12188 lines
445 KiB
Plaintext
12188 lines
445 KiB
Plaintext
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"# Working with Sequences\n",
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":label:`sec_sequence`\n",
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"\n",
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"Up until now, we have focused on models whose inputs\n",
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"consisted of a single feature vector $\\mathbf{x} \\in \\mathbb{R}^d$.\n",
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"The main change of perspective when developing models\n",
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"capable of processing sequences is that we now\n",
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"focus on inputs that consist of an ordered list\n",
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"of feature vectors $\\mathbf{x}_1, \\dots, \\mathbf{x}_T$,\n",
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"where each feature vector $\\mathbf{x}_t$ is\n",
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"indexed by a time step $t \\in \\mathbb{Z}^+$\n",
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"lying in $\\mathbb{R}^d$.\n",
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"\n",
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"Some datasets consist of a single massive sequence.\n",
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"Consider, for example, the extremely long streams\n",
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"of sensor readings that might be available to climate scientists.\n",
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"In such cases, we might create training datasets\n",
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"by randomly sampling subsequences of some predetermined length.\n",
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"More often, our data arrives as a collection of sequences.\n",
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"Consider the following examples:\n",
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"(i) a collection of documents,\n",
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"each represented as its own sequence of words,\n",
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"and each having its own length $T_i$;\n",
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"(ii) sequence representation of\n",
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"patient stays in the hospital,\n",
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"where each stay consists of a number of events\n",
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"and the sequence length depends roughly\n",
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"on the length of the stay.\n",
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"\n",
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"\n",
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"Previously, when dealing with individual inputs,\n",
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"we assumed that they were sampled independently\n",
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"from the same underlying distribution $P(X)$.\n",
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"While we still assume that entire sequences\n",
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"(e.g., entire documents or patient trajectories)\n",
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"are sampled independently,\n",
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"we cannot assume that the data arriving\n",
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"at each time step are independent of each other.\n",
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"For example, the words that likely to appear later in a document\n",
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"depend heavily on words occurring earlier in the document.\n",
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"The medicine a patient is likely to receive\n",
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"on the 10th day of a hospital visit\n",
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"depends heavily on what transpired\n",
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"in the previous nine days.\n",
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"\n",
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"This should come as no surprise.\n",
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"If we did not believe that the elements in a sequence were related,\n",
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"we would not have bothered to model them as a sequence in the first place.\n",
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"Consider the usefulness of the auto-fill features\n",
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"that are popular on search tools and modern email clients.\n",
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"They are useful precisely because it is often possible\n",
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"to predict (imperfectly, but better than random guessing)\n",
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"what the likely continuations of a sequence might be,\n",
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"given some initial prefix.\n",
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"For most sequence models,\n",
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"we do not require independence,\n",
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"or even stationarity, of our sequences.\n",
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"Instead, we require only that\n",
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"the sequences themselves are sampled\n",
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"from some fixed underlying distribution\n",
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"over entire sequences.\n",
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"\n",
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"This flexible approach allows for such phenomena\n",
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"as (i) documents looking significantly different\n",
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"at the beginning than at the end;\n",
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"or (ii) patient status evolving either\n",
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"towards recovery or towards death\n",
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"over the course of a hospital stay;\n",
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"or (iii) customer taste evolving in predictable ways\n",
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"over the course of continued interaction with a recommender system.\n",
