Large datasets provide great opportunities to learn rich statistical representations, for accurate predictions and new scientific insights into our modeling problems. Gaussian processes are promising for large data problems, because they can grow their information capacity with the amount of available data, in combination with automatically calibrated model complexity (Rasmussen and Ghahramani, 2001; Wilson, 2014).
From a Gaussian process perspective, all of the statistical structure in data is learned through a kernel function. Popular kernel functions, such as the RBF kernel, provide smoothing and interpolation, but cannot learn representations necessary for long range extrapolation (Rasmussen and Williams, 2006; Wilson, 2014)
. With smoothing kernels, we can only use the information in a large dataset to learn about noise and length-scale hyperparameters, which tell us only how quickly correlations in our data vary with distance in the input space. If we learn a short length-scale hyperparameter, then by definition we will only make use of a small amount of training data near each testing point. If we learn a long length-scale, then we could subsample the data and make similar predictions.
Therefore to fully use the information in large datasets, we must build kernels with great representational power and useful learning biases, and scale these approaches without sacrificing this representational ability. Indeed many recent approaches have advocated building expressive kernel functions (e.g., Rasmussen and Williams, 2006; Gönen and Alpaydın, 2011; Wilson and Adams, 2013; Wilson, 2014; Lloyd et al., 2014; Yang et al., 2015), and emerging research in this direction takes inspiration from deep learning models (e.g., Wilson et al., 2012; Damianou and Lawrence, 2013; Calandra et al., 2014). However, the scalability, general applicability, and interpretability of such approaches remain a challenge. Recently, Wilson et al. (2016)
proposed simple and scalable deep kernels for single-output regression problems, with promising performance on many experiments. But their approach does not allow for stochastic training, multiple outputs, deep architectures with many output features, or classification. And it is on classification problems, in particular, where we typically have high dimensional input vectors, with little intuition about how these vectors should correlate, and therefore most want tolearn a flexible non-Euclidean similarity metric (Aggarwal et al., 2001).
In this paper, we introduce inference procedures and propose a new deep kernel learning model which enables
(1) classification and non-Gaussian likelihoods; (2) multi-task learning111We follow the GP convention where multi-task learning involves a function mapping a single input to multiple correlated output responses (class probabilities, regression responses, etc.). Unlike NNs which naturally have correlated outputs by sharing hidden basis functions (and multi-task can have a more specialized meaning), most GP models perform multiple binary classification, ignoring correlations between output classes. Even applying a GP to NN features for deep kernel learning does not naturally produce multiple correlated outputs.
output responses (class probabilities, regression responses, etc.). Unlike NNs which naturally have correlated outputs by sharing hidden basis functions (and multi-task can have a more specialized meaning), most GP models perform multiple binary classification, ignoring correlations between output classes. Even applying a GP to NN features for deep kernel learning does not naturally produce multiple correlated outputs.; (3) stochastic gradient mini-batch training; (4) deep architectures with many output features; (5) additive covariance structures; and (5) greatly enhanced scalability.
We propose to use additive base kernels corresponding to Gaussian processes (GPs) applied to subsets of output features of a deep neural architecture. We then linearly mix these Gaussian processes, inducing correlations across multiple output variables. The result is a deep probabilistic neural network, with a hidden layer composed of additive sets of infinite basis functions, linearly mixed to produce correlated output variables. All parameters of the deep architecture and base kernels arejointly learned through a marginal likelihood objective, having integrated away all GPs. For scalability and non-Gaussian likelihoods, we derive stochastic variational inference (SVI) which leverages local kernel interpolation, inducing points, and structure exploiting algebra, and a hybrid sampling scheme, building on Wilson and Nickisch (2015), Wilson et al. (2015), Titsias (2009), Hensman et al. (2013), and Nickson et al. (2015). The resulting approach, SV-DKL, has a complexity of for inducing points and input dimensions, versus the standard for efficient stochastic variational methods.
We achieve good predictive accuracy and scalability over a wide range of classification tasks, while retaining a straightforward, general purpose, and highly practical probabilistic non-parametric representation, with code available at https://people.orie.cornell.edu/andrew/code.
