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AutoAI: Systems, Machine Learning and Mathematics

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\newcommand{\weightedAdjacencyScalar}{a} \newcommand{\weightedAdjacencyVector}{\mathbf{ \weightedAdjacencyScalar}} \newcommand{\onesVector}{\mathbf{1}} \newcommand{\zerosVector}{\mathbf{0}} $$
at Isaac Newton Institute Virtual Christmas Dinner on Dec 16, 2020 [reveal]
Neil D. Lawrence, University of Cambridge

Abstract

Deployed artificial intelligence solutions consist of interacting components often trained as the result of supervised machine learning. Automatic training of these sub-components is known as AutoML. But the real world challenges of deployment consist of the monitoring of system performance in the real world, in terms of accuracy but also for fairness and bias. To make such systems easily maintainable there is a need for automation of the process of monitoring and redeploying models as well as checking the quality of the overall system decomposition. In contrast to AutoML, we call this system-wide approach “Auto AI”. This is the subject of my Turing Fellowship

The Great AI Fallacy

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There is a lot of variation in the use of the term artificial intelligence. I’m sometimes asked to define it, but depending on whether you’re speaking to a member of the public, a fellow machine learning researcher, or someone from the business community, the sense of the term differs.

However, underlying its use I’ve detected one disturbing trend. A trend I’m beginining to think of as “The Great AI Fallacy”.

The fallacy is associated with an implicit promise that is embedded in many statements about Artificial Intelligence. Artificial Intelligence, as it currently exists, is merely a form of automated decision making. The implicit promise of Artificial Intelligence is that it will be the first wave of automation where the machine adapts to the human, rather than the human adapting to the machine.

How else can we explain the suspension of sensible business judgment that is accompanying the hype surrounding AI?

This fallacy is particularly pernicious because there are serious benefits to society in deploying this new wave of data-driven automated decision making. But the AI Fallacy is causing us to suspend our calibrated skepticism that is needed to deploy these systems safely and efficiently.

The problem is compounded because many of the techniques that we’re speaking of were originally developed in academic laboratories in isolation from real-world deployment.

Figure: We seem to have fallen for a perspective on AI that suggests it will adapt to our schedule, rather in the manner of a 1930s manservant.

Intellectual Debt

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Figure: Jonathan Zittrain’s term to describe the challenges of explanation that come with AI is Intellectual Debt.

In computer systems the concept of technical debt has been surfaced by authors including Sculley et al. (2015). It is an important concept, that I think is somewhat hidden from the academic community, because it is a phenomenon that occurs when a computer software system is deployed.

Separation of Concerns

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To construct such complex systems an approach known as “separation of concerns” has been developed. The idea is that you architect your system, which consists of a large-scale complex task, into a set of simpler tasks. Each of these tasks is separately implemented. This is known as the decomposition of the task.

This is where Jonathan Zittrain’s beautifully named term “intellectual debt” rises to the fore. Separation of concerns enables the construction of a complex system. But who is concerned with the overall system?

  • Technical debt is the inability to maintain your complex software system.

  • Intellectual debt is the inability to explain your software system.

It is right there in our approach to software engineering. “Separation of concerns” means no one is concerned about the overall system itself.

FIT Models to FIT Systems

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Zittrain points out the challenge around the lack of interpretability of individual ML models as the origin of intellectual debt. In machine learning I refer to work in this area as fairness, interpretability and transparency or FIT models. To an extent I agree with Zittrain, but if we understand the context and purpose of the decision making, I believe this is readily put right by the correct monitoring and retraining regime around the model. A concept I refer to as “progression testing”. Indeed, the best teams do this at the moment, and their failure to do it feels more of a matter of technical debt rather than intellectual, because arguably it is a maintenance task rather than an explanation task. After all, we have good statistical tools for interpreting individual models and decisions when we have the context. We can linearise around the operating point, we can perform counterfactual tests on the model. We can build empirical validation sets that explore fairness or accuracy of the model.

