A library of scalable Bayesian generalised linear models with fancy features

## A library of scalable Bayesian generalised linear models with *fancy* features

This library implements various Bayesian linear models (Bayesian linear regression) and generalised linear models. A few features of this library are:

- A fancy basis functions/feature composition framework for combining basis functions like radial basis function, sigmoidal basis functions, polynomial basis functions etc.
- Basis functions that can be used to approximate Gaussian processes with shift invariant covariance functions (e.g. square exponential) when used with linear models [1], [2], [3].
- Non-Gaussian likelihoods with Bayesian generalised linear models using a modified version of the nonparametric variational inference algorithm presented in [4].
- Large scale learning using stochastic gradient descent (ADADELTA).

### Quickstart

To install, simply run `setup.py`:

```
$ python setup.py install
```

or install with `pip`:

```
$ pip install git+https://github.com/nicta/revrand.git
```

Refer to docs/installation.rst for advanced installation instructions.

Have a look at some of the demos, e.g.:

```
$ python demos/demo_regression.py
```

Or,

```
$ python demos/demo_glm.py
```

#### Bayesian Linear Regression Example

Here is a very quick example of how to use Bayesian linear regression with
optimisation of the likelihood noise, regulariser and basis function
parameters. Assuming we already have training noisy targets `y`, inputs
`X`, and some query inputs `Xs` (as well as the true noiseless function
`f`):

import matplotlib.pyplot as pl import numpy as np from revrand.basis_functions import LinearBasis, RandomRBF from revrand.regression import learn, predict from revrand.btypes import Parameter, Positive ... # Concatenate a linear basis and a Random radial basis (GP approx) init_lenscale = Parameter(1.0, Positive()) # init val and bounds basis = LinearBasis(onescol=True) \ + RandomRBF(nbases=300, Xdim=X.shape[1], init_lenscale) # Learn regression parameters and predict params = regression.learn(X, y, basis) Eys, Vfs, Vys = regression.predict(Xs, basis, *params) # Training/Truth pl.plot(X, y, 'k.', label='Training') pl.plot(Xs, f, 'k-', label='Truth') # Plot Regressor Sys = np.sqrt(Vys) pl.plot(Xs, Eys, 'g-', label='Bayesian linear regression') pl.fill_between(Xs, Eys - 2 * Sys, Eys + 2 * Sys, facecolor='none', edgecolor='g', linestyle='--', label=None) pl.legend() pl.grid(True) pl.title('Regression demo') pl.ylabel('y') pl.xlabel('x') pl.show()

This script will output something like the following,

#### Bayesian Generalised Linear Model Example

This example is very similar to that above, but now let’s assume our targets
`y` are drawn from a Poisson likelihood, or observation, distribution which
is a function of the inputs, `X`. The task here is to predict the mean of the
Poisson distribution for query inputs `Xs`, as well as the uncertainty
associated with the prediction.

import matplotlib.pyplot as pl import numpy as np from revrand.basis_functions import RandomRBF from revrand.glm import learn, predict_moments, predict_interval ... # Random radial basis (GP approx) init_lenscale = Parameter(1.0, Positive()) # init val and bounds basis = RandomRBF(nbases=100, Xdim=X.shape[1], init_lenscale) # Set up the likelihood of the GLM llhood = likelihoods.Poisson(tranfcn='exp') # log link # Learn regression parameters and predict params = learn(X, y, llhood, basis) Eys, _, _, _ = predict_moments(Xs, llhood, basis, *params) y95n, y95x = predict_interval(0.95, Xs, llhood, basis, *params) # Training/Truth pl.plot(X, y, 'k.', label='Training') pl.plot(Xs, f, 'k-', label='Truth') # Plot GLM SGD Regressor pl.plot(Xs, Eys, 'b-', label='GLM mean.') pl.fill_between(Xs, y95n, y95x, facecolor='none', edgecolor='b', linestyle='--', label=None) pl.legend() pl.grid(True) pl.title('Regression demo') pl.ylabel('y') pl.xlabel('x') pl.show()

