{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Theano 循环：scan（详解）" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Using gpu device 1: Tesla C2075 (CNMeM is disabled)\n" ] } ], "source": [ "import theano, time\n", "import theano.tensor as T\n", "import numpy as np\n", "\n", "def floatX(X):\n", " return np.asarray(X, dtype=theano.config.floatX)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`theano` 中可以使用 `scan` 进行循环，常用的 `map` 和 `reduce` 操作都可以看成是 `scan` 的特例。 \n", "\n", "`scan` 通常作用在一个序列上，每次处理一个输入，并输出一个结果。\n", "\n", "`sum(x)` 函数可以看成是 `z + x(i)` 函数在给定 `z = 0` 的情况下，对 `x` 的一个 `scan`。\n", "\n", "通常我们可以将一个 `for` 循环表示成一个 `scan` 操作，其好处如下：\n", "\n", "- 迭代次数成为符号图结构的一部分\n", "- 最小化 GPU 数据传递\n", "- 序列化梯度计算\n", "- 速度比 `for` 稍微快一些\n", "- 降低内存使用" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## scan 的使用" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "函数的用法如下：\n", "\n", " theano.scan(fn, \n", " sequences=None, \n", " outputs_info=None, \n", " non_sequences=None, \n", " n_steps=None, \n", " truncate_gradient=-1, \n", " go_backwards=False, \n", " mode=None, \n", " name=None, \n", " profile=False, \n", " allow_gc=None, \n", " strict=False)\n", " \n", "主要参数的含义：\n", "\n", "- `fn`\n", " - 一步 `scan` 所进行的操作\n", "- `sequences`\n", " - 输入的序列\n", "- `outputs_info`\n", " - 前一步输出结果的初始状态\n", "- `non_sequences`\n", " - 非序列参数\n", "- `n_steps`\n", " - 迭代步数\n", "- `go_backwards`\n", " - 是否从后向前遍历\n", " \n", "输出为一个元组 `(outputs, updates)`：\n", "\n", "- `outputs`\n", " - 从初始状态开始，每一步 `fn` 的输出结果\n", "- `updates`\n", " - 一个字典，用来记录 `scan` 过程中用到的共享变量更新规则，构造函数的时候，如果需要更新共享变量，将这个变量当作 `updates` 的参数传入。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## scan 和 map" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "这里实现一个简单的 `map` 操作，将向量 $\\mathbf x$ 中的所有元素变成原来的两倍：\n", "\n", "```python\n", "map(lambda t: t * 2, x)\n", "```" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0. 2. 4. 6. 8. 10. 12. 14. 16. 18.]\n" ] } ], "source": [ "x = T.vector()\n", "\n", "results, _ = theano.scan(fn = lambda t: t * 2,\n", " sequences = x)\n", "x_double_scan = theano.function([x], results)\n", "\n", "print x_double_scan(range(10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "之前我们说到，`theano` 中的 `map` 是 `scan` 的一个特例，因此 `theano.map` 的用法其实跟 `theano.scan` 十分类似。\n", "\n", "由于不需要考虑前一步的输出结果，所以 `theano.map` 的参数中没有 `outputs_info` 这一部分。\n", "\n", "我们用 `theano.map` 实现相同的效果：" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 0. 2. 4. 6. 8. 10. 12. 14. 16. 18.]\n" ] } ], "source": [ "result, _ = theano.map(fn = lambda t: t * 2,\n", " sequences = x)\n", "x_double_map = theano.function([x], result)\n", "\n", "print x_double_map(range(10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## scan 和 reduce" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "这里一个简单的 `reduce` 操作，求和：\n", "\n", "```python\n", "reduce(lambda a, b: a + b, x)\n", "```" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "45.0\n" ] } ], "source": [ "result, _ = theano.scan(fn = lambda t, v: t + v,\n", " sequences = x,\n", " outputs_info = floatX(0.))