1 Introduction and Motivation
Least-Mean-Squares (LMS) solvers are the family of fundamental optimization problems in machine learning and statistics that include linear regression, Principle Component Analysis (PCA), Singular Value Decomposition (SVD), Lasso and Ridge regression, Elastic net, and many moreGolub und Reinsch (1971); Jolliffe (2011); Hoerl und Kennard (1970); Seber und Lee (2012); Zou und Hastie (2005); Tibshirani (1996); Safavian und Landgrebe (1991). See formal definition below. First closed form solutions for problems such as linear regression were published by e.g. Pearson Pearson (1900)
around 1900 but were probably known before. Nevertheless, today they are still used extensively as building blocks in both academy and industry for normalizationLiang u. a. (2013); Kang u. a. (2011); Afrabandpey u. a. (2016)2015), graph theory Zhang und Rohe (2018), prediction Copas (1983); Porco u. a. (2015), dimensionality reduction Laparra u. a. (2015)2017) and many more; see more examples in Golub und Van Loan (2012).
Least-Mean-Squares solver in this paper is an optimization problem that gets as input an real matrix , and another -dimensional real vector (possibly the zero vector). It aims to minimize the sum of squared distances from the rows (points) of
to some hyperplane that is represented by its normal or vector ofcoefficients , that is constrained to be in a given set :
Here, is called a regularization term. For example: in linear regression , and for every . In Lasso and for every and . Such LMS solvers can be computed via the covariance matrix . For example, the solution to linear regression of minimizing is .
1.1 Related work
While there are many LMS solvers and implementations, there is always a trade-off between their accuracy and running time; see comparison table in Bauckhage (2015) with references therein. The reason is related to the fact that computing the covariance matrix of can be done essentially in two ways: (i) summing the outer product of the th row of over every , . This is due to the fact that , or (ii) factorization of
, e.g. using SVD or the QR decomposition.
Method (i) is easy to implement for streaming rows of by maintaining only entries of the covariance matrix for the vectors seen so far, or maintaining its inverse as explained e.g. in Golub und Van Loan (2012). This takes for each vector insertion. However, every such addition may introduce another numerical error which accumulates over time. This error increases significantly when running the algorithms using 32 bit floating point representation, which is common for GPU computations. This solution is similar to maintaining the set of rows of the matrix , where is the SVD of , which is not a subset of the original input matrix but has the same covariance matrix . A common problem is that to compute the matrix must be invertible. This may not be the case due to numerical issues. In algorithms such as Lasso, the input cannot be a covariance matrix, but can be its Cholesky decomposition Bjorck (1967) . However, Cholesky decomposition can be applied only on positive-definite matrices, which is not the case even for small numerical errors that are added to . See Section 4 for more details and empirical evidence.
Method (ii) above utilizes factorizations such as SVD, i.e., to compute the covariance matrix via or the QR decomposition to compute . This approach is known to be much more stable. However, it is much more time consuming: while in theory the running time is as in the first method, the constants that are hidden in the notation are significantly larger. Moreover, unlike Method (i), it is impossible to compute such factorizations exactly for streaming data Clarkson und Woodruff (2009).
Caratheodory’s Theorem Carathéodory (1907)
states that every point contained in the convex hull of points in can be represented as a convex combination of a subset of at most points, which we call the Caratheodory set; see Section 2 and Fig. 1. This implies that we can maintain a weighted (scaled) set of points (rows) whose covariance matrix is the same as , since is the mean of matrices and thus in the convex hull of their corresponding points in ; see Algorithm 3. The fact that we maintain such a small sized subset of points instead of updating linear combinations of all the points seen so far, significantly reduces the numerical errors as shown in Fig. 9(b)–9(a). Unfortunately, computing this set from Caratheodory’s Theorem takes or time via calls to an LMS solver. This fact makes it non-practical to use in an LMS solver, as we aim to do in this paper, and may explain the lack of source code for this algorithm on the web.
Approximations via Coresets and Sketches.
In the recent decades numerous approximation and data summarization algorithms were suggested to approximate the covariance matrix or its singular values. This is by computing a small matrix whose covariance approximates, in some sense, the covariance matrix of the input data . The term coreset is usually used when is a weighted (scaled) subset of rows from the rows of the input matrix. The matrix is sometimes called a sketch if each rows in is a linear combination of few or all its rows, i.e. for some matrix . However, those coresets and sketches usually yield -multiplicative approximations for by where the matrix is of rows and may be any vector, or the smallest/largest singular vector of or ; see lower bounds in Feldman u. a. (2010). Moreover, a -approximation to by
does not guarantee an approximation to the actual entries or eigenvectors ofby that may be very different; see Drineas u. a. (2006).
