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3.2. Connectome extraction: inverse covariance for direct connections

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3.3. Extracting resting-state networks: ICA and related

3.2.3.1. Group-sparse covariance estimation

This page gives technical information on the group_sparse_covariance function and related. This is mainly useful for developers or people that want to know more about implementation.

3.2.3.1.1. Description

group_sparse_covariance, and GroupSparseCovariance are two different interfaces to an implementation of the algorithm described in this article:

Jean Honorio and Dimitris Samaras. “Simultaneous and Group-Sparse Multi-Task Learning of Gaussian Graphical Models”. arXiv:1207.4255 (17 July 2012). http://arxiv.org/abs/1207.4255.

The goal of the algorithm is to take a set of K covariance matrices as input, and estimate a set of K sparse precision matrices, using a penalized maximum-likelihood criterion. The penalization has been devised to enforce a common sparsity pattern in all precision matrices. The structure is of a block coordinate descent, with a line search as innermost loop.

The present implementation relies solely on NumPy, SciPy and Scikit-Learn. Nilearn contains only Python code.

In addition to the basic algorithm described in the article, several additions were implemented:

  • computation of bounds for the regularization parameter
  • several stopping criteria
  • an ad-hoc cross-validation algorithm
  • signals synthesis for testing purposes

These are described in the rest of this page. An overview of the design choices and the history of the development is also given.

3.2.3.1.2. Numerical stability

The algorithm proved to be rather numerically stable for a wide range of inputs. It turned out that the condition numbers of the input matrices do not have any significant effect on numerical stability. What is relevant is:

  • covariance matrix symmetry: input covariances matrices in group_sparse_covariance must be as symmetric as possible. This is true in general: a small discrepancy in symmetry tends to be amplified. For this reason, our functions computing covariances ensure symmetry.
  • covariance matrix normalization: using correlation matrices or signals with unit variance is mandatory when a large number of signals is to be processed.
  • normalization of the number of samples: the objective to be optimized contains a sum of terms weighted by the number of samples available for each subject. The sum of these weights must be normalized to a small constant number (1 in the current implementation). Failing to do this leads quickly to instability, because too large numbers are used in the computation.
  • an on-line computation of an inverse is performed in function _update_submatrix. For large problems, this is faster than computing the full inverse each time, but gives unfortunately less precision. In particular, symmetry is not always perfect, that’s why it is enforced at the end on the final result.
  • the Newton-Raphson tolerance value has no influence on numerical stability, unless very large values (like 0.5) are used.

The debug keyword in group_sparse_covariance activates a set of numerical consistency checks (mainly that matrices are s.p.d.) that can be useful to track down numerical instability problems.

3.2.3.1.3. Execution time

The line profiler from Robert Kern was used to locate execution time bottlenecks. Its overhead proved not to be negligible (around 50% more execution time when activated), and not evenly distributed in code lines. Global execution times have also been measured to ensure that the findings were valid. As the code in group_sparse_covariance is highly serial, and rather low-level, some lines have to be executed a very large number of times (10^6 times is easily reached), one of the bottlenecks is thus the Python interpreter overhead. Optimizing then boils down to reducing the number of code lines and function calls in the most executed parts: the Newton-Raphson (line search) loop. It is for this reason that it has been written inline instead of calling Scipy’s version (it saves a lot of lines and calls). A lot of small optimizations of this kind have been done. It is possible that some of these optimizations give less numerical precision that the naive operation. But the original author could not tell.

Speed optimization has been performed by checking the wall-clock time required to get to a given precision, and not the number of iterations. This is what “fast” means in practice: short overall execution time. Tuning of the Newton-Raphson (NR) loop gives a good example of the validity of this approach: the goal was to set the tolerance on the result. Using a large value reduces the number of iterations for NR, saving a lot of time. On the other hand, a loose tolerance increases the number of iterations in the coordinate descent loop, therefore increasing the overall execution time. Measurement proved that tight tolerances were leading to faster convergence rates.

