Deep learning has had enormous recent success in fields such as speech recognition and computer vision. In these problems, our goal is to predict well and we are typically less interested in the uncertainty behind the predictions. However, deep learning is now becoming increasingly popular in applications such as robotics and medical diagnostics, where accurate measures of uncertainty are crucial for reliable decisions. For example, uncertainty estimates are important for physicians who use automated diagnosis systems to choose effective and safe treatment options. Lack of such estimates may lead to decisions that have disastrous consequences.
The goal of Bayesian deep learning is to provide uncertainty estimates by integrating over the posterior distribution of the parameters. Unfortunately, the complexity of deep learning models makes it infeasible to perform the integration exactly. Sampling methods such as stochastic-gradient Markov chain Monte Carlochen2014stochastic have been applied to deep models, but they usually converge slowly. They also require a large memory to store the samples and often need large preconditioners to mix well ahn2012bayesian ; balan2015bayesian ; simsekli2016stochastic . In contrast, variational inference (VI) methods require much less memory and can scale to large problems by exploiting stochastic gradient methods (blundell2015weight, ; graves2011practical, ; ranganath2013black, ). However, they often make crude simplifications, like the mean-field approximation, to reduce the memory and computation cost. This can result in poor uncertainty estimates turner2011two . Fast and accurate estimation of uncertainty for large models remains a challenging problem in Bayesian deep learning.
In this paper, we propose a new variational inference method to estimate Gaussian approximations with a diagonal plus low-rank covariance structure. This gives more accurate and flexible approximations than the mean-field approach. Our method also enables fast estimation by using an approximate natural-gradient algorithm that builds the covariance estimate solely based on the back-propagated gradients of the network log-likelihood. We call our method stochastic low-rank approximate natural-gradient (SLANG). SLANG requires strictly less gradient computations than methods that require gradients of the variational objective obtained using the reparameterization trick miller2017variational ; ong2017gaussian ; titsias2014doubly . Our empirical comparisons demonstrate the improvements obtained over mean-field methods (see Figure 1 for an example) and show that SLANG gives comparable results to the state-of-the-art on standard benchmarks.
The code to reproduce the experimental results in this paper is available at https://github.com/aaronpmishkin/SLANG.
1.1 Related Work
Gaussian variational distributions with full covariance matrices have been used extensively for shallow models barber1998ensemble ; Challis:11 ; Jaakkola96b ; marlin2011piecewise ; miller2017variational ; seeger2000bayesian ; tan2018gaussian ; titsias2014doubly . Several efficient ways of computing the full covariance matrix are discussed by Seeger seeger2010gaussian . Other works have considered various structured covariance approximations, based on the Cholesky decomposition Challis:11 ; titsias2014doubly , sparse covariance matrices tan2018gaussian and low-rank plus diagonal structure barber1998ensemble ; ong2017gaussian ; seeger2000bayesian . Recently, several works miller2017variational ; ong2017gaussian
have applied stochastic gradient descent on the variational objective to estimate such a structure. These methods often employ an adaptive learning rate method, such as Adam or RMSprop, which increases the memory cost. All of these methods have only been applied to shallow models, and it remains unclear how they will perform (and whether they can be adapted) for deep models. Moreover, a natural-gradient method is preferable to gradient-based methods when optimizing the parameters of a distributionamari1998natural ; hoffman2013stochastic ; khan2018fast . Our work shows that a natural-gradient method not only has better convergence properties, but also has lower computation and memory cost than gradient-based methods.
For deep models, a variety of methods have been proposed based on mean-field approximations. These methods optimize the variational objective using stochastic-gradient methods and differ from each other in the way they compute those gradients (blundell2015weight, ; graves2011practical, ; hernandez15pbp, ; khan2017vprop, ; ranganath2013black, ; zhang2017noisy, ). They all give poor uncertainty estimates in the presence of strong posterior correlations and also shrink variances turner2011two . SLANG is designed to add extra covariance structure and ensure better performance than mean-field approaches.
A few recent works have explored structured covariance approximations for deep models. In zhang2017noisy , the Kronecker-factored approximate curvature (K-FAC) method is applied to perform approximate natural-gradient VI. Another recent work has applied K-FAC to find a Laplace approximation ritter2018scalable . However, the Laplace approximation can perform worse than variational inference in many scenarios, e.g., when the posterior distribution is not symmetric nickisch2008approximations . Other types of approximation methods include Bayesian dropout yarin16dropout and methods that use matrix-variate Gaussians louizos2016structured ; sun2017learning . All of these approaches make structural assumptions that are different from our low-rank plus diagonal structure. However, similarly to our work, they provide new ways to improve the speed and accuracy of uncertainty estimation in deep learning.