Throughout this paper, we assume we have access to vectorial input-output pairs , where each is related to through a Gaussian process and observation model. For example, in regression, one could model , where is a latent vector of independent Gaussian processes , and is a noise covariance matrix.
The computational bottleneck in working with Gaussian processes typically involves computing and over an covariance matrix evaluated at training inputs . Standard procedure is to compute the Cholesky decomposition of , which incurs computations and storage, after which predictions cost per test point. Gaussian processes are thus typically limited to at most a few thousand training points. Many promising approaches to scalability have been explored, for example, involving randomized methods (Rahimi and Recht, 2007; Le et al., 2013; Yang et al., 2015) , and low rank approximations (Silverman, 1985; Quiñonero-Candela and Rasmussen, 2005). Wilson and Nickisch (2015) recently introduced the KISS-GP approximate kernel matrix , which admits fast computations, given the exact kernel matrix evaluated on a latent multidimensional lattice of inducing inputs , and , a sparse interpolation matrix. Without requiring any grid structure in , decomposes into a Kronecker product of Toeplitz matrices, which can be approximated by circulant matrices (Wilson et al., 2015). Exploiting such structure in combination with local kernel interpolation enables one to use many inducing points, resulting in near-exact accuracy in the kernel approximation, and inference. Unfortunately, this approach does not typically apply to dimensional inputs (Wilson et al., 2015).
Moreover, the Gaussian process marginal likelihood does not factorize, and thus stochastic gradient descent does not ordinarily apply. To address this issue,Hensman et al. (2013) extended the variational approach from Titsias (2009) and derived a stochastic variational GP posterior over inducing points for a regression model which does have the required factorization for stochastic gradient descent. Hensman et al. (2015b), Hensman et al. (2015a), and Dezfouli and Bonilla (2015)
further combine this with a sampling procedure for estimating non-conjugate expectations. These methods havesampling complexity which becomes prohibitive where many inducing points are desired for accurate approximation. Nickson et al. (2015) consider Kronecker structure in the stochastic approximation of Hensman et al. (2013) for regression, but do not leverage local kernel interpolation or sampling.
To address these limitations, we introduce a new deep kernel learning model for multi-task classification, mini-batch training, and scalable kernel interpolation which does not require low dimensional input spaces. In this paper, we view scalability and flexibility as two sides of one coin: we most want the flexible models on the largest datasets, which contain the necessary information to discover rich statistical structure. We show that the resulting approach can learn very expressive and interpretable kernel functions on large classification datasets, containing millions of training points.
3 Deep Kernel Learning for Multi-task Classification
We propose a new deep kernel learning approach to account for classification and non-Gaussian likelihoods, multiple correlated outputs, additive covariances, and stochastic gradient training.
We propose to build a probabilistic deep network as follows: 1) a deep non-linear transformation, parametrized by weights , is applied to the observed input variable , to produce features at the final layer , ; 2) Gaussian processes, with base kernels , are applied to subsets of these features, corresponding to an additive GP model (e.g., Durrande et al., 2011). The base kernels can thus act on relatively low dimensional inputs, where local kernel interpolation and learning biases such as similarities based on Euclidean distance are most natural; 3) these GPs are linearly mixed by a matrix , and transformed by an observation model, to produce the output variables . The mixing of these variables through produces correlated multiple outputs, a multi-task property which is uncommon in Gaussian processes or SVMs. The structure of this network is illustrated in Figure 1. Critically, all of the parameters in the model (including base kernel hyperparameters) are trained through optimizing a marginal likelihood, having integrated away the Gaussian processes, through the variational inference procedures described in section 4.
For classification, we consider a special case of this architecture. Let be the number of classes, and we have data , where is a one-shot encoding of the class label. We use the softmax observation model:
where is a vector of independent Gaussian processes followed by a linear mixing layer ; and is the indicator vector with the th element being 1 and the rest 0.
For the th Gaussian process in the additive GP layer, let be the latent functions on the input data features. By introducing a set of latent inducing variables indexed by inducing inputs , we can write (e.g., Quiñonero-Candela and Rasmussen, 2005)
Substituting the local interpolation approximation of Wilson and Nickisch (2015) into Eq. (5), we find ; it therefore follows that . In section 4 we exploit this deterministic relationship between and , governed by the sparse matrix , to derive a particularly efficient stochastic variational inference procedure.
