# Principled control of approximate programs

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## Abstract

In conventional computing, most programs are treated as implementations of mathematical functions for which there is an exact output that must computed from a given input. However, in many problem domains, it is sufficient to produce some approximation of this output. For example, when rendering a scene in graphics, it is acceptable to take computational short-cuts if human beings cannot tell the difference in the rendered scene. In other problem domains like machine learning, programs are often implementations of heuristic approaches to solving problems and therefore already compute approximate solutions to the original problem.

This is the key insight for the new research area, approximate computing, which attempts to trade-off such approximations against the cost of computational resources such as program execution time, energy consumption, and memory usage. We believe that approximate computing is an important step towards a more fundamental and comprehensive goal that we call information-efficiency. Current applications compute more information (bits) than are needed to produce their outputs, and since producing and transporting bits of information inside a computer requires energy/computation time/memory usage, information-inefficient computing leads directly to resources inefficiency.

Although there is now a fairly large literature on approximate computing, system researchers have focused mostly on what we can call the forward problem; that is, they have explored different ways in both hardware and software to introduce approximations in a program and have demonstrated that these approximations can enable significant execution speedups and energy savings with some quality degradation of the result. However, these efforts do not provide any guarantee on the amount of the quality degradation. Since the acceptable amount of degradation usually depends on the scenario in which the application is deployed, it is very important to be able to control the degree of approximation. In this dissertation, we refer to this problem as the inverse problem. Relatively little is known about how to solve the inverse problem in a disciplined way.

This dissertation makes two contributions towards solving the inverse problem. First, we investigate a large set of approximate algorithms from a variety of domains in order to understand how approximation is used in real-world applications. From this investigation, we determine that many approximate programs are tunable approximate programs. Tunable approximate programs have one or more parameters called knobs that can be changed to vary the quality of the output of the approximate computation as well as the corresponding cost. For example, an iterative linear equation solver can vary the number of iterations to trade quality of the solution versus the execution time, a Monte Carlo path tracer can change the number of sampling light paths to trade the quality of the resulting image against execution time, etc. Tunable approximate programs provide many opportunities for trading accuracy versus cost. By carefully analyzing these algorithms, we have found a set of patterns for how approximation is applied in tunable programs. Our classification can be used to identify new approximation opportunities in programs.

A second contribution of this dissertation is an approach to solving the inverse problem for tunable approximate programs. Concretely, the problem is to determine knob settings to minimize the cost while keeping the quality degradation within a given bound. There are four challenges: i) for real-world applications, the quality and cost are usually complex non-linear functions of the knobs and these functions are usually hard to express analytically; ii) the quality and the cost for an application vary greatly for different inputs; iii) when an acceptable quality degradation bound is presented, determining the knob setting has to be very efficient so that the extra overhead incurred by the identification will not exceed the cost saved by the approximation; and iv) the approach should be general so that it can be applied to many applications.

To meet these requirements, we formulate the inverse problem as a constrained optimization problem and solve it using a machine learning based approach. We build a system which uses machine learning techniques to learn cost and quality models for the program by profiling the program with a set of representative inputs. Then, when a quality degradation bound is presented, the system searches these error and cost models to identify the knob settings which can achieve the best cost savings while simultaneously guaranteeing the quality degradation bound statistically. We evaluate the system with a set of real world applications, including a social network graph partitioner, an image search engine, a 2-D graph layout engine, a 3-D game physics engine, a SVM solver and a radar signal processing engine. The experiments showed great savings in execution time and energy savings for a variety of quality bounds.