Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality

Introduction

A successful approach to statistical estimation and statistical learning suggests to estimate the object of interest by solving an optimization problem, for instance motivated by maximum likelihood, or empirical risk minimization. In modern applications, the unknown object is often combinatorial, e.g. a sparse vector in high-dimensional regression or a partition in clustering. In these cases, the resulting optimization problem is computationally intractable and convex relaxations have been a method of choice for obtaining tractable and yet statistically efficient estimators.

In this paper we consider the following specific semidefinite program

as well as some of its generalizations. This SDP famously arises as a convex relaxation of the MaxCut problemIn the MaxCut problem, we are given a graph $G=(V,E)$ and want to partition the vertices in two sets as to maximize the number of edges across the partition., whereby the matrix $A$ is the opposite of the adjacency matrix of the graph to be cut. In a seminal paper, Goemans and Williamson [GW95] proved that this SDP provides a $0.878$ approximation of the combinatorial problem. Under the unique games conjecture, this approximation factor is optimal for polynomial time algorithms [KKMO07].

Provided that $k\geq\sqrt{2n}$, the solution of (MC-SDP) corresponds to the global maximum of ($k$-Ncvx-MC-SDP) [Bar95, Pat98, BM03]. Recently, [BVB16] proved that, as long as $k\geq\sqrt{2n}$, for almost all matrices $A$, the problem ($k$-Ncvx-MC-SDP) has a unique local maximum which is also the global maximum. This paper proposed to use the Riemannian trust-region method to solve the non-convex SDP problem, and provided computational complexity guarantees on the resulting algorithm.

As mentioned above, we extend our analysis beyond the MaxCut type problem ($k$-Ncvx-MC-SDP) to treat an optimization problem motivated by ${\rm SO}(d)$ synchronization. ${\rm SO}(d)$ synchronization (with $d=3$) has applications to computer vision [ANKKS+12] and cryo-electron microscopy (cryo-EM) [SS11]. A natural SDP relaxation of the maximum likelihood estimator is given by the problem

By imposing the rank constraint ${\rm rank}(X)\leq k$, we obtain a non-convex analogue of (OC-SDP), namely:

According to the result in [BM03], as long as $k\geq(d+1)\sqrt{m}$, the global maximum of the problem ($k$-Ncvx-OC-SDP) coincides with the maximum of the problem (OC-SDP). As proved in [BVB16], with the same value of $k$ for almost all matrices $A$, the non-convex problem has no local maximum other than the global maximum. [Bou15] proposed to choose the rank $k$ adaptively: as $k$ is not large enough, increase $k$ to find a better solution. However, none of these works considers $k={\mathcal{O}}(1)$, which is the focus of the present paper (under the assumption that $d$ is of order one as well).

A main result of our paper is a Grothendieck-type inequality that generalizes and strengthens the preliminary technical result of [Mon16]. Namely, we prove that for any $\varepsilon$-approximate concave point $\sigma$ of the rank-$k$ non-convex SDP ($k$-Ncvx-MC-SDP), we have

where ${\textup{\rm SDP}}(A)$ denotes the maximum value of the problem (MC-SDP) and ${f}(\sigma)$ is the objective function in ($k$-Ncvx-MC-SDP). An $\varepsilon$-approximate concave point is a point at which the eigenvalues of the Hessian of ${f}(\,\cdot\,)$ are upper bounded by $\varepsilon$ (see below for formal definitions).

Surprisingly, this result connects a second order local property, namely the highest local curvature of the cost function, to its global position. In particular, all the local maxima (corresponding to $\varepsilon=0$) are within a $1/k$-gap of the SDP value. Namely, for any local maximizer $\sigma^{*}$, we have

All the points outside this gap, with an $n\varepsilon/2$-margin have a direction of positive curvature of at least size $\varepsilon$.

Figure 1 illustrates the landscape of the rank-$k$ non-convex MaxCut SDP problem ($k$-Ncvx-MC-SDP). We show that this structure implies global convergence rates for approximately solving ($k$-Ncvx-MC-SDP). We study the Riemannian trust-region method in Theorem 2. In particular, we show that this algorithm with any initialization returns a $0.878\times(1-O(1/k))$ approximation of the MaxCut of a random $d$-regular graph in $\mathcal{O}(nk^{2})$ iterations, cf. Theorem 3.

