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Gaussian Course of Regression with tfprobability


How do you inspire, or provide you with a narrative round Gaussian Course of Regression on a weblog primarily devoted to deep studying?

Straightforward. As demonstrated by seemingly unavoidable, reliably recurring Twitter “wars” surrounding AI, nothing attracts consideration like controversy and antagonism. So, let’s return twenty years and discover citations of individuals saying, “right here come Gaussian Processes, we don’t have to trouble with these finicky, exhausting to tune neural networks anymore!” And right now, right here we’re; everybody is aware of one thing about deep studying however who’s heard of Gaussian Processes?

Whereas comparable tales inform so much about historical past of science and improvement of opinions, we desire a distinct angle right here. Within the preface to their 2006 guide on Gaussian Processes for Machine Studying (Rasmussen and Williams 2005), Rasmussen and Williams say, referring to the “two cultures” – the disciplines of statistics and machine studying, respectively:

Gaussian course of fashions in some sense carry collectively work within the two communities.

On this submit, that “in some sense” will get very concrete. We’ll see a Keras community, outlined and educated the standard means, that has a Gaussian Course of layer for its most important constituent. The duty will likely be “easy” multivariate regression.

As an apart, this “bringing collectively communities” – or methods of considering, or answer methods – makes for a very good general characterization of TensorFlow Chance as properly.

Gaussian Processes

A Gaussian Course of is a distribution over capabilities, the place the operate values you pattern are collectively Gaussian – roughly talking, a generalization to infinity of the multivariate Gaussian. Moreover the reference guide we already talked about (Rasmussen and Williams 2005), there are a variety of good introductions on the web: see e.g. https://distill.pub/2019/visual-exploration-gaussian-processes/ or https://peterroelants.github.io/posts/gaussian-process-tutorial/. And like on all the pieces cool, there’s a chapter on Gaussian Processes within the late David MacKay’s (MacKay 2002) guide.

On this submit, we’ll use TensorFlow Chance’s Variational Gaussian Course of (VGP) layer, designed to effectively work with “huge knowledge.” As Gaussian Course of Regression (GPR, any more) entails the inversion of a – presumably huge – covariance matrix, makes an attempt have been made to design approximate variations, typically primarily based on variational rules. The TFP implementation is predicated on papers by Titsias (2009) (Titsias 2009) and Hensman et al. (2013) (Hensman, Fusi, and Lawrence 2013). As a substitute of (p(mathbf{y}|mathbf{X})), the precise likelihood of the goal knowledge given the precise enter, we work with a variational distribution (q(mathbf{u})) that acts as a decrease certain.

Right here (mathbf{u}) are the operate values at a set of so-called inducing index factors specified by the consumer, chosen to properly cowl the vary of the particular knowledge. This algorithm is so much sooner than “regular” GPR, as solely the covariance matrix of (mathbf{u}) needs to be inverted. As we’ll see under, no less than on this instance (in addition to in others not described right here) it appears to be fairly sturdy as to the variety of inducing factors handed.

Let’s begin.

The dataset

The Concrete Compressive Energy Information Set is a part of the UCI Machine Studying Repository. Its internet web page says:

Concrete is a very powerful materials in civil engineering. The concrete compressive energy is a extremely nonlinear operate of age and elements.

Extremely nonlinear operate – doesn’t that sound intriguing? In any case, it ought to represent an fascinating check case for GPR.

Here’s a first look.

library(tidyverse)
library(GGally)
library(visreg)
library(readxl)
library(rsample)
library(reticulate)
library(tfdatasets)
library(keras)
library(tfprobability)

concrete <- read_xls(
  "Concrete_Data.xls",
  col_names = c(
    "cement",
    "blast_furnace_slag",
    "fly_ash",
    "water",
    "superplasticizer",
    "coarse_aggregate",
    "fine_aggregate",
    "age",
    "energy"
  ),
  skip = 1
)

