On the Arithmetical Complexity of Models

Binarized neural networks (BNNs) and batch normalization (BN) have already become typical techniques in artificial intelligence today. Unfortunately, the massive accumulation and multiplication in BNN models bring challenges to field-programmable gate array (FPGA) implementations, because complex arithmetics in BN consume too much computing resources. To relax FPGA resource limitations and speed up the computing process, we propose a BNN accelerator architecture based on consolidation compressed tree scheme by combining both XNOR and accumulation operation of the low bit into a systematic one. During the compression process, we adopt 0-padding (not ±1) to achieve no-accuracy-loss from software modeling to hardware implementation. Moreover, we introduce shift-addition-BN free binarization technique to shorten the delay path and optimize on-chip storage. To sum up, we drastically cut down the hardware consumption while maintaining great speed performance with the same model complexity as the previous design. We evaluate our accelerator on MNIST and CIFAR-10 dataset and implement the whole system on the ARTIX-7 100T FPGA with speed performance of 2052.65 GOP/s and area efficiency of 70.15 GOPS/KLUT.

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly r.e. families with finite teaching dimension, and (2) the class of uniformly r.e. families with finite positive recursive teaching dimension witnessed by a uniformly r.e. teaching sequence. We also derive the arithmetical complexity of several other decision problems in teaching, such as the problem of deciding, given an effective coding \(\{{\mathcal {L}}_0,{\mathcal {L}}_1,{\mathcal {L}}_2,\ldots \}\) of all uniformly r.e. families, any e such that \({\mathcal {L}}_e = \{L^e_0,L^e_1,\ldots ,\}\), any i and d, whether or not the teaching dimension of \(L^e_i\) with respect to \({\mathcal {L}}_e\) is upper bounded by d.