Accurately handling big data.
The algorithms in this paper return accurate coresets (), which is less common in the literature, including computations of the exact covariance matrix . Such coresets can easily support infinite stream of input rows using memory that is linear in their size, and also support dynamic/distributed data in parallel. This is by the useful merge-and-reduce property of coresets that allow them to handle big data; see details e.g. in Agarwal u. a. (2004). Unlike traditional coresets that pay additional logarithmic multiplicative factors due to the usage of merge-reduce trees and increasing error, the suggested weighted subsets in this paper do not introduce additional error to the resulting compression since they preserve the result accurately. The actual numerical errors are measured in the experimental results, with analysis that explain the differences.
A main advantage of a coreset over a sketch is that it preserves sparsity of the input rows Feldman u. a. (2016), which usually reduces theoretical running time. Our experiments show, as expected, that coresets can also be used to significantly improve the numerical stability of existing algorithms, even if the running time is the same. Another advantage is that the same coreset can be used for parameter tuning over a large set of candidates. In addition to other reasons, this significantly reduced the running time of such algorithms in our experiments; see Section 4.
1.2 Our contribution
A natural question that follows from the previous section is: can we maintain the optimal solution for LMS problems both accurately and fast? We answer this question affirmably by suggesting:
the first algorithm that computes the Caratheodory set of input points in time that is linear in the input for asymptotically large , and using only calls to an LMS solver. This is by using a novel approach of coreset/skecthes fusion that is explained in the next section; see Algorithm 2 and Theorem 2.
example applications for boosting the performance of existing solvers by running them on the matrix above or its variants for Linear/Ridge/Lasso Regressions and Elastic-net.
extensive experimental results on synthetic and real-world data for common LMS solvers of Scikit-learn library with either CPython or Intel’s distribution. Either the running time or numerical stability is improved up to two orders of magnitude.
1.3 Novel approach: Coresets meet Sketches
As explained in Section 1.1, the covariance matrix of itself can be considered as a sketch which is relatively less numerically stable to maintain (especially its inverse, as desired by e.g. linear regression). The Caratheodory set that corresponds to the outer products of the rows of is a coreset whose covariance matrix is and, as a weighted subset of the original rows, is more numerically stable but takes much more time to compute; see Theorem 2.2.
We thus suggest a meta-algorithm that combines these two approaches: sketches and coresets. It may be generalized to other, not-necessarily accurate -coresets and sketches (); see Section 5.
The input to our meta-algorithm is 1) a set of items, 2) an integer from to where is maximum accuracy but longest running time, and 3) a pair of coreset and sketch construction schemes for the problem at hand.
The output is a coreset for the problem whose construction time is faster; see Fig. 1.
Step I: Compute a balanced partition of the input set into clusters of roughly the same size. While the correctness holds for any such arbitrary partition (e.g. see Algorithm 3.1), to reduce numerical errors – a partition that minimizes the sum of loss with respect to the problem at hand would be optimal.
Step II: Compute a sketch for each cluster , where , using the input sketch scheme. This step does not return a subset of as desired, and is usually numerically less stable.
Step III: Compute a coreset for the union of sketches from Step II, using the input coreset scheme. Note that is not a subset (or coreset) of .
Step IV: Compute the union of clusters in that correspond to the selected sketches in Step III, i.e. . By definition, is a coreset for the problem at hand.
Step V: Recursively compute a coreset for until a sufficiently coreset is obtained. This step is used to reduce running time, without selecting that is too small.
We then run an existing solver on the coreset to obtain a faster accurate solution for . Algorithm 2 and 3.1 are special cases of this meta-algorithm, where the sketch is simply the sum of a set of points/matrices, and the coreset is the existing (slow) implementation of the Caratheodory set from Theorem 2.2.
In Section 2 we give our notations, definitions and the current state-of-the-art result. Section 3 presents our main algorithms for efficient computation of the Caratheodory (core-)set and a subset that preserves the inputs covariance matrix, their theorems of correctness and proofs. Section 4 demonstrates the applications of those algorithms to common LMS solvers, with extensive experimental results on both real-world and synthetic data via the Scikit-learn library with either CPython or Intel’s Python distributions. We conclude the paper with open problems and future work in Section 5.