concrete %>% glimpse()
Observations: 1,030
Variables: 9
$ cement             <dbl> 540.0, 540.0, 332.5, 332.5, 198.6, 266.0, 380.0, 380.0, …
$ blast_furnace_slag <dbl> 0.0, 0.0, 142.5, 142.5, 132.4, 114.0, 95.0, 95.0, 114.0,…
$ fly_ash            <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ water              <dbl> 162, 162, 228, 228, 192, 228, 228, 228, 228, 228, 192, 1…
$ superplasticizer   <dbl> 2.5, 2.5, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0…
$ coarse_aggregate   <dbl> 1040.0, 1055.0, 932.0, 932.0, 978.4, 932.0, 932.0, 932.0…
$ fine_aggregate     <dbl> 676.0, 676.0, 594.0, 594.0, 825.5, 670.0, 594.0, 594.0, …
$ age                <dbl> 28, 28, 270, 365, 360, 90, 365, 28, 28, 28, 90, 28, 270,…
$ energy           <dbl> 79.986111, 61.887366, 40.269535, 41.052780, 44.296075, 4…

It isn’t that huge – just a bit greater than 1000 rows –, however nonetheless, we can have room to experiment with totally different numbers of inducing factors.

We’ve got eight predictors, all numeric. Except age (in days), these characterize lots (in kg) in a single cubic metre of concrete. The goal variable, energy, is measured in megapascals.

Let’s get a fast overview of mutual relationships.

Checking for a doable interplay (one {that a} layperson might simply consider), does cement focus act otherwise on concrete energy relying on how a lot water there’s within the combination?

cement_ <- lower(concrete$cement, 3, labels = c("low", "medium", "excessive"))
match <- lm(energy ~ (.) ^ 2, knowledge = cbind(concrete[, 2:9], cement_))
abstract(match)

visreg(match, "cement_", "water", gg = TRUE) + theme_minimal()

To anchor our future notion of how properly VGP does for this instance, we match a easy linear mannequin, in addition to one involving two-way interactions.

# scale predictors right here already, so knowledge are the identical for all fashions
concrete[, 1:8] <- scale(concrete[, 1:8])

# train-test cut up 
set.seed(777)
cut up <- initial_split(concrete, prop = 0.8)
practice <- coaching(cut up)
check <- testing(cut up)

# easy linear mannequin with no interactions
fit1 <- lm(energy ~ ., knowledge = practice)
fit1 %>% abstract()
Name:
lm(method = energy ~ ., knowledge = practice)

Residuals:
    Min      1Q  Median      3Q     Max 
-30.594  -6.075   0.612   6.694  33.032 

Coefficients:
                   Estimate Std. Error t worth Pr(>|t|)    
(Intercept)         35.6773     0.3596  99.204  < 2e-16 ***
cement              13.0352     0.9702  13.435  < 2e-16 ***
blast_furnace_slag   9.1532     0.9582   9.552  < 2e-16 ***
fly_ash              5.9592     0.8878   6.712 3.58e-11 ***
water               -2.5681     0.9503  -2.702  0.00703 ** 
superplasticizer     1.9660     0.6138   3.203  0.00141 ** 
coarse_aggregate     1.4780     0.8126   1.819  0.06929 .  
fine_aggregate       2.2213     0.9470   2.346  0.01923 *  
age                  7.7032     0.3901  19.748  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual commonplace error: 10.32 on 816 levels of freedom
A number of R-squared:  0.627, Adjusted R-squared:  0.6234 
F-statistic: 171.5 on 8 and 816 DF,  p-value: < 2.2e-16
# two-way interactions
fit2 <- lm(energy ~ (.) ^ 2, knowledge = practice)
fit2 %>% abstract()
Name:
lm(method = energy ~ (.)^2, knowledge = practice)

Residuals:
     Min       1Q   Median       3Q      Max 
-24.4000  -5.6093  -0.0233   5.7754  27.8489 