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"\n",
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"\n",
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"We sometimes wish to predict a fixed target $y$\n",
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"given sequentially structured input\n",
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"(e.g., sentiment classification based on a movie review).\n",
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"At other times, we wish to predict a sequentially structured target\n",
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"($y_1, \\ldots, y_T$)\n",
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"given a fixed input (e.g., image captioning).\n",
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"Still other times, our goal is to predict sequentially structured targets\n",
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"based on sequentially structured inputs\n",
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"(e.g., machine translation or video captioning).\n",
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"Such sequence-to-sequence tasks take two forms:\n",
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"(i) *aligned*: where the input at each time step\n",
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"aligns with a corresponding target (e.g., part of speech tagging);\n",
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"(ii) *unaligned*: where the input and target\n",
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"do not necessarily exhibit a step-for-step correspondence\n",
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"(e.g., machine translation).\n",
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"\n",
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"Before we worry about handling targets of any kind,\n",
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"we can tackle the most straightforward problem:\n",
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"unsupervised density modeling (also called *sequence modeling*).\n",
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"Here, given a collection of sequences,\n",
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"our goal is to estimate the probability mass function\n",
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"that tells us how likely we are to see any given sequence,\n",
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"i.e., $p(\\mathbf{x}_1, \\ldots, \\mathbf{x}_T)$.\n"
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]
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"tab": [
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"pytorch"
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]
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},
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"outputs": [],
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"source": [
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"%matplotlib inline\n",
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"import torch\n",
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"from torch import nn\n",
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"from d2l import torch as d2l"
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]
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},
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"cell_type": "markdown",
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"source": [
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"## Autoregressive Models\n",
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"\n",
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"\n",
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"Before introducing specialized neural networks\n",
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"designed to handle sequentially structured data,\n",
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"let's take a look at some actual sequence data\n",
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"and build up some basic intuitions and statistical tools.\n",
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"In particular, we will focus on stock price data\n",
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"from the FTSE 100 index (:numref:`fig_ftse100`).\n",
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"At each *time step* $t \\in \\mathbb{Z}^+$, we observe\n",
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"the price, $x_t$, of the index at that time.\n",
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"\n",
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"\n",
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"\n",
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":width:`400px`\n",
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":label:`fig_ftse100`\n",
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"\n",
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"\n",
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"Now suppose that a trader would like to make short-term trades,\n",
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"strategically getting into or out of the index,\n",
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"depending on whether they believe\n",
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"that it will rise or decline\n",
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"in the subsequent time step.\n",
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"Absent any other features\n",
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"(news, financial reporting data, etc.),\n",
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"the only available signal for predicting\n",
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"the subsequent value is the history of prices to date.\n",
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"The trader is thus interested in knowing\n",
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"the probability distribution\n",
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"\n",
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"$$P(x_t \\mid x_{t-1}, \\ldots, x_1)$$\n",
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"\n",
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"over prices that the index might take\n",
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"in the subsequent time step.\n",
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"While estimating the entire distribution\n",
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"over a continuously valued random variable\n",
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"can be difficult, the trader would be happy\n",
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"to focus on a few key statistics of the distribution,\n",
