{"title": "A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families", "book": "Advances in Neural Information Processing Systems", "page_first": 7723, "page_last": 7735, "abstract": "We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points.  Specifically, we present an algorithm which, given $n$ points in $\\mathbb{R}^d$ and an accuracy parameter $\\eps>0$, runs in time $\\poly(n,d,1/\\eps),$ and returns a log-concave distribution which, with high probability, has the property that the likelihood of the $n$ points under the returned distribution is at most an additive $\\eps$ less than the maximum likelihood that could be achieved via any log-concave distribution.  This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task.  Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, ``locally'' possesses key properties of exponential families.  This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method.  Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.", "full_text": "A Polynomial Time Algorithm for Log-Concave\nMaximum Likelihood via Locally Exponential\n\nFamilies\n\nBrian Axelrod\n\nDepartment of Computer Science\n\nStanford University\n\nbaxelrod@cs.stanford.edu\n\nAnastasios Sidiropoulos\n\nDepartment of Computer Science\nUniversity of Illinois at Chicago\n\nsidiropo@gmail.com\n\nIlias Diakonikolas\n\nDepartment of Computer Science\nUniversity of Wisconsin-Madison\nilias.diakonikolas@gmail.com\n\nAlistair Stewart\nWeb3 Foundation\n\nstewart.al@gmail.com\n\nGregory Valiant\n\nDepartment of Computer Science\n\nStanford University\n\ngvaliant@stanford.edu\n\nAbstract\n\nWe consider the problem of computing the maximum likelihood multivariate log-\nconcave distribution for a set of points. Speci\ufb01cally, we present an algorithm which,\ngiven n points in Rd and an accuracy parameter \u270f> 0, runs in time poly(n, d, 1/\u270f),\nand returns a log-concave distribution which, with high probability, has the property\nthat the likelihood of the n points under the returned distribution is at most an\nadditive \u270f less than the maximum likelihood that could be achieved via any log-\nconcave distribution. This is the \ufb01rst computationally ef\ufb01cient (polynomial time)\nalgorithm for this fundamental and practically important task. Our algorithm rests\non a novel connection with exponential families: the maximum likelihood log-\nconcave distribution belongs to a class of structured distributions which, while not\nan exponential family, \u201clocally\u201d possesses key properties of exponential families.\nThis connection then allows the problem of computing the log-concave maximum\nlikelihood distribution to be formulated as a convex optimization problem, and\nsolved via an approximate \ufb01rst-order method. Ef\ufb01ciently approximating the (sub)\ngradients of the objective function is a main technical challenge in this work.\n\n1\n\nIntroduction\n\nA distribution on Rd is log-concave if the logarithm of its probability density function is concave:\nDe\ufb01nition 1 (Log-concave Density). A probability density function f : Rd ! R+, d 2 Z+, is called\nlog-concave if there exists an upper semi-continuous concave function  : Rd ! [1,1) such that\nf (x) = e(x) for all x 2 Rd. We will denote by Fd the set of upper semi-continuous, log-concave\ndensities with respect to the Lebesgue measure on Rd.\n\nThis paper merges two independent works [4, 35]. Authors are in alphabetical order.\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fLog-concave densities form a broad nonparametric family encompassing a wide range of fundamental\ndistributions, including the uniform, normal, exponential, logistic, extreme value, Laplace, Weibull,\nGamma, Chi and Chi-Squared, and Beta distributions (see, e.g., [5]). Log-concave probability\nmeasures have been extensively investigated in several scienti\ufb01c disciplines, including economics,\nprobability theory and statistics, computer science, and geometry (see, e.g., [60, 3, 54, 62, 59]). The\nproblem of density estimation for log-concave distributions is of central importance in the area of\nnon-parametric estimation (see, e.g., [62, 59, 58]) and has received signi\ufb01cant attention during the\npast decade in statistics [22, 38, 36, 21, 49, 6, 45] and computer science [18, 19, 2, 15, 32, 33, 16].