Guo et al. study calibration of deep neural networks as post-processing step. Here, calibration means a correction of the predicted confidence scores as these are commonlz too overconfident in recent deep networks. They consider several state-of-the-art post-processing steps for calibration, but surprisingly, they show that a simple linear mapping, or even scaling, works surprisingly well. So if $z_i$ are the logits of the network, then (the network being fixed) a parameter $T$ is found such that

$\sigma(\frac{z_i}{T})$

is calibrated and minimized the NLL loss on a held-out validation set. Here, the temeratur $T$ either softens or roughens the probability distribution over classes. Interestingly, finding $T$ by optimizing the same training loss helps to reduce over-confidence.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

Neelakantan et al. study gradient noise for improving neural network training. In particular, they add Gaussian noise to the gradients in each iteration:

$\tilde{\nabla}f = \nabla f + \mathcal{N}(0, \sigma^2)$

where the variance $\sigma^2$ is adapted throughout training as follows:

$\sigma^2 = \frac{\eta}{(1 + t)^\gamma}$

where $\eta$ and $\gamma$ are hyper-parameters and $t$ the current iteration. In experiments, the authors show that gradient noise has the potential to improve accuracy, especially given optimization.

Also find this summary at [davidstutz.de](https://davidstutz.de/category/reading/).

This paper proposes a neural architecture that allows to backpropagate gradients though a procedure that can go through a variable and adaptive number of iterations. These "iterations" for instance could be the number of times computations are passed through the same recurrent layer (connected to the same input) before producing an output, which is the case considered in this paper. 

This is essentially achieved by pooling the recurrent states and respective outputs computed by each iteration. The pooling mechanism is essentially the same as that used in the really cool Neural Stack architecture of Edward Grefenstette, Karl Moritz Hermann, Mustafa Suleyman and Phil Blunsom \cite{conf/nips/GrefenstetteHSB15}. It relies on the introduction of halting units, which are sigmoidal units computed at each iteration and which gives a soft weight on whether the computation should stop at the current iteration.

Crucially, the paper introduces a new ponder cost $P(x)$, which is a regularization cost that penalizes what is meant to be a smooth upper bound on the number of iterations $N(t)$ (more on that below).

The paper presents experiment on RNNs applied on sequences where, at each time step t (not to be confused with what I'm calling computation iterations, which are indexed by n) in the sequence the RNN can produce a variable number $N(t)$ of intermediate states and outputs. These are the states and outputs that are pooled, to produce a single recurrent state and output for the time step t. During each of the $N(t)$ iterations at time step t, the intermediate states are connected to the same time-step-t input. After the $N(t)$ iterations, the RNN pools the $N(t)$ intermediate states and outputs, and then moves to the next time step $t+1$. To mark the transitions between time steps, an extra binary input is appended, which is 1 only for the first intermediate computation iteration. 

Results are presented on a variety of synthetic problems and a character prediction problem.

The paper discusses a number of extensions to the Skip-gram model previously proposed by Mikolov et al (citation [7] in the paper): which learns linear word embeddings that are particularly useful for analogical reasoning type tasks. The extensions proposed (namely, negative sampling and sub-sampling of high frequency words) enable extremely fast training of the model on large scale datasets. This also results in significantly improved performance as compared to previously proposed techniques based on neural networks. The authors also provide a method for training phrase level embeddings by slightly tweaking the original training algorithm. 

This paper proposes 3 improvements for the skip-gram model which allows for learning embeddings for words. The first improvement is subsampling frequent word, the second is the use of a simplified version of noise constrastive estimation (NCE) and finally they propose a method to learn idiomatic phrase embeddings. In all three cases the improvements are somewhat ad-hoc. In practice, both the subsampling and negative samples help to improve generalization substantially on an analogical reasoning task. The paper reviews related work and furthers the interesting topic of additive compositionality in embeddings. 

The article does not propose any explanation as to why the negative sampling produces better results than NCE which it is suppose to loosely approximate. In fact it doesn't explain why besides the obvious generalization gain the negative sampling scheme should be preferred to NCE since they achieve similar speeds. 

### 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.