4 Structure Exploiting Stochastic Variational Inference
Exact inference and learning in Gaussian processes with a non-Gaussian likelihood is not analytically tractable. Variational inference is an appealing approximate technique due to its automatic regularization to avoid overfitting, and its ability to be used with stochastic gradient training, by providing a factorized approximation to the Gaussian process marginal likelihood. We develop our stochastic variational method equipped with a fast sampling scheme for tackling any intractable marginalization.
Let be the collection of the inducing variables of the additive GPs. We assume a variational posterior over the inducing variables . By Jensen’s inequality we have
where we have omitted the mixing weights for clarity. The KL divergence term can be interpreted as a regularizer encouraging the approximate posterior to be close to the prior . We aim at tightening the marginal likelihood lower bound which is equivalent to minimizing the KL divergence from to the true posterior.
Since the likelihood function typically factorizes over data instances: , we can optimize the lower bound with stochastic gradients. In particular, we specify for the independent GPs, and iteratively update the variational parameters and the kernel and deep network parameters using a noisy approximation of the gradient of the lower bound on minibatches of the full data. Henceforth we omit the index for clarity.
Unfortunately, for general non-Gaussian likelihoods the expectation in Eq (7) is usually intractable. We develop a sampling method for tackling this intractability which is highly efficient with structured reparameterization, local kernel interpolation, and structure exploiting algebra.
Using local kernel interpolation, the latent function is expressed as a deterministic local interpolation of the inducing variables (section 3). This result allows us to work around any difficult approximate posteriors on which typically occur in variational approaches for GPs. Instead, our sampler only needs to account for the uncertainty on . The direct parameterization of yields a straightforward and efficient sampling procedure. The latent function samples (indexed by ) are then computed directly through interpolation .
As opposed to conventional mean-field methods, which assume a diagonal variational covariance matrix, we use the Cholesky decomposition for reparameterizing in order to preserve structures within the covariance. Specifically, we let , resulting in the following sampling procedure:
Each step of the above standard sampler has complexity of , where is the number of inducing points. Due to the matrix vector product, this sampling procedure becomes prohibitive in the presence of many inducing points, which are required for accuracy on large datasets with multidimensional inputs – particularly if we have an expressive kernel function (Wilson and Nickisch, 2015).
We scale up the sampler by leveraging the fact that the inducing points are placed on a grid (taking advantage of both Toeplitz and circulant structure), and additionally imposing a Kronecker decomposition on , where is the input dimension of the base kernel. With the fast Kronecker matrix-vector products, we reduce the above sampling cost of to . Our approach thus greatly improves over previous stochastic variational methods which typically scale with complexity, as discussed shortly.
Note that the KL divergence term between the two Gaussians in Eq (7) has a closed form without the need for Monte Carlo estimation. Computing the KL term and its derivatives, with the Kronecker method, is . With samples of and a minibatch of data points of size , we can estimate the marginal likelihood lower bound as
and the derivatives w.r.t the model hyperparameters and the variational parameters can be taken similarly. We provide the detailed derivation in the supplement.
Although a small body of pioneering work has developed stochastic variational methods for Gaussian processes, our approach distinctly provides the above representation-preserving variational approximation, and exploits algebraic structure for significant advantages in scalability and accuracy. In particular, a similar variational lower bound as in Eq (7) was proposed in Titsias (2009); Hensman et al. (2013) for a sparse GP, which were extended to non-conjugate likelihoods, with the intractable integrals estimated using Gaussian quadrature as in the KLSP-GP Hensman et al. (2015a) or univariate Gaussian samples as in the SAVI-GP Dezfouli and Bonilla (2015). Hensman et al. (2015b) estimates nonconjugate expectations with a hybrid Monte Carlo sampler (denoted as MC-GP). The computations in these approaches can be costly, with complexity, due to a complicated variational posterior over as well as the expensive operations on the full inducing point matrix. In addition to its increased efficiency, our sampling scheme is much simpler, without introducing any additional tuning parameters. We empirically compare with these methods and show the practical significance of our algorithm in section 5.