\nOne reason the class of log-concave distributions has attracted this attention, both from the theoretical\nand practical communities, is that log-concavity is a very natural \u201cshape constraint,\u201d which places\nsigni\ufb01cantly fewer assumptions on the distribution in question than most parameterized classes\nof distributions. In extremely high-dimensional settings when the amount of available data is not\ntoo much larger than the dimensionality, \ufb01tting a multivariate Gaussian (or some other parametric\ndistribution) to the data might be all one can hope to do. For many practical settings, however, the\ndimensionality is modest (e.g., 5-20) and the amount of data is signi\ufb01cantly larger (e.g., hundreds of\nthousands or millions). In such settings, making a strong assumption on the parametric form of the\nunderlying distribution is unnecessary\u2014there is suf\ufb01cient data to \ufb01t a signi\ufb01cantly broader class of\ndistributions, and log-concave distributions are one of the most natural such classes. From a practical\nperspective, even in the univariate setting, computing the log-concave density that maximizes the\nlikelihood of the available data is a useful primitive, with the R implementation of Ru\ufb01bach and\nDuembgen having over 39,000 downloads [39]. As we discuss below, the amount of data required to\nlearn a log-concave distribution scales exponentially in the dimension, in contrast to most parametric\nclasses of distributions. Nevertheless, for the many practical settings with modest dimensionality and\nlarge amounts of data, there is suf\ufb01cient data to learn. The question now is computational: how does\none compute the best-\ufb01t log-concave distribution? We focus on this algorithmic question:\n\nIs there an ef\ufb01cient algorithm to compute the log-concave MLE for datapoints in Rd?\n\nObtaining an understanding of the above algorithmic question is of interest for a number of reasons.\nFirst, the log-concave MLE is the prototypical statistical estimator for the class, is fully automatic (in\ncontrast to kernel-based estimators, for example), and was very recently shown to achieve the minimax\noptimal sample complexity for the task of learning a log-concave distribution (up to logarithmic\nfactors) [16, 23]. The log-concave MLE also has an intriguing geometry that is of interest from\na purely theoretical standpoint [22, 57]. Developing an ef\ufb01cient algorithm for computing the log-\nconcave MLE is of signi\ufb01cant theoretical interest, and would also allow this general non-parametric\nclass of distributions to be leveraged in the many practical settings where the dimensionality is\nmoderate and the amount of data is large. We refer the reader to the recent survey [58] for a more\nthorough justi\ufb01cation for why the log-concave MLE is a desirable distribution to compute.\n\n1.1 Our Results and Techniques\nThe main result of this paper is the \ufb01rst ef\ufb01cient algorithm to compute the multivariate log-concave\nMLE. For concreteness, we formally de\ufb01ne the log-concave MLE:\n\nis the density bfn 2F d which maximizes the log-likelihood `(f )\n\nDe\ufb01nition 2 (Log-concave MLE). Let X1, . . . , Xn 2 Rd. The log-concave MLE, bfn =\nbfn(X1, . . . , Xn),\nPn\ni=1 ln(f (Xi)) over f 2F d.\nAs shown in [22], the log-concave MLE bfn exists and is unique. Our main result is the \ufb01rst ef\ufb01cient\nalgorithm to compute it up to any desired accuracy.\nTheorem 1 (Main Result). Fix d 2 Z+ and 0 <\u270f,\u2327 < 1. There is an algorithm that, on input any\nset of points X1, . . . , Xn in Rd, and 0 <\u270f , \u2327< 1, runs in poly(n, d, 1/\u270f, log(1/\u2327 )) time and with\nprobability at least 1  \u2327 outputs a succinct description of a log-concave density h\u21e4 2F d such that\n`(h\u21e4)  `(bfn)  \u270f.\n\ndef=\n\nOur algorithm does not require that the input points X1, . . . , Xn in Rd are i.i.d. samples from a\nlog-concave density, i.e., it ef\ufb01ciently solves the MLE optimization problem for any input set of\npoints. We also note that the succinct output description of h\u21e4 allows for both ef\ufb01cient evaluation\nand ef\ufb01cient sampling. That is, we can ef\ufb01ciently approximate the density at a given point (within\n\n2\n\n\fmultiplicative accuracy), and ef\ufb01cient sample from a distribution that is close in total variation\ndistance.\nRecent work [16, 23] has shown that the log-concave MLE is minimax optimal, within a logarithmic\nfactor, with respect to squared Hellinger distance. In particular, the minimax rate of convergence\n\nwith n samples is \u02dc\u21e5dn2/(d+1). Combining this sample complexity bound with our Theorem 1,\n\nwe obtain the \ufb01rst sample near-optimal and computationally ef\ufb01cient proper learning algorithm for\nmultivariate log-concave densities. See Theorem 4 in Appendix D.\n\nTechnical Overview Here we provide an overview of our algorithmic approach. Notably, our\nalgorithm does not require the assumption that the input points are samples from a log-concave\ndistribution. It runs in poly(n, d, 1/\u270f) on any set of input points and outputs an \u270f-accurate solution to\nthe log-concve MLE. Our algorithm proceeds by convex optimization: We formulate the problem of\ncomputing the log-concave MLE of a set of n points in Rd as a convex optimization problem that\nwe solve via an appropriate \ufb01rst-order method. It should be emphasized that one needs to overcome\nseveral non-trivial technical challenges to implement this plan.