Introduction Neural Nets as Approximators Implementation Evaluation Summary Neural Network-Based Accelerators for Transcendental Function Approximation Schuyler Eldridge ∗ Florian Raudies † David Zou ∗ Ajay Joshi ∗ ∗ Department of Electrical and Computer Engineering, Boston University † Center for Computational Neuroscience and Neural Technology, Boston University schuye@bu.edu May 22, 2014 This work was supported by a NASA Office of the Chief Technologist’s Space Technology Research Fellowship. Eldridge, Raudies, Zou, and Joshi Boston University 1/19
Introduction Neural Nets as Approximators Implementation Library-Level Approximation Overview Evaluation Summary Technology Scaling Trends Figure 1: Trends in CMOS technology [Moore et al., 2011 Salishan ] Eldridge, Raudies, Zou, and Joshi Boston University 2/19
Introduction Neural Nets as Approximators Implementation Library-Level Approximation Overview Evaluation Summary Accelerators to the Rescue? Energy Efficient Accelerators... Lessen the utilization crunch of Dark Silicon Are cheap due to plentiful transistor counts Are typically special-purpose Approaches to General Purpose Acceleration QsCores – Dedicated hardware for frequent code patterns [Venkatesh et al., 2011 MICRO ] NPU – Neural network-based approximation of code regions [Esmaeilzadeh et al., 2012 MICRO ] Eldridge, Raudies, Zou, and Joshi Boston University 3/19
Introduction Neural Nets as Approximators Implementation Library-Level Approximation Overview Evaluation Summary Neural Networks (NNs) as General-Purpose Accelerators The good and the bad... NNs are general-purpose approximators [Cybenko, 1989 Math. Control Signal , Hornik, 1991 Neural Networks ] But... NNs are still approximate Approximation may be acceptable Modern recognition, mining, and synthesis (RMS) benchmarks are robust [Chippa et al., 2013 DAC ] Eldridge, Raudies, Zou, and Joshi Boston University 4/19
Introduction Neural Nets as Approximators Implementation Library-Level Approximation Overview Evaluation Summary Library-Level Approximation with NN-Based Accelerators Big Idea Use NNs to approximate library-level functions cos, exp, log, pow, and sin Explore the design space of NN topologies Define and use an energy–delay–error product (EDEP) metric Evaluate energy–performance improvements Use an energy–delay product (EDP) metric Evaluate accuracy of... NN-based accelerators vs. a traditional approach Applications using NN-based accelerators Eldridge, Raudies, Zou, and Joshi Boston University 5/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Multilayer Perceptron (MLP) NN Primer Y 1 Y o y Output O 1 O o � n . . . � φ � x k w k k =1 w 1 w 2 w 3 w n . . . x 1 x 2 x 3 x n Hidden Figure 3: One neuron H 1 H 2 H h bias . . . Equations � n � � y = φ x k w k Input I 1 I i bias . . . k =1 1 X 1 X i φ sigmoid = 1 + e − 2 x φ linear = x Figure 2: NN with i × h × o nodes. Eldridge, Raudies, Zou, and Joshi Boston University 6/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary NN-Based Approximation Requires Input–Output Scaling Approximating Unbounded Functions on Bounded Domains NNs cannot handle unbounded inputs Input–output scaling can extend the effective domain and range of the approximated function This approach is suitable when... A small region is representative of the whole function There exist easy a operations to scale inputs and outputs Specifically, we use the CORDIC [Volder, 1959 IRE Tran. Comput. ] scalings identified by Walther [Walther, 1971 AFIPS ] a By “easy”, I mean multiplication with a constant, addition, bitshifts, and rounding. Eldridge, Raudies, Zou, and Joshi Boston University 7/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 Other Domains Scaled to Neural Network Domain 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function Neural Network Domain 4 3 Scale outputs onto full range 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function 4 3 Scale outputs x = q log 2 − d ˆ onto full range 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function 4 3 Scale outputs x = q log 2 − d ˆ onto full range − d 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function 4 3 Scale outputs x = q log 2 − d ˆ onto full range exp NN ( − d ) 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function 4 3 Scale outputs x = q log 2 − d ˆ onto full range 2 q exp NN ( − d ) 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators NN Overview Implementation NN-Based Input–Output Scaling Evaluation Summary Walther’s Scaling Approach [Walther, 1971 AFIPS ] for exp x y = exp x Scaling Steps 8 exp NN (ˆ x ) 1 Scale inputs onto exp( q log 2 − d ) = 2 q exp( − d ) NN domain x q = ⌊ log 2 + 1 ⌋ d = x − q log 2 2 NN approximates function 4 3 Scale outputs x = q log 2 − d ˆ onto full range 2 Similar Scalings Exist 1 cos x and sin x x log x − 3 log 2 − log 2 log 2 3 log 2 Figure 4: Graphical scaling for exp x Eldridge, Raudies, Zou, and Joshi Boston University 8/19
Introduction Neural Nets as Approximators Architecture Implementation Design Space Exploration Evaluation Summary Fixed Point Accelerator Architecture for 1 × 3 × 1 NN Figure 5: Block diagram of an NN-based accelerator architecture Eldridge, Raudies, Zou, and Joshi Boston University 9/19
Introduction Neural Nets as Approximators Architecture Implementation Design Space Exploration Evaluation Summary NN Topology Evaluation – Design Space Exploration NN Evaluation Criteria Candidate NN Topologies Energy Fixed point Performance 1–15 hidden nodes Accuracy 6–10 fractional bits Energy–Delay–Error Product (EDEP) Optimal NN topology minimizes EDEP EDEP = energy × latency in cycles × mean squared error frequency Eldridge, Raudies, Zou, and Joshi Boston University 10/19
Introduction Neural Nets as Approximators Architecture Implementation Design Space Exploration Evaluation Summary NN Topology Evaluation – Results Table 1: MSE and energy consumption of the NN-based accelerator implementation of transcendental functions. Func. NN MSE ( × 10 − 4 ) Energy (pJ) Area (um 2 ) Freq. (MHz) cos 9 8 1300 340 h1 b6 sin 7 8 1300 340 h1 b6 exp 2 25 3600 340 h3 b7 log 1 25 3600 340 h3 b7 pow 432 102 3600 340 h3 b7 Evaluation Notes Evaluated with a 45nm predictive technology model (PTM) pow computed using exp and log: a b = exp ( b log a ) Eldridge, Raudies, Zou, and Joshi Boston University 11/19
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