So, this is where, my understanding of intellectual debt in ML systems departs, I believe from John Zittrain’s. The long-term challenge is not in the individual model. We have excellent statistical tools for validating what any individual model, the long-term challenge is the complex interaction between different components in the decomposed system, where the original intent of each component has been forgotten (except perhaps by Lancelot) and each service has been repurposed. We need to move from FIT models to FIT systems.

How to address these challenges? With collaborators I’ve been working towards a solution that contains broadly two parts. The first part is what we refer to as “Data-Oriented Architectures”. The second part is “meta modelling”, machine learning techniques that help us model the models.

Buying System

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An example of a complex decision making system might be an automated buying system. In such a system, the idea is to match demand for products to supply of products.

The matching of demand and supply is a repetetive theme for decision making systems. Not only does it occur in automated buying, but also in the allocation of drivers to riders in a ride sharing system. Or in the allocation of compute resource to users in a cloud system.

The components of any of these system include: predictions of the demand for the product, or the drivers or the compute. Then predictions of the supply. Decisions are then made for how much material to keep in stock, or how many drivers to have on the road, or how much computer capacity to have in your data centres. These decisions have cost implications. The optimal amount of product will depend on the cost of making it available. For a buying system this is the storage costs.

Decisions are made on the basis of the supply and demand to make new orders, to encourage more drivers to come into the system or to build new data centers or rent more computational power.

Figure: The components of a putative automated buying system

Monolithic System

The classical approach to building these systems was a ‘monolithic system’. Built in a similar way to the successful applicaitons software such as Excel or Word, or large operating systems, a single code base was constructed. The complexity of such code bases run to many lines.

In practice, shared dynamically linked libraries may be used for aspects such as user interface, or networking, but the software often has many millions of lines of code. For example, the Microsoft Office suite is said to contain over 30 millions of lines of code.

Figure: A potential path of models in a machine learning system.

Service Oriented Architecture

Such software is not only difficult to develop, it is difficult to scale when computation demands increase. Amazon’s original website software (called Obidos) was a monolithic design but by the early noughties it was becoming difficult to sustain and maintain. The software was phased out in 2006 to be replaced by a modularized software known as a ‘service oriented architecture’.

In Service Oriented Architecture, or “Software as a Service” the idea is that code bases are modularized and communicate with one another using network requests. A standard approach is to use a REST API. So, rather than a single monolithic code base, the code is developed with individual services that handle the different requests.

Figure: A potential path of models in a machine learning system.

This is the landscape we now find ourselves in with regard to software development. In practice, each of these services is often ‘owned’ and maintained by an individual team. The team is judged by the quality of their service provision. They work to detailed specifications on what their service should output, what its availability should be and other objectives like speed of response. This allows for conditional independence between teams and for faster development.

Buying to Banking

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The same model we consider for buying, can also be considered in the case of, for example, a banking application. In a typical banking application, we receive loan requests from customers. For an individual customer, before making a loan, the bank may wish to make a forecast around their costs (expenditures on food, housing, entertainment etc) and their income (salary, rental income etc). These forecasts would inform the conditions of the loan. For example how much the bank is willing to lend, and under what interest rates and repayment conditions. These terms will be based on previous experience of loaning, but also constrained by regulatory conditions often imposed by a financial regulator.

Figure: A potential path of models in a machine learning system where a decision about a loan is being made on the basis of (potentially personal) data from a customer.

In many regulatory environments, the bank will be restricted in terms of what information they are allowed to use in dictating loan terms. For example, with in the EU there are prohibited characteristics such as race, gender, sexuality, religion and health status which cannot be used (even indirectly) for making the loan. Along with stipulating these characteristics, the badly-named GDPR1 also gives particular stipulations for rights individuals have for explanation around consequential decisions, such as obtaining a loan.

