## Student Perspectives: Are larger models always better?

A post by Emma Ceccherini, PhD student on the Compass programme.

In December 2023, I attended NeurIPS, a machine learning conference, with some COMPASS colleagues. There, I attended a tutorial titled “Reconsidering Overfitting in the Age of Overparameterized Models”. The findings the speakers presented overturn some traditional statistical concepts, so I’d like to share some of these innovative ideas with the COMPASS blog readers.

Classical statistician vs deep learning practitioners
Classical statisticians argue that small models have high bias but large variance (Figure 1 (left)) and large models have low bias but high variance (Figure 1 (right)). This is called the bias-variance trade-off and is a crucial notion that can be found in all traditional statistic textbooks. Large, over-parameterised models perfectly interpolate the data points by fitting noise and they have a near-zero training error, but an increasing test error. This phenomenon is called overfitting and causes poor performances on unseen data. Overfitting implies low generalisation, which can be thought of as the model’s performance on newly generated data at test time.

Therefore, statistics textbooks recommend avoiding overfitting and improving generalization by finding a balance in the bias-variance trade-off, either by reducing the number of parameters or using regularisation (Figure 1 (centre)).

However, as available computational power has increased, practitioners have made larger and larger models. For example, neural networks have millions of parameters, more than enough to fit noise, but they generalize very well in practice, performing significantly better than small models. These large over-parametrised models exceed the so-called interpolation threshold that is when the training error is approximately zero. Several theoretical statisticians are trying to infer what happens after this threshold. While we now have some answers, many questions are still up for debate!

The double descent

Nakkiran et al. [2019] show that in the under-parameterised regime, neural networks test errors exhibit the classical u-shape from the bias-variance trade-off, while in the over-parameterised regime, after the interpolation threshold, the test error decreases again creating the so-called double descent (see Figure 3). Figure 4 shows the test error of a neural network classifier on CIFAR-10, a standard image data set. The plot shows a double descent in the test error for neural networks trained until convergence (purple line).

The authors make two more innovative observations: harmless interpolation and good generalisation for large models. It can be observed from Figure 4 that regularisation, equivalent to early stopping (red line), is substantially beneficial around the interpolation threshold. However, as the model size grows the test error for optimal early stopped neural networks (red line) and the one of neural networks trained until convergence test (purple line) overlap. Therefore, For large models, interpolation (trained until convergence) is not worse than regularisation (optimal early stopped), that is interpolation is harmless. Finally, Figure 4 shows that the test error is low as the size of the model grows. Hence, for large models, we can achieve reasonably good test accuracy, namely as a result of good generalisation.

Simple maths for linear models
Given these groundbreaking experimental results, statisticians seek to use theoretical analysis to understand when these three phenomena occur. Although neural networks were the initial motivation of this work, they are hard to analyse even for shallow networks. And so statisticians resorted to understanding these phenomena starting from the well-known linear models.

Over-parameterisation in linear models of the form $\mathbf{Y} = \mathbf{X}\theta^* + \mathbf{W}$ means there are more features $d$ than number of samples $n$, i.e. $d >n$ for an input matrix $\mathbf{X}$ of dimension $n \times d$. Then the system $\mathbf{X}\hat{\theta} = \mathbf{Y}$ has infinite solutions, thus consider the solution with minimum norm $\hat{\theta} = \text{arg min}||\hat{\theta}||_2$.

After the interpolation threshold, the variance is dominating (see Figure 3) so it needs to go down for the test error to go down. Indeed, Bartlett et al. [2020] show that in this setup the variance decreases as $d \gg n$, precisely $$\text{variance} \asymp \frac{\sigma^2n}{d}.$$

It can be shown that data is approximately orthogonal when $d \gg n$, namely $<X_i, X_j> \approx 0$ for $i \neq 0$, so the noise “energy” is spread out along the $d$ dimensions, hence as $d$ grows the noise contribution decreases.
However, Bartlett et al. [2020] also show that the bias increases with $d$, precisely $$\text{bias} \asymp (1-\frac{n}{d})||\theta^*||_2^2.$$ This is because the signal “energy” as well is spread out along $d$ dimensions.

Eventually, the bias will dominate and the test error will increase again, see Figure 5 (left). Therefore under this framework, the double descent and harmless interpolation can be achieved but good generalisation cannot.

Finally, Bartlett et al. [2020] show that in the special case where the $k$ features are “upweighted”, all three phenomena are observed. Assuming a spiked covariance $$\Sigma = \mathbb{E}[\mathbf{X}\mathbf{X}^T] = \begin{bmatrix} R\mathbf{I}_k & \mathbf{0} \\ \mathbf{0} & \mathbf{I}_{d-k} \end{bmatrix},$$ it can be shown that the variance and the bias will go to zero as $d \rightarrow \infty$ provided that $R \gg \frac{d}{n}$, therefore the double descent, harmless interpolation and good generalization are achieved (see Figure 5 (right)).

Many unanswered questions remain
Similar results to the ones described for linear models have been obtained for linear classification [Muthukumar et al., 2021]. While these types of results for neural networks [Frei et al., 2022] are still limited. Moreover, there are still many open questions on benign overfitting for neural networks. For example, the existing result focuses on $d \gg n$ regimes for neural networks, but there are no results on neural networks over-parameterised in low dimensions by increasing their width. Theoretical statisticians still have plenty of work to do to fully understand these phenomena!

References

Peter L. Bartlett, Philip M. Long, G´abor Lugosi, and Alexander Tsigler. Benign overfitting in linear
regression. Proceedings of the National Academy of Sciences, 117(48):30063–30070, April 2020. ISSN
1091-6490. doi: 10.1073/pnas.1907378117. URL http://dx.doi.org/10.1073/pnas.1907378117.

Spencer Frei, Gal Vardi, Peter L. Bartlett, Nathan Srebro, and Wei Hu. Implicit bias in leaky relu
networks trained on high-dimensional data, 2022.

Vidya Muthukumar, Adhyyan Narang, Vignesh Subramanian, Mikhail Belkin, Daniel Hsu, and Anant
Sahai. Classification vs regression in overparameterized regimes: Does the loss function matter?,
2021.

Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tristan Yang, Boaz Barak, and Ilya Sutskever. Deep
double descent: Where bigger models and more data hurt, 2019.

## Student Perspectives: SPREE Methods for Small Area Estimation

A post by Codie Wood, PhD student on the Compass programme.

This blog post is an introduction to structure preserving estimation (SPREE) methods. These methods form the foundation of my current work with the Office for National Statistics (ONS), where I am undertaking a six-month internship as part of my PhD. During this internship, I am focusing on the use of SPREE to provide small area estimates of population characteristic counts and proportions.

# Small area estimation

Small area estimation (SAE) refers to the collection of methods used to produce accurate and precise population characteristic estimates for small population domains. Examples of domains may include low-level geographical areas, or population subgroups. An example of an SAE problem would be estimating the national population breakdown in small geographical areas by ethnic group [2015_Luna].

Demographic surveys with a large enough scale to provide high-quality direct estimates at a fine-grain level are often expensive to conduct, and so smaller sample surveys are often conducted instead.

SAE methods work by drawing information from different data sources and similar population domains in order to obtain accurate and precise model-based estimates where sample counts are too small for high quality direct estimates. We use the term small area to refer to domains where we have little or no data available in our sample survey.

SAE methods are frequently relied upon for population characteristic estimation, particularly as there is an increasing demand for information about local populations in order to ensure correct allocation of resources and services across the nation.

## Structure preserving estimation

Structure preserving estimation (SPREE) is one of the tools used within SAE to provide population composition estimates. We use the term composition here to refer to a population break down into a two-way contingency table containing positive count values. Here, we focus on the case where we have a population broken down into geographical areas (e.g. local authority) and some subgroup or category (e.g. ethnic group or age).

Orginal SPREE-type estimators, as proposed in [1980_Purcell], can be used in the case when we have a proxy data source for our target composition, containing information for the same set of areas and categories but that may not entirely accurately represent the variable of interest. This is usually because the data are outdated or have a slightly different variable definition than the target.