2 Notation and Preliminaries
For integers , we denote by the set of real matrices, and . To avoid abuse of notation, we use the big O notation where is a set Cormen u. a. (2009). A weighted set is a pair where is an ordered finite set in , and is a positive weights function. A linear system solver is an algorithm that solves a system of linear equations with variables, i.e., return such that for a given and , assuming there is such a solution.
Given a point inside the convex hull of a set of points , Caratheodory’s Theorem proves that there a subset of at most points in whose convex hull also contains . This geometric definition can be formulated as follows.
Definition 2.1 (Caratheodory set).
Let be a weighted set of points in such that . A weighted set is called a Caratheodory Set for if: (i) its size is , (ii) its weighted mean is the same, , and (iii) its sum of weights .
Caratheodory’s Theorem suggests a constructive proof for computing this set in time Carathéodory (1907); Cook und Webster (1972). This is implemented in Algorithm 1, which takes as input a weighted set such that and computes a Caratheodory set for in time. However, as observed e.g. in Nasser u. a. (2015) it can be computed only for the first points, and then be updated point by point in time per point, to obtain overall time. This still takes calls to a linear system solver that solves for a given matrix and vector , each of rows and columns, in time per call.
Theorem 2.2 (Carathéodory (1907)).
A Caratheodory set can be computed for any weighted set where in time.
3 Faster Caratheodory Set
In this section, we present our main algorithm that reduces the running time for computing a Caratheodory set from in Theorem 2.2 or Nasser u. a. (2015) to for sufficiently large ; see Theorem 3.1. A visual illustration of the corresponding Algorithm 2 is shown in Fig 1. We also present a second algorithm, called Caratheodory-Matrix, which computes a small weighted subset of a the given input that has the same covariance matrix as the input data; see Algorithm 3.
Theorem 3.1 (Caratheodory-Set Booster).
We use the notation and variable names as defined in Algorithm 2.
Identify the input set and the set that is computed at Line 2 of Algorithm 2 as . We will first prove that the weighted set computed in Lines 2–2 at the current (some) iteration is a Caratheodory set for , i.e., , and .
For every , let .
Let be the pair computed at the current iteration at Line 2. By the definition of Caratheodory, we have that the pair computed at Line 2 is a Caratheodory set of the weighted set , i.e., it satisfies that
Observe that by the definition of for every at Line 2 we have that
We now have that
We also have that the new sum of weights is equal to
Combining (3) and (4) yields that the weighted computed before the recursive call at Line 2 of the algorithm is a Caratheodory set for the weighted input set . Since at each iteration we either return such a Caratheodory set at Line 2 or return the input weighted set itself at Line 2, by induction we get that the output weighted set of a call to is a Caratheodory set for the original input .
Let be a matrix, and be an integer. Let be the output of a call to ; see Algorithm 3. Let be the computation time of Caratheodory given point in . Then satisfies that . Furthermore, can be computed in time.
We use the notation and variable names as defined in Algorithm 3.
(ii) since , and (iii) is computed in time.
Combining (5) with the fact that is simply the concatenation of the entries of , we get that
By the definition of on Line 3, we have that
We also have that
where the second derivation holds since .
4 Experimental Results
In this section we apply our fast construction of the Carathoodory Set from previous section to boost the running time of common LMS solvers in Table 1 by a factor of ten to hundreds, or to improve their numerical accuracy by a similar factor to support, e.g. 32 bit floating point representation. This is by running the given solver as a black box on the small matrix that is returned by Algorithms 5–8, which is based on . That is, our algorithm does not compete with existing solvers but relies on them, which is why we called it "booster". Open code for our algorithms is provided Maalouf u. a. (2019).
From Caratheodory Matrix to LMS solvers.
As stated in Theorem 3.2, Algorithm 3 gets an input matrix and an integer , and returns a matrix of the same covariance , where is a parameter for setting the desired accuracy. To "learn" a given label vector , Algorithm 4 partitions the matrix into partitions, computes using Algorithm 3 a subset for each partition that preserves its covariance matrix, and returns the union of subsets as a pair where and . For , it is easy to see that for every ,
where the third and fourth equalities are by Theorem 3.2 and the construction of , respectively. This enables us to replace the original pair by the smaller pair for the solvers in Table 1 as in Algorithms 5–8. A scaling factor is also needed in Algorithms 7–8.
To select the value of the regularization term , the existing Python solvers we used partition the rows of into folds (subsets) and run the solver times for every fold and to select the desired ; see Kohavi u. a. (1995) for details. For consistency, Algorithm 4 computes a coreset for each of these folds in Line 4 and concatenate them in Line 4. Thus, (9) holds similarly for .
