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BACKGROUND i← 1 Below we provide some background of four topics. par- for i≤ m do allel programming models (Section II-A ), MCTS (Section n← select(r) II-B ), the game of Hex (Section II-C ), and the architecture n← expand(n) of the Xeon Phi (Section II-D). ∆ ←playout(n) A. Parallel Programming Models backup(n,∆) end for Some parallel programming models provide programmers return with thread pools, relieving them of the need to manage end function their parallel tasks explicitly [11][12][13]. Creating threads each time that a program needs them can be undesirable. To Figure 1. The general MCTS algorithm prevent overhead, the program has to manage the lifetime of the thread objects and to determine the number of threads appropriate to the problem and to the current hardware. sequential semantic. When a thread runs out of tasks, it steals The ideal scenario would be that the program could just the deepest half of the stack of another (randomly selected) (1) divide the code into the smallest logical pieces that can thread [11] [17]. In Cilk Plus, thieves steal continuations. be executed concurrently (called tasks), (2) pass them over 2) Threading Building Blocks: Threading Building to the compiler and library, in order to parallelize them. Blocks (TBB) is a C++ template library developed by Intel This approach uses the fact that the majority of threading for writing software programs that take advantage of a libraries do not destroy the threads once created, so they can multicore processor [18]. TBB implements work-stealing be resumed much more quickly in subsequent use. This is to balance a parallel workload across available processing known as creating a thread pool. cores in order to increase core utilization and therefore A thread pool is a group of shared threads [14]. Tasks that scaling. The TBB work-stealing model is similar to the work can be executed concurrently are submitted to the pool, and stealing model applied in Cilk, although in TBB, thieves are added to a queue of pending work. Each task is then steal children [18]. taken from the queue by one of the worker threads, that execute the task before looping back to take another task B. Monte Carlo Tree Search from the queue. The user specifies the number of worker threads. MCTS is a tree search method that has been successfully Thread pools use either a a work-stealing or a work- applied in games such as Go, Hex and other applications sharing scheduling method to balance the work load. Ex- with a large state space [3][19][20]. It works by selectively amples of parallel programming models with work-stealing building a tree, expanding only branches it deems worth- scheduling are TBB [15] and Cilk Plus [12]. The work- while to explore. MCTS consists of four steps [21]. (1) In sharing method is used in the OpenMP programming the selection step, a leaf (or a not fully expanded node) is model [13]. selected according to some criterion. (2) In the expansion Below we discuss two threading libraries: Cilk Plus and step, a random unexplored child of the selected node is TBB. added to the tree. (3) In the simulation step (also called 1) Cilk Plus: Cilk Plus is an extension to C and C++ playout), the rest of the path to a final node is completed designed to offer a quick and easy way to harness the power using random child selection. At the end a score ∆ is of both multicore and vector processing. Cilk Plus is based obtained that signifies the score of the chosen path through on MIT’s research on Cilk [16]. Cilk Plus provides a simple the state space. (4) In the backprogagation step (also called yet powerful model for parallel programming, while runtime backup step), this value is propagated back through the tree, and template libraries offer a well-tuned environment for which affects the average score (win rate) of a node. The building parallel applications [17]. tree is built iteratively by repeating the four steps. In the Function calls can be tagged with the first keyword games of Hex and Go, each node represents a player move cilk spawn which indicates that the function can be executed and in the expansion phase the game is played out, in basic concurrently. The calling function uses the second keyword implementations, by random moves. Figure 1 shows the cilk sync to wait for the completion of all the functions general MCTS algorithm. In many MCTS implementations it spawned. The third keyword is cilk for which converts the Upper Confidence Bounds for Trees (UCT) is chosen a simple for loop into a parallel for loop. The tasks are as the selection criterion [5][22] for the trade-off between executed by the runtime system within a provably efficient exploitation and exploration that is one of the hallmarks of work-stealing framework. Cilk Plus uses a double ended the algorithm. queue per thread to keep track of the tasks to execute and The UCT algorithm addresses the problem of balancing uses it as a stack during regular operations conserving a exploitation and exploration in the selection phase of the MCTS algorithm [5]. A child node j is selected to maximize:
ln(n) UCT (j)= Xj + Cp (1) s nj
wj where Xj = , wj is the number of wins in child j, nj nj is the number of times child j has been visited, n is the number of times the parent node has been visited, and Cp ≥ 0 is a constant. The first term in the UCT equation is for exploitation of known parts of the tree and the second one is for exploration of unknown parts [22]. The level of exploration of the UCT algorithm can be adjusted by the Cp constant. Figure 2. Abstract of Intel Xeon Phi microarchitecture C. Hex Game Previous scalability studies used artificial trees [7], simu- lated hardware [23], or a large and complex program [24]. All of these approaches suffered from some drawback for performance profiling and analysis. For this reason, we de- veloped from scratch a program with the goal of generating realistic trees while being sufficiently transparent to allow △ △ good performance analysis, using the game of Hex. Hex is a game with a board of hexagonal cells [20]. Each Figure 3. Tree parallelism with local lock. The curly arrows represent player is represented by a color (White or Black). Players threads. The grey nodes are locked ones. The dark nodes are newly added to the tree. take turns by alternatively placing a stone of their color on a cell of the board. The goal for each player is to create a connected chain of stones between the opposing sides of the parallelism one MCTS tree is shared among several threads board marked by their colors. The first player to complete that are performing simultaneous tree-traversal starting from this path wins the game. the root [4]. Figure 3 shows the tree parallelism algorithm In our implementation of the Hex game, a disjoint-set data with local locks. structure is used to determine the connected stones. Using It is often hard to find sufficient parallelism in an appli- this data structure we have an efficient representation of the cation when there is a large number of cores available as in board position to find the player who won the game [25]. Xeon Phi. To find enough parallelism, we will adapt MCTS D. Architecture of the Intel Xeon Phi to use logical parallelism, called tasks (see Figure 4). In the We will now provide a brief overview of the Xeon Phi MCTS algorithm, iterations can be divided into chunks to be co-processor architecture (see Figure 2). A Xeon Phi co- executed serially. A chunk is a sequential collection of one processor board consists of up to 61 cores based on the Intel or more iterations. The maximum size of a chunk is called 64-bit ISA. Each of these cores contains vector processing grain size. Controlling the number of tasks (nTasks) allows units (VPU) to execute 512 bits of 8 double-precision to control the grain size (m) of our algorithm. The runtime floating point elements or 16 single-precision floats or 32-bit can allocate the tasks to threads for execution. With this integers at the same time, 4-way SMT, and dedicated L1 and technique we can create more tasks than threads i.e., fine- fully coherent L2 caches [26]. The tag directories (TD) are grained parallelism. By reducing the grain size we expose used to look up cache data distributed among the cores. The more parallelism to the compiler and threading library [27]. connection between cores and other functional units such Finding the right balance is the key to achieve a good as memory controllers (MC) is through a bidirectional ring scaling. The grain size should not be too small because then interconnect. There are 8 distributed memory controllers as spawn overhead reduces performance. It also should not be interface between the ring burst and main memory which is too large because that reduces parallelism and load balance up to 16 GB. (see Table I). In order to change the grain size, a wrapper function III. GRAIN SIZE CONTROLLED PARALLEL MCTS needs to be created. In this function, the maximum number The literature describes three main methods for paralleliz- of playouts is divided by the number of desired tasks. ing MCTS on shared memory machines: leaf parallelism, Each individual task has nPlayouts/nTasks iterations. root parallelism, and tree parallelism [4][6][24]. In this paper Then, the UCTSearch is run for the specified number of an algorithm based on tree parallelism is proposed. In tree iterations as a separate task. The grain size could be as Table I THECONCEPTUALEFFECTOFGRAINSIZE. 