Variational methods have also been used in GP regression for stochastic inference (e.g., (Nickson et al., 2015; Hensman et al., 2013)), and some of the most recent work in this area applied variational auto-encoders (Kingma and Welling, 2013) for coupled variational updates (aka back constraints) Dai et al. (2015); Bui and Turner (2015). We note that these techniques are orthogonal and complementary to our inference approach, and can be leveraged for further enhancements.
We evaluate our proposed approach, stochastic variational deep kernel learning (SV-DKL), on a wide range of classification problems, including an airline delay task with over 5.9 million data points (section 5.1), a large and diverse collection of classification problems from the UCI repository (section 5.2), and image classification benchmarks (section 5.3). Empirical results demonstrate the practical significance of our approach, which provides consistent improvements over stand-alone DNNs, while preserving a GP representation, and dramatic improvements in speed and accuracy over modern state of the art GP models. We use classification accuracy when comparing to DNNs, because it is a standard for evaluating classification benchmarks with DNNs. However, we also compute the negative log probability (NLP) values (supplement), which show similar trends.
All experiments were performed on a Linux machine with eight 4.0GHz CPU cores, one Tesla K40c GPU, and 32GB RAM. We implemented deep neural networks with Caffe(Jia et al., 2014).
For our deep kernel learning model, we used deep neural networks which produce -dimensional top-level features. Here is the number of classes. We place a Gaussian process on each dimension of these features. We used RBF base kernels. The additive GP layer is then followed by a linear mixing layer . We initialized
to be an identity matrix, and optimized in the joint learning procedure to recover cross-dimension correlations from data.
We first train a deep neural network using SGD with the softmax loss objective, and rectified linear activation functions. After the neural network has been pre-trained, we fit an additive KISS-GP layer, followed by a linear mixing layer, using the top-level features of the deep network as inputs. Using this pre-traininginitialization, our joint SV-DKL model of section 3 is then trained through the stochastic variational method of section 4 which jointly optimizes all the hyperparameters
of the deep kernel (including all network weights), as well as the variational parameters, by backpropagating derivatives through the proposed marginal likelihood lower bound of the additive Gaussian process in section4. In all experiments, we use a relatively large mini-batch size (specified according to the full data size), enabled by the proposed structure exploiting variational inference procedures. We achieve good performance setting the number of samples in Eq. 4 for expectation estimation in variational inference, which provides additional confirmation for a similar observation in (Kingma and Welling, 2013).
5.1 Airline Delays
We first consider a large airline dataset consisting of flight arrival and departure details for all commercial flights within the US in 2008. The approximately 5.9 million records contain extensive information about the flights, including the delay in reaching the destination. Following Hensman et al. (2015a), we consider the task of predicting whether a flight was subject to delay based on 8 features (e.g., distance to be covered, day of the week, etc).
, a recent scalable variational GP classifier as discussed in section4; 2) stand-alone deep neural network (DNN); 3) DNN+, a stand-alone DNN with an extra fully-connected hidden layer with , defined as in Figure 1; 4) DNN+GP which is a GP applied to a pre-trained DNN (with same architecture as in 2); and 5) our stochastic variational DKL method (SV-DKL) (same DNN architecture as in 2). For DNN, we used a fully-connected architecture with layers d-1000-1000-500-50-c.222We obtained similar results with other DNN architectures (e.g., d-1000-1000-500-50-20-c).
The DNN component of the SV-DKL model has the exact same architecture. The SV-DKL joint training was conducted using a large minibatch size of 50,000 to reduce the variance of the stochastic gradient. We can use such a large minibatch in each iteration (which is daunting for regular GP even as a whole dataset) due to the efficiency of our inference strategy within each mini-batch, leveraging structure exploiting algebra.
From the table we see that SV-DKL outperforms both the alternative variational GP model (KLSP-GP) and the stand-alone deep network. DNN+GP outperforms stand-alone DNNs, showing the non-parametric flexibility of kernel methods. By combining KISS-GP with DNNs as part of a joint SV-DKL procedure, we obtain better results than DNN and DNN+GP. Besides, both the plain DNN and SV-DKL notably improve on stand-alone GPs, indicating a superior capacity of deep architectures to learn representations from large but finite training sets, despite the asymptotic approximation properties of Gaussian processes. By contrast, adding an extra hidden layer, as in DNN+, does not improve performance.
