This paper explores the use of so-called Monte Carlo objectives for training directed generative models with latent variables. Monte Carlo objectives take the form of the logarithm of a Monte Carlo estimate (i.e. an average over samples) of the marginal probability $P(x)$. One important motivation for using Monte Carlo objectives is that they can be shown (see the Importance Weighted Variational Autoencoder paper \cite{journals/corr/BurdaGS15} and my notes on it) to correspond to bounds on the true likelihood of the model, and one can tighten the bound simply by drawing more samples in the Monte Carlo objective. Currently, the most successful application of Monte Carlo objectives is based on an importance sampling estimate, which involves training a proposal distribution $Q(h|x)$ in addition to the model $P(x,h)$.

This paper considers the problem of training with gradient descent on such objectives, in the context of a model to which the reparametrization trick cannot be used (e.g. for discrete latent variables). They analyze the sources of variance in the estimation of the gradients (see Equation 5) and propose a very simple approach to reducing the variance of a sampling-based estimator of these gradients. 

First, they argue that gradients with respect to the $P(x,h)$ parameters are less susceptible to problems due to high variance gradients.

Second, and most importantly, they derive a multi-sample estimate of the gradient that is meant to reduce the variance of gradients on the proposal distribution parameters $Q(h|x)$. The end result is the gradient estimate of Equations 10-11. It is based on the observation that the first term of the gradient of Equation 5 doesn't distinguish between the contribution of each sampled latent hi. The key contribution is this: they notice that one can incorporate a variance reducing baseline for each sample hi, corresponding to the Monte Carlo estimate of the log-likelihood when removing hi from the estimate (see Equation 10). The authors show that this is a proper baseline, in that using it doesn't introduce a bias in the estimation for the gradients.

Experiments show that this approach yields better performance than training based on Reweighted Wake Sleep \cite{journals/corr/BornscheinB14} or the use of NVIL baselines \cite{conf/icml/MnihG14}, when training sigmoid belief networks as generative models or as structured output prediction (image completion) models on binarized MNIST.

### Introduction

* *Curriculum Learning*  - When training machine learning models, start with easier subtasks and gradually increase the difficulty level of the tasks.
* Motivation comes from the observation that humans and animals seem to learn better when trained with a curriculum like a strategy.
* [Link](http://ronan.collobert.com/pub/matos/2009_curriculum_icml.pdf) to the paper.

### Contributions of the paper

* Explore cases that show that curriculum learning benefits machine learning.
* Offer hypothesis around when and why does it happen.
* Explore relation of curriculum learning with other machine learning approaches.

### Experiments with convex criteria

* Training perceptron where some input data is irrelevant(not predictive of the target class).
* Difficulty can be defined in terms of the number of irrelevant samples or margin from the separating hyperplane.
* Curriculum learning model outperforms no-curriculum based approach.
* Surprisingly, in the case of difficulty defined in terms of the number of irrelevant examples, the anti-curriculum strategy also outperforms no-curriculum strategy.

### Experiments on shape recognition with datasets having different variability in shapes

* Standard(target) dataset - Images of rectangles, ellipses, and triangles.
* Easy dataset - Images of squares, circles, and equilateral triangles.
* Start performing gradient descent on easy dataset and switch to target data set at a particular epoch (called *switch epoch*). 
* For no-curriculum learning, the first epoch is the *switch epoch*.
* As *switch epoch* increases, the classification error comes down with the best performance when *switch epoch* is half the total number of epochs.
* Paper does not report results for higher values of *switch epoch*.

### Experiments on language modeling

* Standard data set is the set of all possible windows of the text of size 5 from Wikipedia where all words in the window appear in 20000 most frequent words.
* Easy dataset considers only those windows where all words appear in 5000 most frequent words in vocabulary.
* Each word from the vocabulary is embedded into a *d* dimensional feature space using a matrix **W** (to be learnt).
* The model predicts the score of next word, given a window of words.
* Expected value of ranking loss function is minimized to learn **W**.
* Curriculum Learning-based model overtakes the other model soon after switching to the target vocabulary, indicating that curriculum-based model quickly learns new words.