We evaluated the algorithms in Table 1 using the common Python’s solvers in its right two columns. Most of these experiments were repeated twice: using the default CPython distribution Wikipedia contributors (2019a) and Intel’s distribution LTD (2019) of Python. All the experiments were conducted on a standard Lenovo Z70 laptop with an Intel i7-5500U CPU @ 2.40GHZ and 16GB RAM. We used the following real-world datasets in our experiments:
3D Road Network (North Jutland, Denmark) Data Set Kaul u. a. (2013). It contains records. We used the attributes: (Web Mercaptor (Google format) longitude [real], Web Mercaptor (Google format) latitude [real]) to predict the attribute (Height in meters [real]).
Individual household electric power consumption Data Set dataset:power (2012). It contains records. We used the attributes: (global active power [kilowatt - real], global reactive power [kilowatt - real]) to predict the attribute (voltage [volt - real]).
House Sales in King County, USA dataset:sales (2015). It contains records. We used the following attributes: (bedrooms [integer], sqft living [integer], sqft lot [integer], floors [integer], waterfront [boolean], sqft above [integer], sqft basement [integer], year built [integer]) to predict the (house price [integer]) attribute.
The synthetic data is an matrix and vector of length , both of random entries in . As expected by the analysis, since our compression introduces no error to the computation accuracy, the actual values of the data had no affect on the results, unlike the size of the input which affects the computation time. Table 2 summarizes the experimental results.
Why running time is faster?
The number of rows in the reduced matrix is , which is usually much smaller than the original matrix . This also explains why some coresets (dashed red line) failed for small values of in Fig. 1(b),1(c),3(b) and 3(c). The construction of takes . Solving Linear Regression takes the same time, with or without the coreset. However, the constant are much smaller since the time for computing becomes neglected for large values of , as shown in Fig. 11. We emphasize that, unlike common coresets, there is no accuracy loss due to the use of our coreset, ignoring additive errors/improvements. The improvement factor in running time due to our booster is in order of up to x10. The contribution of the coreset is much significant, already for smaller values of , when it boosts other solvers that use cross validation for parameter tuning as explained above. In this case, the time complexity reduced by a factor of since the coreset is computed only once for each of the folds, regardless of the size . In this case, the running time improvement is between x10–x100.
As shown in the graphs, the computations via Intel’s Python distribution reduced the running times by 15-40%, with or without the booster, probably due to its tailored implementation for our hardware.
Why numerical stability is better?
A sketch that simply sums the -rank matrices of outer products of rows in the input matrix yields its covariance matrix . The Cholesky decomposition then returns a small matrix that can be plugged to the solvers, similarly to our coreset. This algorithm which we call SKETCH + CHOLESKY is so simple and there is no hope to improve its running time via our much more involved booster. Nevertheless, it is numerically unstable for the reasons that are explained in Section 1.1. In fact, on most of our experiments we could not even apply this technique at all using 32-bit floating point representation. This is because the resulting approximation to was not a positive definite matrix as required by the Cholesky Decomposition, and we could not compute the matrix at all. In case of success, the running time of the booster was slower by at most a factor of but even in these cases numerical accuracy is improved up to orders of magnitude; See Fig. 9(b)–9(a) for histogram of errors using such 32-bit float representation which is especially common in GPUs for saving memory, running time and power Wikipedia contributors (2019b). For the special case of Linear regression, we can avoid Cholesky decomposition and compute the solution directly after maintaining such a sketch for and . However, this sketch wich we call SKETCH + INVERSE still has large numerical issues compared to our coreset computation as shown in Fig. 9(b)–9(a).
5 Conclusion and Future Work
We presented a novel framework that combines sketches and coresets. As an example application, we proved that the set from the Caratheodory Theorem can be computed in overall time for sufficiently large (for a set of points in ). This is instead of time as in the original theorem. We then generalized the result for a matrix whose rows are a weighted subset of the input matrix and their covariance matrix is the same. Our experimental results section shows how to significantly boost the numerical stability or running time of existing LMS solvers by applying them on . Future work includes: (a) applications of our framework to combine other sketch-coreset pairs e.g. as listed in Phillips (2016)1977); Mandic (2004), and (c) experiments with higher dimensional data: we may compute each of the entries in the covariance matrix by calling our algorithm with and the corresponding pair of columns in the columns of the input matrix.
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