Programming Abstractions for Approximate Computing • Michael Carbin • with SasaMisailovic, Hank Hoffmann, DeokhwanKim, SteliosSidiroglou, Martin Rinard • MIT
Challenges for Programming • Expression (specifying approximations) • Reasoning (verifying resulting program) • Debugging (reproducing failures) • Deployment (changes in assumptions)
Challenges for Programming • Expression (specifying approximations) • Reasoning (verifying resulting program) • Debugging (reproducing failures) • Deployment (changes in assumptions)
Fundamental Questions • Scope • What do we approximate and how? • Specifications of “Correctness” • What is correctness? • Reasoning • How do we reason about correctness?
[Misailovic, Carbin, Achour, Qi, Rinard MIT-TR 14] What do we approximate? • Domains have inherent uncertainty • Execution time dominated by a fraction of code
Approximation Transformations • Approximate HardwarePCMOS, Palem et al. 2005; Narayanan et al., DATE ’10; Liu et al. ASPLOS ’11; Sampson et al, PLDI ’11; Esmaeilzadeh et al. , ASPLOS ’12, MICRO’ 12 • Function Substitution Hoffman et al., APLOS ’11; Ansel et al., CGO ’11; Zhu et al., POPL ‘12 • Approximate Memoization Alvarez et al., IEEE TOC ’05; Chaudhuri et al., FSE ’12; Samadi et al., ASPLOS ’14 • Relaxed Synchronization (Lock Elision) Renganarayanaet al., RACES ’12; Rinard, HotPar ‘13; Misailovic, et al., RACES ’12 • Code PerforationRinard, ICS ‘06; Baek et al., PLDI 10; Misailovic et al., ICSE ’10; Sidiroglou et al., FSE ‘11; Misailovic et al., SAS ‘11; Zhu et al., POPL ‘12; Carbin et al. PEPM ’13;
Example: Loop Perforation floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n;
Example: Loop Perforation floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n; float avg = (sum * 2) / n; • Skip iterations (or truncate or random subset) • Potentially add extrapolations to reduce bias i += 2
What is correctness? .c .c Traditional Transformation ≡
What is correctness? .c .c Approximate Transformation
What is correctness? • Safety: satisfies standard unary assertions (type safety, memory safety, partial functionality) • Accuracy: program satisfies relational assertions that constrain difference in results
Relational Assertions • Contribution: program logic and verification system that supports verifying relationships between implementations • x<o>: value of x in original implementation • x<a>: value of x in approximate implementation relate |x<o> - x<a>| / x<o> <= .1; [Carbin, Kim, Misailovic, RinardPLDI’12, PEPM ‘13]
Example: Loop Perforation floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n; float avg = (sum * 2) / n; i += 2
Example: Loop Perforation • Worst-case error: floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n; float avg = (sum * 2) / n; i += 2
Novel Safety Verification Concept • Assume validity of assertions in original program • Use relations to prove that approximation does not change validity of assertions • Lowerverification complexity than verifying an assertion outright p<o> == p<a> ∧ safe(*p<o>) safe(*p<a>) assert (safe(*p))
Specifications and Reasoning • Worst-case (for all inputs) • Non-interference [Sampson et al., PLDI ‘11] • Assertions [Carbin et al., PLDI ‘12; PEPM ‘13] • Accuracy [Carbin et al., PLDI ‘12] • Statistical (with some probability) • Assertions [Sampson et al., PLDI ‘14] • Reliability [Carbin, Misailovic, and Rinard, OOPSLA ‘13] • Expected Error [Zhu, Misailovicet al., POPL ‘13] • Probabilistic Error Bounds [Misailovic et al., SAS ‘13]
Questions from Computer Science
Question #1: • “Programmers will never do this.” • - Unnamed Systems and PL Researchers
Challenge Full FunctionalCorrectness Partial Specifications Reasoning Complexity Types Expressivity
Challenge • Ordering of unary assertions still true (improved by relational verification) Full FunctionalCorrectness Partial Specifications Reasoning Complexity Types Expressivity
Challenge Probabilistic Accuracy Bounds Probabilistic Assertions Reliability Worst-case Accuracy Assertions Expected Error Error Distributions Reasoning Complexity Expressivity Performance/Energy Benefit
Example: Loop Perforation(Misailovic et al., SAS ‘11) floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n; float avg = (sum * 2) / n; i += 2
Example: Loop Perforation(Misailovic et al., SAS ‘11) • Worst-case error: • Probabilistic Error Bound (.95): floatsum= 0; for (inti = 0; i < n; i += 1) { sum = sum+a[i]; } float avg= sum / n; float avg = (sum * 2) / n; i += 2
Question #2: • “There’s no hope for building approximate hardware.” • - Unnamed Computer Architect
Challenge • Approx. PL community must work with hardware community to collaborate on models (not the PL community’s expertise) • Approx. hardware community faces major challenge in publishing deep architectural changes: simulation widely panned • Moving forward may require large joint effort
Question #3: • “I believe your techniques are fundamentally flawed.” • - Unnamed Numerical Analyst
Challenge • Numerical analysts have been wronged • Programming systems have failed to provide support for making the statements they desire • Approximate computing community risks repeating work done by numerical analysis • Opportunity for collaboration • New research opportunities • It’s no longer just about floating-point
Broadly Accessible Motivations • Programming with uncertainty (uncertain operations and data) • Programming unrealizable computation (large scale numerical simulations) • Useful computation from non-digital fabrics(analog, quantum, biological, and human) Opportunity to build reliable and resilient computing systems built upon anything
End
Conclusion • Adoption hinges on tradeoff between expressivity, complexity and benefit • Opportunity/necessity for tighter integration of PL and hardware communities
Open Challenges and Directions • Problem: benefits (performance/energy) may vary between different types of guarantees Probabilistic Accuracy Bounds Worst-case Accuracy Probabilistic Assertions Reasoning Complexity Reliability Error Distributions Expected Accuracy Expressivity
Open Challenges and Directions Expressivity and Reasoning Complexity Type Checking Partial Specifications Full FunctionalCorrectness
Experimental Results
Open Challenges Directions • Traditional Tradeoff Dynamic Analysis Type Checking Partial Specifications Full FunctionalCorrectness Performance/Energy Reasoning Complexity
