Report

Programming Abstractions for Approximate Computing Michael Carbin with Sasa Misailovic, Hank Hoffmann, Deokhwan Kim, Stelios Sidiroglou, 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 Benchmark Domain LOC LOC (Kernel) Time in Kernel (%) sor numerical 173 23 82.30% blackscholes finance 494 88 65.11% 532 62 99.20% 532 93 98.86% 218 88 93.43% dct idct scale image processing • Execution time dominated by a fraction of code Approximation Transformations • Approximate Hardware PCMOS, 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) Renganarayana et al., RACES ’12; Rinard, HotPar ‘13; Misailovic, et al., RACES ’12 • Code Perforation Rinard, 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 float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; } float avg = sum / n; Example: Loop Perforation float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; i += 2 } float avg = sum / n; float avg = (sum * 2) / n; • Skip iterations (or truncate or random subset) • Potentially add extrapolations to reduce bias 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 relate |x<o> - x<a>| / x<o> <= .1; • x<o>: value of x in original implementation • x<a>: value of x in approximate implementation [Carbin, Kim, Misailovic, Rinard PLDI’12, PEPM ‘13] Example: Loop Perforation float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; i += 2 } float avg = sum / n; float avg = (sum * 2) / n; Example: Loop Perforation float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; i += 2 } float avg = sum / n; float avg = (sum * 2) / n; b-a • Worst-case error: 2 Novel Safety Verification Concept • Assume validity of assertions in original program • Use relations to prove that approximation does not change validity of assertions p<o> == p<a> ∧ safe(*p<o>) ⊨ safe(*p<a>) assert (safe(*p)) • Lower verification complexity than verifying an assertion outright 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, Misailovic et 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 Reasoning Complexity Challenge Partial Specifications Types Expressivity Full Functional Correctness Reasoning Complexity Challenge Partial Specifications Full Functional Correctness Types Expressivity • Ordering of unary assertions still true (improved by relational verification) Challenge Probabilistic Accuracy Bounds Probabilistic Assertions Reliability Expected Error Reasoning Complexity Assertions Expressivity Performance/Energy Benefit Worst-case Accuracy Error Distributions Example: Loop Perforation (Misailovic et al., SAS ‘11) float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; i += 2 } float avg = sum / n; float avg = (sum * 2) / n; Example: Loop Perforation (Misailovic et al., SAS ‘11) float sum = 0; for (int i = 0; i < n; i += 1) { sum = sum + a[i]; i += 2 } float avg = sum / n; float avg = (sum * 2) / n; b-a • Worst-case error: 2 0.6 *(b - a) • Probabilistic Error Bound (.95): n 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 Reasoning Complexity Open Challenges and Directions Probabilistic Accuracy Bounds Worst-case Accuracy Probabilistic Assertions Reliability Expected Accuracy Error Distributions Expressivity Problem: benefits (performance/energy) may vary between different types of guarantees Open Challenges and Directions Expressivity and Reasoning Complexity Type Checking Partial Specifications Full Functional Correctness Experimental Results Benchmark LOC LOC (Kernel) Time in Kernel (%) sor 23 173 82.30% blackscholes 88 494 65.11% dct 62 532 99.20% idct 93 532 98.86% scale 88 218 93.43% Open Challenges Directions • Traditional Tradeoff Dynamic Analysis Type Checking Partial Specifications Full Functional Correctness Performance/Energy Reasoning Complexity