The challenge of Intellectual Debt means that it’s possible for a bank to produce an automated loan decision system, which even the bank itself doesn’t understand, which makes it rather hard to conform to the intent of the GDPR which requires the bank to explain to customers the reasoning behind decisions based on personal data.

Statistical Emulation

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Figure: The UK Met office runs a shared code base for its simulations of climate and the weather. This plot shows the different spatial and temporal scales used.

In many real world systems, decisions are made through simulating the environment. Simulations may operate at different granularities. For example, simulations are used in weather forecasts and climate forecasts. Interestingly, the UK Met office uses the same code for both, it has a “Unified Model” approach, but they operate climate simulations one at greater spatial and temporal resolutions.

Figure: Real world systems consist of simulators that capture our domain knowledge about how our systems operate. Different simulators run at different speeds and granularities.

Figure: A statistical emulator is a system that reconstructs the simulation with a statistical model.

A statistical emulator is a data-driven model that learns about the underlying simulation. Importantly, learns with uncertainty, so it ‘knows what it doesn’t know’. In practice, we can call the emulator in place of the simulator. If the emulator ‘doesn’t know’, it can call the simulator for the answer.

Figure: A statistical emulator is a system that reconstructs the simulation with a statistical model. As well as reconstructing the simulation, a statistical emulator can be used to correlate with the real world.

As well as reconstructing an individual simulator, the emulator can calibrate the simulation to the real world, by monitoring differences between the simulator and real data. This allows the emulator to characterise where the simulation can be relied on, i.e. we can validate the simulator.

Similarly, the emulator can adjudicate between simulations. This is known as multi-fidelity emulation. The emulator characterizes which emulations perform well where.

If all this modelling is done with judiscious handling of the uncertainty, the computational doubt, then the emulator can assist in desciding what experiment should be run next to aid a decision: should we run a simulator, in which case which one, or should we attempt to acquire data from a real world intervention.

Figure: The author standing outside the Newton Institute on 2nd August 1997, just after arriving for “Generalisation in Neural Networks and Machine Learning”, see page 28 of this report.

The first tutorial I saw on Gaussian processes was given by Chris Williams at the Newton Institute in August 1997. The school was part of a program on Generalisation in Neural Networks and Machine Learning organised by my PhD supervisor, Chris Bishop (now Director of Microsoft Research in Cambridge).

Bayesian Inference by Rejection Sampling

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One view of Bayesian inference is to assume we are given a mechanism for generating samples, where we assume that mechanism is representing on accurate view on the way we believe the world works.

This mechanism is known as our prior belief.

We combine our prior belief with our observations of the real world by discarding all those samples that are inconsistent with our prior. The likelihood defines mathematically what we mean by inconsistent with the prior. The higher the noise level in the likelihood, the looser the notion of consistent.

The samples that remain are considered to be samples from the posterior.

This approach to Bayesian inference is closely related to two sampling techniques known as rejection sampling and importance sampling. It is realized in practice in an approach known as approximate Bayesian computation (ABC) or likelihood-free inference.

In practice, the algorithm is often too slow to be practical, because most samples will be inconsistent with the data and as a result the mechanism has to be operated many times to obtain a few posterior samples.

However, in the Gaussian process case, when the likelihood also assumes Gaussian noise, we can operate this mechanism mathematically, and obtain the posterior density analytically. This is the benefit of Gaussian processes.

First we will load in two python functions for computing the covariance function.

Next we sample from a multivariate normal density (a multivariate Gaussian), using the covariance function as the covariance matrix.

plot.rejection_samples(kernel=kernel, 
    diagrams='../slides/diagrams/gp')

Figure: One view of Bayesian inference is we have a machine for generating samples (the prior), and we discard all samples inconsistent with our data, leaving the samples of interest (the posterior). This is a rejection sampling view of Bayesian inference. The Gaussian process allows us to do this analytically by multiplying the prior by the likelihood.

