Probabilistically Accurate Program Transformations

Abstract

Abstract. The standard approach to program transformation involves the use of discrete logical reasoning to prove that the transformation does not change the observable semantics of the program. We propose a new approach that, in contrast, uses probabilistic reasoning to justify the application of transformations that may change, within probabilistic accuracy bounds, the result that the program produces. Our new approach produces probabilistic guarantees of the form P(|D| ≥ B) ≤ , ∈ (0, 1), where D is the difference between the results that the transformed and original programs produce, B is an acceptability bound on the absolute value of D, and is the maximum acceptable probability of observing large |D|. We show how to use our approach to justify the application of loop perforation (which transforms loops to execute fewer iterations) to a set of computational patterns.

Managing performance vs. accuracy trade-offs with loop perforation

“…We have identified local computational patterns that interact well with loop perforation [24,32,25]. Examples include the Sum pattern (which computes the sum of a set of numbers) and the Argmin pattern (which computes the index and value of a minimum array element).…”

“…Examples include the Sum pattern (which computes the sum of a set of numbers) and the Argmin pattern (which computes the index and value of a minimum array element). An analysis of these patterns delivers a probabilistic guarantee that (under appropriate assumptions) the perforated computation is highly likely to produce a result that is close to the result that the original computation would have produced [24,25].…”

“…The second column presents the percentage of time spent in each loop (before perforation). The third column presents both the global (as discussed in Section 1) and local (as presented in [25]) computation patterns for the loop. For example, the entry for the first loop in x264 is SSE/Argmin, which indicates that the global pattern for the loop is the Search Space Enumeration pattern, while the local pattern is the Argmin pattern.…”

“…All of the tunable loops in x264 appear in the motion estimation computation. Two of these loops (x264_mb_analyze_inter_p16x16 and x264_search_ref) are instances of the Search Space Enumeration global pattern and the Argmin local pattern (which computes the index and value of the minimum element in an array of elements [25]). The first (x264_mb_analyze_inter_p16x16) searches previously encoded reference frames to find a block that is a good match for the block that x264 is currently encoding.…”

“…All of the remaining loops are instances of the Search Metric for both Selection and Termination global pattern and the Sum local pattern (which computes the sum of a set of numbers [25]). These loops all compute a metric that measures how well the current block matches a previously encoded block.…”