We also incorporate benchmark estimates of the row and column totals for our composition of interest, taken from trusted, quality assured data sources and treated as known values. This ensures consistency with higher level known population estimates. SPREE then adjusts the proxy data to the estimates of the row and column totals to obtain the improved estimate of the target composition.

An illustration of the data required to produce SPREE-type estimates.

In an extension of SPREE, known as generalised SPREE (GSPREE) [2004_Zhang], the proxy data can also be supplemented by sample survey data to generate estimates that are less subject to bias and uncertainty than it would be possible to generate from each source individually. The survey data used is assumed to be a valid measure of the target variable (i.e. it has the same definition and is not out of date), but due to small sample sizes may have a degree of uncertainty or bias for some cells.

The GSPREE method establishes a relationship between the proxy data and the survey data, with this relationship being used to adjust the proxy compositions towards the survey data.

An illustration of the data required to produce GSPREE estimates.

GSPREE is not the only extension to SPREE-type methods, but those are beyond the scope of this post. Further extensions such as Multivariate SPREE are discussed in detail in [2016_Luna].

### Original SPREE methods

First, we describe original SPREE-type estimators. For these estimators, we require only well-established estimates of the margins of our target composition.

We will denote the target composition of interest by $\mathbf{Y} = (Y{aj})$, where $Y{aj}$ is the cell count for small area $a = 1,\dots,A$ and group $j = 1,\dots,J$. We can write $\mathbf Y$ in the form of a saturated log-linear model as the sum of four terms,

$$\log Y_{aj} = \alpha_0^Y + \alpha_a^Y + \alpha_j^Y + \alpha_{aj}^Y.$$

There are multiple ways to write this parameterisation, and here we use the centered constraints parameterisation given by $$\alpha_0^Y = \frac{1}{AJ}\sum_a\sum_j\log Y_{aj},$$ $$\alpha_a^Y = \frac{1}{J}\sum_j\log Y_{aj} – \alpha_0^Y,$$ $$\alpha_j^Y = \frac{1}{A}\sum_a\log Y_{aj} – \alpha_0^Y,$$ $$\alpha_{aj}^Y = \log Y_{aj} – \alpha_0^Y – \alpha_a^Y – \alpha_j^Y,$$

which satisfy the constraints $\sum_a \alpha_a^Y = \sum_j \alpha_j^Y = \sum_a \alpha_{aj}^Y = \sum_j \alpha_{aj}^Y = 0.$

Using this expression, we can decompose $\mathbf Y$ into two structures:

1. The association structure, consisting of the set of $AJ$ interaction terms $\alpha_{aj}^Y$ for $a = 1,\dots,A$ and $j = 1,\dots,J$. This determines the relationship between the rows (areas) and columns (groups).
2. The allocation structure, consisting of the sets of terms $\alpha_0^Y, \alpha_a^Y,$ and $\alpha_j^Y$ for $a = 1,\dots,A$ and $j = 1,\dots,J$. This determines the size of the composition, and differences between the sets of rows (areas) and columns (groups).

Suppose we have a proxy composition $\mathbf X$ of the same dimensions as $\mathbf Y$, and we have the sets of row and column margins of $\mathbf Y$ denoted by $\mathbf Y_{a+} = (Y_{1+}, \dots, Y_{A+})$ and $\mathbf Y_{+j} = (Y_{+1}, \dots, Y_{+J})$, where $+$ substitutes the index being summed over.

We can then use iterative proportional fitting (IPF) to produce an estimate $\widehat{\mathbf Y}$ of $\mathbf Y$ that preserves the association structure observed in the proxy composition $\mathbf X$. The IPF procedure is as follows:

1. Rescale the rows of $\mathbf X$ as $$\widehat{Y}_{aj}^{(1)} = X_{aj} \frac{Y_{+j}}{X_{+j}},$$
2. Rescale the columns of $\widehat{\mathbf Y}^{(1)}$ as $$\widehat{Y}_{aj}^{(2)} = \widehat{Y}_{aj}^{(1)} \frac{Y_{a+}}{\widehat{Y}_{a+}^{(1)}},$$
3. Rescale the rows of $\widehat{\mathbf Y}^{(2)}$ as $$\widehat{Y}_{aj}^{(3)} = \widehat{Y}_{aj}^{(2)} \frac{Y_{+j}}{\widehat{Y}_{+j}^{(2)}}.$$

Steps 2 and 3 are then repeated until convergence occurs, and we have a final composition estimate denoted by $\widehat{\mathbf Y}^S$ which has the same association structure as our proxy composition, i.e. we have $\alpha_{aj}^X = \alpha_{aj}^Y$ for all $a \in \{1,\dots,A\}$ and $j \in \{1,\dots,J\}.$ This is a key assumption of the SPREE implementation, which in practise is often restrictive, motivating a generalisation of the method.

### Generalised SPREE methods

If we can no longer assume that the proxy composition and target compositions have the same association structure, we instead use the GSPREE method first introduced in [2004_Zhang], and incorporate survey data into our estimation process.

The GSPREE method relaxes the assumption that $\alpha_{aj}^X = \alpha_{aj}^Y$ for all $a \in \{1,\dots,A\}$ and $j \in \{1,\dots,J\},$ instead imposing the structural assumption $\alpha_{aj}^Y = \beta \alpha_{aj}^X$, i.e. the association structure of the proxy and target compositions are proportional to one another. As such, we note that SPREE is a particular case of GSPREE where $\beta = 1$.

Continuing with our notation from the previous section, we proceed to estimate $\beta$ by modelling the relationship between our target and proxy compositions as a generalised linear structural model (GLSM) given by
$$\tau_{aj}^Y = \lambda_j + \beta \tau_{aj}^X,$$ with $\sum_j \lambda_j = 0$, and where \begin{align} \tau_{aj}^Y &= \log Y_{aj} – \frac{1}{J}\sum_j\log Y_{aj},\\ &= \alpha_{aj}^Y + \alpha_j^Y, \end{align} and analogously for $\mathbf X$.

It is shown in [2016_Luna] that fitting this model is equivalent to fitting a Poisson generalised linear model to our cell counts, with a $\log$ link function. We use the association structure of our proxy data, as well as categorical variables representing the area and group of the cell, as our covariates. Then we have a model given by $$\log Y_{aj} = \gamma_a + \tilde{\lambda}_j + \tilde{\beta}\alpha_{aj}^X,$$ with $\gamma_a = \alpha_0^Y + \alpha_a^Y$, $\tilde\lambda_j = \alpha_j^Y$ and $\tilde\beta \alpha_{aj}^X = \alpha_{aj}^Y.$

When fitting the model we use survey data $\tilde{\mathbf Y}$ as our response variable, and are then able to obtain a set of unbenchmarked estimates of our target composition. The GSPREE method then benchmarks these to estimates of the row and column totals, following a procedure analagous to that undertaken in the orginal SPREE methodology, to provide a final set of estimates for our target composition.

# ONS applications

The ONS has used GSPREE to provide population ethnicity composition estimates in intercensal years, where the detailed population estimates resulting from the census are outdated [2015_Luna]. In this case, the census data is considered the proxy data source. More recent works have also used GSPREE to estimate counts of households and dwellings in each tenure at the subnational level during intercensal years [2023_ONS].

My work with the ONS has focussed on extending the current workflows and systems in place to implement these methods in a reproducible manner, allowing them to be applied to a wider variety of scenarios with differing data availability.

# References

[1980_Purcell] Purcell, Noel J., and Leslie Kish. 1980. ‘Postcensal Estimates for Local Areas (Or Domains)’. International Statistical Review / Revue Internationale de Statistique 48 (1): 3–18. https://doi.org/10/b96g3g.

[2004_Zhang] Zhang, Li-Chun, and Raymond L. Chambers. 2004. ‘Small Area Estimates for Cross-Classifications’. Journal of the Royal Statistical Society Series B: Statistical Methodology 66 (2): 479–96. https://doi.org/10/fq2ftt.

[2015_Luna] Luna Hernández, Ángela, Li-Chun Zhang, Alison Whitworth, and Kirsten Piller. 2015. ‘Small Area Estimates of the Population Distribution by Ethnic Group in England: A Proposal Using Structure Preserving Estimators’. Statistics in Transition New Series and Survey Methodology 16 (December). https://doi.org/10/gs49kq.