2 Gaussian Approximation with Natural-Gradient Variational Inference
Our goal is to estimate the uncertainty in deep models using Bayesian inference. Givendata examples , a Bayesian version of a deep model can be specified by using a likelihood parametrized by a deep network with parameters and a prior distribution
. For simplicity, we assume that the prior is a Gaussian distribution, such as an isotropic Gaussianwith the scalar precision parameter . However, the methods presented in this paper can easily be modified to handle many other types of prior distributions. Given such a model, Bayesian approaches compute an estimate of uncertainty by using the posterior distribution: . This requires computation of the marginal likelihood , which is a high-dimensional integral and difficult to compute.
Variational inference (VI) simplifies the problem by approximating with a distribution . In this paper, our focus is on obtaining approximations that have a Gaussian form, i.e., with mean and covariance . The parameters and are referred to as the variational parameters and can be obtained by maximizing a lower bound on called the evidence lower bound (ELBO),
denotes the Kullback-Leibler divergence.
A straightforward and popular approach to optimize is to use stochastic gradient methods miller2017variational ; ong2017gaussian ; ranganath2013black ; titsias2014doubly . However, natural-gradients are preferable when optimizing the parameters of a distribution amari1998natural ; hoffman2013stochastic ; khan2018fast . This is because natural-gradient methods perform optimization on the Riemannian manifold of the parameters, which can lead to a faster convergence when compared to gradient-based methods. Typically, natural-gradient methods are difficult to implement, but many easy-to-implement updates have been derived in recent works hoffman2013stochastic ; khan2017conjugate ; khan2017vprop ; zhang2017noisy . We build upon the approximate natural-gradient method proposed in khan2017vprop and modify it to estimate structured covariance-approximations.
where is the iteration number, are learning rates, , is the back-propagated gradient obtained on the ’th data example, is an Empirical Fisher (EF) matrix, and is a minibatch of data examples. This update is an approximate natural-gradient update obtained by using the EF matrix as an approximation of the Hessian martens2014new in a method called Variational Online Newton (VON) khan2017vprop . This is explained in more detail in Appendix A. As discussed in khan2017vprop , the VOGN method is an approximate Natural-gradient update which may not have the same properties as the exact natural-gradient update. However, an advantage of the update (2) is that it only requires back-propagated gradients, which is a desirable feature when working with deep models.
The update (2) is computationally infeasible for large deep models because it requires the storage and inversion of the covariance matrix. Storage takes memory space and inversion requires computations, which makes the update very costly to perform for large models. We cannot form or invert it when is in millions. Mean-field approximations avoid this issue by restricting to be a diagonal matrix, but they often give poor Gaussian approximations. Our idea is to estimate a low-rank plus diagonal approximation of that reduces the computational cost while preserving some off-diagonal covariance structure. In the next section, we propose modifications to the update (2) to obtain a method whose time and space complexity are both linear in .
3 Stochastic, Low-rank, Approximate Natural-Gradient (SLANG) Method
Our goal is to modify the update (2) to obtain a method whose time and space complexity is linear in . We propose to approximate the inverse of by a “low-rank plus diagonal” matrix:
where is a matrix with and is a diagonal matrix. The cost of storing and inverting this matrix is linear in and reasonable when is small. We now derive an update for and such that the resulting closely approximates the update shown in (2). We start by writing an approximation to the update of where we replace covariance matrices by their structured approximations:
This update cannot be performed exactly without potentially increasing the rank of the low-rank component , since the structured components on the right hand side are of rank at most , where is the size of the minibatch. This is shown in (5) below where we have rearranged the left hand side of (4) as the sum of a structured component and a diagonal component. To obtain a rank approximation to the left hand side of (5
), we propose to approximate the structured component by an eigenvalue decomposition as shown in (6) below,
where is a matrix containing the first
leading eigenvectors ofand is an diagonal matrix containing the corresponding eigenvalues. Figure 2 visualizes the update from (5) to (6).