\nThe \ufb01rst dif\ufb01culty lies in choosing the right (convex) formulation. Previous work [22] considered a\nconvex formulation of the problem, though that formulation seems to inherently lead to an exponential\ntime algorithm. Given our convex formulation, a second dif\ufb01culty arises: we do not have direct access\nto the (sub-)gradients of the objective function and the naive algorithm to compute a subgradient at a\npoint takes exponential time. Hence, a second challenge is how to obtain an ef\ufb01cient algorithm for\nthis task. One of our main contributions is a randomized polynomial time algorithm to approximately\ncompute a subgradient of the objective function. Our algorithm for this task leverages structural\nresults on log-concave densities established in [16] combined with classical algorithmic results on\napproximating the volume of convex bodies and uniformly sampling from convex sets [48, 53, 52].\nWe now proceed to explain our convex optimization formulation. Our starting point is a key structural\n\nproperty of the log-concave MLE, shown in [22]: The logarithm of the log-concave MLE lnbfn, is a\n\n\u201ctent\u201d function, whose parameters are the values y1, . . . , yn of the log density at the n input points\nx(1), . . . , x(n), and whose log-likelihoods correspond to polyhedra. Our conceptual contribution\nlies in observing that while tent distributions are not an exponential family, they \u201clocally\u201d retain\nmany properties of exponential families (De\ufb01nition 4). This high-level similarity can be leveraged to\nobtain a convex formulation of the log-concave MLE that is similar in spirit to the standard convex\nformulation of the exponential family MLE [61]. Speci\ufb01cally, we seek to maximize the log-likelihood\nof the probability density function obtained by normalizing the log-concave function whose logarithm\nis the convex hull of the log densities at the samples. This objective function is a concave function of\nthe parameters, so we end up with a (non-differentiable) convex optimization problem. The crucial\nobservation is that the subgradient of this objective at a given point y is given by an expectation under\nthe current hypothesis density at y.\nGiven our convex formulation, we would like to use a \ufb01rst-order method to ef\ufb01ciently \ufb01nd an \u270f-\napproximate optimum. We note that the objective function is not differentiable everywhere, hence we\nneed to work with subgradients. We show that the subgradient of the objective function is bounded in\n`2-norm at each point, i.e., the objective function is Lipschitz. Another important structural result\n(Lemma 2) allows us to essentially restrict the domain of our optimization problem to a compact\nconvex set of appropriately bounded diameter D = poly(n, d). This is crucial for us, as the diameter\nbound implies an upper bound on the number of iterations of a \ufb01rst-order method. Given the above,\nwe can in principle use a projected subgradient method to \ufb01nd an approximate optimum to our\noptimization problem, i.e., \ufb01nd a log-concave density whose log-likelihood is \u270f-optimal.\nIt remains to describe how we can ef\ufb01ciently compute a subgradient of our objective function. Note\nthat the log density of our hypothesis can be considered as an unbounded convex polytope. The\nprevious approach to calculate the subgradient in [22] relied on decomposing this polytope into faces\nand obtaining a closed form for the underlying integral over these faces (that gives their contribution\nto the subgradient). However, this convex polytope is given by n vertices in d dimensions, and\ntherefore the number of its faces can be n\u2326(d). So, such an algorithm cannot run in polynomial time.\nInstead, we note that we can use a linear program (see proof of Lemma 1) to evaluate a function\nproportional to the hypothesis density at a point in time polynomial in n and d. To use this oracle for\nthe density in order to produce samples from the hypothesis density, we use Markov Chain Monte\nCarlo (MCMC) methods. In particular, we use MCMC to draw samples from the uniform distribution\n\n3\n\n\fon super-level sets and estimate their volumes. With appropriate rejection sampling, we can use these\nsamples to obtain samples from a distribution that is close to the hypothesis density. See Lemma 3.\n(We note that it does not suf\ufb01ce to simply run a standard log-concave density sampling technique\nsuch as hit-and-run [51]. These random walks require a hot start which is no easier than the sampling\ntechnique we propose.)\nSince the subgradient of the objective can be expressed as an expectation over this density, we can use\nthese samples to sample from a distribution whose expectation is close to a subgradient. We then use\nstochastic subgradient descent to \ufb01nd an approximately optimal solution to the convex optimization\nproblem. The hypothesis density this method outputs has log-likelihood close to the maximum.