This script will output something like the following,

#### Large-scale Learning with Stochastic Gradients

By default the GLM uses stochastic gradients to learn all of its parameters/hyperparameters and does not require any matrix inversion, and so it can be used to learn from large datasets with lots of features (regression.learn uses L-BFGS and requires a matrix inversion). We can also use the GLM to approximate and scale up regular Bayesian linear regression. For instance, if we modify the Bayesian linear regression example from before,

... from revrand import glm, likelihoods ... # Set up the likelihood of the GLM llhood = likelihoods.Gaussian(var_init=Parameter(1., Positive())) # Learn regression parameters and predict params = glm.learn(X, y, llhood, basis) Ey_g, Vf_g, Eyn, Eyx = glm.predict_moments(Xtest, llhood, base, *params) ... # Plot GLM SGD Regressor Sy_g = np.sqrt(Vy_g) pl.plot(Xpl_s, Ey_g, 'm-', label='GLM') pl.fill_between(Xs, Ey_g - 2 * Sy_g, Ey_g + 2 * Sy_g, facecolor='none', edgecolor='m', linestyle='--', label=None) ...

This script will output something like the following,

We can see the approximation from the GLM is pretty good - this is because it uses a mixture of diagonal Gaussians posterior (thereby avoiding a full matrix inversion) to approximate the full Gaussian posterior covariance over the weights. This also has the advantage of allowing the model to learn multi-modal posterior distributions when non-Gaussian likelihoods are required.

#### Feature Composition Framework

We have implemented an easy to use and extensible feature-building framework within revrand. You have already seen the basics demonstrated in the above examples, i.e. concatenation of basis functions,

>>> X = np.random.randn(100, 5) >>> N, d = X.shape >>> base = LinearBasis(onescol=True) + RandomRBF(Xdim=d, nbases=100) >>> lenscale = 1. >>> Phi = base(X, lenscale) >>> Phi.shape (100, 206)

There are a few things at work in this example:

- Both
`LinearBasis`and`RandomRBF`are applied to all of`X`, and the result is concatenated. `LinearBasis`has pre-pended a column of ones onto`X`so a subsequent algorithm can learn a “bias” term.`RandomRBF`is actually approximating a radial basis*kernel*function, [3], so we can approximate how a kernel machine functions with a basis function! This also outputs`2 * nbases`number of basis functions.- Hence the resulting basis function has a shape of
`(N, d + 1 + 2 * nbases)`.

We can also use *partial application* of basis functions, e.g.

>>> base = LinearBasis(onescol=True, apply_ind=slice(0, 2)) \ + RandomRBF(Xdim=d, nbases=100, apply_ind=slice(2, 5)) >>> Phi = base(X, lenscale) >>> Phi.shape (100, 203)

Now the basis functions are applied to seperate dimensions of the input, `X`.
That is, `LinearBasis` takes dimensions 0 and 1, and `RandomRBF` takes the
rest, and again the results are concatenated.

Finally, if we use these basis functions with any of the algorithms in this
revrand, *the parameters of the basis functions are learned* as well! So
really in the above example `lenscale = 1.` is just an initial value for
the kernel function length-scale!

### Useful Links

- Home Page
- http://github.com/nicta/revrand
- Documentation
- http://nicta.github.io/revrand
- Issue tracking
- https://github.com/nicta/revrand/issues

### Bugs & Feedback

For bugs, questions and discussions, please use Github Issues.

### References

[1] | Yang, Z., Smola, A. J., Song, L., & Wilson, A. G. “A la Carte – Learning Fast Kernels”. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, pp. 1098-1106, 2015. |

[2] | Le, Q., Sarlos, T., & Smola, A. “Fastfood-approximating kernel expansions in loglinear time.” Proceedings of the international conference on machine learning. 2013. |

[3] | (1, 2) Rahimi, A., & Recht, B. “Random features for large-scale kernel
machines.” Advances in neural information processing systems. 2007. |

[4] | Gershman, S., Hoffman, M., & Blei, D. “Nonparametric variational inference”. arXiv preprint arXiv:1206.4665 (2012). |

### Copyright & License

Copyright 2015 National ICT Australia.

Licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

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