Deep Emulation

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Figure: A potential path of models in a machine learning system.

As a solution we can use of emulators. When constructing an ML system, software engineers, ML engineers, economists and operations researchers are explicitly defining relationships between variables of interest in the system. That implicitly defines a joint distribution, $p(\dataVector^*, \dataVector)$. In a decomposable system any sub-component may be defined as $p(\dataVector_\mathbf{i}|\dataVector_\mathbf{j})$ where $\dataVector_\mathbf{i}$ and $\dataVector_\mathbf{j}$ represent sub-sets of the full set of variables $\left\{\dataVector^*, \dataVector \right\}$. In those cases where the relationship is deterministic, the probability density would collapse to a vector-valued deterministic function, $\mappingFunctionVector_\mathbf{i}\left(\dataVector_\mathbf{j}\right)$.

Inter-variable relationships could be defined by, for example a neural network (machine learning), an integer program (operational research), or a simulation (supply chain). This makes probabilistic inference in this joint density for real world systems is either very hard or impossible.

Emulation is a form of meta-modelling: we construct a model of the model. We can define the joint density of an emulator as $s(\dataVector*, \dataVector)$, but if this probability density is to be an accurate representation of our system, it is likely to be prohibitively complex. Current practice is to design an emulator to deal with a specific question. This is done by fitting an ML model to a simulation from the the appropriate conditional distribution, $p(\dataVector_\mathbf{i}|\dataVector_\mathbf{j})$, which is intractable. The emulator provides an approximated answer of the form $s(\dataVector_\mathbf{i}|\dataVector_\mathbf{j})$. Critically, an emulator should incorporate its uncertainty about its approximation. So the emulator answer will be less certain than direct access to the conditional $p(\dataVector_i|\dataVector_j)$, but it may be sufficiently confident to act upon. Careful design of emulators to answer a given question leads to efficient diagnostics and understanding of the system. But in a complex interacting system an exponentially increasing number of questions can be asked. This calls for a system of automated construction of emulators which selects the right structure and redeploys the emulator as necessary. Rapid redeployment of emulators could exploit pre-existing emulators through transfer learning.

Automatically deploying these families of emulators for full system understanding is highly ambitious. It requires advances in engineering infrastructure, emulation and Bayesian optimization. However, the intermediate steps of developing this architecture also allow for automated monitoring of system accuracy and fairness. This facilitates AutoML on a component-wise basis which we can see as a simple implementation of AutoAI. The proposal is structured so that despite its technical ambition there is a smooth ramp of benefits to be derived across the programme of work.

In Applied Mathematics, the field studying these techniques is known as uncertainty quantification. The new challenge is the automation of emulator creation on demand to answer questions of interest and facilitate the system design, i.e. AutoAI through BSO.

At design stage, any particular AI task could be decomposed in multiple ways. Bayesian system optimization will assist both in determining the large-scale system design through exploring different decompositions and in refinement of the deployed system.

So far, most work on emulators has focussed on emulating a single component. Automated deployment and maintenance of ML systems requires networks of emulators that can be deployed and redeployed on demand depending on the particular question of interest. Therefore, the technical innovations we require are in the mathematical composition of emulator models (Damianou and Lawrence 2013; Perdikaris et al. 2017). Different chains of emulators will need to be rapidly composed to make predictions of downstream performance. This requires rapid retraining of emulators and propagation of uncertainty through the emulation pipeline a process we call deep emulation.

Recomposing the ML system requires structural learning of the network. By parameterizing covariance functions appropriately this can be done through Gaussian processes (e.g. (Damianou et al., n.d.)), but one could also consider Bayesian neural networks and other generative models, e.g. Generative Adversarial Networks (Goodfellow et al. 2014).

Figure: A potential path of models in a machine learning system.

Figure: A potential path of models in a machine learning system.

Figure: A potential path of models in a machine learning system.

