Enable JS to Render. # Deep Learning for Beginners Notes for "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville. ## Machine Learning * Machine learning is a branch of statistics that uses samples to approximate functions. * We have a true underlying function or distribution that generates data, but we don't know what it is. * We can sample this function, and these samples form our training data. * Example image captioning: * Function: $f^\star(\text{image}) \rightarrow \text{description}$. * Samples: $\text{data} \in (\text{image}, \text{description})$. * Note: since there are many valid descriptions, the description is a distribution in text space: $\text{description} \sim \text{Text}$. * The goal of machine is to find models that: * Have enough representation power to closely approximate the true function. * Have an efficient algorithm that uses training data to find good approximations of the function. * And the approximation must generalize to return good outputs for unseen inputs. * Possible applications of machine learning: * Convert inputs into another form - learn "information", extract it and express it. eg: image classification, image captioning. * Predict the missing or future values of a sequence - learn "causality", and predict it. * Synthesise similar outputs - learn "structure", and generate it. ## Generalization and Overfitting. <img src="overfitting.png" style="margin: 0; width: 20em"> * Overfitting is when you find a good model of the training data, but this model doesn't generalize. * For example: a student who has memorized the answers to training tests will score well on a training test, but might scores badly on the final test. * There are several tradeoffs: * Model representation capacity: a weak model cannot model the function but a powerful model is more prone to overfitting. * Training iterations: training too little doesn't give enough time to fit the function, training too much gives more time to overfit. * You need to find a middle ground between a weak model and an overfitted model. * The standard technique is to do cross validation: * Set aside "test data" which is never trained upon. * After all training is complete, we run the model on the final test data. * You cannot tweak the model after the final test (of course you can gather more data). * If training the model happens in stages, you need to withhold test data for each stage. * Deep learning is one branch of machine learning techniques. It is a powerful model that has also been successful at generalizing. ## Feedforward Networks Feedforward networks represents $y = f^\star(x)$ with a function family: $$ u = f(x; \theta) $$ * $\theta$ are the model parameters. This could be thousands or millions of parameters $\theta_1 \ldots \theta_T$. * $f$ is a family of functions. $f(x; \theta)$ is a single function of $x$. $u$ is the output of the model. * You can imagine if you chose a sufficiently general family of functions, chances are, one of them will resemble $f^\star$. * For example: let the parameters represent a matrix and a vector: $f(\vec{x}; \theta)() = \begin{bmatrix}\theta_0 & \theta_1 \\\\ \theta_2 & \theta_3\end{bmatrix} \vec{x} + \begin{bmatrix}\theta_4 \\\\ \theta_5 \end{bmatrix}$ ## Designing the Output Layer. The most common output layer is: $$f(x; M, b) = g(Mx+b)$$ * The parameters in $\theta$ are used as $M$ and $b$. * The linear part $Mx+b$ ensures that your output depends on all inputs. * The nonlinear part $g(x)$ allows you to fit the distributon of $y$. * For example for input of photos, the output distribution could be: * Linear: $y \in \mathbb{R}$. eg cuteness of the photo * Sigmoid: $y \in [0, 1]$. eg probability its a cat * Softmax: $y \in \mathbb{R}^C$ and $\sum y = 1$. eg. probability its one of $C$ breeds of cats * To ensure $g(x)$ fits the distribution, you can use: * Linear: $g(x) = x$. * Sigmoid: $g(x) = \frac{1}{1+e^{-x}}$. <img src="sigmoid.svg" style="margin: 0; width: 10em"> * Softmax: $g(x)_c = \frac{e^{x_c}}{\sum_i e^{x_i}}$. * Softmax