Let be a polytope where has nonzero, finite Euclidean volume. The classical theorem of [Joh48] states that if is the ellipsoid of maximal volume contained in , then , where represents a dilation of the ellipsoid by a factor of about its center. Moreover, if is symmetric, then . The maximal volume inscribed ellipsoid (MVIE) is called the John Ellipsoid, and we are interested in the problem of approximating when the polytope is centrally symmetric, i.e. can be expressed as where and has rank .
The problem of computing the ellipsoid of maximal volume inside polytope given by a set of inequalities has a wealth of different applications, including sampling and integration [Vem05, CDWY18], linear bandits [BCBK12, HK16]LS14], cutting plane methods [KTE88] and differential privacy [NTZ13].
Computing the John Ellipsoid additionally has applications in the field of experimental design, a classical problem in statistics [Atw69]. Specifically, the D-optimal design problem wants to maximize the determinant of the Fisher information matrix [KW60, Atw69], which turns out to be equivalent to finding the John Ellipsoid of a symmetric polytope. While this equivalence is known, e.g. [Tod16], we include it in Section 2
for completeness. The problem of D-optimal design has received recent attention in the machine learning community, e.g.[AZLSW17, WYS17, LFN18].
1.1 Our Contribution
Our main contribution is to develop an approximation algorithm to computing the John Ellipsoid inside a centrally symmetric polytope given by a set of inequalities. Previously, for solving the MVIE problem or its dual equivalent D-optimal design problem, researchers have developed various algorithms, such as first-order methods [Kha96, KY05, DAST08], and second-order interior-point methods [NN94, SF04]. Instead of using traditional optimization methods, we apply a very simple fixed point iteration. The analysis is also simple and clean, yet the convergence rate is very fast. We state our main result as follows.
Theorem 1.1 (Informal).
Given , let be a centrally symmetric polytope defined as . For , there is an algorithm (Algorithm 1) that runs in time , returning an ellipsoid so that .
In Lemma 2.3, we show that our ellipsoid is -close to the John Ellipsoid in a certain sense. However, if we want to get -approximation to the maximal volume, we shall set 111For details, see Lemma 2.3., then Algorithm 1 runs in time , and when is constant, this is comparable with the best known results [KY05, TY07].
Furthermore, we use sketching ideas from randomized linear algebra to speed up the algorithm so that the running time does not depend on explicitly. This will make sense if is a sparse matrix. Our result is stated as follows.
Theorem 1.2 (Informal).
let be a centrally symmetric polytope defined as .
For and ,
there is an algorithm (Algorithm 2) that runs within many iterations,
returning an ellipsoid so that with probability at least
so that with probability at least, . Moreover, each iteration involves in solving linear systems of the form where is some diagonal matrix.
Algorithm 2 is near optimal, because in order to verify the correctness of the result, we need to compute the leverage scores of some weighted version of . The best known algorithm for approximating leverage scores needs to solve many linear systems [SS11, DMIMW12, CW13, NN13]. One key advantage of our algorithm is that it reduces the problem of computing John ellipsoid to a relatively small number of linear systems. Therefore, it allows the user to apply the linear systems solver tailored for the given matrix . For example, if is tall, one can apply sketching technique to solve the linear systems in nearly linear time [Woo14]; if each row of A has only two non-zeros, one can apply Laplacian solvers [DS08, KOSZ13, CKM14, KS16, KLP16]. In the code we provided, we used the Cholesky decomposition which is very fast for many sparse matrices in practice.
1.2 Related Works
There is a long line of research on computing the maximal volume ellipsoid inside polytopes given by a list of linear inequalities. We note that [KT93] presented a linear time reduction from the problem of computing a minimum volume enclosing ellipsoid (MVEE) of a set of points to the maximal volume inscribed ellipsoid problem; therefore, these algorithms also hold for approximating the John Ellipsoid.
Using an interior-point algorithm, [NN94] showed that a approximation of MVEE can be computed in time . [KT93] subsequently improved the runtime to . Later on, [Nem99] and [Ans02] independently obtained an algorithm. To the best of authors’ knowledge, currently the best algorithms by [KY05, TY07] run in time . We refer readers to [Tod16] for a comprehensive introduction and overview.