Care has been taken to use proper ordering of data in arrays. In particular, three-dimensional arrays containing precision matrices are in Fortran order, to get prec[..., k] contiguous for any k. This is important to avoid copies by lapack/atlas functions, such as matrix inverse or dot product. It is also consistent with arrays returned by nibabel.load.

An optimization that can be performed, but couldn’t be implemented short of having proper linalg functions for it is to process only half of each matrix: all are symmetric. This would improve numerical stability while saving some execution time. Part of this could be done with versions of Scipy that weren’t available on the targeted systems at the time of writing (Ubuntu 10.04 and 12.04).

Memory optimization hasn’t been performed, because all functions process covariance matrices only, that are quite small compared to the signals from which they are generated.

3.2.3.1.4. Synthetic dataset

For testing purposes, a function for synthesis of signals based on sparse precision matrices has been written: nilearn._utils.testing.generate_group_sparse_gaussian_graphs. Synthesizing such signals is a hard problem that wasn’t solved in the present implementation. It is hopefully good enough.

This function generates n_subjects time, n_features signals with a variable number of samples. Every subject has the same number of features (i.e. signals), for a given subject every signal has the same number of samples, but between two subjects, the sample number can differ. This structure is close to what is available in practice.

Here is how signals are generated:

  • a “topology” matrix containing only zero and ones is generated. This will govern the sparsity pattern of the precision matrices.
  • for each subject, a precision matrix is generated by replacing every 1 in the topology matrix by a random positive number, then multiplying the resulting matrix by its transpose to get a positive definite matrix. This is a way to get a sparse definite positive matrix.
  • inverting precision matrices gives covariance matrices, that are in turn used to generate signals.

The hardest part is generating sparse symmetric positive definite matrices, while controling the sparsity level. With the present scheme, only the location of zeros in the square root of the precision matrices can be specified. Therefore the final sparsity level depends not only on the initial sparsity level, but also on the precise location of zeros. Two different sparsity patterns with the same number of zeros can lead to two significantly different sparsity level in precision matrices. In practice, it means that for a given value of the density parameter in nilearn._utils.testing.generate_group_sparse_gaussian_graphs, the actual number of zeros in the precision matrices can fluctuate widely depending on the random number generation.

The condition number of the precision matrices depends on the range of numbers used to fill the off-diagonal part. The shorter the range (and the closer to zero) the lower the condition number.

This generator is useful for debugging and testing. However, the signals obtained are significantly different from those from experimental data. Some unrealistic features: each signal has a perfectly white spectrum (any two samples are decorrelated), and there is no global additive noise (no confounds whatsoever).

3.2.3.1.5. Stopping criteria

As with any iterative algorithm, iteration should be stopped at some point, which is still mostly an open problem. Several heuristic techniques have been tested and implemented.

3.2.3.1.5.1. Maximum number of iterations

The simplest way of stopping optimization is to always execute a fixed number of iterations. This is simple but most of the time gives slow or bad results. The convergence rate highly depends on the number of features (size of one covariance matrix), and on the value of the regularization parameter (high values give fast convergence, and low values slow convergence). If the requested iteration number is too low, large or weakly regularized problems will be far from the optimum. On the other hand, if the requested iteration number is too large, a lot of time is wasted for almost no gain.

3.2.3.1.5.2. Duality gap

A better way to stop iteration is to use an upper bound on the duality gap value, since the problem is convex. This is performed in group_sparse_covariance_costs. The article by Honorio & Samaras gives the formula for the dual cost, and proves that the derived bound at optimum is tight (strong duality holds). However, the dual problem is not solved by this algorithm, thus bounding the duality gap away from the optimum implies finding a feasible dual point. This proved to be quite hard in practice, because one has to compute positive semi-definite matrices under a norm constraint.