is actually under-constrained, and often $x_0$ is set to 1. In this case sigmoid is just softmax in 2 variables. * There is theory behind why these $g$'s are good choices, but there are many different choices. ## Finding $\theta$ Find $\theta$ by solving the following optimization problem for $J$ the cost function: $$ \min_{\theta \in \text{models}} J\big( y, f(x; \theta) \big) $$ * Deep learning is successful because there is a good family of algorithms to calculate $\min$. * That algorithms are all variations of gradient descent: ``` theta = initial_random_values() loop { xs = fetch_inputs() ys = fetch_outputs() us = model(theta)(xs) cost = J(ys, us) if cost < threshold: exit; theta = theta - gradient(cost) } ``` * Intuitively, at every $\theta$ you chose the direction that reduces the cost the most. * This requires you to compute the gradient $\frac{d\text{cost}}{d\theta_t}$. * You don't want the gradient to be near $0$ because you learn too slowly or near $\inf$ because it is not stable. * This is a greedy algorithm, and thus might converge but into a local minimum. ## Choosing the Cost Function * This cost function could be anything: * Sum of absolute errors: $J = \sum|y - u|$. * Sum of square errors: $J = \sum(y - u)^2$. * As long as the minimum occurs when the distributions are the same, in theory it would work. * One good idea is that $u$ represents the parameters of the distribution of $y$. * Rationale: often natural processes are fuzzy, and any input might have a range of outputs. * This approach also gives a smooth measure of how accurate we are. * The maximum likelihood principle says that: $\theta\_\text{ML} = \arg\max_\theta p(y; u)$ * Thus we want to minimize: $J = -p(y; u)$ * For $i$ samples: $J = -\prod_i p(y_i; u)$ * Taking log both sides: $J' = -\sum_i \log p(y_i; u)$. * This is called cross-entropy. * Applying the idea for: $y \sim \text{Gaussian}(\text{center}=u)$: * $p(y; u) = e^{-(y-u)^2}.$ * $J = -\sum \log e^{-(y-u)^2} = \sum(y-u)^2$ * This motivates sum of squares as a good choice. ## Regularization * Regularization techniques are methods that attempt to reduce generalization error. * It is not meant to improve the training error. * Prefer smaller $\theta$ values: * By adding some function of $\theta$ into $J$ we can encourage small parameters. * $L^2$: $J' = J + \sum |\theta|^2$ * $L^1$: $J' = J + \sum |\theta|$ * $L^0$ is not smooth. * Note for $\theta \rightarrow Mx+b$ usually only $M$ is added. * Data augmentation: * Having more examples reduces overfitting. * Also consider generating valid new data from existing data. * Rotation, stretch existing images to make new images. * Injecting small noise into $x$, into layers, into parameters. * Multi-Task learning: * Share a layer between several different tasks. * The layer is forced to choose useful features that is relevant to a general set of tasks. * Early stopping: * Keep a test data set, called the validation set, that is never trained on * Stop training when the cost on the validation set stops decreasing. * You need an extra test set to truly judge the the final. * Parameter sharing: * If you know invariants about your data, encode that into your parameter choice. * For example: images are translationally invariant, so each small patch should have the same parameters. * Dropout: * Randomly turn off some neurons in the layer. * Neurons learn to not take input data for granted. * Adversarial: * Try to make the points near training points constant, by generating adversarial data near these points. ## Deep Feedforward Networks Deep feedforward networks instead use: $$ u = f(x, \theta) = f^N(\ldots f^1(x; \theta^1) \ldots; \theta^N) $$ * This model has $N$ layers. * $f^1 \ldots f^{N-1}$: hidden layers. * $f^N$: output layer. * A deep model sounds like a bad idea because it needs more parameters. * In practise, it actually needs fewer parameters, and the models perform better (why?). * One possible reason is that each layer learns higher and higher level features of