Mathematically, a deep Gaussian process can be seen as a composite multivariate function,
$$ \mathbf{g}(\inputVector)=\mappingFunctionVector_5(\mappingFunctionVector_4(\mappingFunctionVector_3(\mappingFunctionVector_2(\mappingFunctionVector_1(\inputVector))))). $$
Or if we view it from the probabilistic perspective we can see that a deep Gaussian process is specifying a factorization of the joint density, the standard deep model takes the form of a Markov chain.


$$ p(\dataVector|\inputVector)= p(\dataVector|\mappingFunctionVector_5)p(\mappingFunctionVector_5|\mappingFunctionVector_4)p(\mappingFunctionVector_4|\mappingFunctionVector_3)p(\mappingFunctionVector_3|\mappingFunctionVector_2)p(\mappingFunctionVector_2|\mappingFunctionVector_1)p(\mappingFunctionVector_1|\inputVector) $$

Figure: Probabilistically the deep Gaussian process can be represented as a Markov chain. Indeed they can even be analyzed in this way (Dunlop et al., n.d.).

Figure: More usually deep probabilistic models are written vertically rather than horizontally as in the Markov chain.

Why Composition?

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If the result of composing many functions together is simply another function, then why do we bother? The key point is that we can change the class of functions we are modeling by composing in this manner. A Gaussian process is specifying a prior over functions, and one with a number of elegant properties. For example, the derivative process (if it exists) of a Gaussian process is also Gaussian distributed. That makes it easy to assimilate, for example, derivative observations. But that also might raise some alarm bells. That implies that the marginal derivative distribution is also Gaussian distributed. If that’s the case, then it means that functions which occasionally exhibit very large derivatives are hard to model with a Gaussian process. For example, a function with jumps in.

A one off discontinuity is easy to model with a Gaussian process, or even multiple discontinuities. They can be introduced in the mean function, or independence can be forced between two covariance functions that apply in different areas of the input space. But in these cases we will need to specify the number of discontinuities and where they occur. In otherwords we need to parameterise the discontinuities. If we do not know the number of discontinuities and don’t wish to specify where they occur, i.e. if we want a non-parametric representation of discontinuities, then the standard Gaussian process doesn’t help.

Stochastic Process Composition

The deep Gaussian process leads to non-Gaussian models, and non-Gaussian characteristics in the covariance function. In effect, what we are proposing is that we change the properties of the functions we are considering by composing stochastic processes. This is an approach to creating new stochastic processes from well known processes.

Additionally, we are not constrained to the formalism of the chain. For example, we can easily add single nodes emerging from some point in the depth of the chain. This allows us to combine the benefits of the graphical modelling formalism, but with a powerful framework for relating one set of variables to another, that of Gaussian processes

Figure: More generally we aren’t constrained by the Markov chain. We can design structures that respect our belief about the underlying conditional dependencies. Here we are adding a side note from the chain.

Figure: Gaussian process fit to the step function data. Note the large error bars and the over-smoothing of the discontinuity. Error bars are shown at two standard deviations.

Step Function

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Next we consider a simple step function data set.

num_low=25
num_high=25
gap = -.1
noise=0.0001
x = np.vstack((np.linspace(-1, -gap/2.0, num_low)[:, np.newaxis],
              np.linspace(gap/2.0, 1, num_high)[:, np.newaxis]))
y = np.vstack((np.zeros((num_low, 1)), np.ones((num_high,1))))
scale = np.sqrt(y.var())
offset = y.mean()
yhat = (y-offset)/scale

Step Function Data

Figure: Simulation study of step function data artificially generated. Here there is a small overlap between the two lines.

GPy: A Gaussian Process Framework in Python

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Gaussian processes are a flexible tool for non-parametric analysis with uncertainty. The GPy software was started in Sheffield to provide a easy to use interface to GPs. One which allowed the user to focus on the modelling rather than the mathematics.