\n", "\n", "# 因为每一步的输出值都会被记录到最后的 result 中，所以最后的和是 result 的最后一个元素。\n", "x_sum_scan = theano.function([x], result[-1])\n", "\n", "# 计算 1 + 2 + ... + 10\n", "print x_sum_scan(range(10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`theano.reduce` 也是 `scan` 的一个特例，使用 `theano.reduce` 实现相同的效果： " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "45.0\n" ] } ], "source": [ "result, _ = theano.reduce(fn = lambda t, v: t + v,\n", " sequences = x,\n", " outputs_info = 0.)\n", "\n", "x_sum_reduce = theano.function([x], result)\n", "\n", "# 计算 1 + 2 + ... + 10\n", "print x_sum_reduce(range(10))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`reduce` 与 `scan` 不同的地方在于，`result` 包含的内容并不是每次输出的结果，而是最后一次输出的结果。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## scan 的使用" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 输入与输出" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "`fn` 是一个函数句柄，对于这个函数句柄，它每一步接受的参数是由 `sequences, outputs_info, non_sequence` 这三个参数所决定的，并且按照以下的顺序排列：\n", "\n", "- `sequences` 中第一个序列的值\n", "- ...\n", "- `sequences` 中最后一个序列的值\n", "- `outputs_info` 中第一个输出之前的值\n", "- ...\n", "- `outputs_info` 中最后一个输出之前的值\n", "- `non_squences` 中的参数\n", "\n", "这些序列的顺序与在参数 `sequences, outputs_info` 中指定的顺序相同。\n", "\n", "默认情况下，在第 `k` 次迭代时，如果 `sequences` 和 `outputs_info` 中给定的值不是字典（`dictionary`）或者一个字典列表（`list of dictionaries`），那么 \n", "\n", "- `sequences` 中的序列 `seq` 传入 `fn` 的是 `seq[k]` 的值\n", "- `outputs_info` 中的序列 `output` 传入 `fn` 的是 `output[k-1]` 的值\n", "\n", "`fn` 的返回值有两部分 `(outputs_list, update_dictionary)`，第一部分将作为序列，传入 `outputs` 中，与 `outputs_info` 中的**初始输入值的维度一致**（如果没有给定 `outputs_info` ，输出值可以任意。）\n", "\n", "第二部分则是更新规则的字典，告诉我们如何对 `scan` 中使用到的一些共享的变量进行更新：\n", "```python\n", "return [y1_t, y2_t], {x:x+1}\n", "```\n", "\n", "这两部分可以任意，即顺序既可以是 `(outputs_list, update_dictionary)`, 也可以是 `(update_dictionary, outputs_list)`，`theano` 会根据类型自动识别。\n", "\n", "两部分只需要有一个存在即可，另一个可以为空。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 例子分析" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "例如，在我们的第一个例子中\n", "\n", "```python\n", "theano.scan(fn = lambda t: t * 2,\n", " sequences = x)\n", "```\n", "\n", "在第 `k` 次迭代的时候，传入参数 `t` 的值为 `x[k]`。\n", "\n", "再如，在我们的第二个例子中：\n", "\n", "```python\n", "theano.scan(fn = lambda t, v: t + v,\n", " sequences = x,\n", " outputs_info = floatX(0.))\n", "```\n", "\n", "`fn` 接受了两个参数，初始迭代时，按照规则，`t` 接受的参数为 `x[0]`，`v` 接受的参数为我们传入 `outputs_info` 的第一个初始值即 `0` （认为是 `outputs[-1]`），他们的结果 `t+v` 将作为 `outputs[0]` 的值传入下一次迭代以及最终 `scan` 输出的 `outputs` 值中。