[2016_Luna] Luna Hernández, Ángela. 2016. ‘Multivariate Structure Preserving Estimation for Population Compositions’. PhD thesis, University of Southampton, School of Social Sciences. https://eprints.soton.ac.uk/404689/.

[2023_ONS] Office for National Statistics (ONS), released 17 May 2023, ONS website, article, Tenure estimates for households and dwellings, England: GSPREE compared with Census 2021 data

## Student Perspectives: Semantic Search

A post by Ben Anson, PhD student on the Compass programme.

## Semantic Search

Semantic search is here. We already see it in use in search engines [13], but what is it exactly and how does it work?

Search is about retrieving information from a corpus, based on some query. You are probably using search technology all the time, maybe $\verb|ctrl+f|$, or searching on google. Historically, keyword search, which works by comparing the occurrences of keywords between queries and documents in the corpus, has been surprisingly effective. Unfortunately, keywords are inherently restrictive – if you don’t know the right one to use then you are stuck.

Semantic search is about giving search a different interface. Semantic search queries are provided in the most natural interface for humans: natural language. A semantic search algorithm will ideally be able to point you to a relevant result, even if you only provided the gist of your desires, and even if you didn’t provide relevant keywords.

Figure 1 illustrates a concrete example where semantic search might be desirable. The query ‘animal’ should return both the dog and cat documents, but because the keyword ‘animal’ is not present in the cat document, the keyword model fails. In other words, keyword search is susceptible to false negatives.

Transformer neural networks turn out to be very effective for semantic search [1,2,3,10]. In this blog post, I hope to elucidate how transformers are tuned for semantic search, and will briefly touch on extensions and scaling.

### The search problem, more formally

Suppose we have a big corpus $\mathcal{D}$ of documents (e.g. every document on wikipedia). A user sends us a query $q$, and we want to point them to the most relevant document $d^*$. If we denote the relevance of a document $d$ to $q$ as $\text{score}(q, d)$, the top search result should simply be the document with the highest score,

$$d^* = \mathrm{argmax}_{d\in\mathcal{D}}\, \text{score}(q, d).$$

This framework is simple and it generalizes. For $\verb|ctrl+f|$, let $\mathcal{D}$ be the set of individual words in a file, and $\text{score}(q, d) = 1$ if $q=d$ and $0$ otherwise. The venerable keyword search algorithm BM25 [4], which was state of the art for decades [8], uses this score function.

For semantic search, the score function is often set as the inner product between query and document embeddings: $\text{score}(q, d) = \langle \phi(q), \phi(d) \rangle$. Assuming this score function actually works well for finding relevant documents, and we use a simple inner product, it is clear that the secret sauce lies in the embedding function $\phi$.

### Transformer embeddings

We said above that a common score function for semantic search is $\text{score}(q, d) = \langle \phi(q), \phi(d) \rangle$. This raises two questions:

• Question 1: what should the inner product be? For semantic search, people tend to use the cosine similarity for their inner product.
•  Question 2: what should $\phi$ be? The secret sauce is to use a transformer encoder, which is explained below.

#### Quick version

Transformers magically gives us a tunable embedding function $\phi: \text{“set of all pieces of text”} \rightarrow \mathbb{R}^{d_{\text{model}}}$, where $d_{\text{model}}$ is the embedding dimension.

#### More detailed version

See Figure 2 for an illustration of how a transformer encoder calculates an embedding for a piece of text. In the figure we show how to encode “cat”, but we can encode arbitrary pieces of text in a similar way. The transformer block details are out of scope here; though, for these details I personally found Attention is All You Need [9] helpful, the crucial part being the Multi-Head Attention which allows modelling dependencies between words.

The transformer encoder is very flexible, with almost every component parameterized by a learnable weight / bias – this is why it can be used to model the complicated semantics in natural language. The pooling step in Figure 2, where we map our sequence embedding $X’$ to a fixed size, is not part of a ‘regular’ transformer, but it is essential for us. It ensures that our score function $\langle \phi(q), \phi(d) \rangle$ will work when $q$ and $d$ have different sizes.

### Making the score function good for search

There is a massive issue with transformer embedding as described above, at least for our purposes – there is no reason to believe it will satisfy simple semantic properties, such as,

$\text{score}(\text{“busy place”}, \text{“tokyo”}) > \text{score}(\text{“busy place”}, \text{“a small village”})$

‘But why would the above not work?’ Because, of course, transformers are typically trained to predict the next token in a sequence, not to differentiate pieces of text according to their semantics.

The solution to this problem is not to eschew transformer embeddings, but rather to fine-tune them for search. The idea is to encourage the transformer to give us embeddings that place semantically dissimilar items far apart. E.g. let $q=$’busy place’, then we want $d^+=$’tokyo’ to be close to $q$ and $d^-=$’a small village’ to be far away.

This semantic separation can be achieved by fine-tuning with a contrastive loss [1,2,3,10],

$$\text{maximize}_{\theta}\,\mathcal{L} = \log \frac{\exp(\text{score}(q, d^+))}{\exp(\text{score}(q, d^+)) + \exp(\text{score}(q, d^-))},$$

where $\theta$ represents the transformer parameters. The $\exp$’s in the contastive loss are to ensure we never divide by zero. Note that we can interpret the contrastive loss as doing classification since we can think of the argument to the logarithm as $p(d^+ | q)$.

That’s all we need, in principle, to turn a transformer encoder into a text embedder! In practice, the contrastive loss can be generalized to include more positive and negative examples, and it is indeed a good idea to have a large batch size [11] (intuitively it makes the separation of positive and negative examples more difficult, resulting in a better classifier). We also need a fine-tuning dataset – a dataset of positive/negative examples. OpenAI showed that it is possible to construct one in an unsupervised fashion [1]. However, there are also publicly available datasets for supervised fine-tuning, e.g. MSMARCO [12].

### Extensions

One really interesting avenue of research is training of general purposes encoders. The idea is to provide instructions alongside the queries/documents [2,3]. The instruction could be $\verb|Embed this document for search: {document}|$ (for the application we’ve been discussing), or $\verb|Embed this document for clustering: {document}|$ to get embeddings suitable for clustering, or $\verb|Embed this document for sentiment analysis: {document}|$ for embeddings suitable for sentiment analysis. The system is fine-tuned end-to-end with the appropriate task, e.g. a contrastive learning objective for the search instruction, a classification objective for sentiment analysis, etc., leaving us with an easy way to generate embeddings for different tasks.

### A note on scaling

The real power of semantic (and keyword) search comes when a search corpus is too large for a human to search manually. However if the corpus is enormous, we’d rather avoid looking at every document each time we get a query. Thankfully, there are methods to avoid this by using specially tailored data structures: see Inverted Indices for keyword algorithms, and Hierarchical Navigable Small World graphs [5] for semantic algorithms. These both reduce search time complexity from $\mathcal{O}(|\mathcal{D}|)$ to $\mathcal{O}(\log |\mathcal{D}|)$, where $|\mathcal{D}|$ is the corpus size.

There are many startups (Pinecone, Weviate, Milvus, Chroma, etc.) that are proposing so-called vector databases – databases in which embeddings are stored, and queries can be efficiently performed. Though, there is also work contesting the need for these types of database in the first place [7].

### Summary

We summarised search, semantic search, and how transformers are fine-tuned for search with a contrastive loss. I personally find this a very nice area of research with exciting real-world applications – please reach out (ben.anson@bristol.ac.uk) if you’d like to discuss it!