Coefficients:
                                    Estimate Std. Error t worth Pr(>|t|)    
(Intercept)                          40.7908     0.8385  48.647  < 2e-16 ***
cement                               13.2352     1.0036  13.188  < 2e-16 ***
blast_furnace_slag                    9.5418     1.0591   9.009  < 2e-16 ***
fly_ash                               6.0550     0.9557   6.336 3.98e-10 ***
water                                -2.0091     0.9771  -2.056 0.040090 *  
superplasticizer                      3.8336     0.8190   4.681 3.37e-06 ***
coarse_aggregate                      0.3019     0.8068   0.374 0.708333    
fine_aggregate                        1.9617     0.9872   1.987 0.047256 *  
age                                  14.3906     0.5557  25.896  < 2e-16 ***
cement:blast_furnace_slag             0.9863     0.5818   1.695 0.090402 .  
cement:fly_ash                        1.6434     0.6088   2.700 0.007093 ** 
cement:water                         -4.2152     0.9532  -4.422 1.11e-05 ***
cement:superplasticizer              -2.1874     1.3094  -1.670 0.095218 .  
cement:coarse_aggregate               0.2472     0.5967   0.414 0.678788    
cement:fine_aggregate                 0.7944     0.5588   1.422 0.155560    
cement:age                            4.6034     1.3811   3.333 0.000899 ***
blast_furnace_slag:fly_ash            2.1216     0.7229   2.935 0.003434 ** 
blast_furnace_slag:water             -2.6362     1.0611  -2.484 0.013184 *  
blast_furnace_slag:superplasticizer  -0.6838     1.2812  -0.534 0.593676    
blast_furnace_slag:coarse_aggregate  -1.0592     0.6416  -1.651 0.099154 .  
blast_furnace_slag:fine_aggregate     2.0579     0.5538   3.716 0.000217 ***
blast_furnace_slag:age                4.7563     1.1148   4.266 2.23e-05 ***
fly_ash:water                        -2.7131     0.9858  -2.752 0.006054 ** 
fly_ash:superplasticizer             -2.6528     1.2553  -2.113 0.034891 *  
fly_ash:coarse_aggregate              0.3323     0.7004   0.474 0.635305    
fly_ash:fine_aggregate                2.6764     0.7817   3.424 0.000649 ***
fly_ash:age                           7.5851     1.3570   5.589 3.14e-08 ***
water:superplasticizer                1.3686     0.8704   1.572 0.116289    
water:coarse_aggregate               -1.3399     0.5203  -2.575 0.010194 *  
water:fine_aggregate                 -0.7061     0.5184  -1.362 0.173533    
water:age                             0.3207     1.2991   0.247 0.805068    
superplasticizer:coarse_aggregate     1.4526     0.9310   1.560 0.119125    
superplasticizer:fine_aggregate       0.1022     1.1342   0.090 0.928239    
superplasticizer:age                  1.9107     0.9491   2.013 0.044444 *  
coarse_aggregate:fine_aggregate       1.3014     0.4750   2.740 0.006286 ** 
coarse_aggregate:age                  0.7557     0.9342   0.809 0.418815    
fine_aggregate:age                    3.4524     1.2165   2.838 0.004657 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual commonplace error: 8.327 on 788 levels of freedom
A number of R-squared:  0.7656,    Adjusted R-squared:  0.7549 
F-statistic: 71.48 on 36 and 788 DF,  p-value: < 2.2e-16

We additionally retailer the predictions on the check set, for later comparability.

linreg_preds1 <- fit1 %>% predict(check[, 1:8])
linreg_preds2 <- fit2 %>% predict(check[, 1:8])

examine <-
  knowledge.body(
    y_true = check$energy,
    linreg_preds1 = linreg_preds1,
    linreg_preds2 = linreg_preds2
  )

With no additional preprocessing required, the tfdatasets enter pipeline finally ends up good and brief:

create_dataset <- operate(df, batch_size, shuffle = TRUE) {
  
  df <- as.matrix(df)
  ds <-
    tensor_slices_dataset(record(df[, 1:8], df[, 9, drop = FALSE]))
  if (shuffle)
    ds <- ds %>% dataset_shuffle(buffer_size = nrow(df))
  ds %>%
    dataset_batch(batch_size = batch_size)
  
}

# only one doable alternative for batch measurement ...
batch_size <- 64
train_ds <- create_dataset(practice, batch_size = batch_size)
test_ds <- create_dataset(check, batch_size = nrow(check), shuffle = FALSE)

And on to mannequin creation.