Figure: GPy is a BSD licensed software code base for implementing Gaussian process models in Python. It is designed for teaching and modelling. We welcome contributions which can be made through the Github repository https://github.com/SheffieldML/GPy

GPy is a BSD licensed software code base for implementing Gaussian process models in python. This allows GPs to be combined with a wide variety of software libraries.

The software itself is available on GitHub and the team welcomes contributions.

The aim for GPy is to be a probabilistic-style programming language, i.e. you specify the model rather than the algorithm. As well as a large range of covariance functions the software allows for non-Gaussian likelihoods, multivariate outputs, dimensionality reduction and approximations for larger data sets.

The documentation for GPy can be found here.

Step Function Data GP

We can fit a Gaussian process to the step function data using GPy as follows.

m_full = GPy.models.GPRegression(x,yhat)
_ = m_full.optimize() # Optimize parameters of covariance function

Where GPy.models.GPRegression() gives us a standard GP regression model with exponentiated quadratic covariance function.

The model is optimized using m_full.optimize() which calls an L-BGFS gradient based solver in python.

Figure: Gaussian process fit to the step function data. Note the large error bars and the over-smoothing of the discontinuity. Error bars are shown at two standard deviations.

The resulting fit to the step function data shows some challenges. In particular, the over smoothing at the discontinuity. If we know how many discontinuities there are, we can parameterize them in the step function. But by doing this, we form a semi-parametric model. The parameters indicate how many discontinuities are, and where they are. They can be optimized as part of the model fit. But if new, unforeseen, discontinuities arise when the model is being deployed in practice, these won’t be accounted for in the predictions.

The deep Gaussian process code we are using is research code by Andreas Damianou.

To extend the research code we introduce some approaches to initialization and optimization that we’ll use in examples. These approaches can be found in the deepgp_tutorial.py file.

Deep Gaussian process models also can require some thought in the initialization. Here we choose to start by setting the noise variance to be one percent of the data variance.


from deepgp_tutorial import initialize

Secondly, we introduce a staged optimization approach.

Optimization requires moving variational parameters in the hidden layer representing the mean and variance of the expected values in that layer. Since all those values can be scaled up, and this only results in a downscaling in the output of the first GP, and a downscaling of the input length scale to the second GP. It makes sense to first of all fix the scales of the covariance function in each of the GPs.

Sometimes, deep Gaussian processes can find a local minima which involves increasing the noise level of one or more of the GPs. This often occurs because it allows a minimum in the KL divergence term in the lower bound on the likelihood. To avoid this minimum we habitually train with the likelihood variance (the noise on the output of the GP) fixed to some lower value for some iterations.

Next an optimization of the kernel function parameters at each layer is performed, but with the variance of the likelihood fixed. Again, this is to prevent the model minimizing the Kullback-Leibler divergence between the approximate posterior and the prior before achieving a good data-fit.

Finally, all parameters of the model are optimized together.


from deepgp_tutorial import staged_optimize

The next code is for visualizing the intermediate layers of the deep model. This visualization is only appropriate for models with intermediate layers containing a single latent variable.


from deepgp_tutorial import visualize

The pinball visualization is to bring the pinball-analogy to life in the model. It shows how a ball would fall through the model to end up in the right pbosition. This visualization is only appropriate for models with intermediate layers containing a single latent variable.


from deepgp_tutorial import visualize_pinball

The posterior_sample code allows us to see the output sample locations for a given input. This is useful for visualizing the non-Gaussian nature of the output density.


from deepgp_tutorial import posterior_sample

Finally, we bind these methods to the DeepGP object for ease of calling.

Step Function Data Deep GP

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First we initialize a deep Gaussian process with three latent layers (four layers total). Within each layer we create a GP with an exponentiated quadratic covariance (GPy.kern.RBF).