the data. * Residual models: $f'^n(x) = f^n(x) + x^{n-1}$. * Data can come from the past, we add on some more details to it. ## Designing Hidden Layers. The most common hidden layer is: $$f^n(x) = g(Mx+b)$$ * The hidden layers have the same structure as the output layer. * However the $g(x)$ which work well for the output layer don't work well for the hidden layers. * The simplest and most successful $g$ is the rectified linear unit (ReLU): $g(x) = \max(0, x)$. * Compared to sigmoid, the gradients of ReLU does not approach zero when x is very big. * Other common non-linear functions include: * Modulated ReLU: $g(x) = \max(0, x) + \alpha\min(0, x)$. * Where alpha is -1, very small, or a model parameter itself. * The intuition is that this function has a slope for x &lt; 0. * In practise there is no absolute winner between this and ReLU. * Maxout: $g(x)\_i = \max_{j \in G(i)} x_j$ * $G$ partitions the range $[1 .. I]$ into subsets $[1 .. m], [m+1 .. 2m], [I-m .. I]$. * For comparison ReLU is $\mathbb{R}^n \rightarrow \mathbb{R}^n$, and maxout is $\mathbb{R}^n \rightarrow \mathbb{R}^\frac{n}{m}$. * It is the max of each bundle of $m$ inputs, think of it as $m$ piecewise linear sections. * Linear: $g(x) = x$ * After multiplying with the next layer up, it is equivalent to: $f^n(x) = g'(NMx+b')$ * It's useful because you can use use narrow $N$ and $M$, which has less parameters. ## Optimizaton Methods * The methods we use is based on stochastic gradient descent: * Choose a subset of the training data (a minibatch), and calculate the gradient from that. * Benefit: does not depend on training set size, but on minibatch size. * There are many ways to do gradient descent (using: gradient $g$, learning rate $\epsilon$, gradient update $\Delta$) * Gradient descent - use gradient: $\Delta = \epsilon g$. * Momentum - use exponential decayed gradient: $\Delta = \epsilon \sum e^{-t} g_t$. * Adaptive learning rate where $\epsilon = \epsilon_t$: * AdaGrad - slow learning on gradient magnitude: $\epsilon_t = \frac{\epsilon}{\delta + \sqrt{\sum g_t^2}}$. * RMSProp - slow learning on exponentially decayed gradient magnitude: $\epsilon_t = \frac{\epsilon}{\sqrt{\delta + \sum e^{-t} g_t^2}}$. * Adam - complicated. * Newton's method: it's hard to apply due to technical reasons. * Batch normalization is a layer with the transform: $y = m\frac{x - \mu}{\sigma} + b$ * $m$ and $b$ are learnable, while $\mu$ and $\sigma$ are average and standard deviation. * This means that the layers can be fully independent (assume the distribution of the previous layer). * Curriculum learning: provide easier things to learn first then mix harder things in. ## Simplifying the Network * At this point, we have enough basis to design and optimize deep networks. * However, these models are very general and large. * If your network has $N$ layers each with $S$ inputs/outputs, the parameter space is $|\theta| = O(NS^2)$. * This has two downsides: overfitting, and longer training time. * There are many methods to reduce parameter space: * Find symmetries in the problem and choose layers that are invariant about the symmetry. * Create layers with lower output dimensionality, the network must summarize information into a more compact representation. ## Convolution Networks A convolutional network simplifies some layers by using convolution instead of matrix multiply (denoted with a star): $$ f^n(x) = g(\theta^n \ast x) $$ * It is used for data that is spatially distributed, and works for 1d, 2d and 3d data. * 1d: $(\theta \ast x)\_i = \sum\_a \theta\_a x\_{i+a}$ * 2d: $(\theta \ast x)\_{ij} = \sum\_a \sum\_b \theta\_{ab} x\_{ab+ij}$ * It's slightly from the mathematical definition, but has the same idea: the output at each point is a weighted sum of nearby points. * Benefits: * Captures the notion of locality, if $\theta$ is zero, except in a window $w$ wide near $i=0$. * Captures the notion of translation invariance, as the same $\theta$ are used for each point. * Reduces the number of model parameters from $O(S^2)$, to $O(w^2)$. * If there are $n$ layers of convolution, one base value will be able to influence the outputs in a $wn$ radius. * Practical considerations: * Pad the edges with 0, and how far to pad. * Tiled convolutions (you rotate between different convolutions). ## Pooling A common layer used in unison with convnets is max pooling: $$ f^n(x)\_i = \max\_{j \in G(i)} x_j $$ * It is the same structure as maxout, and equivalent in 1d. * For higher dimensions $G$ partitions the input space into tiles. * This reduces the size of the input data, and can be considered as collapsing a local region of the inputs into a summary. * It is also invariant to small translations. ## Recurrent Networks Recurrent networks use previous outputs as inputs, forming a recurrence: $$ s^{(t)} = f(s^{(t-1)}, x^{(t)}; \theta) $$ * The state $s$ contains a summary of the past, while $x$ is the inputs that arrive at each step. * It is a simpler model than a fully dynamic: $s^{(t)} = d^{(t)}(x^{(t)}, ... x^{(1)}; \theta')$ * All the $\theta$'s are shared across time - the recurrent network assumes time invariance. * A RNN can learn for any input length, while a fully dynamic model needs a different $g$ for each input length. * Output: the model may return $y^{(t)}$ at each time step: * No output during steps, only the final state matters. Eg: sentiment analysis. * $y^{(t)} = s^{(t)}$, the model has no internal state and thus less powerful. But it is easier to train, since the training data $y$ is just $s$. * $y^{(t)} = o(s^{(t)})$, use an output layer to transform (and hide) the internal state. But training is more indirect and harder. * As always we prefer to think of $y$ as the parameters to a distribution. * The output chosen may be fed back to $f$ as extra inputs. If not fed in, the $y$ are conditionally independent of each other. * When generating a sentence, we need conditional dependence between words, eg: A-A and B-B might be valid, but A-B might be invalid. * Completion: * Finish when input ends. This works for $x^{(t)} \rightarrow y^{(t)}$. * Extra output $y_\text{end}^{(t)}$ with the probability the output has completed. * Extra output $y_\text{length}^{(t)}$ with the length of output remaining/total. * Optimization is done using the same gradient descent class of methods. * Gradients are calculated by expanding the recurrence to a flat formula, called back-propagation through time (BPTT). * One difficult aspect of BPTT is the gradient $\Delta = \frac{\partial}{\partial s^t}$: * $\Delta > 0$: the state explodes, and provide unstable gradient. The solution is to clip the gradient updates to a reasonable size during descent. * $\Delta \approx 0$: this allows the state to persist for a long time, howevr the gradient descent method needs a gradient to work. * $\Delta < 0$: the RNN is in a constant state of information loss. * There are variants of RNN that impose a simple prior to help preserve state $s^{(t)} \approx s^{(t-1)}$: * $s^{(t)} = f_t s^{(t-1)} + f(...)$: we get a direct first derivative $\frac{\partial}{\partial x}$ * It lets us pass along a gradient from previous steps, even when $f$ itself has zero gradient. * Long short-term memory (LSTM) model input, output and forgetting: $s^{(t)} = f\_t s^{(t-1)} + i\_t f(s^{(t-1)}, x^{(t)}; \theta)$ * The ouput is: $y^{(t)} = o\_t s^{(t)}$ * It uses probabilities (known as gates): $o_t$ output, $f_t$ forgetting, $i_t$ input. * The gates are usually a sigmoid layer: $o_t = g(Mx+b) = \frac{1}{1+e^{Mx+b}}$. * Long term information is preserved, because generating new data $g$ and using it $i$ are decoupled. * Gradients are preserved more as there is a direct connection between the past and future. * Gated recurrent unit (GRU) are a simpler model: $s^{(t)} = (1-u\_t)s^{(t-1)} + u\_tf(r_t s^{(t-1)}, x^{(t)}; \theta)$ * The gates: $u_t$ update, $r_t$ reset. * There is no clear winner. * For dropout, prefer $d(f(s, x; \theta))$ not storing information, over $d(s)$ losing information. * Memory Networks and attention