically usingtopological field theory.2


MEMCOMPUTING ADVANTAGE (2/2)2


DEMONSTRATED PERFORMANCE (1/2)

In order to demonstrate the extraordinaryperformance of MemComputing technology, wecompared Falcon© with the best solvers of the 2016MaxSAT competition. Max-SAT problems arenotoriously harder than traditional satisfiability (SAT)problems. Max-SAT problems cannot be satisfied.Instead, a solution must be found that identifies themaximum/largest number of clauses/constraints thatcan be satisfied.It is worth noting that while Falcon© was tested as asingle threaded interpreted MatLab application it wascompared to the best solvers from the MaxSATcompetition, CCLS and DeciLS. These solvers weredesigned using high level languages that arecompiled software solutions designed for thegreatest efficiency. Interpreted MatLab is notoriouslyslow when compared to compiled software.However, despite the anticipated performance gapsbetween the two, MemComputing offered betterapproximations in the least amount of timeregardless of the increasing number of variables.Further, Falcon© was shown to scale linearly vs.exponentially, which only becomes more evident asthe number of inputs increases.

The Challenge

Three types of problems tested (in order ofincreasing difficulty) were the random-Max-E3SAT,hyper-Max-E3SAT, and delta-Max-E3SAT. Twocompeting solvers were used as benchmarks forFalcon’s© performance: CCLS and DeciLS. Thesewere the best solvers in the 2016 Max-SATcompetition.

Since these problems are considered NP-hard, Max-SAT algorithms can experience an inapproximabilitygap which says that no approximation can surpass

a fraction of the global optimal solution withoutincurring exponential overhead. Particularly for Max-E3SAT, the best approximation, under the conditionsassociated with classical computer algorithms whereit is assumed that NP ≠ P, is that a maximum of ⅞ ofthe optimal number of constraints can be satisfied.3

The Procedure

Step 1: Construct a Boolean circuit that representsthe optimization problem. All optimization problemscan be written in Boolean format and any standardBoolean formula can be written in conjunctivenormal form (CNF)

Step 2: Replace traditional uni-directional Boolean gates with SOLGsStep 3: Feed the appropriate terminals with therequired output in CNF form (for instance, the logical1 if we are interested in checking its satisfiability)Step 4: The circuit is represented by non-linearordinary differential equations which can be solvednumerically to find the equilibrium points. Given thecollective behavior of the circuit, the equilibria comeextremely close to the global optimum

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The Result

With a threshold of 1.5% of unsatisfiable clauses, all three solvers were tested for the length of time itwould take to surpass the limit with an increasing number of clauses for the hardest cases, namely thoseof the delta-Max-E3SAT.

The exponential blowup of both CCLS and DeciLS is immediately apparent in Fig. 1. In comparison,Falcon’s non-combinatorial approach scaled linearly for up to 2 X 10⁶ variables which necessitated 10⁴seconds (a little more than a couple of hours) to reach the unsatisfiable clause limit at 1.5%. Had CCLSand DeciLS been required to compute 2 X 10⁶ variables, the computation time would, at best, run for10²⁵⁰⁰ seconds which is 10²⁴⁸⁰ times the estimated age of the universe.

FIG. 1. Time comparison between CCLS, DeciLS, and Falcon for the delta-Max-E3SAT problem witha set threshold of unsatisfiable clauses at 1.5%. The projected times for CCLS and DeciLS wereestimated using a linear regression of the Log₁₀(time) versus the number of variables.


DEMONSTRATED PERFORMANCE (2/2)3


CONCLUSION

In conclusion, we have shown empirical evidencethat a non-combinatorial approach utilizing DMMsto solve hard combinatorial optimization problemsexponentially outperforms heuristics designed tosolve such problems.

Specifically, compared to the best solvers in the2016 Max-SAT competition, DMMs:

• Found closer approximations to the globaloptimum

• Accelerated computation to a few hours (vs.the age of the universe) on a single thread ofan Intel Xeon E5-2680

• Scaled linearly in time and memory

It is important to note that the linear scalability ofDMMs does not prove the availability of polynomialsolutions to NP-hard problems. Nevertheless, theoutstanding performance of a physics-basedapproach to solving complex optimizationproblems demonstrates a promising trajectory foradvancements in optimization computation.

[1] Traversa, Fabio L., Pietro Cicotti, ForrestSheldon, and Massimiliano Di Ventra. Evidenceof an exponential speed-up in the solution of hardoptimization problems. arXiv preprintarXiv:1710.09278 (2017).

[2] Di Ventra, M., Traversa, F. L. & Ovchinnikov,I. V. Topological field theory and computing withinstantons. Ann. Phys. (Berlin) 1700123 (2017).

[3] Hastad, J. Some optimal inapproximabilityresults. Journal of the ACM 48, 798-859 (2001).

THE INFORMATION IN THIS DOCUMENT IS PROVIDED “AS IS” WITHOUT ANY WARRANTY,EXPRESS OR IMPLIED, INCLUDING WITHOUT ANY WARRANTIES OF MERCHANTABILITY, FITNESSFOR A PARTICULAR PURPOSE AND ANY WARRANTY OR CONDITION OF NON-INFRINGEMENT.

REFERENCES4 5


MemComputing Inc.4250 Executive Square, Suite 200,La Jolla, CA 92037

http://[email protected]

MemComputing, Inc.’s disruptive coprocessor technology is accelerating the time to findfeasible solutions to the most challenging operations research problems in all industries.Using physics principles, this novel software architecture is based on the logic and reasoningfunctions of the human brain.

MemComputing enables companies to analyze huge amounts of data and make informeddecisions quickly, bringing efficiencies to areas of operations research such as Big Dataanalytics, scheduling of resources, routing of vehicles, network and cellular traffic, geneticassembly and sequencing, portfolio optimization, drug discovery and oil and gas exploration.

The company was formed by serial entrepreneur, John Beane and the inventors, PhDPhysicists Massimiliano Di Ventra & Fabio Traversa


http://www.memcpu.com/

mailto:[email protected]

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