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"particularly the expected value and the variance.\n",
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"One simple strategy for estimating the conditional expectation\n",
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"\n",
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"$$\\mathbb{E}[(x_t \\mid x_{t-1}, \\ldots, x_1)],$$\n",
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"\n",
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"would be to apply a linear regression model\n",
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"(recall :numref:`sec_linear_regression`).\n",
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"Such models that regress the value of a signal\n",
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"on the previous values of that same signal\n",
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"are naturally called *autoregressive models*.\n",
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"There is just one major problem: the number of inputs,\n",
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"$x_{t-1}, \\ldots, x_1$ varies, depending on $t$.\n",
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"In other words, the number of inputs increases\n",
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"with the amount of data that we encounter.\n",
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"Thus if we want to treat our historical data\n",
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"as a training set, we are left with the problem\n",
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"that each example has a different number of features.\n",
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"Much of what follows in this chapter\n",
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"will revolve around techniques\n",
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"for overcoming these challenges\n",
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"when engaging in such *autoregressive* modeling problems\n",
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"where the object of interest is\n",
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"$P(x_t \\mid x_{t-1}, \\ldots, x_1)$\n",
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"or some statistic(s) of this distribution.\n",
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"\n",
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"A few strategies recur frequently.\n",
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"First of all,\n",
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"we might believe that although long sequences\n",
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"$x_{t-1}, \\ldots, x_1$ are available,\n",
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"it may not be necessary\n",
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"to look back so far in the history\n",
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"when predicting the near future.\n",
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"In this case we might content ourselves\n",
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"to condition on some window of length $\\tau$\n",
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"and only use $x_{t-1}, \\ldots, x_{t-\\tau}$ observations.\n",
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"The immediate benefit is that now the number of arguments\n",
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"is always the same, at least for $t > \\tau$.\n",
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"This allows us to train any linear model or deep network\n",
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"that requires fixed-length vectors as inputs.\n",
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"Second, we might develop models that maintain\n",
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"some summary $h_t$ of the past observations\n",
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"(see :numref:`fig_sequence-model`)\n",
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"and at the same time update $h_t$\n",
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"in addition to the prediction $\\hat{x}_t$.\n",
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"This leads to models that estimate not only $x_t$\n",
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"with $\\hat{x}_t = P(x_t \\mid h_{t})$\n",
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"but also updates of the form\n",
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"$h_t = g(h_{t-1}, x_{t-1})$.\n",
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"Since $h_t$ is never observed,\n",
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"these models are also called\n",
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"*latent autoregressive models*.\n",
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"\n",
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"\n",
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":label:`fig_sequence-model`\n",
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"\n",
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"To construct training data from historical data, one\n",
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"typically creates examples by sampling windows randomly.\n",
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"In general, we do not expect time to stand still.\n",
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"However, we often assume that while\n",
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"the specific values of $x_t$ might change,\n",
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"the dynamics according to which each subsequent\n",
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"observation is generated given the previous observations do not.\n",
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"Statisticians call dynamics that do not change *stationary*.\n",
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"\n",
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"\n",
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"\n",
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"## Sequence Models\n",
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"\n",
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"Sometimes, especially when working with language,\n",
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"we wish to estimate the joint probability\n",
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"of an entire sequence.\n",
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"This is a common task when working with sequences\n",
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"composed of discrete *tokens*, such as words.\n",
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"Generally, these estimated functions are called *sequence models*\n",