### Curriculum as a continuation method

* Continuation methods start with a smoothed objective function and gradually move to less smoothed function.
* Useful in the case where the objective function in non-convex.
* Consider a family of cost functions $C_\lambda (\theta)$ such that  $C_0(\theta)$ can be easily optimized and $C_1(\theta)$ is the actual objective function.
* Start with $C_0 (\theta)$ and increase $\lambda$, keeping $\theta$ at a local minimum of $C_\lambda (\theta)$.
* Idea is to move $\theta$ towards a dominant (if not global) minima of $C_1(\theta)$.
* Curriculum learning can be seen as a sequence of training criteria starting with an easy-to-optimise objective and moving all the way to the actual objective.
* The paper provides a mathematical formulation of curriculum learning in terms of a target training distribution and a weight function (to model the probability of selecting anyone training example at any step).

### Advantages of Curriculum Learning

* Faster training in the online setting as learner does not try to learn difficult examples when it is not ready.
* Guiding training towards better local minima in parameter space, specifically useful for non-convex methods.

### Relation to other machine learning approaches

* **Unsupervised preprocessing** - Both have a regularizing effect and lower the generalization error for the same training error.
* **Active learning** - The learner would benefit most from the examples that are close to the learner's frontier of knowledge and are neither too hard nor too easy.
* **Boosting Algorithms** - Difficult examples are gradually emphasised though the curriculum starts with a focus on easier examples and the training criteria do not change.
* **Transfer learning** and **Life-long learning** - Initial tasks are used to guide the optimisation problem.

### Criticism

* Curriculum Learning is not well understood, making it difficult to define the curriculum.
* In one of the examples, anti-curriculum performs better than no-curriculum. Given that curriculum learning is modeled on the idea that learning benefits when examples are presented in order of increasing difficulty, one would expect anti-curriculum to perform worse.

This paper presents a variational approach to the maximisation of mutual information in the context of a reinforcement learning agent. Mutual information in this context can provide a learning signal to the agent that is "intrinsically motivated", because it relies solely on the agent's state/beliefs and does not require from the ("outside") user an explicit definition of rewards.

Specifically, the learning objective, for a current state s, is the mutual information between the sequence of K actions a proposed by an exploration distribution $w(a|s)$ and the final state s' of the agent after performing these actions. To understand what the properties of this objective, it is useful to consider the form of this mutual information as a difference of conditional entropies:

$$I(a,s'|s) = H(a|s) - H(a|s',s)$$

Where $I(.|.)$ is the (conditional) mutual information and $H(.|.)$ is the (conditional) entropy. This objective thus asks that the agent find an exploration distribution that explores as much as possible (i.e. has high $H(a|s)$ entropy) but is such that these actions have predictable consequences (i.e. lead to predictable state s' so that $H(a|s',s)$ is low). So one could think of the agent as trying to learn to have control of as much of the environment as possible, thus this objective has also been coined as "empowerment".

The main contribution of this work is to show how to train, on a large scale (i.e. larger state space and action space) with this objective, using neural networks. They build on a variational lower bound on the mutual information and then derive from it a stochastic variational training algorithm for it. The procedure has 3 components: the exploration distribution $w(a|s)$, the environment $p(s'|s,a)$ (can be thought as an encoder, but which isn't modeled and is only interacted with/sampled from) and the planning model $p(a|s',s)$ (which is modeled and can be thought of as a decoder). The main technical contribution is in how to update the exploration distribution (see section 4.2.2 for the technical details).

This approach exploits neural networks of various forms. Neural autoregressive generative models are also used as models for the exploration distribution as well as the decoder or planning distribution. Interestingly, the framework allows to also learn the state representation s as a function of some "raw" representation x of states. For raw states corresponding to images (e.g. the pixels of the screen image in a game), CNNs are used.

This paper builds on top of a bunch of existing ideas for building neural conversational agents so as to control against generic and repetitive responses. 

Their model is the sequence-to-sequence model with attention (Bahdanau et al.), first trained with the usual MLE loss and fine-tuned with policy gradients to optimize for specific conversational properties. Specifically, they define 3 rewards:

1. Ease of answering — Measured as the likelihood of responding to a query with a list of hand-picked dull responses (more negative log likelihood is higher reward). 
2. Information flow — Consecutive responses from the same agent (person) should have different information, measured as negative of log cosine distance (more negative is better).
3. Semantic coherence — Mutual information between source and target (the response should make sense wrt query). $P(a|q) + P(q|a)$ where a is answer, q is question.

