Georgina Hall

Rajiv Sambharya, Georgina Hall, Brandon Amos et al. · princeton

We introduce a machine-learning framework to warm-start fixed-point optimization algorithms. Our architecture consists of a neural network mapping problem parameters to warm starts, followed by a predefined number of fixed-point iterations. We propose two loss functions designed to either minimize the fixed-point residual or the distance to a ground truth solution. In this way, the neural network predicts warm starts with the end-to-end goal of minimizing the downstream loss. An important feature of our architecture is its flexibility, in that it can predict a warm start for fixed-point algorithms run for any number of steps, without being limited to the number of steps it has been trained on. We provide PAC-Bayes generalization bounds on unseen data for common classes of fixed-point operators: contractive, linearly convergent, and averaged. Applying this framework to well-known applications in control, statistics, and signal processing, we observe a significant reduction in the number of iterations and solution time required to solve these problems, through learned warm starts.

Amir Ali Ahmadi, Georgina Hall

We devise a scheme for solving an iterative sequence of linear programs (LPs) or second order cone programs (SOCPs) to approximate the optimal value of any semidefinite program (SDP) or sum of squares (SOS) program. The first LP and SOCP-based bounds in the sequence come from the recent work of Ahmadi and Majumdar on diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (SDSOS) polynomials. We then iteratively improve on these bounds by pursuing better bases in which more relevant SOS polynomials admit a DSOS or SDSOS representation. Different interpretations of the procedure from primal and dual perspectives are given. While the approach is applicable to SDP relaxations of general polynomial programs, we apply it to two problems of discrete optimization: the maximum independent set problem and the partition problem. We further show that some completely trivial instances of the partition problem lead to strictly positive polynomials on the boundary of the sum of squares cone and hence make the SOS relaxation fail.

Amir Ali Ahmadi, Sanjeeb Dash, Georgina Hall

We develop algorithms for inner approximating the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear and integer programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.

Amir Ali Ahmadi, Georgina Hall

In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellensätze from the late 20th century (e.g., due to Stengle, Putinar, or Schmüdgen) that certify positivity of a polynomial on an arbitrary closed basic semialgebraic set. In this paper, we show that such hierarchies could in fact be designed from much more limited Positivstellensätze dating back to the early 20th century that only certify positivity of a polynomial globally. More precisely, we show that any inner approximation to the cone of positive homogeneous polynomials that is arbitrarily tight can be turned into a converging hierarchy of lower bounds for general polynomial minimization problems with compact feasible sets. This in particular leads to a semidefinite programming-based hierarchy that relies solely on Artin's solution to Hilbert's 17th problem. We also use a classical result of Polyá on global positivity of even forms to construct an "optimization-free" converging hierarchy for general polynomial minimization problems. This hierarchy only requires polynomial multiplication and checking nonnegativity of coefficients of certain fixed polynomials. As a corollary, we obtain new linear programming and second-order cone programming-based hierarchies for polynomial minimization problems that rely on the recently introduced concepts of dsos and sdsos polynomials. We remark that the scope of this paper is theoretical at this stage as our hierarchies-though they involve at most two sum of squares constraints or only basic arithmetic at each level-require the use of bisection and increase the number of variables (resp. degree) of the problem by the number of inequality constraints plus three (resp. by a factor of two).

Amir Ali Ahmadi, Georgina Hall

We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial in $n$ variables of constant degree (as low as three) and outputs a point whose gradient has Euclidean norm at most $2^n$ whenever the polynomial has a critical point, then P=NP. The algorithm is permitted to return an arbitrary point when no critical point exists. We also prove hardness results for approximate computation of critical points under additional structural assumptions, including settings in which existence and uniqueness of a critical point are guaranteed, the function is lower bounded, and approximation is measured in terms of distance to a critical point. Overall, our results stand in contrast to the commonly-held belief that, in nonconvex optimization, approximate computation of critical points is a tractable task.

Amir Ali Ahmadi, Cemil Dibek, Georgina Hall

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but negative already for bivariate quartic SPQ polynomials. We use our decomposition result for convex SPQ polynomials to show that convex SPQ polynomial optimization problems can be solved by "small" semidefinite programs. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end by presenting applications of SPQ polynomials to upper bounding sparsity of solutions to linear programs, polynomial regression problems in statistics, and a generalization of Newton's method which incorporates separable higher-order derivative information.

Georgina Hall, Laurent Massoulié

In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy version of the well-known graph isomorphism problem and appears in many applications, including social network deanonymization and cellular biology. Our focus here is on partial recovery, i.e., we look for a one-to-one mapping which is correct on a fraction of the nodes of the graph rather than on all of them, and we assume that the two input graphs to the problem are correlated Erdős-Rényi graphs of parameters $(n,q,s)$. Our main contribution is then to give necessary and sufficient conditions on $(n,q,s)$ under which partial recovery is possible with high probability as the number of nodes $n$ goes to infinity. In particular, we show that it is possible to achieve partial recovery in the $nqs=Θ(1)$ regime under certain additional assumptions.

Anirudha Majumdar, Georgina Hall, Amir Ali Ahmadi

Historically, scalability has been a major challenge to the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this paper, we survey recent approaches for addressing this challenge including (i) approaches for exploiting structure (e.g., sparsity and symmetry) in a problem, (ii) approaches that produce low-rank approximate solutions to semidefinite programs, (iii) more scalable algorithms that rely on augmented Lagrangian techniques and the alternating direction method of multipliers, and (iv) approaches that trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs). For each class of approaches we provide a high-level exposition, an entry-point to the corresponding literature, and examples drawn from machine learning, control, or robotics. We also present a list of software packages that implement many of the techniques discussed in the paper. Our hope is that this paper will serve as a gateway to the rich and exciting literature on scalable semidefinite programming for both theorists and practitioners.

Amir Ali Ahmadi, Georgina Hall

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity only over a compact region, most commonly a box (i.e., hyper-rectangle). In this paper, we show that this problem is also strongly NP-hard, in fact for polynomials of degree as low as three. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control.

Amir Ali Ahmadi, Georgina Hall, Ameesh Makadia et al.

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distances between two such sets, separation of two convex basic semalgebraic sets that overlap, and tight containment of the union of several basic semialgebraic sets with a single convex one. We use algebraic techniques from sum of squares optimization that reduce all these tasks to semidefinite programs of small size and present numerical experiments in realistic scenarios.

Amir Ali Ahmadi, Georgina Hall

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex-concave procedure (CCP). We prove, however, that optimizing over the entire set of dcds is NP-hard.