The low-rank component can now be updated to mirror the low-rank component of (6),
and the diagonal can be updated to match the diagonal of the left and right sides of (4), i.e.,
This gives us the following update for using a diagonal correction ,
This step is cheap since computing the diagonal of the EF matrix is linear in .
The new covariance approximation can now be used to update according to (2) as shown below:
The above update uses a stochastic, low-rank covariance estimate to approximate natural-gradient updates, which is why we use the name SLANG.
When , is full rank and SLANG is identical to the approximate natural-gradient update (2). When , SLANG produces matrices with diagonals matching (2) at every iteration. The diagonal correction ensures that no diagonal information is lost during the low-rank approximation of the covariance. A formal statement and proof is given in Appendix D.
We also tried an alternative method where is learned using an exponential moving-average of the eigendecompositions of . This previous iteration of SLANG is discussed in Appendix B, where we show that it gives worse results than the SLANG update.
Next, we give implementation details of SLANG.
3.1 Details of the SLANG Implementation
At every iteration, we generate a sample . This is implemented in line 4 of Algorithm 1 using the subroutine fast_sample. Pseudocode for this subroutine is given in Algorithm 3. This function uses the Woodbury identity and to compute the square-root matrix ambikasaran2014fast . The sample is then computed as , where . The function fast_sample requires computations in to generate samples, which is linear in . More details are given in Appendix C.4.
Given a sample, we need to compute and store all the individual stochastic gradients for all examples in a minibatch
. The standard back-propagation implementation does not allow this. We instead use a version of the backpropagation algorithm outlined in a note by Goodfellowgoodfellow2015efficient , which enables efficient computation of the gradients . This is shown in line 6 of Algorithm 1, where a subroutine backprop_goodfellow is used (see details of this subroutine in Appendix C.1).
In line 7, we compute the eigenvalue decomposition of by using the fast_eig subroutine. The subroutine fast_eig implements a randomized eigenvalue decomposition method discussed in halko2011finding . It computes the top- eigendecomposition of a low-rank matrix in . More details on the subroutine are given in Appendix C.2. The matrix and are updated using the eigenvalue decomposition in lines 8, 9 and 10.
In lines 11 and 12, we compute the update vector, which requires solving a linear system. We use the subroutine fast_inverse shown in Algorithm 2. This subroutine uses the Woodbury identity to efficiently compute the inverse with a cost . More details are given in Appendix C.3. Finally, in line 13, we update .
The overall computational complexity of SLANG is and its memory cost is . Both are linear in and . The cost is quadratic in , but since (e.g., 5 or 10), this only adds a small multiplicative constant in the runtime. SLANG reduces the cost of the update (2) significantly while preserving some posterior correlations.
In this section, our goal is to show experimental results in support of the following claims: (1) SLANG gives reasonable posterior approximations, and (2) SLANG performs well on standard benchmarks for Bayesian neural networks. We present evaluations on several LIBSVM datasets, the UCI regression benchmark, and MNIST classification. SLANG beats mean-field methods on almost all tasks considered and performs comparably to state-of-the-art methods. SLANG also converges faster than mean-field methods.
4.1 Bayesian Logistic Regression
|Mean-Field Methods||SLANG||Full Gaussian|
|Dataset||Metrics||EF||Hess.||Exact||L = 1||L = 5||L = 10||EF||Hess.||Exact|
|KL ( )||2.240||2.030||0.195||0.033||0.008||0.002||0.000||0.000||0.000|
|KL ( )||8.019||9.071||7.771||0.911||0.842||0.638||0.637||0.002||0.000|
Results on Bayesian logistic regression where we compare SLANG to three full-Gaussian methods and three mean-field methods. We measure negative ELBO, test log-loss, and symmetric KL-divergence between each approximation and the Full-Gaussian Exact method (last column). Lower values are better. SLANG nearly always gives better results than the mean-field methods, and withis comparable to Full-Gaussian methods. This shows that our structured covariance approximation is reasonably accurate for Bayesian logistic regression.
We considered four benchmark datasets for our comparison: USPS 3vs5, Australian, Breast-Cancer, and a1a. Details of the datasets are in Table 8 in Appendix E.2 along with the implementation details of the methods we compare to. We use L-BFGS wright1999numerical to compute the optimal full-Gaussian variational approximation that minimizes the ELBO using the method described in Marlin et al. marlin2011piecewise . We refer to the optimal full-Gaussian variational approximation as the “Full-Gaussian Exact” method. We also compute the optimal mean-field Gaussian approximation and refer to it as “MF Exact”.