68 2 64 4 Large grain size Speedup bounded by tasks 60 8 (nTasks ≪ nCores) (not enough parallelism) 56 16 Right grain size Good speedup 32 Small grain size Spawn and scheduling overhead 52 64 (nTasks ≫ nCores) (reduces performance) 48 128 44 256 function GSCPM(s,nPlayouts) 40 512 m ← nPlayouts/nTasks 36 1024 ← 32 2048 r create a shared root node with state s Speedup 4096 t ← 1 28 8192 24 for t ≤ nTasks do 16384 execute UCTSearch(r,m) as task t 20 end for 16 wait for all tasks to be finished 12 return best child of r 8 end function 4 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 Figure 4. The pseudo-code of GSCPM algorithm. Cores
Figure 5. The scalability profile produced by Cilkview for the GSCPM small as one iteration. We call this approach Grain Size algorithm from Figure 4. The number of tasks are shown. Higher is more Controlled Parallel MCTS (GSCPM).The GSCPM pseudo- fine-grained. code is shown in Figure 4. It is important to study how our method scales as the number of processing cores increases. Figure 5 shows the of children. The values of wj and nj are also defined to be scalability profile produced by Cilkview [28] that results atomic integers (see Section II-B). from a single instrumented serial run of the GSCPM al- The main approach to parallelize the algorithm is to create gorithm for different numbers of tasks. The curves show a number of parallel tasks equal to nTasks. By increasing the the amount of available parallelism in our algorithm; they value of nTasks, each of the tasks contains a lower number of are lower bounds indicating an estimation of the potential iterations. Figure 6 shows how the algorithm is parallelized program speedup with the given grain size. As can be with different threading libraries. seen, fine-grained parallelism (many tasks) is needed for Cilk Plus and TBB are designed to efficiently balance MCTS to achieve good intrinsic parallelism. Using 16384 the load for fork-join parallelism automatically, using a tasks shows near perfect speedup on 61 cores. The actual technique called work-stealing. In a basic work-stealing performance of a parallel application is determined not only scheduler, each thread is called a worker. Each worker by its intrinsic parallelism, but also by the performance of maintains its own double-ended queue (deque) of tasks. Cilk the runtime scheduler. Therefore, it is important to find an Plus and TBB differ in their concept of what is a stealable efficient scheduler. task. For each spawned UCTSearch, there are two conceptual tasks: (1) the continuation of executing the loop around A. Implementation Details UCTSearch, (2) a child task UCTSearch. This task called Parallelization of the computational loop in the UCT- continuation. A key difference between Cilk Plus and TBB Search function (see Figure 1) across processing units (e.g., is that in Cilk Plus, thieves steal continuations. In TBB, cores and threads) is rather simple. We assume the absence thieves steal children. In TPFIFO the tasks are put in a of any logical dependencies between iterations.1 In our queue. It implements work-sharing; but the order that the implementation of tree parallelism, a lock is used in the tasks are executed is similar to child stealing. The first task expansion phase of the MCTS algorithm in order to avoid that enters the queue is the first task that gets executed. 2 the loss of any information and corruption of the tree data In our thread pool implementation (called TPFIFO) structure [1]. To allocate all children of a given node, a pre- the task functions are executed asynchronously. A task is allocated vector of children is used. When a thread tries to submitted to a FIFO work queue and will be executed as append a new child to a node it increments an atomic integer soon as one of the pool’s threads is idle. Schedule returns variable as the index to the next possible child in the vector immediately and there are no guarantees about when the
1This assumption may affect the quality of the search, but this phe- 2We use the open source library based on Boost C++ libraries which is nomenon, called search overhead, is out of scope of this paper [6] available at http://sourceforge.net/projects/threadpool/ Table II / /C++11 SEQUENTIAL VERSION.TIMEINSECONDS. std :: vector
30
26 10
22 Speedup Speedup