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 输入多个序列" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以一次输入多个序列，这些序列会按照顺序传入 fn 的参数中，例如计算多项式\n", "$$\n", "\\sum_{n=0}^N a_n x^ n\n", "$$\n", "时，我们可以将多项式的系数和幂数两个序列放到一个 `list` 中作为输入参数：" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2301.0\n" ] } ], "source": [ "# 变量 x\n", "x = T.scalar(\"x\")\n", "\n", "# 不为 0 的系数\n", "A = T.vectors(\"A\")\n", "\n", "# 对应的幂数\n", "N = T.ivectors(\"N\")\n", "\n", "# a 对应的是 A， n 对应 N，v 对应 x\n", "components, _ = theano.scan(fn = lambda a, n, v: a * (v ** n),\n", " sequences = [A, N],\n", " non_sequences = x)\n", "\n", "result = components.sum()\n", "\n", "polynomial = theano.function([x, A, N], result)\n", "\n", "# 计算 1 + 3 * 10 ^ 2 + 2 * 10^3 = 2301\n", "print polynomial(floatX(10), \n", " floatX([1, 3, 2]),\n", " [0, 2, 3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 使用序列的多个值" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "默认情况下，我们只能使用输入序列的当前时刻的值，以及前一个输出的输出值。\n", "\n", "事实上，`theano` 会将参数中的序列变成一个有 `input` 和 `taps` 两个键值的 `dict`：\n", "\n", "- `input`：输入的序列\n", "- `taps`：要传入 `fn` 的值的列表\n", " - 对于 `sequences` 参数中的序列来说，默认值为 [0]，表示时间 `t` 传入 `t+0` 时刻的序列值，可以为正，可以为负。\n", " - 对于 `outputs_info` 参数中的序列来说，默认值为 [-1]，表示时间 `t` 传入 `t-1` 时刻的序列值，只能为负值，如果值为 `None`，表示这个输出结果不会作为参数传入 `fn` 中。\n", " \n", "传入 `fn` 的参数也会按照 `taps` 中的顺序来排列，我们考虑下面这个例子：\n", "```python\n", "scan(fn, sequences = [ dict(input= Sequence1, taps = [-3,2,-1])\n", " , Sequence2\n", " , dict(input = Sequence3, taps = 3) ]\n", " , outputs_info = [ dict(initial = Output1, taps = [-3,-5])\n", " , dict(initial = Output2, taps = None)\n", " , Output3 ]\n", " , non_sequences = [ Argument1, Argument2])\n", "```\n", "首先是 `Sequence1` 的 `[-3, 2, -1]` 被传入，然后 `Sequence2` 不是 `dict`， 所以传入默认值 `[0]`，`Sequence3` 传入的参数是 `3`，所以 `fn` 在第 `t` 步接受的前几个参数是：\n", "```\n", "Sequence1[t-3]\n", "Sequence1[t+2]\n", "Sequence1[t-1]\n", "Sequence2[t]\n", "Sequence3[t+3]\n", "```\n", "\n", "然后 `Output1` 传入的是 `[-3, -5]`（**传入的初始值的形状应为 `shape (5,)+`**），`Output2` 不作为参数传入，`Output3` 传入的是 `[-1]`，所以接下的参数是：\n", "```\n", "Output1[t-3]\n", "Output1[t-5]\n", "Output3[t-1]\n", "Argument1\n", "Argument2\n", "```\n", "\n", "总的说来上面的例子中，`fn` 函数按照以下顺序最多接受这样 10 个参数：\n", "```\n", "Sequence1[t-3]\n", "Sequence1[t+2]\n", "Sequence1[t-1]\n", "Sequence2[t]\n", "Sequence3[t+3]\n", "Output1[t-3]\n", "Output1[t-5]\n", "Output3[t-1]\n", "Argument1\n", "Argument2\n", "```\n", "\n", "例子，假设 $x$ 是我们的输入，$y$ 是我们的输出，我们需要计算 $y(t) = tanh\\left[W_{1} y(t-1) + W_{2} x(t) + W_{3} x(t-1)\\right]$ 的值：" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "X = T.matrix(\"X\")\n", "Y = T.vector(\"y\")\n", "\n", "W_1 = T.matrix(\"W_1\")\n", "W_2 = T.matrix(\"W_2\")\n", "W_3 = T.matrix(\"W_3\")\n", "\n", "# W_yy 和 W_xy 作为不变的参数可以直接使用\n", "results, _ = theano.scan(fn = lambda x, x_pre, y: T.tanh(T.dot(W_1, y) + T.dot(W_2, x) + T.dot(W_3, x_pre)), \n", " # 0 对应 x，-1 对应 x_pre\n", " sequences = dict(input=X, taps=[0, -1]), \n", " outputs_info = Y)\n", "\n", "Y_seq = theano.function(inputs = [X, Y, W_1, W_2, W_3], \n", " outputs = results)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "测试小矩阵计算：" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "theano running time 0.0537 s\n", "numpy running time 0.0197 s\n", "the max difference of the first 10 results is 1.25780650354e-06\n" ] } ], "source": [ "# 测试\n", "t = 1001\n", "x_dim = 