Figure 1 shows a qualitative comparison of the estimated posterior means, variances, and covariances for the USPS-3vs5 dataset (, ). The figure on the left compares covariance approximations obtained with SLANG to the Full-Gaussian Exact method. Only off-diagonal entries are shown. We see that the approximation becomes more and more accurate as the rank is increased. The figures on the right compare the means and variances. The means match closely for all methods, but the variance is heavily underestimated by the MF Exact method; we see that the variances obtained under the mean-field approximation estimate a high variance where Full-Gaussian Exact has a low variance and vice-versa. This “trend-reversal” is due to the typical shrinking behavior of mean-field methods turner2011two . In contrast, SLANG corrects the trend reversal problem even when . Similar results for other datasets are shown in Figure 7 in Appendix E.1.
The complete results for Bayesian logistic regression are summarized in Table 1, where we also compare to four additional methods called “Full-Gaussian EF”, “Full-Gaussian Hessian”, “Mean-Field EF”, and “Mean-Field Hessian”. The Full-Gaussian EF method is the natural-gradient update (2) which uses the EF matrix , while the Full-Gaussian Hessian method uses the Hessian instead of the EF matrix (the updates are given in (12) and (13) in Appendix A). The last two methods are the mean-field versions of the Full-Gaussian EF and Full-Gaussian Hessian methods, respectively. We compare negative ELBO, test log-loss using cross-entropy, and symmetric KL-divergence between the approximations and the Full-Gaussian Exact method. We report averages over 20 random 50%-50% training-test splits of the dataset. Variances and results for SLANG with are omitted here due to space constraints, but are reported in Table 6 in Appendix E.1.
We find that SLANG with nearly always produces better approximations than the mean-field methods. As expected, increasing improves the quality of the variational distribution found by SLANG according to all three metrics. We also note that Full-Gaussian EF method has similar performance to the Full-Gaussian Hessian method, which indicates that the EF approximation may be acceptable for Bayesian logistic regression.
The left side in Figure 4 shows convergence results on the USPS 3vs5 and Breast Cancer datasets. The three methods SLANG(1, 2, 3) refer to SLANG with . We compare these three SLANG methods to Mean-Field Hessian and Full-Gaussian Hessian. SLANG converges faster than the mean-field method, and matches the convergence of the full-Gaussian method when is increased.
4.2 Bayesian Neural Networks (BNNs)
An example for Bayesian Neural Networks on a synthetic regression dataset is given in Appendix F.1, where we illustrate the quality of SLANG’s posterior covariance.
The right side in Figure 4 shows convergence results for the USPS 3vs5 and Breast Cancer datasets. Here, the three methods SLANG(1, 2, 3) refer to SLANG with . We compare SLANG to a mean-field method called Bayes by Backprop blundell2015weight . Similar to the Bayesian logistic regression experiment, SLANG converges much faster than the mean-field method. However, the ELBO convergence for SLANG shows that the optimization procedure does not necessarily converge to a local minimum. This issue does not appear to affect the test log-likelihood. While it might only be due to stochasticity, it is possible that the problem is exacerbated by the replacement of the Hessian with the EF matrix. We have not determined the specific cause and it warrants further investigation in future work.
|Test RMSE||Test log-likelihood|
|Boston||3.43 0.20||2.97 0.19||3.21 0.19||-2.66 0.06||-2.46 0.06||-2.58 0.05|
|Concrete||6.16 0.13||5.23 0.12||5.58 0.19||-3.25 0.02||-3.04 0.02||-3.13 0.03|
|Energy||0.97 0.09||1.66 0.04||0.64 0.03||-1.45 0.10||-1.99 0.02||-1.12 0.01|
|Kin8nm||0.08 0.00||0.10 0.00||0.08 0.00||1.07 0.00||0.95 0.01||1.06 0.00|
|Naval||0.00 0.00||0.01 0.00||0.00 0.00||4.61 0.01||3.80 0.01||4.76 0.00|
|Power||4.21 0.03||4.02 0.04||4.16 0.04||-2.86 0.01||-2.80 0.01||-2.84 0.01|
|Wine||0.64 0.01||0.62 0.01||0.65 0.01||-0.97 0.01||-0.93 0.01||-0.97 0.01|
|Yacht||1.13 0.06||1.11 0.09||1.08 0.06||-1.56 0.02||-1.55 0.03||-1.88 0.01|
Next, we present results on the UCI regression datasets which are common benchmarks for Bayesian neural networks hernandez15pbp . We repeat the setup111We use the data splits available at https://github.com/yaringal/DropoutUncertaintyExps used in Gal and Ghahramani yarin16dropoutto the Bayes By Backprop (BBB) method blundell2015weight and the Bayesian Dropout method of yarin16dropout . For the 5 smallest datasets, we used a mini-batch size of 10 and 4 Monte-Carlo samples during training. For the 3 larger datasets, we used a mini-batch size of 100 and 2 Monte-Carlo samples during training. More details are given in Appendix F.3. We report test RMSE and test log-likelihood in Table 2. SLANG with just one rank outperforms BBB on 7 out of 8 datasets for RMSE and on 5 out of 8 datasets for log-likelihood. Moreover, SLANG shows comparable performance to Dropout.