For ${\rm SO}(d)$ synchronization, we consider the problem ($k$-Ncvx-OC-SDP) and generalize our main Grothendieck-type inequality to this case, cf. Theorem 7. Namely, for any $\varepsilon$-approximate concave point $\sigma$ of the rank-$k$ non-convex Orthogonal-Cut SDP ($k$-Ncvx-OC-SDP), we have

where $k_{d}=2k/(d+1)$, ${\textup{\rm SDP}}_{o}(A)$ denotes the maximum value of the problem (OC-SDP) and ${f}(\sigma)$ is the objective function in ($k$-Ncvx-OC-SDP). We expect that the statistical analysis of local maxima, as well as the analysis of optimization algorithms, should extend to this case as well, but we leave this to future work.

2 Notations

Optimization is performed over the convex set of positive-semidefinite matrices with diagonal entries equal to one, also known as the elliptope. We write ${\textup{Rg}}(A)={\textup{\rm SDP}}(A)+{\textup{\rm SDP}}(-A)$ for the length of the range of the SDP with data $A$ (noticing that for every matrix $X$ in the elliptope, we have ${\textup{\rm SDP}}(A)\geq\langle A,X\rangle\geq-{\textup{\rm SDP}}(-A)$).

For the rank-$k$ non-convex SDP problem ($k$-Ncvx-MC-SDP), we define the manifold ${\mathcal{M}}_{k}$ as

The Hessian is uniquely defined by the following holding for all $u,v$ in the tangent space $T_{\sigma}\mathcal{M}_{k}$:

Main results

First we define the notion of approximate concave point of a function $f$ on a manifold ${\mathcal{M}}$.

Let $f$ be a twice differentiable function on a Riemannian manifold ${\mathcal{M}}$. We say $\sigma\in{\mathcal{M}}$ is an $\varepsilon$-approximate concave point of $f$ on ${\mathcal{M}}$, if $\sigma$ satisfies

where ${\textup{Hess}}f(\sigma)$ denotes the Riemannian (intrinsic) Hessian of $f$ at point $\sigma$, $T_{\sigma}{\mathcal{M}}$ is the tangent space, and $\langle\,\cdot\,,\cdot\,\rangle$ is the scalar product on $T_{\sigma}{\mathcal{M}}$.

Note that an approximate concave point may not be a stationary point, or may not even be an approximate stationary point. Both local maximizers and saddles with largest eigenvalue of the Hessian close to zero are approximate concave points.

The classical Grothendieck inequality relates the global maximum of a non-convex optimization problem to the maximum of its SDP relaxation [Gro96, KN12]. Our main tool is instead an inequality that applies to all approximate concave ponts in the non-convex problem.

For any $\varepsilon$-approximate concave point $\sigma\in{\mathcal{M}}_{k}$ of the rank-$k$ non-convex problem ($k$-Ncvx-MC-SDP), we have

We can use the structural information in Theorem 1, to develop an algorithm that approximately solves the problem ($k$-Ncvx-MC-SDP), and hence the MaxCut SDP (MC-SDP). The algorithm we propose is a variant of the Riemannian trust-region algorithm.

The Riemannian trust-region algorithm (RTR) [ABG07] is a generalization of the trust-region algorithm to manifolds. To maximize the objective function ${f}$ on the manifold ${\mathcal{M}}$, RTR proceeds as follows: at each step, we find a direction $\xi\in T_{\sigma}{\mathcal{M}}$ that maximizes the quadratic approximation of ${f}$ over a ball of small radius $\eta_{\sigma}$

Solving the trust-region problem (RTR-update) exactly is computationally expensive. In order to obtain a faster algorithm, we adopt two variants in the RTR algorithm. First, if the gradient of ${f}$ at the current estimate $\sigma^{t}$ is sufficiently large, we only use gradient information to determine the new direction: we call this a gradient-step; if the gradient is small (i.e. we are at an approximately stationary point), we try to maximize uniquely the Hessian contribution: we call this an eigen-step. Second, in an eigen-step, we only approximately maximize the Hessian contribution. Let us emphasize that these two variants are commonly used and we do not claim they are novel.

Take $\mu_{G}=\infty$, which means that only eigen-steps are used. In this implementation, we take the step size $\eta_{H}^{t}=\langle u^{t},{\textup{Hess}}{f}(\sigma^{t})[u^{t}]\rangle/(100\|A\|_{1})$.