### References

[1]: Text and code embeddings by contrastive pre-training, Neelakantan et al (2022)

[2]: Task-aware Retrieval with Instructions, Asai et al (2022)

[3]: One embedder, any task: Instruction-finetuned text embeddings, Su et al (2022)

[4]: Some simple effective approximations to the 2-poisson model for probabilistic weighted retrieval, Robertson and Walker (1994)

[5]: Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs, https://arxiv.org/abs/1603.09320

[6]: An Image is Worth 16×16 Words: Transformers for Image Recognition at Scale, https://arxiv.org/abs/2010.11929

[7]: Vector Search with OpenAI Embeddings: Lucene Is All You Need, arXiv preprint arXiv:2308.14963

[8]: Complement lexical retrieval model with semantic residual embeddings, Advances in Information Retrieval (2021)

[9]: Attention is all you need, Advances in neural information processing systems (2017)

[10]: Sgpt: Gpt sentence embeddings for semantic search, arXiv preprint arXiv:2202.08904

[11]: Contrastive representation learning: A framework and review, IEEE Access 8 (2020)

[12]: Ms marco: A human generated machine reading comprehension dataset, arXiv preprint arXiv:1611.09268

[13]: AdANNS: A Framework for Adaptive Semantic Search, arXiv preprint arXiv:2305.19435

## Student Perspectives: Impurity Identification in Oligonucleotide Drug Samples

A post by Harry Tata, PhD student on the Compass programme.

# Oligonucleotides in Medicine

Oligonucleotide therapies are at the forefront of modern pharmaceutical research and development, with recent years seeing major advances in treatments for a variety of conditions. Oligonucleotide drugs for Duchenne muscular dystrophy (FDA approved) [1], Huntington’s disease (Phase 3 clinical trials) [2], and Alzheimer’s disease [3] and amyotrophic lateral sclerosis (early-phase clinical trials) [4] show their potential for tackling debilitating and otherwise hard-to-treat conditions. With continuing development of synthetic oligonucleotides, analytical techniques such as mass spectrometry must be tailored to these molecules and keep pace with the field.

Working in conjunction with AstraZeneca, this project aims to advance methods for impurity detection and quantification in synthetic oligonucleotide mass spectra. In this blog post we apply a regularised version of the Richardson-Lucy algorithm, an established technique for image deconvolution, to oligonucleotide mass spectrometry data. This allows us to attribute signals in the data to specific molecular fragments, and therefore to detect impurities in oligonucleotide synthesis.

# Oligonucleotide Fragmentation

If we have attempted to synthesise an oligonucleotide $\mathcal O$ with a particular sequence, we can take a sample from this synthesis and analyse it via mass spectrometry. In this process, molecules in the sample are first fragmented — broken apart into ions — and these charged fragments are then passed through an electromagnetic field. The trajectory of each fragment through this field depends on its mass/charge ratio (m/z), so measuring these trajectories (e.g. by measuring time of flight before hitting some detector) allows us to calculate the m/z of fragments in the sample. This gives us a discrete mass spectrum: counts of detected fragments (intensity) across a range of m/z bins [5].

To get an idea of how much of $\mathcal O$ is in a sample, and what impurities might be present, we first need to consider what fragments $\mathcal O$ will produce. Oligonucleotides are short strands of DNA or RNA; polymers with a backbone of sugars (such as ribose in RNA) connected by linkers (e.g. a phosphodiester bond), where each sugar has an attached base which encodes genetic information [6].

On each monomer, there are two sites where fragmentation is likely to occur: at the linker (backbone cleavage) or between the base and sugar (base loss). Specifically, depending on which bond within the linker is broken, there are four modes of backbone cleavage [7,8].
We include in $\mathcal F$ every product of a single fragmentation of $\mathcal O$ — any of the four backbone cleavage modes or base loss anywhere along the nucleotide — as well as the results of every combination of two fragmentations (different cleavage modes at the same linker are mutually exclusive).

# Sparse Richardson-Lucy Algorithm

Suppose we have a chemical sample which we have fragmented and analysed by mass spectrometry. This gives us a spectrum across n bins (each bin corresponding to a small m/z range), and we represent this spectrum with the column vector $\mathbf{b}\in\mathbb R^n$, where $b_i$ is the intensity in the $i^{th}$ bin. For a set $\{f_1,\ldots,f_m\}=\mathcal F$ of possible fragments, let $x_j$ be the amount of $f_j$ that is actually present. We would like to estimate the amounts of each fragment based on the spectrum $\mathbf b$.

If we had a sample comprising a unit amount of a single fragment $f_j$, so $x_j=1$ and $x_{k\ne j}=0,$ and this produced a spectrum $\begin{pmatrix}a_{1j}&\ldots&a_{nj}\end{pmatrix}^T$, we can say the intensity contributed to bin $i$ by $x_j$ is $a_{ij}.$ In mass spectrometry, the intensity in a single bin due to a single fragment is linear in the amount of that fragment, and the intensities in a single bin due to different fragments are additive, so in some general spectrum we have $b_i=\sum_j x_ja_{ij}.$

By constructing a library matrix $\mathbf{A}\in\mathbb R^{n\times m}$ such that $\{\mathbf A\}_{ij}=a_{ij}$ (so the columns of $\mathbf A$ correspond to fragments in $\mathcal F$), then in ideal conditions the vector of fragment amounts $\mathbf x=\begin{pmatrix}x_1&\ldots&x_m\end{pmatrix}^T$ solves $\mathbf{Ax}=\mathbf{b}$. In practice this exact solution is not found — due to experimental noise and potentially because there are contaminant fragments in the sample not included in $\mathcal F$ — and we instead make an estimate $\mathbf {\hat x}$ for which $\mathbf{A\hat x}$ is close to $\mathbf b$.

Note that the columns of $\mathbf A$ correspond to fragments in $\mathcal F$: the values in a single column represent intensities in each bin due to a single fragment only. We $\ell_1$-normalise these columns, meaning the total intensity (over all bins) of each fragment in the library matrix is uniform, and so the values in $\mathbf{\hat x}$ can be directly interpreted as relative abundances of each fragment.

The observed intensities — as counts of fragments incident on each bin — are realisations of latent Poisson random variables. Assuming these variables are i.i.d., it can be shown that the estimate of $\mathbf{x}$ which maximises the likelihood of the system is approximated by the iterative formula

$\mathbf {\hat{x}}^{(t+1)}=\left(\mathbf A^T \frac{\mathbf b}{\mathbf{A\hat x}^{(t)}}\right)\odot \mathbf{\hat x}^{(t)}.$

Here, quotients and the operator $\odot$ represent (respectively) elementwise division and multiplication of two vectors. This is known as the Richardson-Lucy algorithm [9].

In practice, when we enumerate oligonucleotide fragments to include in $\mathcal F$, most of these fragments will not actually be produced when the oligonucleotide passes through a mass spectrometer; there is a large space of possible fragments and (beyond knowing what the general fragmentation sites are) no well-established theory allowing us to predict, for a new oligonucleotide, which fragments will be abundant or negligible. This means we seek a sparse estimate, where most fragment abundances are zero.

The Richardson-Lucy algorithm, as a maximum likelihood estimate for Poisson variables, is analagous to ordinary least squares regression for Gaussian variables. Likewise lasso regression — a regularised least squares regression which favours sparse estimates, interpretable as a maximum a posteriori estimate with Laplace priors — has an analogue in the sparse Richardson-Lucy algorithm:

$\mathbf {\hat{x}}^{(t+1)}=\left(\mathbf A^T \frac{\mathbf b}{\mathbf{A\hat x}^{(t)}}\right)\odot \frac{ \mathbf{\hat x}^{(t)}}{\mathbf 1 + \lambda},$

where $\lambda$ is a regularisation parameter [10].

# Library Generation

For each oligonucleotide fragment $f\in\mathcal F$, we smooth and bin the m/z values of the most abundant isotopes of $f$, and store these values in the columns of $\mathbf A$. However, if these are the only fragments in $\mathcal F$ then impurities will not be identified: the sparse Richardson-Lucy algorithm will try to fit oligonucleotide fragments to every peak in the spectrum, even ones that correspond to fragments not from the target oligonucleotide. Therefore we also include ‘dummy’ fragments corresponding to single peaks in the spectrum — the method will fit these to non-oligonucleotide peaks, showing the locations of any impurities.

# Results

For a mass spectrum from a sample containing a synthetic oligonucleotide, we generated a library of oligonucleotide and dummy fragments as described above, and applied the sparse Richardson-Lucy algorithm. Below, the model fit is plotted alongside the (smoothed, binned) spectrum and the ten most abundant fragments as estimated by the model. These fragments are represented as bars with binned m/z at the peak fragment intensity, and are separated into oligonucleotide fragments and dummy fragments indicating possible impurities. All intensities and abundances are Anscombe transformed ($x\rightarrow\sqrt{x+3/8}$) for clarity.