Finally, we report results for classification on MNIST. We train a BNN with two hidden layers of 400 hidden units each. The training set consists of 50,000 examples and the remaining 10,000 are used as a validation set. The test set is a separate set which consists of 10,000 examples. We use SLANG with . For each value of , we choose the prior precision and learning rate based on performance on the validation set. Further details can be found in Appendix F.4. The test accuracies are reported in Table 3 and compared to the results obtained in blundell2015weight by using BBB. For SLANG, a good performance can be obtained for a moderate . Note that there might be small differences between our experimental setup and the one used in blundell2015weight since BBB implementation is not publicly available. Therefore, the results might not be directly comparable. Nevertheless, SLANG appears to perform well compared to BBB.
|BBB||L = 1||L = 2||L = 4||L = 8||L = 16||L = 32|
We consider the challenging problem of uncertainty estimation in large deep models. For such problems, it is infeasible to form a Gaussian approximation to the posterior distribution. We address this issue by estimating a Gaussian approximation that uses a covariance with low-rank plus diagonal structure. We proposed an approximate natural-gradient algorithm to estimate the structured covariance matrix. Our method, called SLANG, relies only on the back-propagated gradients to estimate the covariance structure, which is a desirable feature when working with deep models. Empirical results strongly suggest that the accuracy of our method is better than those obtained by using mean-field methods. Moreover, we observe that, unlike mean-field methods, our method does not drastically shrink the marginal variances. Experiments also show that SLANG is highly flexible and that its accuracy can be improved by increasing the rank of the covariance’s low-rank component. Finally, our method converges faster than the mean-field methods and can sometimes converge as fast as VI methods that use a full-Gaussian approximation.
The experiments presented in this paper are restricted to feed-forward neural networks. This is partly because existing deep-learning software packages do not support individual gradient computations. Individual gradients, which are required in line 6 of Algorithm1, must be manually implemented for other types of architectures. Further work is therefore necessary to establish the usefulness of our method on other types of network architectures.
SLANG is based on a natural-gradient method that employs the empirical Fisher approximation khan2017vprop . Our empirical results suggest that this approximation is reasonably accurate. However, it is not clear if this is always the case. It is important to investigate this issue to gain better understanding of the effect of the approximation, both theoretically and empirically.
During this work, we also found that comparing the quality of covariance approximations is a nontrivial task for deep neural networks. We believe that existing benchmarks are not sufficient to establish the quality of an approximate Bayesian inference method for deep models. An interesting and useful area of further research is the development of good benchmarks that better reflect the quality of posterior approximations. This will facilitate the design of better inference algorithms.
We thank the anonymous reviewers for their helpful feedback. We greatly appreciate useful discussions with Shun-ichi Amari (RIKEN), Rio Yokota (Tokyo Institute of Technology), Kazuki Oosawa (Tokyo Institute of Technology), Wu Lin (University of British Columbia), and Voot Tangkaratt (RIKEN). We are also thankful for the RAIDEN computing system and its support team at the RIKEN Center for Advanced Intelligence Project, which we used extensively for our experiments.
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Appendix A Derivation of the VOGN Update
We can derive the VOGN update by using the Variational Online Newton (VON) method derived in Appendix D of . The VON updates are given as follows:
where is the iteration number, is the learning rate, , and and are the stochastic gradient and Hessian, defined respectively as follows:
is the back-propagated gradient obtained on the ’th data example, and is a minibatch of data examples.