\n\n1.2 Related Work\nThere are two main strands of research in density estimation. The \ufb01rst one concerns the learnability\nof high-dimensional parametric distributions, e.g., mixtures of Gaussians. The sample complexity\nof learning parametric families is typically polynomial in the dimension and the challenge is to\ndesign computationally ef\ufb01cient algorithms. The second research strand \u2014 which is the focus of this\npaper \u2014 considers the problem of learning a probability distribution under various non-parametric\nassumptions on the shape of the underlying density, typically focusing on the univariate or small\nconstant dimensional regime. There has been a long line of work in this vein within statistics since the\n1950s, dating back to the pioneering work of [42] who analyzed the MLE of a univariate monotone\ndensity. Since then, shape constrained density estimation has been an active research area with a rich\nliterature in mathematical statistics and, more recently, in computer science. The reader is referred\nto [10] for a summary of the early work and to [44] for a recent book on the subject.\nThe standard method used in statistics for density estimation problems of this form is the MLE.\nSee [14, 55, 63, 46, 43, 11, 12, 40, 17, 7, 47, 38, 9, 41, 8, 50, 62, 21, 49, 6, 45, 16] for a partial list\nof works analyzing the MLE for various distribution families. During the past decade, there has\nbeen a body of algorithmic work on shape constrained density estimation in computer science with\na focus on both sample and computational ef\ufb01ciency [24\u201326, 18\u201320, 1, 2, 29, 30, 27, 31, 33, 34].\nThe majority of this literature has studied the univariate (one-dimensional) setting which is by now\nfairly well-understood for a wide range of distributions. On the other hand, the multivariate setting is\nsigni\ufb01cantly more challenging and wide gaps in our understanding remain even for d = 2.\nFor the speci\ufb01c problem of learning a log-concave distribution, a line of work in statistics [22, 38,\n36, 21, 6] has characterized the global consistency properties of the log-concave multivariate MLE.\n\nRegarding \ufb01nite sample bounds, [49, 23] gave a sample complexity lower bound of \u2326d(1/\u270f)(d+1)/2\nfor d 2 Z+ that holds for any estimator, and [49] gave a near-optimal sample complexity upper bound\nfor the log-concave MLE for d \uf8ff 3. [33] established the \ufb01rst \ufb01nite sample complexity upper bound\nfor learning multivariate log-concave densities under global loss functions. Their estimator (which\nis different than the MLE and seems hard to compute in multiple dimensions) learns log-concave\ndensities on Rd within squared Hellinger loss \u270f with \u02dcOd(1/\u270f)(d+5)/2 samples. [16] showed a\nsample complexity upper bound of \u02dcOd(1/\u270f)(d+3)/2 for the multivariate log-concave MLE with\nrespect to squared Hellinger loss, thus obtaining the \ufb01rst \ufb01nite sample complexity upper bound\nfor this estimator in dimension d  4. Building on their techniques, this bound was subsequently\nimproved in [23] to a near-minimax optimal bound of \u02dcOd(1/\u270f)(d+1)/2. Alas, the computational\ncomplexity of the log-concave MLE has remained open in the multivariate case. Finally, we note that\na recent work [28] obtained a non-proper estimator for multivariate log-concave densities with sample\ncomplexity \u02dcOd((1/\u270f)d+2) (i.e., at least quadratic in that of the MLE) and runtime \u02dcOd((1/\u270f)2d+2).\nOn the empirical side, recent work [56] proposed a non-convex optimization approach to the problem\nof computing the log-concave MLE, which seems to exhibit superior performance in practice in\ncomparison to previous implementations (scaling to 6 or higher dimensions). Unfortunately, their\nmethod is of a heuristic nature, in the sense that there is no guarantee that their solution will converge\nto the log-concave MLE.\n\n2 Preliminaries\nNotation. We denote by X1, . . . , Xn 2 Rd the sequence of samples. We denote by Sn =\ni=1) the convex hull of X1, . . . , Xn, and by X the d \u21e5 n matrix with columns vec-\nConv({Xi}n\n\n4\n\n\ftors X1, . . . , Xn. We write\ndenotes the indicator function for Y .\n\nfor the all-ones vector of the appropriate length. For a set Y \u21e2 Z, Y\n\nTent Densities. We start by de\ufb01ning tent functions and tent densities:\nDe\ufb01nition 3 (Tent Function). For y = (y1, . . . , yn) 2 Rn and a set of points X1, . . . , Xn in Rd, we\nde\ufb01ne the tent function hX,y : Rd ! R as follows:\n\nhX,y(x) =\u21e2 max{z 2 R such that (x, z) 2 Conv({(Xi, yi)}n\n\n1\n\ni=1)}\n\nif x 2 Sn\nif x /2 Sn\n\nThe points (Xi, yi) are referred to as tent poles. (See Figure 1 in appendix A for the graph of an\nexample tent function.)