mechanisms. ## Useful Data Sets * There are broad categories of input data, the applications are limitless. * Images vector $[0-1]^{WH}$: image to label, image to description. * Audio vector for each time slice: speech to text. * Text embed each word into vector $[0-1]^N$: translation. * Knowledge Graphs: question answering. ## Autoencoders An autoencoder has two functions, which encode $f$ and decode $g$ from input space to representation space. The objective is: $$ J = L(x, g(f(x))) $$ * $L$ is the loss function, and is low when images are similar. * The idea is that the representation space learns important features. * To prevent overfitting we have some additional regularization tools: * Sparse autoencoders minimize: $J' = J + S(f(x))$. This is a regularizer on representation space. * Denoising autoencoders minimize: $J = L(x, g(f(n(x))))$, where $n$ adds noise. This forces the network to differentiate noise from signal. * Contractive autoencoders minimize: $J' = J + \sum \frac{\partial f}{\partial x}$. This forces the encoder to be smooth: similar inputs get similar outputs. * Predictive autoencoders: $J = L(x, g(h)) + L'(h, f(x))$. Instead of optimizing $g$ and $h$ simultaneously, optimize them alternately. * Another solution is to train a discriminator network $D$ which outputs a scalar representing the probability the input is generated. ## Representation Learning The idea is that instead of optimizing $ u = f(x; \theta) $, we optimize: $$ u = r_o(f(r_i(x); \theta)) $$ * $r_i$ and $r_o$ are the input and output representations, but the idea can apply to CNN, RNN and other models. * For example the encoder half of an autoencoder can be used to represent the input $r_i$. * The hope is that there are other representations that make the task easier. * These representations can be trained on large amounts of data and understand the base data. * For example: words is a very sparse input vector (all zeros, with one active). * There are semantic representations of words that is easier to work with. ## Practical Advice * Have a good measure of your success. * Build a working model as soon as possible. * Instrument and iterate from data. ## Appendix: Probability * Probability is a useful tool because it allows us to model: * Randomness: truely random system (quantum etc). * Hidden variables: deterministic, but we can't see all the critical factors. * Incomplete models: especially relevant in chaotic systems that are sensitive to small perturbations. * It is useful for reading papers and more advanced machine learning, but not as critical for playing around with a network. * Probability: $P(x,y)$ means $P(\text{x} = x, \text{y} = y)$. * Marginal probability: $P(x) = \sum_y P(\text{x} = x, \text{y} = y)$. * Chain rule: $P(x,y) = P(x|y)P(y)$. * If $x$ and $y$ are independent: $P(x,y) = P(x)P(y)$. * Expectation: $\mathbb{E}_{x \sim P}[f(x)] = \sum_x P(x)f(x)$. * Bayes rule: $P(x|y) = \frac{P(x)P(y|x)}{P(y)} = \frac{P(x)P(y|x)}{\sum_x P(x)P(y|x)}$. * Self information: $I(x) = -\log P(x)$. * Shannon entropy: $H(x) = \mathbb{E}_{x \sim P}[I(x)] = -\sum_x P(x) \log P(x)$. * KL divergence: $D\_\text{KL}(P||Q) = \mathbb{E}\_{x \sim P}\big[\log\frac{P(x)}{Q(x)}\big]$. * It is a measure of how similar distributions $P$ and $Q$ are (not true measure, not symmetric). * Cross entropy: $H(P,Q) = H(P) + D\_\text{KL} = -\mathbb{E}\_{x \sim P} \log Q(x)$. * Maximum likelihood: * For $p$ is data and $q$ is model: * $\theta\_\text{ML} = \arg\max\_\theta Q(X; \theta)$. * Assuming iid and using log: $\theta\_\text{ML} = \arg\max\_\theta \sum\_x \log Q(x; \theta)$. * Since each data point is equally likely: $\theta\_\text{ML} = \arg\max\_\theta H(P, Q; \theta)$. * The only component of KL that varies is the entropy: $\theta\_\text{ML} = \arg\max\_\theta D\_\text{KL}(P||Q; \theta)$. * Maximum a posteriori: * $\theta\_\text{MAP} = \arg\max\_\theta Q(\theta | X) = \arg\max\_\theta\log Q(X | \theta) + \log Q(\theta)$. * This is like a regularizing term based on the prior of $Q(\theta)$.