Dealing with Hessians can be difficult because they may not always be positive-definite and may produce invalid Gaussian approximations. Following , we approximate the Hessian by the Empirical Fisher (EF) matrix:
The VON update is an exact natural-gradient method and uses a single learning rate . The VOGN update, on the other hand, is an approximate natural-gradient method because it uses the EF approximation. Due to this approximation, a single learning rate might not give good results and it can be sensible to use different learning rates for the and updates. In (2), we therefore use a different learning rate for (denoted by ).
Appendix B An Alternative Low-Rank Update
We tried an alternative approach to learn the low-rank plus diagonal covariance approximation. We call this method SLANG-OnlineEig. It forms the low-rank term in the precision approximation from an online estimate of the leading eigenvectors of . We now describe this procedure in detail. Experimental results are presented and comparisons are made with SLANG.
b.1 Approximating Natural Gradients by Online Estimation of the Eigendecomposition (SLANG-OnlineEig)
The following eigenvalue decomposition forms the basis of SLANG-OnlineEig:
We emphasize that SLANG-OnlineEig involves the decomposition of , rather than the updated matrix as in SLANG. This is cheaper by a factor of , which is a marginal difference as in most applications. To mimic the update of in (2), we use the following “moving-average“ update for :
where is a scalar learning rate. is a thus an online estimate of weighted eigenvectors of the EF.
Similar to SLANG, the diagonal is updated to capture the curvature information lost in the projection of to , i.e., the remaining eigenvectors:
The updated covariance is then given by
The final step of SLANG-OnlineEig is identical to equation (11):
SLANG-OnlineEig is amenable to the same algorithmic tools as SLANG, which were discussed in 3.1. Goodfellow’s trick can be used to compute the Jacobian needed for the EF and algorithms 2 and 3 are available for fast covariance-vector products and fast sampling, respectively. The overall computational complexity is and memory cost is .
b.2 Comparison with SLANG
SLANG-OnlineEig has several promising properties. It has slightly better computational complexity than SLANG and its update closely resembles the natural gradient update in (2). However, update (18) involves the product approximation,
where the second line is the true product of interest. The update assumes that the online estimate of the factors of the eigendecomposition well-approximates the online estimate of the eigendecomposition itself.
SLANG does not require the product approximation for efficient covariance learning. Instead, is updated exactly before the projection into the space of rank- matrices. This is why SLANG with reduces to 2 using the EF approximation. SLANG-OnlineEig does not posses this property because of the product approximation. It also does not necessarily match precision diagonals with the update, as SLANG does for all .
A final issue is that SLANG-OnlineEig also requires matching stochastic eigenvector estimates to their corresponding online estimates in . This may introduce additional approximation error when is highly stochastic.
b.3 Experimental Results for SLANG-OnlineEig
|Metrics||L = 1||L = 2||L = 5||L = 1||L = 2||L = 5|
Table 4 compares SLANG and SLANG-OnlineEig for logistic regression on the Australian, Breast Cancer, USPS 3vs5, and a1a datasets from LIBSVM. The results presented for SLANG are identical to those in Table 1. SLANG always matches or beats the best results for SLANG-OnlineEig. Furthermore, as is increased, the quality of posterior approximations computed by SLANG improves while SLANG-OnlineEig approximations sometimes degrade. We speculate that this is due to the product approximation.
Figure 5 presents results on the convergence of SLANG-OnlineEig for logistic regression and regression with a Bayesian neural network.
Table 5 shows regression on the UCI datasets using Bayesian neural networks. The setup for this experiment was similar to the experiment in Section 4.2, except that the learning rates were fixed to and for all datasets, both for SLANG-OnlineEig and for the Adam optimizer used for BBB. Moreover, the search spaces for the Bayesian optimization were fixed (using a normalized scale for the noise precision) and not adjusted to the individual datasets. Finally, BBB used 40 MC samples for the 5 smallest datasets and 20 MC samples for the 3 largest datasets in this experiment.