\nLet pX,y(x) = c exp(hX,y(x)) with c chosen such that pX,y(x) integrates to one. We refer to pX,y\nas a tent density and the corresponding distribution as a tent distribution. Note that the support of a\ntent distribution must be within the convex hull of X1, . . . , Xn. For the remainder of the paper, we\nchoose a scaling such that T y = 0. This scaling is arbitrary, and has no signi\ufb01cant effect on either\nthe algorithm or its analysis.\nTent densities are notable because they contain solutions to the log-concave MLE [22]. The solution\nto the log-concave MLE over X1, . . . , Xn is always a tent density, because tent densities with tent\npoles X1, . . . , Xn are the minimal log-concave functions with log densities y1, . . . , yn at points\nX1, . . . , Xn.\nThe algorithm which we present can be thought of as an optimization over tent functions. In Section\n3.1, we will show that tent distributions retain important properties of exponential families which\nwill be useful to establish the correctness of our algorithm.\n\nRegular Subdivisions. Given a tent function hX,y with hX,y(Xi) = yi, its associated regular\nsubdivision X,y of X is a collection of subsets of X1, . . . , Xn 2 Rd whose convex hulls are the\nregions of linearity of hX,y. See Figure 1 in appendix A for an illustration of a tent function and its\nregular subdivision. We refer to these polytopes of linearity as cells. We say that X,y is a regular\ntriangulation of X if every cell is a ddimensional simplex.\nIt is helpful to think of regular subdivisions in the following way: Consider the hyperplane H in\nRd+1 obtained by \ufb01xing the last coordinate. Consider the function hX,y as a polytope and project\neach face onto H. Each cell is a projection of a face, and together the cells partition the convex hull\nof X1, . . . , Xn. Observe that regular subdivisions may vary with y. Figure 2 in appendix A provides\none example of how changing the y vector changes the regular subdivision.\nFor a given regular triangulation , the associated consistent neighborhood N is the set of all\ny 2 Rn, such that X,y = . That is, consistent neighborhoods are the sets of parameters where the\nregular triangulation remains \ufb01xed. Note that these neighborhoods are open and their closures cover\nthe whole space. See Figure 2 in appendix A for an example of how crossing between consistent\nneighborhoods results in different subdivisions. We note that for \ufb01xed X, when y is chosen in general\nposition, X,y is always a regular triangulation.\n\n3 Locally Exponential Convex Programs\n\nIn this section, we lay the foundations for the algorithm presented in the next section. We present\nthe \u201clocally\" exponential form of tent distributions and show it has the necessary properties to enable\nef\ufb01cient computation of the log-concave MLE. Though they form a broader class of distributions,\n\u201clocally\" exponential distributions share some important properties of exponential families. Namely,\nthe log-likelihood optimization is convex, and the expectation of the suf\ufb01cient statistic is a subgradient.\nThis will allows us to formulate a convex program which we will be able to solve in polynomial time.\nDe\ufb01nition 4. Let T be some function (possibly parametrized by y) and let qy =\nexp (hT (x), yi  A(y)) be a family of probability densities parametrized by y with A(y) acting\nto normalize the density so it integrates to 1. We say that the family {qy} is locally-exponential if the\nfollowing hold: (1) A(y) is convex in y, and (2) Ex\u21e0qy [T (x)] 2 @yA(y).\nNote that the above de\ufb01nition differs from an exponential family in that for exponential families T\nmay not depend on y.\n\n5\n\n\fIn this section, we derive a suf\ufb01cient statistic, the polyhedral statistic, that shows that tent distributions\nare in fact locally exponential. More formally, we show:\nLemma 1. For tent poles X1, . . . , Xn, there exists a function TX,y : Rd ! Rn (the polyhedral statis-\ntic) such that pX,y(x) = exp (hTX,y(x), yi  A(y)) corresponds to the family of tent-distributions\nsuch that {pX,y} is locally exponential. Furthermore, TX,y is computable in time poly(n, d).\nSince we know that the log-concave MLE is a tent distribution, and all tent-distributions are log-\nconcave, we know that the optimum of the maximum likelihood convex program in Equation (3.1)\ncorresponds to the log-concave MLE.\n\nMLE of tents = max\n\ny Xi\n\nhX,y(Xi)  logZ exp hX,y(x)dx = max\n\ny Xi\n\nyi  A(y)\n\n(3.1)\n\nCombining the above with the fact that the suf\ufb01cient statistic allows us to compute the stochastic\nsubgradient suggests that Algorithm 1 can compute the log-concave MLE in polynomial time.\n\nAlgorithm 1 ComputeLogConcaveMLE(X1, . . . , Xn,\u270f )\n\n\u270f2\n\ny 0; c 8n2d log(2nd); m 2c2\nfor i 1, m do\n\u2318 c/pi\ns \u21e0 pX,y\ny y + \u2318 1\n\nn  TX,y(s)\n\nreturn y\n\n.T computed via Lemma 1. 