# Lecture 10

## Introduction

Recurrent Networks offer a lot of flexibility: 1. one to one: Vanilla Neural Networks
2. one to many: e.g. Image Captioning (image -> sequence of words)
3. many to one: e.g. Sentiment Classification (sequence of words -> sentiment)
4. many to many:
• e.g. Machine Translation (seq of words -> seq of words)
• e.g. Video classification on frame level

RNN can also do sequential precessing of fix inputs (Multiple Object Recognition with Visual Attention, Ba et al.) or fixed outputs (DRAW: A Recurrent Neural Network For Image Generation, Gregor et al.).

## Recurrent Neural Network

### Concept

Usually we want to predict a vector at some time steps. To achieve this goal, we can process a sequence of vectors $x$ by applying a recurrence formula at every time step: Notice: the same function and the same set of parameters are used at every time step. That’s to say, we use shared weights.

(Vanilla) Recurrent Neural Network

The state consists of a single “hidden” vector $h$:

• $h_t = tanh (W_{hh} h_{t-1} + W_{xh} x_t)$
• $y_t = W_{hy} h_t$

### Example: Character-level language model

We have a vocabulary of four characters $\begin{bmatrix} h & e & l & o \end{bmatrix}$, and the example training sequence is “hello”. And we can look its the implement.

Data I/O

  1 2 3 4 5 6 7 8 9 10 11 12 13  """ Minimal character-level Vanilla RNN model. Written by Andrej Karpathy (@karpathy) BSD License """ import numpy as np # data I/O data = open('input.txt', 'r').read() # should be simple plain text file chars = list(set(data)) data_size, vocab_size = len(data), len(chars) print 'data has %d characters, %d unique.' % (data_size, vocab_size) char_to_ix = { ch:i for i,ch in enumerate(chars) } ix_to_char = { i:ch for i,ch in enumerate(chars) }

Initializations

  1 2 3 4 5 6 7 8 9 10 11  # hyperparameters hidden_size = 100 # size of hidden layer of neurons seq_length = 25 # number of steps to unroll the RNN for learning_rate = 1e-1 # model parameters Wxh = np.random.randn(hidden_size, vocab_size)*0.01 # input to hidden Whh = np.random.randn(hidden_size, hidden_size)*0.01 # hidden to hidden Why = np.random.randn(vocab_size, hidden_size)*0.01 # hidden to output bh = np.zeros((hidden_size, 1)) # hidden bias by = np.zeros((vocab_size, 1)) # output bias

Main Loop

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32  n, p = 0, 0 mWxh, mWhh, mWhy = np.zeros_like(Wxh), np.zeros_like(Whh), np.zeros_like(Why) mbh, mby = np.zeros_like(bh), np.zeros_like(by) # memory variables for Adagrad smooth_loss = -np.log(1.0/vocab_size)*seq_length # loss at iteration 0 while True: # prepare inputs (we're sweeping from left to right in steps seq_length long) if p+seq_length+1 >= len(data) or n == 0: hprev = np.zeros((hidden_size,1)) # reset RNN memory p = 0 # go from start of data inputs = [char_to_ix[ch] for ch in data[p:p+seq_length]] targets = [char_to_ix[ch] for ch in data[p+1:p+seq_length+1]] # sample from the model now and then if n % 100 == 0: sample_ix = sample(hprev, inputs, 200) txt = ''.join(ix_to_char[ix] for ix in sample_ix) print '----\n %s \n----' % (txt, ) # forward seq_length characters through the net and fetch gradient loss, dWxh, dWhh, dWhy, dbh, dby, hprev = lossFun(inputs, targets, hprev) smooth_loss = smooth_loss * 0.999 + loss * 0.001 if n % 100 == 0: print 'iter %d, loss: %f' % (n, smooth_loss) # print progress # perform parameter update with Adagrad for param, dparam, mem in zip([Wxh, Whh, Why, bh, by], [dWxh, dWhh, dWhy, dbh, dby], [mWxh, mWhh, mWhy, mbh, mby]): mem += dparam * dparam param += -learning_rate * dparam / np.sqrt(mem + 1e-8) # adagrad update p += seq_length # move data pointer n += 1 # iteration counter

Loss function

• forward pass (compute loss)
• backward pass (compute param gradient)
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35  def lossFun(inputs, targets, hprev): """ inputs,targets are both list of integers. hprev is Hx1 array of initial hidden state returns the loss, gradients on model parameters, and last hidden state """ xs, hs, ys, ps = {}, {}, {}, {} hs[-1] = np.copy(hprev) loss = 0 # forward pass for t in xrange(len(inputs)): xs[t] = np.zeros((vocab_size,1)) # encode in 1-of-k representation xs[t][inputs[t]] = 1 hs[t] = np.tanh(np.dot(Wxh, xs[t]) + np.dot(Whh, hs[t-1]) + bh) # hidden state ys[t] = np.dot(Why, hs[t]) + by # unnormalized log probabilities for next chars ps[t] = np.exp(ys[t]) / np.sum(np.exp(ys[t])) # probabilities for next chars loss += -np.log(ps[t][targets[t],0]) # softmax (cross-entropy loss) # backward pass: compute gradients going backwards dWxh, dWhh, dWhy = np.zeros_like(Wxh), np.zeros_like(Whh), np.zeros_like(Why) dbh, dby = np.zeros_like(bh), np.zeros_like(by) dhnext = np.zeros_like(hs) for t in reversed(xrange(len(inputs))): dy = np.copy(ps[t]) dy[targets[t]] -= 1 # backprop into y. see http://cs231n.github.io/neural-networks-case-study/#grad if confused here dWhy += np.dot(dy, hs[t].T) dby += dy dh = np.dot(Why.T, dy) + dhnext # backprop into h dhraw = (1 - hs[t] * hs[t]) * dh # backprop through tanh nonlinearity dbh += dhraw dWxh += np.dot(dhraw, xs[t].T) dWhh += np.dot(dhraw, hs[t-1].T) dhnext = np.dot(Whh.T, dhraw) for dparam in [dWxh, dWhh, dWhy, dbh, dby]: np.clip(dparam, -5, 5, out=dparam) # clip to mitigate exploding gradients return loss, dWxh, dWhh, dWhy, dbh, dby, hs[len(inputs)-1]