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"and for natural language data, they are called *language models*.\n",
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||
"The field of sequence modeling has been driven so much by natural language processing,\n",
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"that we often describe sequence models as \"language models\",\n",
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"even when dealing with non-language data.\n",
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"Language models prove useful for all sorts of reasons.\n",
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"Sometimes we want to evaluate the likelihood of sentences.\n",
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"For example, we might wish to compare\n",
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"the naturalness of two candidate outputs\n",
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"generated by a machine translation system\n",
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"or by a speech recognition system.\n",
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"But language modeling gives us not only\n",
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"the capacity to *evaluate* likelihood,\n",
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"but the ability to *sample* sequences,\n",
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"and even to optimize for the most likely sequences.\n",
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"\n",
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"While language modeling might not, at first glance, look\n",
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"like an autoregressive problem,\n",
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"we can reduce language modeling to autoregressive prediction\n",
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"by decomposing the joint density of a sequence $p(x_1, \\ldots, x_T)$\n",
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"into the product of conditional densities\n",
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"in a left-to-right fashion\n",
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"by applying the chain rule of probability:\n",
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"\n",
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"$$P(x_1, \\ldots, x_T) = P(x_1) \\prod_{t=2}^T P(x_t \\mid x_{t-1}, \\ldots, x_1).$$\n",
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"\n",
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"Note that if we are working with discrete signals such as words,\n",
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"then the autoregressive model must be a probabilistic classifier,\n",
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"outputting a full probability distribution\n",
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"over the vocabulary for whatever word will come next,\n",
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"given the leftwards context.\n",
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"\n",
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"\n",
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"\n",
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"### Markov Models\n",
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":label:`subsec_markov-models`\n",
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"\n",
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"\n",
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"Now suppose that we wish to employ the strategy mentioned above,\n",
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"where we condition only on the $\\tau$ previous time steps,\n",
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"i.e., $x_{t-1}, \\ldots, x_{t-\\tau}$, rather than\n",
|
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"the entire sequence history $x_{t-1}, \\ldots, x_1$.\n",
|
||
"Whenever we can throw away the history\n",
|
||
"beyond the previous $\\tau$ steps\n",
|
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"without any loss in predictive power,\n",
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"we say that the sequence satisfies a *Markov condition*,\n",
|
||
"i.e., *that the future is conditionally independent of the past,\n",
|
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"given the recent history*.\n",
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"When $\\tau = 1$, we say that the data is characterized\n",
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"by a *first-order Markov model*,\n",
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"and when $\\tau = k$, we say that the data is characterized\n",
|
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"by a $k^{\\textrm{th}}$-order Markov model.\n",
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"For when the first-order Markov condition holds ($\\tau = 1$)\n",
|
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"the factorization of our joint probability becomes a product\n",
|
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"of probabilities of each word given the previous *word*:\n",
|
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"\n",
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"$$P(x_1, \\ldots, x_T) = P(x_1) \\prod_{t=2}^T P(x_t \\mid x_{t-1}).$$\n",
|
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"\n",
|
||
"We often find it useful to work with models that proceed\n",
|
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"as though a Markov condition were satisfied,\n",
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"even when we know that this is only *approximately* true.\n",
|
||
"With real text documents we continue to gain information\n",
|
||
"as we include more and more leftwards context.\n",
|
||
"But these gains diminish rapidly.\n",
|
||
"Thus, sometimes we compromise, obviating computational and statistical difficulties\n",
|
||
"by training models whose validity depends\n",
|
||
"on a $k^{\\textrm{th}}$-order Markov condition.\n",
|
||
"Even today's massive RNN- and Transformer-based language models\n",
|
||
"seldom incorporate more than thousands of words of context.\n",
|
||
"\n",
|
||
"\n",
|
||
"With discrete data, a true Markov model\n",
|
||
"simply counts the number of times\n",
|
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"that each word has occurred in each context, producing\n",
|
||
"the relative frequency estimate of $P(x_t \\mid x_{t-1})$.\n",
|
||
"Whenever the data assumes only discrete values\n",
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"(as in language),\n",
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"the most likely sequence of words can be computed efficiently\n",
|
||
"using dynamic programming.\n",
|
||
"\n",
|