1\n\n. Using Lemma 3\nn follows from Equation (3.1)\n\n3.1 The Polyhedral Suf\ufb01cient Statistic\n\nConsider a regular triangulation  corresponding to tent distribution parametrized by X and y. The\npolyhedral statistic is the function\n\nTX,y(x) : Sn ! [0, 1]n,\n\nhX,y(x) = hTy(x), yi\n\nthat expresses x as a convex combination of corners of the cell containing x in y. That is x =\nXTX,y(x) where ||Ty(x)||1 = 1 and Ty(x)i = 0 if Xi is not a corner of the cell containing x. The\npolyhedral statistic gives an alternative way of writing tent functions and tent densities:\nIf we restrict y such thatPi\n\nevery consistent neighborhood N we have an exponential family of the form\n\npX,y(x) = exp(hTy(x), yi) .\n\npX,y(x)dx, then we can see that for\n\nyi = 0 and de\ufb01ne A(y) = logRx\nexp (hTy(x),\u2713 i  A(y)) for \u2713 2 N .\n\n(3.2)\nWhile Equation (3.2) shows how subsets of tent distributions are exponential families, it also helps\nhighlight why tent distributions are not an exponential family. The suf\ufb01cient statistic depends on y\nthrough the regular subdivision. This means that tent distributions do not admit the same factorized\nform as exponential families since the suf\ufb01cient statistic depends on y.\nNote that we can use any ordering of X1, . . . , Xn to de\ufb01ne the polyhedral suf\ufb01cient statistic every-\nwhere including on regular subdivisions that are not regular triangulations. Also note that, assuming\nn \u2713 = 0 makes each ex-\nthat no Xi = Xj, i 6= j, eliminating the last coordinate using the constraint T\nponential family minimal. In other words, over regions where the regular subdivision does not change\n(for example the consistent neighborhoods), tent distributions are minimal exponential families. This\nmeans the set of tent distribution can be seen as the \ufb01nite union of a set of minimal exponential\nfamilies. We refer to Equation (3.3) as the exponential form for tent densities:\n\npX,y(x) = exp (hTX,y(x), yi  A(y)) Sn(x).\n\n(3.3)\n\nBoth the polyhedral statistic and tent density queries can be computed in polynomial time with\nthe packing linear program presented in Equation (3.4). For a point x, the value of y yields the\n\n6\n\n\flog-density and the vector \u21b5 corresponds to polyhedral statistic.\n\nmax y s.t. (x, y) =Xi\n\n\u21b5i(Xi, yi),Xi\n\n\u21b5i = 1,\u21b5 i  0\n\n(3.4)\n\nNote that the above combined with tent distributions being exponential families on consistent\nneighborhoods gives us that the properties from Lemma 1 hold true on consistent neighborhoods. We\nextend the proof to the full result below.\n\nProof. Convexity follows by iteratively applying known operations that preserve convexity of a\nfunction. Since a sum of convex functions is convex (see, e.g., page 79 of [13]), it suf\ufb01ces to show\n\ny, by de\ufb01nition, exp(hX,y(x)) is log-convex as a function of y. Since an integral of log-convex\n\nthat the function G(y) = ln(R exp(hX,y(x))dx) is convex. Since hX,y(x) is a convex function of\nfunctions is log-convex (see, e.g., page 106 of [13]), it follows thatR exp(hy(x))dx is log-convex.\nTherefore, G is convex. We have therefore established that Equation (3.1) is convex, as desired.\nEx\u21e0pX,y [TX,y(x)] 2 @yA(y): Note that when y is in the interior of a consistent neighborhood, the\npolyhedral statistic LP has a unique solution and Ex\u21e0pX,y [T (x)] 2 @yA(y) (by Fact 3). When y is\non the boundary the solution set to the LP corresponds to the convex hull of solutions corresponding\nto each adjacent consistent neighborhood. This corresponds to the convex hull of limiting gradients\nfrom each neighboring consistent neighborhood and is the set of subgradients.\n\n4 Algorithm and Analysis\n\nRecall that we compute the log-concave MLE via a \ufb01rst-order method on the optimization formulation\npresented in Equation (3.1). The complete method is presented in Algorithm 1. The algorithm is\nbased on the stochastic gradient computation presented in the previous section, a standard application\nof the stochastic gradient method, and a sampler that we describe later in this section. Theorem 1\nfollows from bounding the rate of convergence of the stochastic subgradient method and the ef\ufb01ciency\nof the sampling procedure. We outline these two components below.