As the oligonucleotide in question is proprietary, its specific composition and fragmentation is not mentioned here, and the bins plotted have been transformed (without changing the shape of the data) so that individual fragment m/z values are not identifiable.

We see the data is fit extremely closely, and that the spectrum is quite clean: there is one very pronounced peak roughly in the middle of the m/z range. This peak corresponds to one of the oligonucleotide fragments in the library, although there is also an abundant dummy fragment slightly to the left inside the main peak. Fragment intensities in the library matrix are smoothed, and it may be the case that the smoothing here is inappropriate for the observed peak, hence other fragments being fit at the peak edge. Investigating these effects is a target for the rest of the project.

We also see several smaller peaks, most of which are modelled with oligonucleotide fragments. One of these peaks, at approximately bin 5352, has a noticeably worse fit if excluding dummy fragments from the library matrix (see below). Using dummy fragments improves this fit and indicates a possible impurity. Going forward, understanding and quantification of these impurities will be improved by including other common fragments in the library matrix, and by grouping fragments which correspond to the same molecules.

# References

[1] Junetsu Igarashi, Yasuharu Niwa, and Daisuke Sugiyama. “Research and Development of Oligonucleotide Therapeutics in Japan for Rare Diseases”. In: Future Rare Diseases 2.1 (Mar. 2022), FRD19.

[2] Karishma Dhuri et al. “Antisense Oligonucleotides: An Emerging Area in Drug Discovery and Development”. In: Journal of Clinical Medicine 9.6 (6 June 2020), p. 2004.

[3] Catherine J. Mummery et al. “Tau-Targeting Antisense Oligonucleotide MAPTRx in Mild Alzheimer’s Disease: A Phase 1b, Randomized, Placebo-Controlled Trial”. In: Nature Medicine (Apr. 24, 2023), pp. 1–11.

[4] Benjamin D. Boros et al. “Antisense Oligonucleotides for the Study and Treatment of ALS”. In: Neurotherapeutics: The Journal of the American Society for Experimental NeuroTherapeutics 19.4 (July 2022), pp. 1145–1158.

[5] Ingvar Eidhammer et al. Computational Methods for Mass Spectrometry Proteomics. John Wiley & Sons, Feb. 28, 2008. 299 pp.

[6] Harri Lönnberg. Chemistry of Nucleic Acids. De Gruyter, Aug. 10, 2020.

[7] S. A. McLuckey, G. J. Van Berkel, and G. L. Glish. “Tandem Mass Spectrometry of Small, Multiply Charged Oligonucleotides”. In: Journal of the American Society for Mass Spectrometry 3.1 (Jan. 1992), pp. 60–70.

[8] Scott A. McLuckey and Sohrab Habibi-Goudarzi. “Decompositions of Multiply Charged Oligonucleotide Anions”. In: Journal of the American Chemical Society 115.25 (Dec. 1, 1993), pp. 12085–12095.

[9] Mario Bertero, Patrizia Boccacci, and Valeria Ruggiero. Inverse Imaging with Poisson Data: From Cells to Galaxies. IOP Publishing, Dec. 1, 2018.

[10] Elad Shaked, Sudipto Dolui, and Oleg V. Michailovich. “Regularized Richardson-Lucy Algorithm for Reconstruction of Poissonian Medical Images”. In: 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro. Mar. 2011, pp. 1754–1757.

## Student Perspectives: Density Ratio Estimation with Missing Data

A post by Josh Givens, PhD student on the Compass programme.

Density ratio estimation is a highly useful field of mathematics with many applications.  This post describes my research undertaken alongside my supervisors Song Liu and Henry Reeve which aims to make density ratio estimation robust to missing data. This work was recently published in proceedings for AISTATS 2023.

## Density Ratio Estimation

### Definition

As the name suggests, density ratio estimation is simply the task of estimating the ratio between two probability densities. More precisely for two RVs (Random Variables) $Z^0, Z^1$ on some space $\mathcal{Z}$ with probability density functions (PDFs) $p_0, p_1$ respectively, the density ratio is the function $r^*:\mathcal{Z}\rightarrow\mathbb{R}$ defined by

$r^*(z):=\frac{p_0(z)}{p_1(z)}$.

Density ratio estimation (DRE) is then the practice of using IID (independent and identically distributed) samples from $Z^0$ and $Z^1$ to estimate $r^*$. What makes DRE so useful is that it gives us a way to characterise the difference between these 2 classes of data using just 1 quantity, $r^*$.

### The Density Ratio in Classification

We now give demonstrate this characterisability in the case of classification. To frame this as a classification problem define $Y\sim\text{Bernoulli}(0.5)$ and $Z$ by $Z|Y=y\sim Z^{y}$. The task of predicting $Y$ given $Z$ using some function $\phi:\mathcal{Z}\rightarrow\{0,1\}$ is then our standard classification problem. In classification a common target is the Bayes Optimal Classifier, the classifier $\phi^*$ which maximises $\mathbb{P}(Y=\phi(Z)).$ We can write this classifier in terms of $r^*$  as we know that $\phi^*(z)=\mathbb{I}\{\mathbb{P}(Y=1|Z=z)>0.5\}$ where $\mathbb{I}$ is the indicator function. Then, by the total law of probability, we have

$\mathbb{P}(Y=1|Z=z)=\frac{p_{Z|Y=1}(z)\mathbb{P}(Y=1)}{p_{Z|Y=1}(z)\mathbb{P}(Y=1)+p_{Z|Y=0}(z)\mathbb{P}(Y=0)}$

$=\frac{p_1(z)\mathbb{P}(Y=1)}{p_1(z)\mathbb{P}(Y=1)+p_0(z)\mathbb{P}(Y=0)} =\frac{1}{1+\frac{1}{r}\frac{\mathbb{P}(Y=0)}{\mathbb{P}(Y=1)}}.$

Hence to learn the Bayes optimal classifier it is sufficient to learn the density ratio and a constant. This pattern extends well beyond Bayes optimal classification to many other areas such as error controlled classification, GANs, importance sampling, covariate shift, and others.  Generally speaking, if you are in any situation where you need to characterise the difference between two classes of data, it’s likely that the density ratio will make an appearance.

### Estimation Implementation – KLIEP

Now we have properly introduced and motivated DRE, we need to look at how we can go about performing it. We will focus on one popular method called KLIEP here but there are a many different methods out there (see Sugiyama et al 2012 for some additional examples.)

The intuition behind KLIEP is simple: as $r^* \cdot p_0=p_1$, if $\hat r\cdot p_0$ is “close” to $p_1$ then $\hat r$ is a good estimate of $r^*$. To measure this notion of closeness KLIEP uses the KL (Kullback-Liebler) divergence which measures the distance between 2 probability distributions. We can now formulate our ideal KLIEP objective as follows:

$\underset{r}{\text{min}}~ KL(p_1|p_0\cdot r)$

$\text{subject to:}~ \int_{\mathcal{Z}}r(z)p_0(z)\mathrm{d}z=1$

where $KL(p|p')$ represent the KL divergence from $p$ to $p'$. The constraint  ensures that the right hand side of our KL divergence is indeed a PDF. From the definition of the KL-divergence we can rewrite the solution to this as $\hat r:=\frac{\tilde r}{\mathbb{E}[r(X^0)]}$ where $\tilde r$ is the solution to the unconstrained optimisation

$\underset{r}{\text{min}}~\mathbb{E}[\log (r(Z^1))]-\log(\mathbb{E}[r(Z^0)]).$

As this is now just an unconstrained optimisation over expectations of known transformations of $Z^0, Z^1$, we can approximate this using samples. Given samples $z^0_1,\dotsc,z^0_n$ from $Z_0$ and samples $z^1_1,\dotsc,z^1_n$ from $Z_1$ our estimate of the density ratio will be $\hat r:=\left(\frac{1}{n}\sum_{i=1}^nr(z_i^0)\right)^{-1}\tilde r$  where $\tilde r$ solves

$\underset{r}{\min}~ \frac{1}{n}\sum_{i=1}^n \log(r(z^1_i))-\log\left(\frac{1}{n}\sum_{i=1}^n r(z^0_i)\right).$

Despite KLIEP being commonly used, up until now it has not been made robust to missing not at random data. This is what our research aims to do.