What is done is computing an estimate for a dual point using the formula relating the primal and dual points at optimum. This estimate does not satisfies in general the norm constraint. It is then projected on the corresponding ball. Most of the time, this is enough to ensure the required positive definiteness of another quantity. As the primal point is coming close to the optimal, the estimate for the dual point also comes close to the optimal, and the initial estimate is closer and closer to the norm ball.

But there are cases for which the projection is not enough to get to a feasible point. No solution to this problem (simultaneous projection on a norm ball and on a set of positive definite matrices) has been found. In that case, an easier to compute but non-tight bound is used instead.

In practice, using the duality gap value to stop iteration leads to guaranteed uncertainty on the objective value, in any case. No time is lost on over optimizing rapidly converging problems. However, the duality gap criterion can lead to prohibitive computation time on slowly converging cases. In practice, finding a proper value for the duality gap uncertainty can be tricky, because it is most easily given as an absolute uncertainty on an objective whose value highly depends on input data.

3.2.3.1.5.3. Variation of norm of estimate

Depending on the application at hand, giving an uncertainty on the precision matrices instead of the objective can be useful. This is partly achieved by computing the change of the precision estimate between two iterations. Optimization is stopped once this value goes below a threshold. The maximum norm (maximum of the absolute value of the difference) is used in the current implementation. It ensures that all coefficients vary less than the threshold when optimization is stopped.

This technique it is only a way to stop iterating based on the estimate value instead of the criterion value. It does not ensure a given uncertainty on the estimate. This has been tested on synthetic and real fMRI data: using two different starting points leads to two estimates that can differ (in max norm) by more than the threshold (see next paragraph). However, it has the same property as the duality gap criterion: quickly converging cases use fewer iterations than slower cases. From a performance point of view, this is a good thing.

One advantage of this criterion is that the threshold value does not depend significantly on the input data. Matrix coefficients can be requested to change less than e.g. 0.1 for any size of the input.

3.2.3.1.5.4. Initial estimate value

One of the possible way to reduce the computation time of an iterating algorithm is to start with a initial guess that is as close as possible to the optimum. In the present case, two initializations were tested: using a diagonal matrix (with variance of input signals), or using a Ledoit-Wolf estimate. It turned out that even if the Ledoit-Wolf initialization allows for starting with a better value for the objective, the difference with the diagonal matrix initialization dwindles rather fast. It does not allow any significant speedup in practice.

Only initialization by the diagonal matrix, as in the original paper, has been implemented.

3.2.3.1.5.5. Modifying the stopping criterion

Modifying the stopping criterion is more complicated than specifying the initial estimate, since it requires to gain access to the algorithm internals. This is achieved by a technique close to aspect-oriented programming: a function can be provided by the user, that will be called after each iteration, with all internal values as parameter. If that function returns True, iteration is stopped. Changing the stopping criterion is thus just a matter of writing a function and passing it to group_sparse_covariance. The same feature can be used to study the algorithm convergence properties. An example is the EarlyStopProbe class used by the cross-validation object.

3.2.3.1.6. Cross-validation algorithm

An ad-hoc cross-validation scheme has been implemented in the GroupSparseCovarianceCV class. This implementation is significantly faster than the “naive” cross-validation scheme.

The cross-validating object performs to distinct tasks: the first one is to select a value for the regularization parameter, the second is fitting the precision matrices for the selected parameter. The latter is identical to what has been described in the previous parts, we thus focus only on the former.

3.2.3.1.6.1. Principle of cross-validation

Cross-validation consists in splitting the input samples into two different sets: train and test. For several values of the regularization parameter, a model is fit on the train set, and the generalization performance is assessed on the test set, by computing the unpenalized criterion (log-likelihood) using the precisions matrices obtained on the train set with the empirical covariances of the test set. The chosen regularization parameter is given by the best criterion on the test set.

The simplest scheme is here to fit many models, for many values of the regularization parameter alpha, and pick up the best value afterward. It works in any case, but is very time-consuming. A cleverer scheme is used, that is very close to that used in the graph lasso implementation in Scikit-Learn.