\n\n4.1 The Stochastic Subgradient Method\nRecall that algorithm 1 is simply applying the stochastic subgradient method to the following convex\n\nn n, y\u21b5 A(y). We require a slight strengthening of the following\nprogram with T y = 0: h(y) =\u2326 1\nstandard result, see, e.g., Theorem 3.4.11 in [37]:\nFact 1. Let C be a compact convex set of diameter diam(C) < 1. Suppose that the projections \u21e1C\nare ef\ufb01ciently computable, and there exists M < 1 such that for all y 2C we have that kgk2 \uf8ff M\nfor all stochastic subgradients. Then, after K =\u2326 M \u00b7 diam(C) log(1/\u2327 )/\u270f2 iterations of the\nprojected stochastic subgradient method (for appropriate step sizes), with probability at least 1  \u2327,\nwe have that F\u00afy(K)  miny2C F (y) \uf8ff \u270f, where \u00afy(K) = (1/K)PK\nWe note that Fact 1 assumes that, in each iteration, we can ef\ufb01ciently calculate an unbiased stochastic\nsubgradient, i.e., a vector g(k) such that E[g(k)] 2 @yF (y(k)). Unfortunately, this is not the case in\nour setting, because we can only approximately sample from log-concave densities. However, it\nis straightforward to verify that the conclusion of Fact 1 continues to hold if in each iteration we\n\ni=1 y(i).\n\ncan compute a random vectoreg(k) such that kE[eg(k)]  g(k)k2 < def= \u270f/(2diam(C)), for some\n\ng(k) 2 @yF (y(k)). This slight generalization is the basic algorithm we use in our setting.\nWe now return to the problem at hand. We note that since T represents the coef\ufb01cients of a convex\ncombination ||T (x)|| < 1 for all x, bounding M by 1.\nLemma 2 will show that diam(C) = O(2n2d log(2nd)).\nc = 8n2d log(2nd) and run SGD for 2c2\n\u270f2\nvalue within \u270f of the log-concave MLE.\n\nif we let\niterations, the resulting point will have objective\n\nThis implies that\n\nLemma 2. Let X1, . . . , Xn be a set of points in Rd and \u02c6f be the corresponding log-concave MLE.\n\u02c6f (Xi)\nThen, we have that R1\n\u02c6f (Xi) \uf8ff (2nd)2nd. Converting to an `2 norm yields a bound on\nthe diameter of C: diam(C) \uf8ff 2n2d log(2nd).\n\ndef= maxi2[n]\nmini2[n]\n\n7\n\n\fLet us brie\ufb02y sketch the proof of Lemma 2. The main idea is to show that if R1 were too high, then\n\nbfn would have a lower likelihood than the uniform distribution on the convex hull of the samples\nSn. More speci\ufb01cally, if the maximum value M of the density bfn is large, then the volume of the set\n{x 2 Rd : bfn(x)  M/R} is small. For a \ufb01xed R, this set contains Sn and thus R1 must be large\ncompared to M vol(Sn). Since bfn has likelihood at least as high as the uniform distribution over Sn,\n\nR must be small compared to M vol(Sn). Combining these two observations yields a bound on R.\nWe now proceed with the complete proof.\n\nProof of Lemma 2. Let V = vol(Sn) be the volume of the convex hull of the sample points and\n\nvalue and its global non-zero positive value in one of the points Xi.\n\nM = maxx bfn(x) be the maximum pdf value of the MLE. By basic properties of the log-concave\nMLE (see, e.g., Theorem 2 of [22]), we have that bfn(x) > 0 for all x 2 Sn and bfn(x) = 0 for all\nx 62 Sn. Moreover, by the de\ufb01nition of a tent function, it follows that bfn attains its global maximum\nWe can assume without loss of generality that bfn is not the uniform distribution on Sn, since otherwise\nR1 = 1 and the lemma follows. Under this assumption, we have that R1 > 1 or ln R1 > 0, which\nimplies that M > 1/V . The following fact bounds the volume of upper level sets of any log-concave\ndensity:\nFact 2 (see, e.g., Lemma 8 in [16]). Let f 2F d with maximum value Mf . Then for all w > 0, we\nhave vol(Lf (Mf ew)) \uf8ff wd/Mf .\nBy Fact 2 applied to the MLE bfn, for w = ln R1, we get that vol(Lbfn\n(M/R1)) \uf8ff (ln R1)d/M.\nSince the pdf value of bfn at any point in the convex hull Sn is at least that of the smallest sample point\nXi, i.e., M/R1, it follows that Sn is contained in Lbfn\n(M/R1). Therefore, V \uf8ff (ln R1)d/M.\nOn the other hand, the log-likelihood of bfn is at least the log-likelihood of the uniform distribution\nUSn on Sn. Since at least one sample point Xi has pdf value bfn(Xi) = M/R1 and the other n  1\nsample points have pdf value bfn(Xi) \uf8ff M, we have that ln(M/R1) + (n  1) ln M  `(bfn) \n`(USn) = n ln(1/V ), or n ln M  ln R1  n ln V , and therefore ln(M V )  (ln R1)/n. This\ngives that R1/n\n1 \uf8ff M V. Combining this expression with V \uf8ff (ln R1)d/M from above yields that\nR1 \uf8ff (ln R1)nd.\nSince ln x < x, x 2 R, setting x = R\n(2nd)nd \u00b7 R1/2\ncompletes the proof of Lemma 2.\n\n1 . By the above, we deduce that R1 \uf8ff (2nd)nd \u00b7 R1/2\n\n2nd\n\n1 or (ln R1)nd <\n1 or R1 \uf8ff (2nd)2nd . This\n\n1\n\n2nd\n\n1 gives that ln R1 < 2nd \u00b7 R\n\n1\n\n4.2 Ef\ufb01cient Sampling and Log-Partition Function Evaluation\n\nIn this section, we establish the following result, which gives an ef\ufb01cient algorithm for sampling\nfrom the log-concave distribution computed by our algorithm.\nLemma 3 (Ef\ufb01cient Sampling). There exist algorithms A1 and A2 satisfying the following: Let\n, \u2327 > 0, let X = X1, . . . , Xn 2 Rd, let y 2 Rn be a parameter of a tent-density in exponential\nform. Then the following conditions hold:\n\n(1) On input X, y, , and \u2327, algorithm A1 outputs a random vector Z 2 Rd, distributed\naccording to some probability distribution with densitye, such that ke  pX,yk1 = O(),\nin time poly(n, d,kyk1, 1/, log(1/\u2327 )), with probability at least 1  \u2327.