## Missing Data

Suppose that instead of observing samples from $Z$, we observe samples from some corrupted version of $Z$, $X$. We assume that $\mathbb{P}(\{X=\varnothing\}\cup \{X=Z\})=1$ so that either $X$ is missing or $X$ takes the value of $Z$. We also assume that whether $X$ is missing depends upon the value of $Z$. Specifically we assume $\mathbb{P}(X=\varnothing|Z=z)=\varphi(z)$ with $\varphi(z)$ not constant and refer to $\varphi$ as the missingness function. This type of missingness is known as missing not at random (MNAR) and when dealt with improperly can lead to biased result. Some examples of MNAR data could be readings take from a medical instrument which is more likely to err when attempting to read extreme values or recording responses to a questionnaire where respondents may be more likely to not answer if the deem their response to be unfavourable. Note that while we do not see what the true response would be, we do at least get a response meaning that we know when an observation is missing.

## Missing Data with DRE

We now go back to density ratio estimation in the case where instead of observing samples from  $Z^0,Z^1$  we observe samples from their corrupted versions $X^0, X^1$. We take their respective missingness functions to be $\varphi_0, \varphi_1$ and assume them to be known. Now let us look at what would happen if we implemented KLIEP with the data naively by simply filtering out the missing-values. In this case, the actual density ratio we would be estimating would be

$r'(z):=\frac{p_{X_1|X_1\neq\varnothing}(z)}{p_{X_0|X_o\neq\varnothing}(z)}\propto\frac{(1-\varphi_1(z))p_1(z)}{(1-\varphi_0(z))p_0(z)}\not{\propto}r^*(z)$

and so we would get inaccurate estimates of the density ratio no matter how many samples are used to estimate it. The image below demonstrates this in the case were samples in class $1$ are more likely to be missing when larger and class $0$ has no missingness.

### Our Solution

Our solution to this problem is to use importance weighting. Using relationships between the densities of $X$ and $Z$ we have that

$\mathbb{E}[g(Z)]=\mathbb{E}\left[\frac{\mathbb{I}\{X\neq\varnothing\}g(X)}{1-\varphi(X)}\right].$

As such we can re-write the KLIEP objective to keep our expectation estimation unbiased even when using these corrupted samples. This gives our modified objective which we call M-KLIEP as follows. Given samples $x^0_1,\dotsc,x^0_n$ from $X_0$ and samples $x^1_1,\dotsc,x^1_n$ from $X_1$ our estimate is $\hat r=\left(\frac{1}{n}\sum_{i=1}^n\frac{\mathbb{I}\{x_i^0\neq\varnothing\}r(x_i^0)}{1-\varphi_o(x_i^o)}\right)^{-1}\tilde r$ where $\tilde r$ solves

$\underset{r}{\min}~\frac{1}{n}\sum_{i=1}^n\frac{\mathbb{I}\{x_i^1\neq\varnothing\}\log(r(x_i^1))}{1-\varphi_1(x_i^1)}-\log\left(\frac{1}{n}\sum_{i=1}^n\frac{\mathbb{I}\{x_i^0\neq\varnothing\}r(x_i^0)}{1-\varphi_0(x_i^0)}\right).$

This objective will now target $r^*$ even when used on MNAR data.

### Application to Classification

We now apply our density ratio estimation on MNAR data to estimate the Bayes optimal classifier. Below shows a plot of samples alongside the true Bayes optimal classifier and estimated classifiers from the samples via our method M-KLIEP and a naive method CC-KLIEP which simply ignores missing points. Missing data points are faded out.

As we can see, due to not accounting for the MNAR nature of the data, CC-KLIEP underestimates the true number of class 1 samples in the top left region and therefore produces a worse classifier than our approach.

As well as this modified objective our paper provides the following additional contributions:

• Theoretical finite sample bounds on the accuracy of our modified procedure.
• Methods for learning the missingness functions $\varphi_1,\varphi_0$.
• Expansions to partial missingness via a Naive-Bayes framework.
• Downstream implementation of our method within Neyman-Pearson classification.
• Adaptations to Neyman-Pearson classification itself making it robust to MNAR data.

For more details see our paper and corresponding github repository. If you have any questions on this work feel free to contact me at josh.givens@bristol.ac.uk.

### References

Givens, J., Liu, S., & Reeve, H. W. J. (2023). Density ratio estimation and neyman pearson classification with missing data. In F. Ruiz, J. Dy, & J.-W. van de Meent (Eds.), Proceedings of the 26th international conference on artificial intelligence and statistics (Vol. 206, pp. 8645–8681). PMLR.
Sugiyama, M., Suzuki, T., & Kanamori, T. (2012). Density Ratio Estimation in Machine Learning. Cambridge University Press.

## Compass students attending AISTATS 2023 in Valencia

We (Ed Davis, Josh Givens, Alex Modell, and Hannah Sansford) attended the 2023 AISTATS conference in Valencia in order to explore the interesting research being presented as well as present some of our own work. While we talked about our work being published at the conference in this earlier blog post, having now attended the conference, we thought we’d talk about our experience there. We’ll spotlight some of the talks and posters which interested us most and talk about our highlights of Valencia as a whole.

## Talks & Posters

### Mode-Seeking Divergences: Theory and Applications to GANs

One especially interesting talk and poster at the conference was presented by Cheuk Ting Li on their work in collaboration with Farzan Farnia. This work aims to set up a formal classification for various probability measure divergences (such as f-Divergences, Wasserstein distance, etc.) in terms of there mode-seeking or mode-covering properties. By mode-seeking/mode-covering we mean the behaviour of the divergence when used to fit a unimodal distribution to a multi-model target. Specifically a mode-seeking divergence will encourage the target distribution to fit just one of the modes ignoring the other while a mode covering divergence will encourage the distribution to cover all modes leading to less accurate fitting on an individual mode but better covering the full support of the distribution. While these notions of mode-seeking and mode-covering divergences had been discussed before, up to this point there seems to be no formal definition for these properties, and disagreement on the appropriate categorisation of some divergences. This work presents such a definition and uses it to categorise many of the popular divergence methods. Additionally they show how an additive combination a mode seeking f-divergence and the 1-Wasserstein distance retain the mode-seeking property of the f-divergence while being implementable using only samples from our target distribution (rather than knowledge of the distribution itself) making it a desirable divergence for use with GANs.

### Using Sliced Mutual Information to Study Memorization and Generalization inDeep Neural Networks

The benefit of attending large conferences like AISTATS is having the opportunity to hear talks that are not related to your main research topic. This was the case with a talk by Wongso et. al. was one such talk. Although it did not overlap with any of our main research areas, we all found this talk very interesting.
The talk was on the topic of tracking memorisation and generalisation in deep neural networks (DNNs) through the use of /sliced mutual information/. Mutual information (MI) is commonly used in information theory and represents the reduction of uncertainty about one random variable, given the knowledge of the other. However, MI is hard to estimate in high dimensions, which makes it a prohibitive metric for use in neural networks.
Enter sliced mutual information (SMI). This metric is the average of the MI terms between their one-dimensional projections. The main difference between SMI and MI is that SMI is scalable to high dimensions and scales faster than MI.
Next, let’s talk about memorisation. Memorisation is known to occur in DNNs and is where the DNN fits random labels in training as it has memorised noisy labels in training, leading to bad generalisation. The authors demonstrate this behaviour by fitting a multi-layer perceptron to the MNIST dataset with various amounts of label noise. As the noise increased, the difference between the training and test accuracy became greater.
As the label noise increases, the MI between the features and target variable does not change, meaning that MI did not track the loss in generalisation. However, the authors show that the SMI did track the generalisation. As the label noise increased, the SMI decreased significantly as the MLP’s generalisation got worse. Their main theorem showed that SMI is lower-bounded by a term which includes the spherical soft-margin separation, a quantity which is used to track memorisation and generalisation!
In summary, unlike MI, SMI can track memorization and generalisation in DNNs. If you’d like to know more, you can find the full paper here: https://proceedings.mlr.press/v206/wongso23a.html.