|Test RMSE||Test log-likelihood|
|Boston||4.04 0.28||2.97 0.19||3.17 0.17||-2.75 0.07||-2.46 0.06||-2.61 0.06|
|Concrete||6.16 0.14||5.23 0.12||5.79 0.13||-3.22 0.02||-3.04 0.02||-3.19 0.02|
|Energy||0.86 0.04||1.66 0.04||0.59 0.01||-1.20 0.05||-1.99 0.02||-1.05 0.01|
|Kin8nm||0.09 0.00||0.10 0.00||0.08 0.00||0.97 0.01||0.95 0.01||1.13 0.00|
|Naval||0.00 0.00||0.01 0.00||0.00 0.00||5.34 0.07||3.80 0.01||5.09 0.08|
|Power||4.28 0.03||4.02 0.04||4.09 0.04||-2.87 0.01||-2.80 0.01||-2.83 0.01|
|Wine||0.66 0.01||0.62 0.01||0.64 0.01||-0.99 0.01||-0.93 0.01||-0.98 0.01|
|Yacht||1.07 0.08||1.11 0.09||0.72 0.04||-1.45 0.03||-1.55 0.03||-1.41 0.01|
Appendix C Additional Algorithmic Details for SLANG
The following section gives more detail about the individual components of the algorithm required to leverage the low-rank plus diagonal structure for computational efficiency. In general, we obtain efficient algorithms by operating only on or matrices. This avoids the storage cost and the computational cost of working in the space.
c.1 Fast Computation of Individual Gradients
Most deep-learning automatic differentiation packages, such as PyTorch  and TensorFlow , are optimized to return the overall gradient of a minibatch, not individual gradients for each example passed through the network. It is true that the naïve option of doing a forward and backward pass for each example has a similar computational complexity as a fully parallel version. However, in practice most of the computations involved can be either reused across examples, or sped up drastically by batching them. Implementations that use matrix-matrix multiplications instead of repeatedly doing matrix-vector operations are far more efficient on GPUs.
Ian Goodfellow’s note 
outlines a method for efficiently computing per-example gradients. It suggests saving the neuron activations and the linear combinations of activations computed during the minibatch’s forward pass through the neural network. These values are then used in a manual implementation of the per-example gradients, which avoids the summation defined by the cost function. While much more efficient than the sequential approach, Goodfellow’s approach requires more implementation effort; a separate implementation is required to handle each type of layer used. This is partly why the experiments presented here are limited to standard Multi-Layer Perceptrons. We hope to improve upon this in future implementations.
c.2 Fast Top- Eigendecomposition
The goal is to get the top- eigenvalues and eigenvectors of the matrix defined in (6). Since we do not want to compute the matrix explicitly, we use the low-rank structure of the update matrix to compute matrix-vector or matrix-matrix products with computations in . This can be seen by rewriting the matrix as follows:
These products can be used to compute eigendecomposition efficiently by using a randomized algorithm.
The main idea behind the randomized eigendecomposition is to project a matrix onto a randomly selected subspace by sampling vectors , each entry being selected uniformly at random, and computing , where is larger than . A traditional eigendecomposition can then be performed on the matrix to recover the top eigenvectors, with acting as a precision-computation tradeoff parameter. More details on randomized eigenvalue methods can be found in .
closely; starting with a random matrix, multiplies it by
and applies a QR decomposition on the result. This process is repeated on the resulting matrix for a few iterations to improve stability, similarly to the Lanczos iterations. As all operations are done on the smallermatrix, using as recommended in 2), the computational cost of the QR decomposition and eigendecomposition are in , leading to an algorithm overall.
c.3 Fast multiplication by inverse of low-rank + diagonal
To implement the natural gradient update (11), we need to be able to multiply an arbitrary vector by , given . Woodbury’s identity can be used to do so without forming the matrix and doing the costly inversion. The identity gives
The only inversions remaining involve diagonal or matrices. Correct ordering of the operations allows the storage cost to be avoided when computing the product and ensures that we only need to store or diagonal matrices,
yielding a algorithm.
c.4 Fast sampling
To generate a sample from , it is sufficient to generate a sample and compute , where . can be factorized efficiently by exploiting its "low-rank plus diagonal" structure:
Letting be a symmetric factor for , we then have that is a symmetric factor for . Such a factorization can be found using the work of , which showed that by taking
is a symmetric factorization for . We can then use Woodbury’s Identity to avoid taking the inverse in the space,
and careful ordering of operations, as above, leads to a complexity. This subroutine is implemented in Algorithm 3.
Appendix D Diagonal Correction
In this section, we prove that the diagonal of the precision computed by SLANG when is identical to the diagonal computed when .