The mannequin

Mannequin definition is brief as properly, though there are some things to broaden on. Don’t execute this but:

mannequin <- keras_model_sequential() %>%
  layer_dense(models = 8,
              input_shape = 8,
              use_bias = FALSE) %>%
  layer_variational_gaussian_process(
    # variety of inducing factors
    num_inducing_points = num_inducing_points,
    # kernel for use by the wrapped Gaussian Course of distribution
    kernel_provider = RBFKernelFn(),
    # output form 
    event_shape = 1, 
    # preliminary values for the inducing factors
    inducing_index_points_initializer = initializer_constant(as.matrix(sampled_points)),
    unconstrained_observation_noise_variance_initializer =
      initializer_constant(array(0.1))
  )

Two arguments to layer_variational_gaussian_process() want some preparation earlier than we are able to really run this. First, because the documentation tells us, kernel_provider ought to be

a layer occasion geared up with an @property, which yields a PositiveSemidefiniteKernel occasion”.

In different phrases, the VGP layer wraps one other Keras layer that, itself, wraps or bundles collectively the TensorFlow Variables containing the kernel parameters.

We will make use of reticulate’s new PyClass constructor to satisfy the above necessities. Utilizing PyClass, we are able to instantly inherit from a Python object, including and/or overriding strategies or fields as we like – and sure, even create a Python property.

bt <- import("builtins")
RBFKernelFn <- reticulate::PyClass(
  "KernelFn",
  inherit = tensorflow::tf$keras$layers$Layer,
  record(
    `__init__` = operate(self, ...) {
      kwargs <- record(...)
      tremendous()$`__init__`(kwargs)
      dtype <- kwargs[["dtype"]]
      self$`_amplitude` = self$add_variable(initializer = initializer_zeros(),
                                            dtype = dtype,
                                            identify = 'amplitude')
      self$`_length_scale` = self$add_variable(initializer = initializer_zeros(),
                                               dtype = dtype,
                                               identify = 'length_scale')
      NULL
    },
    
    name = operate(self, x, ...) {
      x
    },
    
    kernel = bt$property(
      reticulate::py_func(
        operate(self)
          tfp$math$psd_kernels$ExponentiatedQuadratic(
            amplitude = tf$nn$softplus(array(0.1) * self$`_amplitude`),
            length_scale = tf$nn$softplus(array(2) * self$`_length_scale`)
          )
      )
    )
  )
)

The Gaussian Course of kernel used is one among a number of accessible in tfp.math.psd_kernels (psd standing for constructive semidefinite), and doubtless the one which involves thoughts first when considering of GPR: the squared exponential, or exponentiated quadratic. The model utilized in TFP, with hyperparameters amplitude (a) and size scale (lambda), is

[k(x,x’) = 2 a exp (frac{- 0.5 (x−x’)^2}{lambda^2}) ]

Right here the fascinating parameter is the size scale (lambda). When we’ve got a number of options, their size scales – as induced by the educational algorithm – replicate their significance: If, for some characteristic, (lambda) is giant, the respective squared deviations from the imply don’t matter that a lot. The inverse size scale can thus be used for automated relevance dedication (Neal 1996).

The second factor to maintain is selecting the preliminary index factors. From experiments, the precise decisions don’t matter that a lot, so long as the info are sensibly lined. For example, an alternate means we tried was to assemble an empirical distribution (tfd_empirical) from the info, after which pattern from it. Right here as a substitute, we simply use an – pointless, admittedly, given the supply of pattern in R – fancy strategy to choose random observations from the coaching knowledge:

num_inducing_points <- 50

sample_dist <- tfd_uniform(low = 1, excessive = nrow(practice) + 1)
sample_ids <- sample_dist %>%
  tfd_sample(num_inducing_points) %>%
  tf$solid(tf$int32) %>%
  as.numeric()
sampled_points <- practice[sample_ids, 1:8]

One fascinating level to notice earlier than we begin coaching: Computation of the posterior predictive parameters entails a Cholesky decomposition, which might fail if, attributable to numerical points, the covariance matrix is not constructive particular. A adequate motion to absorb our case is to do all computations utilizing tf$float64:

Now we outline (for actual, this time) and run the mannequin.