At each layer we use 20 inducing points for the variational approximation.

layers = [y.shape[1], 1, 1, 1,x.shape[1]]
inits = ['PCA']*(len(layers)-1)
kernels = []
for i in layers[1:]:
    kernels += [GPy.kern.RBF(i)]
    
m = deepgp.DeepGP(layers,Y=yhat, X=x, 
                  inits=inits, 
                  kernels=kernels, # the kernels for each layer
                  num_inducing=20, back_constraint=False)

Once the model is constructed we initialize the parameters, and perform the staged optimization which starts by optimizing variational parameters with a low noise and proceeds to optimize the whole model.

m.initialize()
m.staged_optimize()

We plot the output of the deep Gaussian process fitted to the stpe data as follows.

The deep Gaussian process does a much better job of fitting the data. It handles the discontinuity easily, and error bars drop to smaller values in the regions of data.

Figure: Deep Gaussian process fit to the step function data.

Step Function Data Deep GP

The samples of the model can be plotted with the helper function from teaching_plots.py, model_sample

The samples from the model show that the error bars, which are informative for Gaussian outputs, are less informative for this model. They make clear that the data points lie, in output mainly at 0 or 1, or occasionally in between.

Figure: Samples from the deep Gaussian process model for the step function fit.

The visualize code allows us to inspect the intermediate layers in the deep GP model to understand how it has reconstructed the step function.

Figure: From top to bottom, the Gaussian process mapping function that makes up each layer of the resulting deep Gaussian process.

A pinball plot can be created for the resulting model to understand how the input is being translated to the output across the different layers.

Figure: Pinball plot of the deep GP fitted to the step function data. Each layer of the model pushes the ‘ball’ towards the left or right, saturating at 1 and 0. This causes the final density to be be peaked at 0 and 1. Transitions occur driven by the uncertainty of the mapping in each layer.

  • Automate the process of explaining and maintaining ML Systems.

  • Challenge is these systems are highly complex:

    • Flaws need rapid addressing.
  • This becomes apparent in operational science

  • Been working with the DELVE Group

  • One perspective on challenge facing government:

    • Intellectual Debt for the whole country
    • Ideas about explaining a complex system much wider applicability.
  • Implications for digital twins

Thanks!

For more information on these subjects and more you might want to check the following resources.

References

Damianou, Andreas, Carl Henrik Ek, Michalis K. Titsias, and Neil D. Lawrence. n.d. “Manifold Relevance Determination.” In.

Damianou, Andreas, and Neil D. Lawrence. 2013. “Deep Gaussian Processes.” In, 31:207–15.

Dunlop, Matthew M., Mark A. Girolami, Andrew M. Stuart, and Aretha L. Teckentrup. n.d. “How Deep Are Deep Gaussian Processes?” Journal of Machine Learning Research 19 (54): 1–46. http://jmlr.org/papers/v19/18-015.html.

Goodfellow, Ian, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. 2014. “Generative Adversarial Nets.” In Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, 2672–80. Curran Associates, Inc.

Perdikaris, Paris, Maziar Raissi, Andreas Damianou, Neil D. Lawrence, and George Em Karnidakis. 2017. “Nonlinear Information Fusion Algorithms for Data-Efficient Multi-Fidelity Modelling.” Proc. R. Soc. A 473 (20160751). https://doi.org/10.1098/rspa.2016.0751.

Sculley, D., Gary Holt, Daniel Golovin, Eugene Davydov, Todd Phillips, Dietmar Ebner, Vinay Chaudhary, Michael Young, Jean-François Crespo, and Dan Dennison. 2015. “Hidden Technical Debt in Machine Learning Systems.” In Advances in Neural Information Processing Systems 28, edited by Corinna Cortes, Neil D. Lawrence, Daniel D. Lee, Masashi Sugiyama, and Roman Garnett, 2503–11. Curran Associates, Inc. http://papers.nips.cc/paper/5656-hidden-technical-debt-in-machine-learning-systems.pdf.


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