### Invited Speakers and the Test of Time Award

As well as the talks on papers that had been selected for oral presentation, each day began with a (longer) invited talk which, for many of us, were highlights of each day. The invited speakers were extremely engaging and covered varied and interesting topics; from Arthur Gretton (UCL) presenting ‘Causal Effect Estimation with Context and Confounders’ to Shakir Mohamed (DeepMind) presenting ‘Elevating our Evaluations: Technical and Sociotechnical Standards of Assessment in Machine Learning’. A favourite amongst us was a talk from Tamara Broderick (MIT) titled ‘An Automatic Finite-Sample Robustness Check: Can Dropping a Little Data Change Conclusions?’. In this talk she addressed a worry that researchers might have when the goal is to analyse a data sample and apply any conclusions to a new population: was a small proportion of the data sample instrumental to the original conclusion? Tamara and collorators propose a method to assess the sensitivity of statistical conclusions to the removal of a very small fraction of the data set. They find that sensitivity is driven by a signal-to-noise ratio in the inference problem, does not disappear asymptotically, and is not decided by misspecification. In experiments they find that many data analyses are robust, but that the conclusions of severeal influential economics papers can be changed by removing (much) less than 1% of the data! A link to the talk can be found here: https://youtu.be/QYtIEqlwLHE

This year, AISTATS featured a Test of Time Award to recognise a paper from 10 years ago that has had a prominent impact in the field. It was awarded to Andreas Damianou and Neil Lawrence for the paper ‘Deep Gaussian Processes’, and their talk was a definite highlight of the conferece. Many of us had seen Neil speak at a seminar at the University of Bristol last year and, being the engaging speaker he is, we were looking forward to hearing him speak again. Rather than focussing on the technical details of the paper, Neils talk concentrated on his (and the machine learning community’s) research philosophy in the years preceeding writing the paper, and how the paper came about – a very interesting insight, and a refreshing break from the many technical talks!

## Valencia

There was so much to like about Valencia even from our short stay there. We’ll try and give you a very brief highlight of our favourite things.

### Food & Drink:

Obviously Valencia is renowned for being the birthplace of paella and while the paella was good we sampled many other delights our stay. Our collective highlight was the nicest Burrata any of us had ever had which, in a stunning display of individualism, all four of us decided to get on our first day at the conference.

### Beach:

About half an hours tram ride from the conference centre are the beaches of Valencia. These stretch for miles as well as having a good 100m in depth with (surprisingly hot) sand covering the lot. We visited these after the end of the conference on the Thursday and despite it being the only cloudy day of the week it was a perfect way to relax at the end of a hectic few days with the pleasantly temperate water being an added bonus.

### Architecture:

Valencia has so much interesting architecture scattered around the city centre. One of the most memorable remarkable places was the San Nicolás de Bari and San Pedro Mártir (Church of San Nicolás) which is known as the Sistine chapel of Valencia (according to the audio-guide for the church at least) with its incredible painted ceiling and live organ playing.

## Student Perspectives: Intro to Recommendation Systems

A post by Hannah Sansford, PhD student on the Compass programme.

# Introduction

Like many others, I interact with recommendation systems on a daily basis; from which toaster to buy on Amazon, to which hotel to book on booking.com, to which song to add to a playlist on Spotify. They are everywhere. But what is really going on behind the scenes?

Recommendation systems broadly fit into two main categories:

1) Content-based filtering. This approach uses the similarity between items to recommend items similar to what the user already likes. For instance, if Ed watches two hair tutorial videos, the system can recommend more hair tutorials to Ed.

2) Collaborative filtering. This approach uses the the similarity between users’ past behaviour to provide recommendations. So, if Ed has watched similar videos to Ben in the past, and Ben likes a cute cat video, then the system can recommend the cute cat video to Ed (even if Ed hasn’t seen any cute cat videos).

Both systems aim to map each item and each user to an embedding vector in a common low-dimensional embedding space $E = \mathbb{R}^d$. That is, the dimension of the embeddings ($d$) is much smaller than the number of items or users. The hope is that the position of these embeddings captures some of the latent (hidden) structure of the items/users, and so similar items end up ‘close together’ in the embedding space. What is meant by being ‘close’ may be specified by some similarity measure.

# Collaborative filtering

In this blog post we will focus on the collaborative filtering system. We can break it down further depending on the type of data we have:

1) Explicit feedback data: aims to model relationships using explicit data such as user-item (numerical) ratings.

2) Implicit feedback data: analyses relationships using implicit signals such as clicks, page views, purchases, or music streaming play counts. This approach makes the assumption that: if a user listens to a song, for example, they must like it.

The majority of the data on the web comes from implicit feedback data, hence there is a strong demand for recommendation systems that take this form of data as input. Furthermore, this form of data can be collected at a much larger scale and without the need for users to provide any extra input. The rest of this blog post will assume we are working with implicit feedback data.

## Problem Setup

Suppose we have a group of $n$ users $U = (u_1, \ldots, u_n)$ and a group of $m$ items $I = (i_1, \ldots, i_m)$. Then we let $\mathbf{R} \in \mathbb{R}^{n \times m}$ be the ratings matrix where position $R_{ui}$ represents whether user $u$ interacts with item $i$. Note that, in most cases the matrix $\mathbf{R}$ is very sparse, since most users only interact with a small subset of the full item set $I$. For any items $i$ that user $u$ does not interact with, we set $R_{ui}$ equal to zero. To be clear, a value of zero does not imply the user does not like the item, but that they have not interacted with it. The final goal of the recommendation system is to find the best recommendations for each user of items they have not yet interacted with.

## Matrix Factorisation (MF)

A simple model for finding user emdeddings, $\mathbf{X} \in \mathbb{R}^{n \times d}$, and item embeddings, $\mathbf{Y} \in \mathbb{R}^{m \times d}$, is Matrix Factorisation. The idea is to find low-rank embeddings such that the product $\mathbf{XY}^\top$ is a good approximation to the ratings matrix $\mathbf{R}$ by minimising some loss function on the known ratings.

A natural loss function to use would be the squared loss, i.e.

$L(\mathbf{X}, \mathbf{Y}) = \sum_{u, i} \left(R_{ui} - \langle X_u, Y_i \rangle \right)^2.$

This corresponds to minimising the Frobenius distance between $\mathbf{R}$ and its approximation $\mathbf{XY}^\top$, and can be solved easily using the singular value decomposition $\mathbf{R} = \mathbf{U S V}^\top$.

Once we have our embeddings $\mathbf{X}$ and $\mathbf{Y}$, we can look at the row of $\mathbf{XY}^\top$ corresponding to user $u$ and recommend the items corresponding to the highest values (that they haven’t already interacted with).

## Logistic MF

Minimising the loss function in the previous section is equivalent to modelling the probability that user $u$ interacts with item $i$ as the inner product $\langle X_u, Y_i \rangle$, i.e.

$R_{ui} \sim \text{Bernoulli}(\langle X_u, Y_i \rangle),$

and maximising the likelihood over $\mathbf{X}$ and $\mathbf{Y}$.

In a research paper from Spotify [3], this relationship is instead modelled according to a logistic function parameterised by the sum of the inner product above and user and item bias terms, $\beta_u$ and $\beta_i$,

$R_{ui} \sim \text{Bernoulli} \left( \frac{\exp(\langle X_u, Y_i \rangle + \beta_u + \beta_i)}{1 + \exp(\langle X_u, Y_i \rangle + \beta_u + \beta_i)} \right).$

## Relation to my research

A recent influential paper [1] proved an impossibility result for modelling certain properties of networks using a low-dimensional inner product model. In my 2023 AISTATS publication [2] we show that using a kernel, such as the logistic one in the previous section, to model probabilities we can capture these properties with embeddings lying on a low-dimensional manifold embedded in infinite-dimensional space. This has various implications, and could explain part of the success of Spotify’s logistic kernel in producing good recommendations.