Figure 2 further studies how performance changes as data size increases. We observe that the proposed SV-DKL classifier trained on 1/50 of the data already can reach a competitive accuracy as compared to the KLSP-GP model trained on the full dataset. As the number of the training points increases, the SV-DKL and DNN models continue to improve. This experiment demonstrates the value of expressive kernel functions on large data problems, which can effectively capture the extra information available as seeing more training instances. Furthermore, SV-DKL consistently provides better performance over the plain DNN, through its non-parametric flexibility.
We next measure the scalability of our variational DKL in terms of the number of inducing points in each GP. Figure 2 shows the runtimes in seconds, as a function of , for evaluating the objective and any relevant derivatives. We compare our structure exploiting variational method with the scalable variational inference in KLSP-GP, and the MCMC-based variational method in MC-GP Hensman et al. (2015b). We see that our inference approach is far more efficient than previous scalable algorithms. Moreover, when the number of inducing points is not too large (e.g., ), the added time for SV-DKL over DNN is reasonable (e.g., 0.39s vs. 0.27s), especially considering the gains in performance and expressive power. Figure 2 shows the runtime scaling of different variational methods as grows. We can see that the runtime of our approach increases only slowly in a wide range of (), greatly enhancing the scalability over the other methods. This empirically validates the improved time complexity of our new inference method as presented in section 4.
We next investigate the total training time of the models. Table 1, right panel, lists the time cost of training KLSP-GP, DNN, and SV-DKL; and Figure 2 shows how the training time of SV-DKL and DNN changes as more training data is presented. We see that on the full dataset DKL, as a GP model, saves over 60% time as compared to the modern state of the art KLSP-GP, while at the same time achieving over an 18% improvement in predictive accuracy. Generally, the training time of SV-DKL increases slowly with growing data sizes, and has only modest additional overhead compared to stand-alone architectures, justified by improvements in performance, and the general benefits of a non-parametric probabilistic representation. Moreover, the DNN was fully trained on a GPU, while in SV-DKL the base kernel hyperparameters and variational parameters were optimized on a CPU. Since most updates of the SV-DKL parameters are computed in matrix forms, we believe the already modest time gap between SV-DKL and DNNs can be almost entirely closed by deploying the whole SV-DKL model on GPUs.
|Datasets||n||d||c||Accuracy||Total Training Time (h)|
|KLSP-GP Hensman et al. (2015a)||DNN||DNN+||DNN+GP||SV-DKL||KLSP-GP||DNN||SV-DKL|
one standard deviation. Since the code of KLSP-GP is not publicly available we directly show the results fromHensman et al. (2015a).
5.2 UCI Classification Tasks
The second evaluation of our proposed algorithm (SV-DKL) is conducted on a number of commonly used UCI classification tasks of varying sizes and properties. Table 2 lists the classification accuracy of SVM, DNN, DNN+ (a stand-alone DNN with an extra fully-connected hidden layer with , defined as in Figure 1), DNN+GP (a GP trained on the top level features of a trained DNN without the extra hidden layer), and SV-DKL (same architecture as DNN).
The plain DNN, which learns salient features effectively from raw data, gives notably higher accuracy compared to an SVM, the mostly widely used kernel method for classification problems. We see that the extra layer in DNN+GP can sometimes harm performance. By contrast, non-parametric flexibility of DNN+GP consistently improves upon DNN. And SV-DKL, by training a DNN through a GP marginal likelihood objective, consistently provides further enhancements (with particularly notable performance on the Connect4 and Covtype datasets).