The model is pre-trained with the usual supervised objective function, taking source as concatenation of two previous utterances. Then they have two stages of policy gradient training, first with just a mutual information reward and then with a combination of all three. The policy network (sequence-to-sequence model) produces a probability distribution over actions (responses) given state (previous utterances). To estimate the gradient in an iteration, the network is frozen and responses are sampled from the model, the rewards for which are then averaged and gradients are computed for first L tokens of response using MLE and remaining T-L tokens with policy gradients, with L being gradually annealed to zero (moving towards just the long-term reward).

Evaluation is done based on length of dialogue, diversity (distinct unigram, bigrams) and human studies on

1. Which of two outputs has better quality (single turn)
2. Which of two outputs is easier to respond to, and
3. Which of two conversations have better quality (multi turn).

## Strengths

- Interesting results
    - Avoids generic responses
    - 'Ease of responding' reward encourages responses to be question-like
- Adding in hand-engineereed approximate reward functions based on conversational properties and using those to fine-tune a pre-trained network using policy gradients is neat.
- Policy gradient training also encourages two dialogue agents to interact with each other and explore the complete action space (space of responses), which seems desirable to identify modes of the distribution and not converge on a single, high-scoring, generic response.

## Weaknesses / Notes

- Evaluating conversational agents is hard. BLEU / perplexity are intentionally avoided as they don't necessarily reward desirable conversational properties.

The goal of this work is to perform transfer learning among numerous tasks and to discover visual relationships among them. Specifically, while we intiutively might guess the depth of an image and surface normals are related, this work takes a step forward and discovers a beneficial relationship among 26 tasks in terms of task transferability - many of them are not obvious. This is important for scenarios when an insufficient budget is available for target task for annotation, thus, learned representation from the 'cheaper' task could be used along with small dataset for the target task to reach sufficient performance on par with fully supervised training on a large dataset.

The basis of the approach is to compute an affinity matrix among tasks based on whether the solution for one task can be sufficiently easily used for another task. This approach does not impose human intuition about the task relationships and chooses task transferability based on the quality of a transfer operation in a fully computational manner.

The task taxonomy (i.e. **taskonomy**) is a computationally found directed hypergraph that captures the notion of task transferability over any given task dictionary. It performed using a four-step process depicted in the figure below: ![Process overview. The steps involved in creating the taxonomy.](http://taskonomy.stanford.edu/img/svg/Process.svg)

-  In stage I (**Task-specific Modelling**), a task-specific network is trained in a fully supervised manner. The network is composed of the encoder (modified ResNet-50), and fully convolutional decoder for pixel-to-pixel tasks, or 2-3 FC layers for low-dimensional tasks. Dataset consists of 4 million images of indoor scenes from about 600 buildings; every image has an annotation for every task.

- In stage II (**Transfer modeling**), all feasible transfers between sources and targets are trained (multiple inputs task to single target transfer is also considered). Specifically, after the task-specific networks are trained in stage I, the weights of an encoder are fixed (frozen network is used to extract representations only) and the representation from the encoder is used to train a small readout network (similar to a decoder from stage I) with a new task as a target (i.e. ground truth is available). In total, about 3000 transfer possibilities are trained.

- In stage III (**Taxonomy solver**), the task affinities acquired from the transfer functions performance are normalized. This is needed because different tasks lie in different spaces and transfer function scale. This is performed using ordinal normalization - Analytical Hierarchy Process (details are in the paper - Section 3.3). This results in an affinity matrix where a complete graph of relationships is completely normalized and this graph quantifies a pair-wise set of tasks evaluated in terms of a transfer function (i.e. task dependency).

- In stage IV (**Computed Taxonomy**), a hypergraph which can predict the performance of any transfer policy and optimize for the optimal one is synthesized. This is solved using Binary Integer Program as a subgraph selection problem where tasks are nodes and transfers are edges. After the optimization process, the solution devices a connectivity that solves all target tasks, maximizes their collective performance while using only available source tasks under user-specified constraints (e.g. budget).

So, if you want to train your network on an unseen task, you can obtain pretrained weights for existing tasks from the [project page](https://github.com/StanfordVL/taskonomy/tree/master/taskbank), train readout functions against each task (as well as combination of multiple inputs), build an affinity matrix to know where your task is positioned against the other ones, and through subgraph selection procedure observe what tasks have favourable influence on your task. Consequently, you can train your task with much less data by utilizing representations from the existing tasks which share visual significance with your task. Magnificent!