Mizera, Andrzej

Pang, Jun

Yuan, Qixia

Open Repository and Bibliography - Luxembourg

GPU-accelerated Steady-State Analysis ofProbabilistic Boolean NetworksAndrzej Mizera1, Jun Pang1,2, and Qixia Yuan1?1 Faculty of Science, Technology and CommunicationUniversity of Luxembourg, Luxembourg2 Interdisciplinary Centre for Security, Reliability and TrustUniversity of Luxembourg, Luxembourgfirstname.lastname@uni.luProblem statement. Steady-state computation is important for analysing bi-ological systems modelled as probabilistic Boolean networks (PBNs). Since thestate-space is exponential in the number of nodes, the use of statistical meth-ods and Monte Carlo methods remain the only feasible approach to address theproblem for large PBNs (e.g., with more than 50 nodes) [5, 2]. Such methodsusually rely on long simulations of a PBN; hence the simulation speed becomescritical. For large PBNs with high precision requirements, a slow simulationspeed becomes an obstacle of computing the steady-state probabilities. Intu-itively, parallelising the simulation process can be an ideal way to accelerate thecomputation process.Our approach. We propose to parallel the simulation of PBNs using multiplegraphics processing unit (GPU) cores. A GPU usually contains hundreds orthousands of cores. It uses data parallelism, i.e., the same instruction is runin different cores with different data. The memories provided by GPU can bedivided into two types based on the access speed: fast-memory and slow-memory.Accessing fast-memory is highly efficient, but the size of fast-memory is verylimited. A GPU program is executed in parallel by the GPU threads. Usuallythousands of threads are launched in parallel to hide latency. Due to the specificarchitecture of GPUs, parallelising a process in a GPU has to be treated carefully.A discussion of two particular issues follows.Firstly, synchronisation between cores is very time expensive in a GPU. Toavoid it, we let each GPU core handle all the nodes in a PBN. Instead of simu-lating one trajectory, we simulate multiple trajectories in parallel. Samples frommultiple trajectories can be combined to compute steady-state probabilities us-ing a combination of the two-state Markov chain approach with the Gelman &Rubin method [1, 3].Secondly, the performance of a GPU is highly related to how well the latencyis hidden. Latency can be hidden via the use of more threads, more blocks,and/or fast-memory. More threads/blocks require more fast-memory, but thesize of fast-memory is very limited. Therefore, a trade-off between the numberof threads/blocks and the use of fast-memory is required. We first optimize our? Supported by the National Research Fund, Luxembourg (grant 7814267).node # CPU time (s) GPU time (s) speedup100 635.27 1.84 345×200 424.18 1.84 231×500 1567.77 5.80 270×91 905.10 3.54 256×Table 1. GPU speedup for computing four steady-state probabilities.data structure to minimize the use of memory and then follow the rule that thefrequently accessed information should be put in fast-memory whenever possi-ble since the latency caused by accessing slow-memory is relatively large. Tobetter understand the optimization, we briefly review what a PBN is. A PBNis composed of a set of binary-valued nodes, each of which has a certain num-ber of Boolean functions. The process of simulating a PBN consists of select-ing a Boolean function for each node and updating its value in accordance theselected function (see [4, 6] for details). The state (value of all nodes) and theBoolean functions (stored as a truth table) are repeatedly used in the simulationprocess and require much memory to store, hence we optimize the data structureto represent the state of a PBN and the truth table. We use the primitive integertype (32 bits) instead of Boolean arrays to store a state of a PBN. The integertype is used due to the following two reasons: 1) it reduces the memory usageby 4 times comparing to Boolean arrays; 2) operations on 32 bits data are fasterthan or equal to those on non-32 bits data in our GPU architecture. The truthtable is optimized similarly as the state, i.e., it is also stored using integers. Afteroptimization, we store the state in registers (fast-memory), if possible. In thecases that a PBN is extremely large and registers are not enough, the slow globalmemory is used. However, accessing this slow global memory is reduced by 32times using an intermediate register. The truth table as well as other frequentlyaccessed arrays (e.g., the selection probabilities) are stored in shared memory(fast-memory). Frequently accessed single variables are stored in registers. Afterarranging all variables in memory, we compute the optimal number of threadsand blocks to be launched based on the usage of fast-memory to hide latency asmuch as possible.Preliminary results. We have evaluated the proposed GPU-based simulationof PBNs for computing steady-state probabilities of both randomly generatednetworks and of a real biological network using the approach in [3]. On randomlygenerated networks, our proposed GPU-based parallelised approach showed morethan two orders of magnitude speedups compared to the sequential CPU ver-sion. The evaluation on a real biological network was performed by analysing anapoptosis network with 91 nodes [2]. The speedups for computing steady-stateprobabilities for 3 randomly generated networks and the real 91-node networkare shown in Table 1. All experiments were conducted on a high-performancecomputing (HPC) node, which contained Intel Xeon E5-2680 v3 @2.5 GHz anda NVIDIA Tesla K80 Graphic Card with 2496 cores @824MHz.References1. Gelman, A., Rubin, D.: Inference from iterative simulation using multiple sequences.Statistical Science 7(4), 457–472 (1992)2. Mizera, A., Pang, J., Yuan, Q.: Reviving the two-state markov chain approach(technical report) (2015), available online at http://arxiv.org/abs/1501.017793. Mizera, A., Pang, J., Yuan, Q.: Parallel approximate steady-state analysis of largeprobabilistic Boolean networks. In: Proc. 31st ACM Symposium on Applied Com-puting. pp. 1–8 (2016)4. Shmulevich, I., Dougherty, E., Zhang, W.: From Boolean to probabilistic Booleannetworks as models of genetic regulatory networks. Proceedings of the IEEE 90(11),1778–1792 (2002)5. Trairatphisan, P., Mizera, A., Pang, J., Tantar, A.A., Sauter, T.: optPBN: An op-timisation toolbox for probabilistic boolean networks. PLOS ONE 9(7) (2014)6. Trairatphisan, P., Mizera, A., Pang, J., Tantar, A.A., Schneider, J., Sauter, T.:Recent development and biomedical applications of probabilistic Boolean networks.Cell Communication and Signaling 11, 46 (2013)

GPU-accelerated steady-state analysis of probabilistic Boolean networks

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GPU-accelerated steady-state analysis of probabilistic Boolean networks

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