3.2.3.1.6.2. Bounds on alpha

The simplest and fastest thing is to get bounds for the value of alpha. Above a critical value, the optimal precision matrices are fully sparse (i.e. diagonal). This critical value depends on the input covariance matrices, and can be obtained by compute_alpha_max. The formula for computing this critical value can be obtained with techniques presented in:

Duchi, John, Stephen Gould, and Daphne Koller. ‘Projected Subgradient Methods for Learning Sparse Gaussians’. ArXiv E-prints 1206 (1 June 2012): 3249.

This very same method can be also used for determining a lower critical value, for which the optimal precision matrices are fully dense (no zero values). In practice, this critical value is zero if there is a zero in the input matrices. For this reason, the second value returned by compute_alpha_max is that under which all coefficients that can be non-zero are non-zero in the optimal precision matrices.

3.2.3.1.6.4. Warm restart

During each step of the grid search, a set of regularization parameters has to been tested. The straighforward strategy consists of running independently each fit, each optimization being started with basically the same initial value (diagonal matrices). Execution time can be reduced by running all optimizations sequentially, and using the final result of one as initial value for the next. This goes faster because it saves part of the optimization trajectory starting with the second one. However, there is a real gain in execution time only if the parameter values are ordered from the largest to the smallest (and not the other way).

The usefulness of this scheme depends on several things. First, using warm restart does not gives exactly the same result as running independant optimizations, because optimization paths are not the same. This is not an issue for cross-validation, since there are many other larger sources of fluctuations. It has been checked that in practice, the selected value does not change. Second, using warm restart forces running all optimization one after another: there is no parallelism at all. However, this is true only for a given fold: when n folds are used, n such evaluations can be run in parallel. Thus, the fact that warm restart gives faster evaluation compared to fixed initialization depends on the number of folds, and the number of computation cores. No more cores that the number of folds can be used at the same time. Thus, if the number of folds is much smaller than the number of usable cores, warm restart slows down computation (note that if the goal is energy efficiency, not speed, warm restart is always a good idea.) This argument is also valid for the iterative grid search: if many cores are available, the brute-force grid search is faster than the iterative scheme, just because every point can be explored simultaneously, without waiting for the previous step to finish. Many more evaluations are performed, but the overall running time is limited of by the slowest evaluation. The choice of these schemes (iterative grid search and warm restart) has been made in the present implementation because the targeted hardware is a commodity computer, with a moderate number of cores (4 to 16). More cores (and memory) will probably be available in future years, these schemes could be removed easily.

3.2.3.1.6.5. Stopping criterion

Finding the regularization parameter optimal value is equivalent to finding a maximum. But since only the location of the maximum (not its value) is of interest, any curve that peaks at the same location than the log-likelihood can be used.

Implicitely, the curve whose maximum is sought is supposed to be obtained after convergence for any value of alpha. This is never the case in practice: a stopping criterion has to be used. In the present implementation, the variation criterion gives results that seem to be consistent with what would be obtained at convergence (that is: the log-likelihood-vs-alpha curve seems to be close to convergence). This can be pushed one step further: any stopping criterion that gives the same maximum location can be used instead. We stress that only the location is important: the curve can be anything apart from that.

It was found that stopping iteration just after the log-likelihood has reached a maximum works in most cases. The obtained log-likelihood vs alpha curve is different, but its maximum is the same as with the variation criterion stopping. It is also faster (2 times to 4 times in our tests).

In more detail: for a given value of alpha, start optimization. After each step, compute the log-likelihood on the test set. If the current value is smaller than the previous one, then stop. The variation criterion is also computed for the rare cases when the log-likelihood never decreases, and a maximum number of iterations is enforced, to limit the time spent optimizing in any case.

It is possible to disable the first criterion with the early_stopping keyword in GroupSparseCovarianceCV. In that case, only the two latter criteria are used. This provides a mean to test for the validity of the heuristic.