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Randomized Accuracy Aware Program Transformations for Efficient Approximate Computations Sasa Misailovic Joint work with Zeyuan Allen ZhuJonathan KelnerMartin Rinard MIT CSAIL

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Nodes represent computation Edges represent flow of data

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Functions process individual data Reduction nodes aggregate data

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min avg Functions process individual data Reduction nodes aggregate data

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min avg f2f2 f3f3 f1f1

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min avg

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Sampling inputs of reduction nodes Reductions consume fewer inputs min avg

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min avg Sampling inputs of reduction nodes Reductions consume fewer inputs

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Tradeoff Space Error Time

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Tradeoff Space Error Time

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Optimal Tradeoff Curve Error Time Using the tradeoff curve: Minimize time subject to error bound Minimize error subject to time bound

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Our Result AnalysisOptimization Original program Error bound Optimized program Transformations

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Outline Model of Computation Tradeoff Curve Construction Optimized Program Selection Related Work

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Model of Computation ff t uv w t uv w ffgg gg min avg

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Model of Computation ff t uv w t uv w ffgg gg min avg min 1 n n n f g t uv w avg mm

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Structure of Computation Computation nodes DAGs of functions Functions: arbitrary code Process individual inputs Reduction nodes Aggregation functions Average, min, max, sum Computation Tree Computation nodes and reduction nodes min 1 n n n f g t uv w avg mm

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min Accuracy-Aware Transformations 1 n n n avg mm

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min Accuracy-Aware Transformations 1 n n n avg mm

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min Accuracy-Aware Transformations 1 n n n avg mm

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min Accuracy-Aware Transformations 1 n n n avg mm

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min Program Configuration Vector Defines transformed program Functions: probability of executing each version Reductions: number of elements to sample 1 n n n avg mm

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min Configuration Vector Specifies program version Functions: probability of executing each version Reductions: number of elements to sample 1 n n n avg mm Find optimal program = Find configuration vector that achieves optimal accuracy vs. performance tradeoff Find optimal program = Find configuration vector that achieves optimal accuracy vs. performance tradeoff

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min 1 n n n avg mm Tradeoff Curve Construction: Algorithm

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min Tradeoff Curve Construction: Algorithm 1 n n n avg mm

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Tradeoff Curve Construction: Algorithm 1 mm n n n min avg

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Tradeoff Curve Construction: Algorithm 1 mm n n n min avg

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Tradeoff Curve Construction: Algorithm 1 n n n min avg m

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Tradeoff Curve Construction: Algorithm 1 n n n min

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Tradeoff Curve Construction: Algorithm 1 n n n min

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Tradeoff Curve Construction: Algorithm 1 n n n min

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Tradeoff Curve Construction: Algorithm 1 n min

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Tradeoff Curve Construction: Algorithm

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Computation Node Optimization (E 2,T 2 ) (E 0,T 0 )(E 1,T 1 ) Linear program

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Computation Node Optimization (E 2,T 2 ) (E 0,T 0 )(E 1,T 1 ) Linear program

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Computation Node Optimization (E 2,T 2 ) (E 0,T 0 )(E 1,T 1 ) Linear program

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Computation Node Optimization (E 2,T 2 ) (E 0,T 0 )(E 1,T 1 ) Linear program

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The Algorithm: Reduction Nodes m avg

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The Algorithm: Reduction Nodes m avg

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The Algorithm: Reduction Nodes m avg

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The Algorithm: Reduction Nodes m avg

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Approximate Tradeoff Curve Error Time Bidimensional Discretization

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Approximating Tradeoff Curve Error Time Bidimensional Discretization

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Approximating Tradeoff Curve Error Time

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Approximating Tradeoff Curve Error Time

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Approximating Tradeoff Curve Error Time

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Approximating Tradeoff Curve Error Time

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Properties of the Algorithm

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Obtaining Optimized Programs

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Related Work Accuracy-aware transformations Empirical justification: training/test input set [Rinard ICS 06, Rinard OOPSLA 07, Ansel et al. PLDI 09, Misailovic et al. ICSE 10, Baek & Chilimbi PLDI 10 Hoffmann et al. ASPLOS 11, Sidiroglou et al. FSE 11] Probabilistic accuracy analysis for loop perforation [Misailovic et al. SAS 11, Chaudhuri et al. FSE 11]

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Related Work Accuracy-aware transformations Empirical justification: training/test input set [Rinard ICS 06, Rinard OOPSLA 07, Ansel et al. PLDI 09, Misailovic et al. ICSE 10, Baek & Chilimbi PLDI 10 Hoffmann et al. ASPLOS 11, Sidiroglou et al. FSE 11] Probabilistic accuracy analysis for loop perforation [Misailovic et al. SAS 11, Chaudhuri et al. FSE 11] Ensuring safety of transformed programs Separating critical and approximate parts of program [Carbin & Rinard ISSTA 10, Sampson et al. PLDI 11] Verifying relaxed semantics of programs [Carbin et al. CSAIL-TR 11] Analytic properties of programs [Majumdar & Saha RTSS 09, Chaudhuri et al. POPL10, Ivancic et al. MEMOCODE 10, Reed & Pierce ICFP 10, Chaudhuri & Solar-Lezama PLDI 10, Chaudhuri et al. FSE 11]

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Summary Model of Computation Accuracy-aware program transformations Effects on overall accuracy and execution time Explore and Exploit Optimal Tradeoffs Approximate optimal tradeoff curve construction Polynomial, dynamic programming algorithm Randomized program configurations to achieve tradeoffs Envisioned Applications Image and video processing, numerical algorithms, queries on big data sets, machine learning, Optimization, fault tolerance, dynamic adaptation

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