10\n", "y_dim = 20\n", "\n", "x = 2 * floatX(np.random.random([t, x_dim])) - 1\n", "y = 2 * floatX(np.zeros(y_dim)) - 1\n", "w_1 = 2 * floatX(np.random.random([y_dim, y_dim])) - 1\n", "w_2 = 2 * floatX(np.random.random([y_dim, x_dim])) - 1\n", "w_3 = 2 * floatX(np.random.random([y_dim, x_dim])) - 1\n", "\n", "tic = time.time()\n", "\n", "y_res_theano = Y_seq(x, y, w_1, w_2, w_3)\n", "\n", "print \"theano running time {:.4f} s\".format(time.time() - tic)\n", "\n", "tic = time.time()\n", "# 与 numpy 的结果进行比较：\n", "y_res_numpy = np.zeros([t, y_dim])\n", "y_res_numpy[0] = y\n", "\n", "for i in range(1, t):\n", " y_res_numpy[i] = np.tanh(w_1.dot(y_res_numpy[i-1]) + w_2.dot(x[i]) + w_3.dot(x[i-1]))\n", "\n", "print \"numpy running time {:.4f} s\".format(time.time() - tic)\n", "\n", "# 这里要从 1 开始，因为使用了 x(t-1)，所以 scan 从第 1 个位置开始计算\n", "print \"the max difference of the first 10 results is\", np.max(np.abs(y_res_theano[0:10] - y_res_numpy[1:11]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "测试大矩阵运算：" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "theano running time 0.0754 s\n", "numpy running time 0.1334 s\n", "the max difference of the first 10 results is 0.000656997077348\n" ] } ], "source": [ "# 测试\n", "t = 1001\n", "x_dim = 100\n", "y_dim = 200\n", "\n", "x = 2 * floatX(np.random.random([t, x_dim])) - 1\n", "y = 2 * floatX(np.zeros(y_dim)) - 1\n", "w_1 = 2 * floatX(np.random.random([y_dim, y_dim])) - 1\n", "w_2 = 2 * floatX(np.random.random([y_dim, x_dim])) - 1\n", "w_3 = 2 * floatX(np.random.random([y_dim, x_dim])) - 1\n", "\n", "tic = time.time()\n", "\n", "y_res_theano = Y_seq(x, y, w_1, w_2, w_3)\n", "\n", "print \"theano running time {:.4f} s\".format(time.time() - tic)\n", "\n", "tic = time.time()\n", "# 与 numpy 的结果进行比较：\n", "y_res_numpy = np.zeros([t, y_dim])\n", "y_res_numpy[0] = y\n", "\n", "for i in range(1, t):\n", " y_res_numpy[i] = np.tanh(w_1.dot(y_res_numpy[i-1]) + w_2.dot(x[i]) + w_3.dot(x[i-1]))\n", "\n", "print \"numpy running time {:.4f} s\".format(time.time() - tic)\n", "\n", "# 这里要从 1 开始，因为使用了 x(t-1)，所以 scan 从第 1 个位置开始计算\n", "print \"the max difference of the first 10 results is\", np.max(np.abs(y_res_theano[:10] - y_res_numpy[1:11]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "值得注意的是，由于 `theano` 和 `numpy` 在某些计算的实现上存在一定的差异，随着序列长度的增加，这些差异将被放大：" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "iter 001, max diff:0.000002\n", "iter 002, max diff:0.000005\n", "iter 003, max diff:0.000007\n", "iter 004, max diff:0.000010\n", "iter 005, max diff:0.000024\n", "iter 006, max diff:0.000049\n", "iter 007, max diff:0.000113\n", "iter 008, max diff:0.000145\n", "iter 009, max diff:0.000334\n", "iter 010, max diff:0.000657\n", "iter 011, max diff:0.001195\n", "iter 012, max diff:0.002778\n", "iter 013, max diff:0.004561\n", "iter 014, max diff:0.004748\n", "iter 015, max diff:0.014849\n", "iter 016, max diff:0.012696\n", "iter 017, max diff:0.043639\n", "iter 018, max diff:0.046540\n", "iter 