\n(2) On input X, y, , and \u2327, algorithm A2 outputs some 0 > 0, such that 0/(1 + O()) \uf8ff\nR exp(hX,y(x))dx \uf8ff 0 \u00b7 (1 + O()), in time poly(n, d,kyk1, 1/, log(1/\u2327 )), with prob-\nability at least 1  \u2327.\n\nThe algorithm used to show lemma 3 is presented in algorithm 2.\nAlgorithm 2 operates in two stages (The formal analysis is presented in appendix B.) First, it slices\nthe tent distribution into level sets and computes their volume. Properties of log-concave distributions\n\n8\n\n\fAlgorithm 2 Algorithm to sample from pX,y\n\nsamplable probability distribution such that\n\ni=1 in Rd, vector y 2 Rn, parameter 0 << 1.\n\nvol(Li)/(1 + ) \uf8ff fvol(Li) \uf8ff vol(Li)(1 + ).\n\nprocedure SAMPLE(X1, . . . , Xn, y)\nInput: Sequence of points X = {Xi}n\nOutput: A random vector Z 2 Rd sampled from a probability distribution with density function\ne, such that ke  pX,yk1 \uf8ff .\nStep 1. Let m = d1 + 2kyk1e. Let M = maxx2Rd exp(hX,y(x)). For any i 2 [m], let\nLi = {x 2 Rd : exp(hX,y(x))  M \u00b7 2i}. For each i 2 [m] compute an estimate fvol(Li) of\nvol(Li) such that\nStep 2. For i 2 [m], let ui be the uniform probability distribution on Li, and leteui be an ef\ufb01ciently\nkeui  uik1 \uf8ff .\nStep 3. Letec =Pm\ni=1 2ifvol(Li) + 2mfvol(Lm).\nStep 4. Let bD be the probability distribution on [m] with\nPrI\u21e0eD[I = i] =( fvol(Li) \u00b7 2i/ec\n2 \u00b7fvol(Lm) \u00b7 2m/ec\nStep 5. Sample I \u21e0 eD and sample Z \u21e0euI.\n\nif i 2{ 1, . . . , m  1}\nif i = m\n\nStep 6. For any x 2 Rd let GX,y(x) = M \u00b7 2blog2(M/ exp(hX,y(x)))c.\nStep 7. With probability 1  exp(hX,y(Z))/GX,y(Z) go to Step 5.\nreturn Z.\n\nallow us to guarantee that we obtain a good approximation of the the density with these slices. We\nthen derive a linear program which can be used as a separation oracle for tent densities. This allows\nus to compute their volume using a classic result for volume estimation [48]. In the second stage we\nsample from the \u201csliced\u201d distribution above. We \ufb01rst draw a single random number to choose a level\nset, weighted by the volume computed in the \ufb01rst stage. We then draw a sample uniformly at random\nfrom the corresponding level set and return that as our sample. Please see appendix B for a complete\nexposition, proof and pseudocode.\n\n5 Conclusions\n\nIn this paper, we gave a poly(n, d, 1/\u270f) time algorithm to compute an \u270f-approximation of the\nlog-concave MLE based on n points in Rd. Ours is the \ufb01rst algorithm for this problem with a sub-\nexponential dependence in the dimension d. We hope that our approach may lead to more practical\nmethods for computing the log-concave MLE in higher dimensions than was previously possible.\nOne concrete open question is whether there exists an algorithm for computing the log-concave\nMLE that runs in time poly(n, d, log(1/\u270f)), instead of the poly(n, d, 1/\u270f) that we achieve. Such\nan algorithm would likely be technically interesting as it may require going beyond the \ufb01rst-order\nmethods we employ. More broadly, it seems worth investigating whether the MLE can be ef\ufb01ciently\ncomputed for other natural classes of non-parametric distributions. Alternately, one could hope that\nthere is a simple set of natural properties such that, if a class of distributions satis\ufb01es those properties,\nthen the MLE can be ef\ufb01ciently computed.\n\nAcknowledgments: Ilias Diakonikolas was supported by NSF Award CCF-1652862 (CAREER)\nand a Sloan Research Fellowship. Alistair Stewart was supported by a USC startup grant. Anas-\ntasios Sidiropoulos was supported by NSF awards CCF-1453472 (CAREER) and CCF-1934915\n(TRIPODS), and NSF grants CCF-1423230 and CCF-1815145. Brian Axelrod was supported by NSF\nFellowship grant DGE-1656518, NSF award CCF-1763299, and a Finch family fellowship. Brian\nAxelrod and Gregory Valiant were supported by NSF awards CCF-1704417, and an ONR Young\nInvestigator Award (N00014-18-1-2295).\n\n9\n\n\fReferences\n[1] J. Acharya, I. Diakonikolas, C. Hegde, J. Li, and L. Schmidt. Fast and near-optimal algorithms\nfor approximating distributions by histograms. 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Math.\n\nStatist., 41:457\u2013471, 2169\u20132174, 1970.\n\n13\n\n\f", "award": [], "sourceid": 4190, "authors": [{"given_name": "Brian", "family_name": "Axelrod", "institution": "Stanford"}, {"given_name": "Ilias", "family_name": "Diakonikolas", "institution": "UW Madison"}, {"given_name": "Alistair", "family_name": "Stewart", "institution": "University of Southern California"}, {"given_name": "Anastasios", "family_name": "Sidiropoulos", "institution": "University of Illinois at Chicago"}, {"given_name": "Gregory", "family_name": "Valiant", "institution": "Stanford University"}]}