mannequin <- keras_model_sequential() %>%
  layer_dense(models = 8,
              input_shape = 8,
              use_bias = FALSE) %>%
  layer_variational_gaussian_process(
    num_inducing_points = num_inducing_points,
    kernel_provider = RBFKernelFn(),
    event_shape = 1,
    inducing_index_points_initializer = initializer_constant(as.matrix(sampled_points)),
    unconstrained_observation_noise_variance_initializer =
      initializer_constant(array(0.1))
  )

# KL weight sums to 1 for one epoch
kl_weight <- batch_size / nrow(practice)

# loss that implements the VGP algorithm
loss <- operate(y, rv_y)
  rv_y$variational_loss(y, kl_weight = kl_weight)

mannequin %>% compile(optimizer = optimizer_adam(lr = 0.008),
                  loss = loss,
                  metrics = "mse")

historical past <- mannequin %>% match(train_ds,
                         epochs = 100,
                         validation_data = test_ds)

plot(historical past)

Curiously, increased numbers of inducing factors (we tried 100 and 200) didn’t have a lot influence on regression efficiency. Nor does the precise alternative of multiplication constants (0.1 and 2) utilized to the educated kernel Variables (_amplitude and _length_scale)

tfp$math$psd_kernels$ExponentiatedQuadratic(
  amplitude = tf$nn$softplus(array(0.1) * self$`_amplitude`),
  length_scale = tf$nn$softplus(array(2) * self$`_length_scale`)
)

make a lot of a distinction to the tip end result.

Predictions

We generate predictions on the check set and add them to the knowledge.body containing the linear fashions’ predictions. As with different probabilistic output layers, “the predictions” are the truth is distributions; to acquire precise tensors we pattern from them. Right here, we common over 10 samples:

yhats <- mannequin(tf$convert_to_tensor(as.matrix(check[, 1:8])))

yhat_samples <-  yhats %>%
  tfd_sample(10) %>%
  tf$squeeze() %>%
  tf$transpose()

sample_means <- yhat_samples %>% apply(1, imply)

examine <- examine %>%
  cbind(vgp_preds = sample_means)

We plot the typical VGP predictions in opposition to the bottom fact, along with the predictions from the straightforward linear mannequin (cyan) and the mannequin together with two-way interactions (violet):

ggplot(examine, aes(x = y_true)) +
  geom_abline(slope = 1, intercept = 0) +
  geom_point(aes(y = vgp_preds, colour = "VGP")) +
  geom_point(aes(y = linreg_preds1, colour = "easy lm"), alpha = 0.4) +
  geom_point(aes(y = linreg_preds2, colour = "lm w/ interactions"), alpha = 0.4) +
  scale_colour_manual("", 
                      values = c("VGP" = "black", "easy lm" = "cyan", "lm w/ interactions" = "violet")) +
  coord_cartesian(xlim = c(min(examine$y_true), max(examine$y_true)), ylim = c(min(examine$y_true), max(examine$y_true))) +
  ylab("predictions") +
  theme(side.ratio = 1) 

Predictions vs. ground truth for linear regression (no interactions; cyan), linear regression with 2-way interactions (violet), and VGP (black).

Determine 1: Predictions vs. floor fact for linear regression (no interactions; cyan), linear regression with 2-way interactions (violet), and VGP (black).

Moreover, evaluating MSEs for the three units of predictions, we see

mse <- operate(y_true, y_pred) {
  sum((y_true - y_pred) ^ 2) / size(y_true)
}

lm_mse1 <- mse(examine$y_true, examine$linreg_preds1) # 117.3111
lm_mse2 <- mse(examine$y_true, examine$linreg_preds2) # 80.79726
vgp_mse <- mse(examine$y_true, examine$vgp_preds)     # 58.49689

So, the VGP does the truth is outperform each baselines. One thing else we could be considering: How do its predictions range? Not as a lot as we’d need, have been we to assemble uncertainty estimates from them alone. Right here we plot the ten samples we drew earlier than:

samples_df <-
  knowledge.body(cbind(examine$y_true, as.matrix(yhat_samples))) %>%
  collect(key = run, worth = prediction, -X1) %>% 
  rename(y_true = "X1")

ggplot(samples_df, aes(y_true, prediction)) +
  geom_point(aes(colour = run),
             alpha = 0.2,
             measurement = 2) +
  geom_abline(slope = 1, intercept = 0) +
  theme(legend.place = "none") +
  ylab("repeated predictions") +
  theme(side.ratio = 1)

Predictions from 10 consecutive samples from the VGP distribution.

