A study of how scan orders influence Mixing time in Gibbs sampling.

This paper is interested in comparing the mixing rates of Gibbs sampling using either systematic scan or random updates. The basic contributions are two: First, in Section 2, a set of cases where 1) systematic scan is polynomially faster than random updates. Together with a previously known case where it can be slower this contradicts a conjecture that the speeds of systematic and random updates are similar. Secondly, (In Theorem 1) a set of mild conditions under which the mixing times of systematic scan and random updates are not "too" different (roughly within squares of each other).

First, following from a recent paper by Roberts and Rosenthal, the authors construct several examples which do not satisfy the commonly held belief that systematic scan is never more than a constant factor slower and a log factor faster than random scan. The authors then provide a result Theorem 1 which provides weaker bounds, which however they verify at least under some conditions. In fact the Theorem compares random scan to a lazy version of the systematic scan and shows that and obtains bounds in terms of various other quantities, like the minimum probability, or the minimum holding probability.

MCMC is at the heart of many applications of modern machine learning and statistics. It is thus important to understand the computational and theoretical performance under various conditions. The present paper focused on examining systematic Gibbs sampling in comparison to random scan Gibbs. They do so first though the construction of several examples which challenge the dominant intuitions about mixing times, and develop theoretical bounds which are much wider than previously conjectured.

#### Introduction

* The paper demonstrates how simple CNNs, built on top of word embeddings, can be used for sentence classification tasks.
* [Link to the paper](https://arxiv.org/abs/1408.5882)
* [Implementation](https://github.com/shagunsodhani/CNN-Sentence-Classifier)

#### Architecture

* Pad input sentences so that they are of the same length.
* Map words in the padded sentence using word embeddings (which may be either initialized as zero vectors or initialized as word2vec embeddings) to obtain a matrix corresponding to the sentence.
* Apply convolution layer with multiple filter widths and feature maps.
* Apply max-over-time pooling operation over the feature map.
* Concatenate the pooling results from different layers and feed to a fully-connected layer with softmax activation.
* Softmax outputs probabilistic distribution over the labels.
* Use dropout for regularisation.

#### Hyperparameters

* RELU activation for convolution layers
* Filter window of 3, 4, 5 with 100 feature maps each.
* Dropout - 0.5
* Gradient clipping at 3
* Batch size - 50
* Adadelta update rule.

#### Variants

* CNN-rand
    * Randomly initialized word vectors.
* CNN-static
    * Uses pre-trained vectors from word2vec and does not update the word vectors.
* CNN-non-static
    * Same as CNN-static but updates word vectors during training.
* CNN-multichannel
    * Uses two set of word vectors (channels).
    * One set is updated and other is not updated.

#### Datasets

* Sentiment analysis datasets for Movie Reviews, Customer Reviews etc.
* Classification data for questions.
* Maximum number of classes for any dataset - 6

#### Strengths

* Good results on benchmarks despite being a simple architecture.
* Word vectors obtained by non-static channel have more meaningful representation. 

#### Weakness

* Small data with few labels.
* Results are not very detailed or exhaustive.

This paper described an algorithm of parametrically adding noise and applying a variational regulariser similar to that in ["Variational Dropout Sparsifies Deep Neural Networks"][vardrop]. Both have the same goal: make neural networks more efficient by removing parameters (and therefore the computation applied with those parameters). Although, this paper also has the goal of giving a prescription of how many bits to store each parameter with as well.

There is a very nice derivation of the hierarchical variational approximation being used here, which I won't try to replicate here. In practice, the difference to prior work is that the stochastic gradient variational method uses hierarchical samples; ie it samples from a prior, then incorporates these samples when sampling over the weights (both applied through local reparameterization tricks). It's a powerful method, which allows them to test two different priors (although they are clearly not limited to just these), and compare both against competing methods. They are comparable, and the choice of prior offers some tradeoffs in terms of sparsity versus quantization.

[vardrop]: http://www.shortscience.org/paper?bibtexKey=journals/corr/1701.05369

Ma et al. detect adversarial examples based on their estimated intrinsic dimensionality. I want to note that this work is also similar to [1] – in both publications, local intrinsic dimensionality is used to analyze adversarial examples. Specifically, the intrinsic dimensionality of a sample is estimated based on the radii $r_i(x)$ of the $k$-nearest neighbors around a sample $x$:

$- \left(\frac{1}{k} \sum_{i = 1}^k \log \frac{r_i(x)}{r_k(x)}\right)^{-1}$.

For details regarding the original, theoretical formulation of local intrinsic dimensionality I refer to the paper. In experiments, the authors show that adversarial examples exhibit a significant higher intrinsic dimensionality than training samples or randomly perturbed examples. This observation allows detection of adversarial examples. A proper interpretation of this finding is, however, missing. It would be interesting to investigate what this finding implies about the properties of adversarial examples.

Simon-Gabriel et al. Study the robustness of neural networks with respect to the input dimensionality. Their main hypothesis is that the vulnerability of neural networks against adversarial perturbations increases with the input dimensionality. To support this hypothesis, they provide a theoretical analysis as well as experiments.

The general idea of robustness is that small perturbations $\delta$ of the input $x$ do only result in small variations $\delta \mathcal{L}$ of the loss:

$\delta \mathcal{L} = \max_{\|\delta\| \leq \epsilon} |\mathcal{L}(x + \delta) - \mathcal{L}(x)| \approx \max_{\|\delta\| \leq \epsilon} |\partial_x \mathcal{L} \cdot \delta| = \epsilon \||\partial_x \mathcal{L}\||$

where the approximation is due to a first-order Taylor expansion and $\||\cdot\||$ is the dual norm of $\|\cdot\|$. As a result, the vulnerability of networks can be quantified by considering $\epsilon\mathbb{E}_x\||\partial_x \mathcal{L}\||$. A natural regularizer to increase robustness (i.e. decrease vulnerability) would be $\epsilon \||\partial_x \mathcal{L}\||$ which is a similar regularizer as proposed in [1].

The remainder of the paper studies the norm $\|\partial_x \mathcal{L}\|$ with respect to the input dimension $d$. Specifically, they show that the gradient norm increases monotonically with the input dimension. I refer to the paper for the exact theorems and proofs. This claim is based on the assumption of non-trained networks that have merely been initialized. However, in experiments, they show that the conclusion may hold true in realistic settings, e.g. on ImageNet.

[1] Matthias Hein, Maksym Andriushchenko:
Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation. NIPS 2017: 2263-2273

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