019, max diff:0.083032\n", "iter 020, max diff:0.123678\n" ] } ], "source": [ "for i in xrange(20):\n", " print \"iter {:03d}, max diff:{:.6f}\".format(i + 1, \n", " np.max(np.abs(y_res_numpy[i + 1,:] - y_res_theano[i,:])))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 控制循环次数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "假设我们要计算方阵$A$的$A^k$，$k$ 是一个未知变量，我们可以这样通过 `n_steps` 参数来控制循环计算的次数： " ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 107616. -152192.]\n", " [ -76096. 107616.]]\n", "[[ 107616. -152192.]\n", " [ -76096. 107616.]]\n" ] } ], "source": [ "A = T.matrix(\"A\")\n", "k = T.iscalar(\"k\")\n", "\n", "results, _ = theano.scan(fn = lambda P, A: P.dot(A),\n", " # 初始值设为单位矩阵\n", " outputs_info = T.eye(A.shape[0]),\n", " # 乘 k 次\n", " non_sequences = A,\n", " n_steps = k)\n", "\n", "A_k = theano.function(inputs = [A, k], outputs = results[-1])\n", "\n", "test_a = floatX([[2, -2], [-1, 2]])\n", "\n", "print A_k(test_a, 10)\n", "\n", "# 使用 numpy 进行验证\n", "a_k = np.eye(2)\n", "for i in range(10):\n", " a_k = a_k.dot(test_a)\n", " \n", "print a_k" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 使用共享变量" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "可以在 `scan` 中使用并更新共享变量，例如，利用共享变量 `n`，我们可以实现这样一个迭代 `k` 步的简单计数器：" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.0\n", "10.0\n", "20.0\n" ] } ], "source": [ "n = theano.shared(floatX(0))\n", "k = T.iscalar(\"k\")\n", "\n", "# 这里 lambda 的返回值是一个 dict，因此这个值会被传入 updates 中\n", "_, updates = theano.scan(fn = lambda n: {n:n+1},\n", " non_sequences = n,\n", " n_steps = k)\n", "\n", "counter = theano.function(inputs = [k], \n", " outputs = [],\n", " updates = updates)\n", "\n", "print n.get_value()\n", "counter(10)\n", "print n.get_value()\n", "counter(10)\n", "print n.get_value()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "之前说到，`fn` 函数的返回值应该是 `(outputs_list, update_dictionary)` 或者 `(update_dictionary, outputs_list)` 或者两者之一。\n", "\n", "这里 `fn` 函数返回的是一个字典，因此自动被放入了 `update_dictionary` 中，然后传入 `function` 的 `updates` 参数中进行迭代。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 使用条件语句结束循环" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们可以将 `scan` 设计为 `loop-until` 的模式，具体方法是在 `scan` 中，将 `fn` 的返回值增加一个参数，使用 `theano.scan_module` 来设置停止条件。\n", "\n", "假设我们要计算所有不小于某个值的 2 的幂，我们可以这样定义：" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 2. 4. 8. 16. 32.]\n" ] } ], "source": [ "max_value = T.scalar()\n", "\n", "results, _ = theano.scan(fn = lambda v_pre, max_v: (v_pre * 2, theano.scan_module.until(v_pre * 2 > max_v)), \n", " outputs_info = T.constant(1.),\n", " non_sequences = max_value,\n", " n_steps = 1000)\n", "\n", "# 注意，这里不能取 results 的全部\n", "# 例如在输入值为 40 时，最后的输出可以看成 (64, False)\n", "# scan 发现停止条件满足，停止循环，但是不影响 64 被输出到 results 中，因此要将 64 去掉\n", "power_of_2 = theano.function(inputs = [max_value], outputs = results[:-1])\n", "\n", "print power_of_2(40)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.6" } }, "nbformat": 4, "nbformat_minor": 0 }