||
"\n",
|
||
"### The Order of Decoding\n",
|
||
"\n",
|
||
"You may be wondering why we represented\n",
|
||
"the factorization of a text sequence $P(x_1, \\ldots, x_T)$\n",
|
||
"as a left-to-right chain of conditional probabilities.\n",
|
||
"Why not right-to-left or some other, seemingly random order?\n",
|
||
"In principle, there is nothing wrong with unfolding\n",
|
||
"$P(x_1, \\ldots, x_T)$ in reverse order.\n",
|
||
"The result is a valid factorization:\n",
|
||
"\n",
|
||
"$$P(x_1, \\ldots, x_T) = P(x_T) \\prod_{t=T-1}^1 P(x_t \\mid x_{t+1}, \\ldots, x_T).$$\n",
|
||
"\n",
|
||
"\n",
|
||
"However, there are many reasons why factorizing text\n",
|
||
"in the same direction in which we read it\n",
|
||
"(left-to-right for most languages,\n",
|
||
"but right-to-left for Arabic and Hebrew)\n",
|
||
"is preferred for the task of language modeling.\n",
|
||
"First, this is just a more natural direction for us to think about.\n",
|
||
"After all we all read text every day,\n",
|
||
"and this process is guided by our ability\n",
|
||
"to anticipate which words and phrases\n",
|
||
"are likely to come next.\n",
|
||
"Just think of how many times you have completed\n",
|
||
"someone else's sentence.\n",
|
||
"Thus, even if we had no other reason to prefer such in-order decodings,\n",
|
||
"they would be useful if only because we have better intuitions\n",
|
||
"for what should be likely when predicting in this order.\n",
|
||
"\n",
|
||
"Second, by factorizing in order,\n",
|
||
"we can assign probabilities to arbitrarily long sequences\n",
|
||
"using the same language model.\n",
|
||
"To convert a probability over steps $1$ through $t$\n",
|
||
"into one that extends to word $t+1$ we simply\n",
|
||
"multiply by the conditional probability\n",
|
||
"of the additional token given the previous ones:\n",
|
||
"$P(x_{t+1}, \\ldots, x_1) = P(x_{t}, \\ldots, x_1) \\cdot P(x_{t+1} \\mid x_{t}, \\ldots, x_1)$.\n",
|
||
"\n",
|
||
"Third, we have stronger predictive models\n",
|
||
"for predicting adjacent words than\n",
|
||
"words at arbitrary other locations.\n",
|
||
"While all orders of factorization are valid,\n",
|
||
"they do not necessarily all represent equally easy\n",
|
||
"predictive modeling problems.\n",
|
||
"This is true not only for language,\n",
|
||
"but for other kinds of data as well,\n",
|
||
"e.g., when the data is causally structured.\n",
|
||
"For example, we believe that future events cannot influence the past.\n",
|
||
"Hence, if we change $x_t$, we may be able to influence\n",
|
||
"what happens for $x_{t+1}$ going forward but not the converse.\n",
|
||
"That is, if we change $x_t$, the distribution over past events will not change.\n",
|
||
"In some contexts, this makes it easier to predict $P(x_{t+1} \\mid x_t)$\n",
|
||
"than to predict $P(x_t \\mid x_{t+1})$.\n",
|
||
"For instance, in some cases, we can find $x_{t+1} = f(x_t) + \\epsilon$\n",
|
||
"for some additive noise $\\epsilon$,\n",
|
||
"whereas the converse is not true :cite:`Hoyer.Janzing.Mooij.ea.2009`.\n",
|
||
"This is great news, since it is typically the forward direction\n",
|
||
"that we are interested in estimating.\n",
|
||
"The book by :citet:`Peters.Janzing.Scholkopf.2017` contains more on this topic.\n",
|
||
"We barely scratch the surface of it.\n",
|
||
"\n",
|
||
"\n",
|
||
"## Training\n",
|
||
"\n",
|
||
"Before we focus our attention on text data,\n",
|
||
"let's first try this out with some\n",
|
||
"continuous-valued synthetic data.\n",
|
||
"\n",
|
||
"(**Here, our 1000 synthetic data will follow\n",
|
||
"the trigonometric `sin` function,\n",
|
||
"applied to 0.01 times the time step.\n",
|
||
"To make the problem a little more interesting,\n",
|
||
"we corrupt each sample with additive noise.**)\n",
|
||
"From this sequence we extract training examples,\n",
|
||
"each consisting of features and a label.\n"
|
||
]
|
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|
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"Unfortunately, in this case we fail spectacularly.\n",
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|
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|
||
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|
||
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|
||
"tend to be pretty accurate but beyond that,\n",
|
||
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|
||
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||
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"source": [
|
||
"This clearly illustrates how the quality of the prediction changes\n",
|
||
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|
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"While the 4-step-ahead predictions still look good,\n",
|
||
"anything beyond that is almost useless.\n",
|
||
"\n",
|
||
"## Summary\n",
|
||
"\n",
|
||
"There is quite a difference in difficulty\n",
|
||
"between interpolation and extrapolation.\n",
|
||
"Consequently, if you have a sequence, always respect\n",
|
||
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|
||
"i.e., never train on future data.\n",
|
||
"Given this kind of data,\n",
|
||
"sequence models require specialized statistical tools for estimation.\n",
|
||
"Two popular choices are autoregressive models\n",
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
"is the $k$*-step-ahead prediction*.\n",
|
||
"As we predict further in time by increasing $k$,\n",
|
||
"the errors accumulate and the quality of the prediction degrades,\n",
|
||
"often dramatically.\n",
|
||
"\n",
|
||
"## Exercises\n",
|
||
"\n",
|
||
"1. Improve the model in the experiment of this section.\n",
|
||
" 1. Incorporate more than the past four observations? How many do you really need?\n",
|
||
" 1. How many past observations would you need if there was no noise? Hint: you can write $\\sin$ and $\\cos$ as a differential equation.\n",
|
||
" 1. Can you incorporate older observations while keeping the total number of features constant? Does this improve accuracy? Why?\n",
|
||
" 1. Change the neural network architecture and evaluate the performance. You may train the new model with more epochs. What do you observe?\n",
|
||
"1. An investor wants to find a good security to buy.\n",
|
||
" They look at past returns to decide which one is likely to do well.\n",
|
||
" What could possibly go wrong with this strategy?\n",
|
||
"1. Does causality also apply to text? To which extent?\n",
|
||
"1. Give an example for when a latent autoregressive model\n",
|
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|
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]
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},
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"source": [
|
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"[Discussions](https://discuss.d2l.ai/t/114)\n"
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]
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