### References

[1] Seshadhri, C., Sharma, A., Stolman, A., and Goel, A. (2020). The impossibility of low-rank representations for triangle-rich complex networks. Proceedings of the National Academy of Sciences, 117(11):5631–5637.

[2] Sansford, H., Modell, A., Whiteley, N., and Rubin-Delanchy, P. (2023). Implications of sparsity and high triangle density for graph representation learning. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:5449-5473.

[3] Johnson, C. C. (2014). Logistic matrix factorization for implicit feedback data. Advances in Neural Information Processing Systems, 27(78):1–9.

## Compass students attending the Workshop on Functional Inference and Machine Intelligence (FIMI) at ISM Tokyo

A post by Compass CDT students Edward Milsom, Jake Spiteri, Jack Simons, and Sam Stockman.

### We (Edward Milsom, Jake Spiteri, Jack Simons, Sam Stockman) attended the 2023 Workshop on Functional Inference and Machine Intelligence (FIMI) taking place on the 14, 15 and 16th of March at the Institute of Statistical Mathematics in Tokyo, Japan. Our attendance to the workshop was to further collaborative ties between the two institutions. The in-person participants included many distinguished academics from around Japan as well as our very own Dr Song Liu. Due to the workshops modest size, there was an intimate atmosphere which nurtured many productive research discussions. Whilst staying in Tokyo, we inevitably sampled some Japanese culture, from Izakayas to cherry blossoms and sumo wrestling!

We thought we’d share some of our thoughts and experiences. We’ll first go through some of our most memorable talks, and then talk about some of our activities outside the workshop.

# Talks

## Sho Sonoda – Ridgelet Transforms for Neural Networks on Manifolds and Hilbert Spaces

We particularly enjoyed the talk given by Sho Sonoda, a Research Scientist from the Deep Learning Theory group at Riken AIP on “Ridgelet Transforms for Neural Networks on Manifolds and Hilbert Spaces.” Sonoda’s research aims to demystify the black box nature of neural networks, shedding light on how they work and their universal approximation capabilities. His talk provided valuable insights into the integral representations of neural networks, and how they can be represented using ridgelet transforms. Sonoda presented a reconstruction formula from which we see that if a neural network can be represented using ridgelet transforms, then it is a universal approximator. He went on to demonstrate that various types of networks, such as those on finite fields, group convolutional neural networks (GCNNs), and networks on manifolds and Hilbert spaces, can be represented in this manner and are thus universal approximators. Sonoda’s work improves upon existing universality theorems by providing a more unified and direct approach, as opposed to the previous case-by-case methods that relied on manual adjustments of network parameters or indirect conversions of (G)CNNs into other universal approximators, such as invariant polynomials and fully-connected networks. Sonoda’s work is an important step toward a more transparent and comprehensive understanding of neural networks.

## Greg Yang – The unreasonable effectiveness of mathematics in large scale deep learning

Greg Yang is a researcher at Microsoft Research who is working on a framework for understanding neural networks called “tensor programs”. Similar to Neural Tangent Kernels and Neural Network Gaussian Processes, the tensor program framework allows us to consider neural networks in the infinite-width limit, where it becomes possible to make statements about the properties of very wide networks. However, tensor programs aim to unify existing work on infinite-width neural networks by allowing one to take the infinite limit of a much wider range of neural network architectures using one single framework.

In his talk, Yang discussed his most recent work in this area, concerning the “maximal update parametrisation”. In short, they show that in this parametrisation, the optimal hyperparameters of very wide neural networks are the same as those for much smaller neural networks. This means that hyperparameter search can be done using small, cheap models, and then applied to very large models like GPT-3, where hyperparameter search would be too expensive. The result is summarised in this figure from their paper “Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer”, which shows how this is not possible in the standard parametrisation. This work was only possible by building upon the tensor program framework, thereby demonstrating the value of having a solid theoretical understanding of neural networks.

# Statistical Seismology Seminar Series

In addition to the workshop, Sam attended the 88th Statistical Seismology seminar in the Risk Analysis Research Centre at ISM https://www.ism.ac.jp/~ogata/Ssg/ssg_statsei_seminarsE.html. The Statistical Seismology Research Group at ISM was created by Emeritus Professor Yosihiko Ogata and is one of the leading global research institutes for statistical seismology. Its most significant output has been the Epidemic-Type Aftershock Sequence (ETAS) model, a point process based earthquake forecasting model that has been the most dominant model for forecasting since its creation by Ogata in 1988.

As part of the Seminar series, Sam gave a talk on his most recent work (Forecasting the 2016-2017 Central Apennines Earthquake Sequence with a Neural Point Process’, https://arxiv.org/abs/2301.09948) to the research group and other visiting academics.

Japan’s interest is earthquake science is due to the fact that they record the most earthquakes in the world. The whole country is in a very active seismic area, and they have the densest seismic network. So even though they might not actually have the most earthquakes in the world (which is most likely Indonesia) they certainly document the most. The evening before flying back to the UK, Sam and Jack felt a magnitude 5.2 earthquake 300km north of Tokyo in the Miyagi prefecture. At that distance all that was felt was a small shudder…

# Japan

It’s safe to say that the abundance of delicious food was the most memorable aspect of our trip. In fact, we never had a bad meal! Our taste buds were taken on a culinary journey as we tried a variety of Japanese dishes. From hearty, broth-based bowls of ramen and tsukemen, to fun conveyor-belt sushi restaurants, and satisfying tonkatsu (breaded deep-fried pork cutlet) with sticky rice or spicy udon noodles, we were never at a loss for delicious options. We even had the opportunity to cook our own food at an indoor barbecue!

Aside from the food, we thoroughly enjoyed our time in Tokyo – exploring the array of second-hand clothes shops, relaxing in bath-houses, and trying random things from the abundance of vending machines.

24th May 2023
11:00 - 14:00

## Congratulations to Compass students Josh Givens, Hannah Sansford and Alex Modell who, along with their supervisors have had their papers accepted to be published at AISTATS 2023.

### ‘Implications of sparsity and high triangle density for graph representation learning’

Hannah Sansford, Alexander Modell, Nick Whiteley, Patrick Rubin-Delanchy

Hannah: In this paper we explore the implications of two common characteristics of real-world networks, sparsity and triangle-density, for graph representation learning. An example of where these properties arise in the real-world is in social networks, where, although the number of connections each individual has compared to the size of the network is small (sparsity), often a friend of a friend is also a friend (triangle-density). Our result counters a recent influential paper that shows the impossibility of simultaneously recovering these properties with finite-dimensional representations of the nodes, when the probability of connection is modelled by the inner-product. We, by contrast, show that it is possible to recover these properties using an infinite-dimensional inner-product model, where representations lie on a low-dimensional manifold. One of the implications of this work is that we can ‘zoom-in’ to local neighbourhoods of the network, where a lower-dimensional representation is possible.

The paper has been selected for oral presentation at the conference in Valencia (<2% of submissions).

### Density Ratio Estimation and Neyman Pearson Classification with Missing Data

Josh Givens, Song Liu, Henry W J Reeve

Josh: In our paper we adapt the popular density ratio estimation procedure KLIEP to make it robust to missing not at random (MNAR) data and demonstrate its efficacy in Neyman-Pearson (NP) classification. Density ratio estimation (DRE) aims to characterise the difference between two classes of data by estimating the ratio between their probability densities. The density ratio is a fundamental quantity in statistics appearing in many settings such as classification, GANs, and covariate shift making its estimation a valuable goal. To our knowledge there is no prior research into DRE with MNAR data, a missing data paradigm where the likelihood of an observation being missing depends on its underlying value. We propose the estimator M-KLIEP and provide finite sample bounds on its accuracy which we show to be minimax optimal for MNAR data. To demonstrate the utility of this estimator we apply it the the field of NP classification. In NP classification we aim to create a classifier which strictly controls the probability of incorrectly classifying points from one class. This is useful in any setting where misclassification for one class is much worse than the other such as fault detection on a production line where you would want to strictly control the probability of classifying a faulty item as non-faulty. In addition to showing the efficacy of our new estimator in this setting we also provide an adaptation to NP classification which allows it to still control this misclassification probability even when fit using MNAR data.