Computing the minimum volume enclosing ellipsoid of a set of points is the dual problem of D-optimal design. By generalizing smoothness condition on first order method, [LFN18] managed to solve D-optimal design problem within many iterations. However, in the dense case, their iteration costs time, which leads to larger running time comparing to [KY05, TY07]. [GP18] applied Bregman proximal method on the D-optimal design problem and observe accelerated convergence rate in their numerical experiments; however, they did not prove that their experimental parameter settings satisfy the assumption of their algorithm222We are grateful Gutman and Peña provided us their code for testing, of which D-optimal design was only one application..
A natural version of the D-optimal design problem is to require an integral solution. The integral variant is shown to be -hard [ČH12], although recently approximation algorithms have been developed [AZLSW17, SX18]
. In our context, this means the weight vectoris the integral optimal solution to (2), where the sum of the weights is some specified integral parameter .
Several Markov chains for sampling convex bodies have well understood performance guarantees based upon the roundedness of the convex body. If, then the mixing time of hit-and-run and the ball walk are both steps [LV06, KLS97]. Thus, placing a convex body in John position guarantees the walks mix in steps, and steps if the body is symmetric; this transformation is used in practice with the convex body to be a polytope [HCT17]. Generating the John Ellipsoid, with a fixed center point, has also been employed as a proposal distribution for a Markov chain [CDWY18, GN18].
We build our even faster algorithm via sketching techniques. Sketching has been successfully applied to speed up different problems, such as linear programs [LSZ19], clustering [CEM15, SYZ18], low rank approximations [CW13, NN13, BW14, CW15a, RSW16, SWZ17]CW13, NN13, CLM15, CW15b, PSW17, ALS18], total least regression [DSWY19]
, tensor regression[LHW17, DSSW18] and tensor decomposition [WTSA15, SWZ16, SWZ19]. Readers may refer to [Woo14] for a comprehensive survey on sketching technique. We use sketching techniques to speed up computing leverage scores. This idea was first used in [SS11].
Previous research on the MVEE problem did take advantage of the sparsity of the input matrix , and to the best of our knowledge, our algorithm is the first one that is able to deal with large sparse input. It would be interesting if we can apply sketching techniques to further speed up existing algorithms.
1.2.1 Relation with [Cp15]
We shall mention that our work is greatly inspired by [CP15]. The Lewis Weights for matrix is defined as the unique vector so that for ,
It is known that computing the Lewis Weight is equivalent to computing the maximal volume inscribed ellipsoid. [CP15] proposes an algorithm for approximating Lewis Weights for all . Their algorithm is an iterative algorithm that is very similar to our Algorithm 1, and the convergence is proved by arguing the iteration mapping is contractive. The main difference is that [CP15] outputs the weights in the last round, while our Algorithm 1 takes the average over all rounds and outputs the averaging weights, which allows us to conduct a convexity analysis and deal with the case.
2 Problem Formulation
In this section, we formally define the problem of computing the John Ellipsoid of a symmetric polytope. Let be a symmetric convex polytope, where denotes the set . We assume has full rank. By symmetry, we know that the maximal volume ellipsoid inside the polytope should be centered at the origin. Any ellipsoid centered at the origin can be expressed by , where is a positive definite matrix. Note that the volume of is proportional to , and an ellipsoid is contained in polytope if and only if for , . For any , we can write where . Hence
Therefore, we can compute the John Ellipsoid of by solving the following optimization program:
It turns out that the optimal ellipsoid satisfies , where is the optimal solution of the program
Lemma 2.1 (Optimality criteria, Proposition 2.5 in [Tod16]).
A weight is optimal for program (2) if and only if
We emphasize that program (3) and program (2) are equivalent, in the following sense. By Lemma 2.1, we can rewrite program (2) as minimizing , subject to for and . By setting , we obtain program (3). Thus, optimal solutions to programs (2) and (3) are equivalent up to a multiplicative factor .
We can also talk about an approximate John Ellipsoid.
For , we say is a -approximation of program (2) if satisfies
Lemma 2.3 (-approximation is good rounding).