Determine 2: Predictions from 10 consecutive samples from the VGP distribution.

Dialogue: Function Relevance

As talked about above, the inverse size scale can be utilized as an indicator of characteristic significance. When utilizing the ExponentiatedQuadratic kernel alone, there’ll solely be a single size scale; in our instance, the preliminary dense layer takes of scaling (and moreover, recombining) the options.

Alternatively, we might wrap the ExponentiatedQuadratic in a FeatureScaled kernel. FeatureScaled has an extra scale_diag parameter associated to precisely that: characteristic scaling. Experiments with FeatureScaled (and preliminary dense layer eliminated, to be “truthful”) confirmed barely worse efficiency, and the realized scale_diag values assorted fairly a bit from run to run. For that cause, we selected to current the opposite strategy; nevertheless, we embrace the code for a wrapping FeatureScaled in case readers wish to experiment with this:

ScaledRBFKernelFn <- reticulate::PyClass(
  "KernelFn",
  inherit = tensorflow::tf$keras$layers$Layer,
  record(
    `__init__` = operate(self, ...) {
      kwargs <- record(...)
      tremendous()$`__init__`(kwargs)
      dtype <- kwargs[["dtype"]]
      self$`_amplitude` = self$add_variable(initializer = initializer_zeros(),
                                            dtype = dtype,
                                            identify = 'amplitude')
      self$`_length_scale` = self$add_variable(initializer = initializer_zeros(),
                                               dtype = dtype,
                                               identify = 'length_scale')
      self$`_scale_diag` = self$add_variable(
        initializer = initializer_ones(),
        dtype = dtype,
        form = 8L,
        identify = 'scale_diag'
      )
      NULL
    },
    
    name = operate(self, x, ...) {
      x
    },
    
    kernel = bt$property(
      reticulate::py_func(
        operate(self)
          tfp$math$psd_kernels$FeatureScaled(
            kernel = tfp$math$psd_kernels$ExponentiatedQuadratic(
              amplitude = tf$nn$softplus(array(1) * self$`_amplitude`),
              length_scale = tf$nn$softplus(array(2) * self$`_length_scale`)
            ),
            scale_diag = tf$nn$softplus(array(1) * self$`_scale_diag`)
          )
      )
    )
  )
)

Lastly, if all you cared about was prediction efficiency, you might use FeatureScaled and hold the preliminary dense layer all the identical. However in that case, you’d most likely use a neural community – not a Gaussian Course of – anyway …

Thanks for studying!

Breiman, Leo. 2001. “Statistical Modeling: The Two Cultures (with Feedback and a Rejoinder by the Creator).” Statist. Sci. 16 (3): 199–231. https://doi.org/10.1214/ss/1009213726.
Hensman, James, Nicolo Fusi, and Neil D. Lawrence. 2013. “Gaussian Processes for Massive Information.” CoRR abs/1309.6835. http://arxiv.org/abs/1309.6835.

MacKay, David J. C. 2002. Data Idea, Inference & Studying Algorithms. New York, NY, USA: Cambridge College Press.

Neal, Radford M. 1996. Bayesian Studying for Neural Networks. Berlin, Heidelberg: Springer-Verlag.

Rasmussen, Carl Edward, and Christopher Ok. I. Williams. 2005. Gaussian Processes for Machine Studying (Adaptive Computation and Machine Studying). The MIT Press.

Titsias, Michalis. 2009. “Variational Studying of Inducing Variables in Sparse Gaussian Processes.” In Proceedings of the Twelth Worldwide Convention on Synthetic Intelligence and Statistics, edited by David van Dyk and Max Welling, 5:567–74. Proceedings of Machine Studying Analysis. Hilton Clearwater Seashore Resort, Clearwater Seashore, Florida USA: PMLR. http://proceedings.mlr.press/v5/titsias09a.html.

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