Sampling

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  def sample(h, seed_ix, n): """ sample a sequence of integers from the model h is memory state, seed_ix is seed letter for first time step """ x = np.zeros((vocab_size, 1)) x[seed_ix] = 1 ixes = [] for t in xrange(n): h = np.tanh(np.dot(Wxh, x) + np.dot(Whh, h) + bh) y = np.dot(Why, h) + by p = np.exp(y) / np.sum(np.exp(y)) ix = np.random.choice(range(vocab_size), p=p.ravel()) x = np.zeros((vocab_size, 1)) x[ix] = 1 ixes.append(ix) return ixes

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  # gradient checking from random import uniform def gradCheck(inputs, target, hprev): global Wxh, Whh, Why, bh, by num_checks, delta = 10, 1e-5 _, dWxh, dWhh, dWhy, dbh, dby, _ = lossFun(inputs, targets, hprev) for param,dparam,name in zip([Wxh, Whh, Why, bh, by], [dWxh, dWhh, dWhy, dbh, dby], ['Wxh', 'Whh', 'Why', 'bh', 'by']): s0 = dparam.shape s1 = param.shape assert s0 == s1, 'Error dims dont match: %s and %s.' % (s0, s1) print name for i in xrange(num_checks): ri = int(uniform(0,param.size)) # evaluate cost at [x + delta] and [x - delta] old_val = param.flat[ri] param.flat[ri] = old_val + delta cg0, _, _, _, _, _, _ = lossFun(inputs, targets, hprev) param.flat[ri] = old_val - delta cg1, _, _, _, _, _, _ = lossFun(inputs, targets, hprev) param.flat[ri] = old_val # reset old value for this parameter # fetch both numerical and analytic gradient grad_analytic = dparam.flat[ri] grad_numerical = (cg0 - cg1) / ( 2 * delta ) rel_error = abs(grad_analytic - grad_numerical) / abs(grad_numerical + grad_analytic) print '%f, %f => %e ' % (grad_numerical, grad_analytic, rel_error) # rel_error should be on order of 1e-7 or less

Results

Using Shakespeare’s sonnet as input: ### Example: Image Captioning

We use CNN to recognize objects and use RNN to generate captions. Cut the last two layers from CNN and connect it to RNN: And smaple the output from previous layer to next layer as input: Sampling is stoped when meeting an END Finally, we’ll get a complete sentence (using Microsoft COCO dataset). The first row are good, but the second row may be not satisfactory. Reference:

• Explain Images with Multimodal Recurrent Neural Networks, Mao et al.
• Deep Visual-Semantic Alignments for Generating Image Descriptions, Karpathy and Fei-Fei
• Show and Tell: A Neural Image Caption Generator, Vinyals et al.
• Long-term Recurrent Convolutional Networks for Visual Recognition andDescription, Donahue et al.
• Learning a Recurrent Visual Representation for Image CaptionGeneration, Chen and Zitnick

### More examples

We can also use RNN to generate open source textbooks written in LaTex, or generate C code from Linux source code, or searching for interpretable cells.

## Long Short Term Memory (LSTM)

• Truncated BPTT
• Clip gradients at threshold (something like anti-windup in control science LOL)
• RMSProp to adjust learning rate
• Harder to detect
• Weight Initialization
• ReLU activation functions
• RMSProp
• LSTM, GRUs (<– That’s why we use LSTM)

### Introduction

LSTM is proposed in [Hochreiter et al., 1997]. GRU is a knid of simplified LSTM. ResNet is to PlainNet what LSTM is to RNN, kind of. ### Concept LSTM have two states, one is cell state ($c$), another is hidden state ($h$):

• $i$: input gate, “add to memory”, decides whether do we want to add value to this cell.
• $f$: forget gate, “flush the memory”, decides whether to shut off the cell and reset the counter.
• $o$: output gate, “get from memory”, decides how much do we want to get from this cell.
• $g$: input, decides how much do we want to add to this cell.

## Summary

• RNNs allow a lot of flexibility inarchitecture design
• Vanilla RNNs are simple but don’twork very well
• Common to use LSTM or GRU: theiradditive interactions improve gradient flow
• Backward flow of gradients in RNNcan explode or vanish. Exploding is controlled with gradient clipping.Vanishing is controlled with additive interactions (LSTM)
• Better/simpler architectures are ahot topic of current research
• Better understanding (boththeoretical and empirical) is needed.

(To be improved by adding extra materials…)