18 6 14
10
6 2 2
2 4 8 2 4 8 16 32 64 16 32 64 128 256 512 128 256 512 1024 2048 4096 1024 2048 4096 Number of Tasks Number of Tasks
Figure 7. Speedup on Intel Xeon Phi. Higher is better. Left: coarse-grained Figure 8. Speedup on Intel Xeon CPU. Higher is better. Left: coarse- parallelism. Right: fine-grained parallelism. grained parallelism. Right: fine-grained parallelism. limitation of this approach is that creating larger numbers 68 2 of threads has large overhead. The reduction in speedup for 64 32 60 256 C++11 is shown in Figure 7. 56 512 For cilk for on the Xeon Phi, the best execution time is 52 achieved for fine-grained tasks, when the number of tasks 48 4096 is greater than 2048. The performance of TBB (task group) 44 512 and TPFIFO are quite close on the Xeon Phi. TPFIFO scales 40 256 for up to 4096 tasks and TBB (task group) scales for up to 36 32
2048 tasks. The reason for the similarity between TBB and Speedup 28 TPFIFO on Xeon Phi is explained in III-A. 24 32 20 VI. DISCUSSION AND ANALYSIS 16 12 In analyzing our results on Xeon Phi, we also measured 8 4 the performance of GSCPM on Xeon CPU. Figure 8 depicts 2 the observed time for each threading library as a function 0 of the number of processed tasks, as measured on Xeon 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 Cores CPU. We see three interesting facts. (1) Each method has different behaviors, which depend on its implementation. It is shown that on the Xeon CPU, by doubling the numbers Figure 9. Comparing Cilkview analysis with TPFIFO speedup on Xeon Phi. The dots show the number of tasks used for TPFIFO. The lines shows of tasks the running time becomes almost half for up to the number of tasks used for Cilkview. 32 threads for C++11, Cilk Plus (cilk spawn and cilk for), and TPFIFO. C++11 and TBB (task group) achieve very close performance for up to 32 tasks. (2) For 64 and 128 256 tasks. After that the speedup continues to improve but tasks, the speedup for C++11 is better than for Cilk Plus and not as expected by Cilkview due to overheads. TBB. Cilk Plus speedup is less than the other methods up to 16 threads. (3) The best time for cilk for on Xeon CPU VII. RELATED WORK is observed for coarse-grained tasks, when the numbers of Saule et al. [31] compared the scalability of Cilk Plus, tasks are equal to 32. It shows the optimal task grain size for TBB, and OpenMP for a parallel graph coloring algorithm. Cilk Plus. The measured time for Cilk Plus comes very close They also studied the performance of aforementioned pro- to TBB (task group) on Xeon CPU, while it never reaches to gramming models for a micro-benchmark with irregular TBB (task group) performance on Xeon Phi after 64 tasks. computations. The micro-benchmark is a parallel for loop Figure 9 shows a mapping from TPFIFO speedup to the especially designed to be less memory intensive than graph Cilkview graph for 61 cores. We remark that the results of coloring. The maximum speedup for this micro-benchmark Figure 7 correspond nicely to the Cilkview results for up to on Xeon Phi was 47 and is obtained using 121 threads. Authors in [32] used a thread pool with work-stealing and Scheduling [7][34] and may find application in other best- compared it to OpenMP, Cilk Plus, and TBB. They used first algorithms, e.g., in [35][36]. Therefore, we envision a Fibonacci as an example of unbalanced tasks. In contrast to wider applicability of Grain Size Controlled to other best- our results, their approach shows no improvement over other first searches. Furthermore, the high scaling performance is methods for unbalanced tasks before using 2048 tasks with promising. Our approach with a specially designed clean work-stealing. program allows easy experimentation, and we are working Baudiˇset al. [33] reported the performance of a lock free on extending the thread pool scheduler on the Xeon Phi. tree parallelism for up to 22 threads. They used a different ACKNOWLEDGMENT speedup measure. The strength speedup is good up to 16 cores but the improvement drops after 22 cores. The authors would like to thank Jan Just Keijser for his Yoshizoe et al. [7] study the scalability of MCTS algo- helpful support with the Intel Xeon Phi co-processor. This rithm on distributed systems. They have used artificial game work is supported in part by the ERC Advanced Grant no. trees as the benchmark. Their closest settings to our study 320651, HEPGAME. are 0.1 ms playout time and branching factor of 150 with REFERENCES 72 distributed processors. They showed a maximum of 7.49 [1] M. Enzenberger and M. 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