5.3 Image Classification
We next evaluate the proposed scalable SV-DKL procedure for efficiently handling high-dimensional highly-structured image data. We used a minibatch size of 5,000 for stochastic gradient training of SV-DKL. Table 3 compares SV-DKL with the most recent scalable GP classifiers. Besides KLSP-GP, we also collected the results of the MC-GP Hensman et al. (2015b) which uses a hybrid Monte Carlo sampler to tackle non-conjugate likelihoods, SAVI-GP Dezfouli and Bonilla (2015) which approximates with a univariate Gaussian sampler, as well as the distributed GP latent variable model (denoted as D-GPLVM) Gal et al. (2014). We see that on the respective benchmark tasks, SV-DKL improves over all of the above scalable GP methods by a large margin. We note that these datasets are very challenging for conventional GP methods.
We further compare SV-DKL to stand-alone convolutional neural networks, and GPs applied to fixed pre-trained CNNs (CNN+GP). On the first three datasets in Table3, we used the reference CNN models implemented in Caffe; and for the SVHN dataset, as no benchmark architecture is available, we used the CIFAR10 architecture which turned out to perform quite well. As we can see, the SV-DKL model outperforms CNNs and CNN+GP on all datasets. By contrast, the extra hidden hidden layer CNN+ does not consistently improve performance over CNN.
|MC-GP Hensman et al. (2015b)||SAVI-GP Dezfouli and Bonilla (2015)||D-GPLVM Gal et al. (2014)||KLSP-GP Hensman et al. (2015a)||CNN||CNN+||CNN+GP||SV-DKL|
Classification accuracy on the image classification benchmarks. MNIST-Binary is the task to differentiate between odd and even digits on the MNIST dataset. We followed the standard training-test set partitioning of all these datasets. We have collected recently published results of a variety of scalable GPs. For CNNs, we used the respective benchmark architectures (or with slight adaptations) from Caffe. CNN+ is a stand-alone CNN withfully connected extra hidden layer. See the text for more details, including a comparison with ResNets on CIFAR10.
We also provide brief evaluations of SV-DKL for ResNets and ImageNet, as we believe such exploration will be a promising direction for future research.
ResNet Comparison: Based on one of the best public implementations on Caffe, the ResNet-20 has 0.901 accuracy on CIFAR10, and SV-DKL (with this ResNet base architecture) improves to 0.910.
ImageNet: We randomly selected 20 categories of images with an AlexNet variant as the base NN (Krizhevsky et al., 2012), which has an accuracy of 0.6877, while SV-DKL achieves 0.7067 accuracy.
In Figure 3 we investigate the deep kernels learned on the MNIST dataset by randomly selecting 4 classes and visualizing the covariance matrices of respective dimensions. The covariance matrices are evaluated on the set of test inputs, sorted in terms of the labels of the input images. We see that the deep kernel on each dimension effectively discovers the correlations between the images within the corresponding class. For instance, in the data points between 2k-3k (i.e., images of digit 2) are strongly correlated with each other, and carry little correlation with the rest of the images. Besides, we can also clearly observe that the rest of the data points also form multiple “blocks”, rather than being crammed together without any structure. This validates that the DKL procedure and additive GPs do capture the correlations across different dimensions.
To further explore the learnt dependencies between the output classes and the additive GPs serving as the bases, we visualized the weights of the mixing layer () in Fig. 3
, enabling the correlated multi-output (multi-task) nature of the model. Besides the expected high weights along the diagonal, we find that class 9 is also highly correlated with dimension 0 and 6, which is consistent with the visual similarity between digit “9” and “0”/“6”. Overall, the ability tointerpret the learned deep covariance matrix as discovering an expressive similarity metric across data instances is a distinctive feature of our approach.
We introduced a scalable Gaussian process model which leverages deep learning, stochastic variational inference, structure exploiting algebra, and additive covariance structures. The resulting deep kernel learning approach, SV-DKL, allows for classification and non-Gaussian likelihoods, multi-task learning, and mini-batch training. SV-DKL achieves superior performance over alternative scalable GP models and stand-alone deep networks on many significant benchmarks.
Several fundamental themes emerge from the exposition: (1) kernel methods and deep learning approaches are complementary, and we can combine the advantages of each approach; (2) expressive kernel functions are particularly valuable on large datasets; (3) by viewing neural networks through the lens of metric learning, deep learning approaches become more interpretable.