Let be a -approximation of (2). Define as . Then
Let and suppose . Then, we have that . So,
Since , then and .
On the other hand, for , we have that . Hence
Let the matrix representation of be , then by optimality of and the duality gap, we have
Since is proportional to , we conclude that . ∎
3 Main Algorithm
Let be the function defined as where for ,
Inspired by (5), we use the fixed point iteration for and . has very nice properties. Actually, by setting , we can rewrite as , hence is actually the leverage score of the -th row of the matrix [CLM15]. From the well-known properties of leverage scores, we have
Lemma 3.1 (Properties of leverage scores, e.g. see Section 3.3 of [Clm15]).
For and , we have . Moreover, .
In order to show Algorithm 1 provides a good approximation of the John Ellipsoid, in the sense of Definition 2.2, we need to argue that for the output of Algorithm 1, . Our main result is the following theorem.
Theorem 3.2 (Main Result).
We now analyze the running time of Algorithm 1.
Theorem 3.3 (Performance of Algorithm 1).
For all , we can find a -approximation of John Ellipsoid inside a symmetric convex polytope in time .
The main loop is executed times, and inside each loop, we can first use time to compute , then compute in time. To see why we introduce , observe that . Now we can compute the Cholesky decomposition of in time , and use the Cholesky decomposition to compute in time for each . Finally, we can compute by computing in time . This is valid since . To summarize, in each iteration we use time, hence the overall running time is as stated. ∎
Lemma 3.4 (Convexity).
For , let be the function defined as
Then is convex.
Now that is convex, we can apply Jensen’s inequality to get Lemma 3.5.
Lemma 3.5 (Telescoping).
Now we are ready to prove Theorem 3.2.
Proof of Theorem 3.2.
Set . By Lemma 3.5, we have for ,
where the last step uses the fact that when , . This gives us .
On the other hand, from Lemma 3.1 we have . Hence
We shall mention that it is possible to further improve Algorithm 1 by applying sketching technique from randomized linear algebra. Here we present the performance of our accelerated algorithm, and detailed analysis can be found in Appendix C.
Theorem 3.6 (Performance of Algorithm 2).
For all , we can find a -approximation of the John Ellipsoid inside a symmetric convex polytope within iterations with probability at least . Moreover, each iteration involves solving linear systems of the form for some diagonal matrix .
We thank Zhao Song for his generous help on this paper.
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Appendix A Preliminaries
In this section we introduce notations and preliminaries used in the appendix. We use
to represent the normal distribution with mean
a.1 Multivariate Calculus
Let be a differentiable function. The directional derivative of in the direction is defined as
We can also define a high order directional derivative as
The following two properties of directional derivatives will be useful.
Let where . For , .
a.2 Gamma Function
function is a well-known math object. It is defined as
We need the following result on Gamma function.
Lemma A.2 (Corollary 1 of [Jam13]).
For all and ,
a.3 Tail Bound for Distribution
We need the following version of concentration for distribution.
Appendix B Proof of Lemma 3.4
In this section we provide the proof of Lemma 3.4.
We first prove a strengthened result: for fixed , the function defined as is convex. Here is the set of all positive definite matrices. Notice that is an open set, so we can differentiate .
We argue that it is sufficient to show for all and all , the second order directional derivative is non-negative. This is because . So if for all we have , then , which is precisely the convex condition.
Let us do some computation with Proposition A.1.
By Cauchy-Schwarzt inequality, we have
Hence for all and all .
Now we are ready to work on . For all in the domain of , let and . Then for all ,
So is also convex. ∎
Appendix C Faster Algorithm for Computing John Ellipsoid for Sparse Matrix
Proposition C.1 (Bound on ).
For completeness we define . For and , . Moreover, .
However, is a by matrix, so it is computationally expensive to compute . The trick we use here, which is initially introduced first by [SS11], is to introduce a random Gaussian matrix with rows to speed up the computation. Of course, this will introduce extra error, however we can prove that the overall result has good concentration.
c.1 Approximation Guarantee
Theorem C.2 (Main result).
By scaling so that , we have
Theorem C.3 (Approximation guarantee).
From Theorem C.2, with probability at least , , and . Therefore for all ,