Deep learning is able to obtain good predictive accuracy by automatically learning structure which would be difficult to a priori feature engineer into a model. In the future, we hope deep kernel learning approaches will be particularly helpful both for characterizing uncertainty and for interpreting these learned features, leading to new scientific insights into our modelling problems.
Acknowledgements: We thank NSF IIS-1563887, ONR N000141410684, N000141310721, N000141512791, and ADeLAIDE FA8750-16C-0130-001 grants. We also thank anonymous reviewers for helpful comments.
- Aggarwal et al.  C. C. Aggarwal, A. Hinneburg, and D. A. Keim. On the surprising behavior of distance metrics in high dimensional space. Springer, 2001.
- Bui and Turner  T. D. Bui and R. E. Turner. Stochastic variational inference for Gaussian process latent variable models using back constraints. Black Box Learning and Inference NIPS workshop, 2015.
- Calandra et al.  R. Calandra, J. Peters, C. E. Rasmussen, and M. P. Deisenroth. Manifold Gaussian processes for regression. arXiv preprint arXiv:1402.5876, 2014.
Chang and Lin 
C.-C. Chang and C.-J. Lin.
Libsvm: a library for support vector machines.ACM Transactions on Intelligent Systems and Technology (TIST), 2(3):27, 2011.
- Dai et al.  Z. Dai, A. Damianou, J. González, and N. Lawrence. Variational auto-encoded deep Gaussian processes. arXiv preprint arXiv:1511.06455, 2015.
- Damianou and Lawrence  A. Damianou and N. Lawrence. Deep Gaussian processes. Artificial Intelligence and Statistics, 2013.
- Dezfouli and Bonilla  A. Dezfouli and E. V. Bonilla. Scalable inference for Gaussian process models with black-box likelihoods. In Advances in Neural Information Processing Systems, pages 1414–1422, 2015.
- Durrande et al.  N. Durrande, D. Ginsbourger, and O. Roustant. Additive kernels for Gaussian process modeling. arXiv preprint arXiv:1103.4023, 2011.
- Gal et al.  Y. Gal, M. van der Wilk, and C. Rasmussen. Distributed variational inference in sparse Gaussian process regression and latent variable models. In Advances in Neural Information Processing Systems, pages 3257–3265, 2014.
Gönen and Alpaydın 
M. Gönen and E. Alpaydın.
Multiple kernel learning algorithms.
Journal of Machine Learning Research, 12:2211–2268, 2011.
- Hensman et al.  J. Hensman, N. Fusi, and N. Lawrence. Gaussian processes for big data. In Uncertainty in Artificial Intelligence (UAI). AUAI Press, 2013.
- Hensman et al. [2015a] J. Hensman, A. Matthews, and Z. Ghahramani. Scalable variational Gaussian process classification. In Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, pages 351–360, 2015a.
- Hensman et al. [2015b] J. Hensman, A. G. Matthews, M. Filippone, and Z. Ghahramani. MCMC for variationally sparse Gaussian processes. In Advances in Neural Information Processing Systems, pages 1648–1656, 2015b.
- Jia et al.  Y. Jia, E. Shelhamer, J. Donahue, S. Karayev, J. Long, R. Girshick, S. Guadarrama, and T. Darrell. Caffe: Convolutional architecture for fast feature embedding. arXiv preprint arXiv:1408.5093, 2014.
- Kingma and Welling  D. P. Kingma and M. Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
- Krizhevsky et al.  A. Krizhevsky, I. Sutskever, and G. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, 2012.
- Le et al.  Q. Le, T. Sarlos, and A. Smola. Fastfood-computing Hilbert space expansions in loglinear time. In Proceedings of the 30th International Conference on Machine Learning, pages 244–252, 2013.
- Lloyd et al.  J. R. Lloyd, D. Duvenaud, R. Grosse, J. B. Tenenbaum, and Z. Ghahramani. Automatic construction and Natural-Language description of nonparametric regression models. In Association for the Advancement of Artificial Intelligence (AAAI), 2014.
- Nickson et al.  T. Nickson, T. Gunter, C. Lloyd, M. A. Osborne, and S. Roberts. Blitzkriging: Kronecker-structured stochastic Gaussian processes. arXiv preprint arXiv:1510.07965, 2015.
- Quiñonero-Candela and Rasmussen  J. Quiñonero-Candela and C. Rasmussen. A unifying view of sparse approximate Gaussian process regression. The Journal of Machine Learning Research, 6:1939–1959, 2005.
- Rahimi and Recht  A. Rahimi and B. Recht. Random features for large-scale kernel machines. In Neural Information Processing Systems, 2007.
- Rasmussen and Ghahramani  C. E. Rasmussen and Z. Ghahramani. Occam’s razor. In Neural Information Processing Systems (NIPS), 2001.
- Rasmussen and Williams  C. E. Rasmussen and C. K. I. Williams. Gaussian processes for Machine Learning. The MIT Press, 2006.
- Silverman  B. W. Silverman. Some aspects of the spline smoothing approach to non-parametric regression curve fitting. Journal of the Royal Statistical SocietyB, 47(1):1–52, 1985.
- Titsias  M. K. Titsias. Variational learning of inducing variables in sparse Gaussian processes. In International Conference on Artificial Intelligence and Statistics, pages 567–574, 2009.
- Wilson  A. G. Wilson. Covariance kernels for fast automatic pattern discovery and extrapolation with Gaussian processes. PhD thesis, University of Cambridge, 2014.
- Wilson and Adams  A. G. Wilson and R. P. Adams. Gaussian process kernels for pattern discovery and extrapolation. International Conference on Machine Learning (ICML), 2013.
- Wilson and Nickisch  A. G. Wilson and H. Nickisch. Kernel interpolation for scalable structured Gaussian processes (KISS-GP). International Conference on Machine Learning (ICML), 2015.
- Wilson et al.  A. G. Wilson, D. A. Knowles, and Z. Ghahramani. Gaussian process regression networks. In International Conference on Machine Learning (ICML), Edinburgh, 2012.
- Wilson et al.  A. G. Wilson, C. Dann, and H. Nickisch. Thoughts on massively scalable Gaussian processes. arXiv preprint arXiv:1511.01870, 2015. https://arxiv.org/abs/1511.01870.
- Wilson et al.  A. G. Wilson, Z. Hu, R. Salakhutdinov, and E. P. Xing. Deep kernel learning. Artificial Intelligence and Statistics, 2016.
- Yang et al.  Z. Yang, A. J. Smola, L. Song, and A. G. Wilson. A la carte - learning fast kernels. Artificial Intelligence and Statistics, 2015.
Appendix A Negative Log Probability (NLP) Results
Appendix B Stochastic Variational Inference for Deep Kernel Learning Classification
Recall the SV-DKL classification model
Let . We assume a variational posterior over the inducing variables
By Jensen’s inequality we have
In the following we omit the GP index when there is no ambiguity. Due to the deterministic mapping, we can obtain latent function samples from the samples of :
To sample from , we use the Cholesky decomposition for reparameterizing in order to preserve structures within the covariance. Specifically, we let . This results in the following sampling procedure for :
We further scale up the sampler by leveraging the fact that the inducing points are placed on a grid, and imposing Kronecker decomposition on , where is the input dimension of the base kernel. With the fast Kronecker matrix-vector products, the sampling cost is . Note that
With the samples, then for any , we have
Next we give the derivation of the objective lower bound and its derivatives in detail. In the following we denote for clarity.
Computation of the marginal likelihood lower bound
With the Kronecker product representation of the covariance matrices, all the above matrix operations can be conducted efficiently:
where is the number of inducing points in dimension (we have ); ; and .
Derivatives w.r.t the base kernel hyperparameters
Derivatives w.r.t other model parameters
Derivatives w.r.t the variational parameters
We only show the derivatives w.r.t the variational covariance parameters . The derivatives w.r.t the variational means can be derived similarly.
(1) The derivative of the softmax expectation term of input w.r.t the -th element of , denoted as for clarity, is given by
where is the th row of the interpolation matrix (i.e., the interpolation vector of input ); and . Note that for , we can directly write down the derivatives w.r.t the whole matrix which is efficient for computing:
(2) The derivative of the KL term is (index omitted):
where . Note that the Kronecker factor matrices are small (with size ) and thus the above computations are fast.