Take $\mu_{G}=\|A\|_{2}$. When $\|{\textup{grad}}{f}(\sigma^{t})\|_{F}>\mu_{G}$, we choose the step size $\eta_{G}^{t}=\mu_{G}/(20\|A\|_{1})$. When $\|{\textup{grad}}{f}(\sigma^{t})\|_{F}\leq\mu_{G}$, we choose the step size $\eta_{H}^{t}=\min\{\sqrt{\lambda_{H}^{t}/(216\|A\|_{1})};\lambda_{H}^{t}/(12\|A\|_{2})\}$, where $\lambda_{H}^{t}=\langle u^{t},{\textup{Hess}}{f}(\sigma^{t})[u^{t}]\rangle$.

In each eigen-step, we need to compute a direction $u\in T_{\sigma}{\mathcal{M}}_{k}$ such that $\|u\|_{F}=1$ and $\langle u,{\textup{Hess}}{f}(\sigma)[u]\rangle\geq\lambda_{\max}({\textup{Hess}}{f}(\sigma))/2$. This can be done using the following power method. (Note that the condition $\langle u^{t},{\textup{grad}}f(\sigma^{t})\rangle\geq 0$ can always be ensured eventually by replacing $u^{t}$ by $-u^{t}$.)

The shifting parameter $\mu_{H}$ can be chosen as $4\|A\|_{1}$ which is an upper bound of $\|{\textup{Hess}}{f}(\sigma)\|_{{\rm op}}$. We take the parameter $N_{H}=C\cdot\|A\|_{1}\log n/\lambda_{\max}({\textup{Hess}}{f}(\sigma))$ with a large absolute constant $C$. In practice, when choosing the parameter $N_{H}$, we do not know $\lambda_{\max}({\textup{Hess}}{f}(\sigma))$ for each $\sigma$, but we can replace it by a lower bound, or estimate it using some heuristics. It is a classical result that –with high probability– the power method with this number of iterations finds a solution $u^{t}$ with the required curvature [KW92].

There exists a universal constant $c$ such that, for any matrix $A$ and $\varepsilon>0$, the Fast Riemannian Trust-Region method with step size as described above for each iteration and initialized with any $\sigma_{0}\in\mathcal{M}_{k}$ returns a point $\sigma^{*}\in{\mathcal{M}}_{k}$ with

within the following number of steps with each implementation

Taking $\mu_{G}=\infty$ (i.e. only eigen-steps are used), then it is sufficient to run $T_{H}\leq c\cdot n\|A\|_{1}^{2}/\varepsilon^{2}$ steps.

Taking $\mu_{G}=\|A\|_{2}$, then it is sufficient to run $T=T_{H}+T_{G}$ steps in which there are $T_{H}\leq c\cdot n\max\left(\|A\|_{2}^{2}/\varepsilon^{2},\|A\|_{1}/\varepsilon\right)$ eigen-steps and $T_{G}\leq c\cdot{\textup{Rg}}(A)\|A\|_{1}/\|A\|_{2}^{2}$ gradient-steps.

The gap ${\textup{Rg}}(A)/(k-1)=({\textup{\rm SDP}}(A)+{\textup{\rm SDP}}(-A))/(k-1)$ in Eq. (9), is due to the fact that Theorem 1 does not rule out the presence of local maxima within an interval ${\textup{Rg}}(A)/(k-1)$ from the global maximum. It is therefore natural to set $\varepsilon=2{\textup{Rg}}(A)/(n(k-1))$, to obtain the following corollary.

There exists a universal constant $c$ such that for any matrix $A$, the Fast Riemannian Trust-Region method with step size as described above for each iteration and initialized with any $\sigma_{0}\in\mathcal{M}_{k}$ returns a point $\sigma^{*}\in{\mathcal{M}}_{k}$ with

within the following number of steps with each implementation

Taking $\mu_{G}=\infty$ , then it is sufficient to run $T_{H}\leq c\cdot nk^{2}\left(n\|A\|_{1}/{\textup{Rg}}(A)\right)^{2}$ eigen-steps.

Taking $\mu_{G}=\|A\|_{2}$, then it is sufficient to run $T=T_{H}+T_{G}$ steps in which there are $T_{H}\leq c\cdot n\max\left(n^{2}k^{2}\|A\|_{2}^{2}/{\textup{Rg}}(A)^{2},nk\|A\|_{1}/{\textup{Rg}}(A)\right)$ eigen-steps and $T_{G}\leq c\cdot{\textup{Rg}}(A)\|A\|_{1}/\|A\|_{2}^{2}$ gradient-steps.

In order to develop some intuition on these complexity bounds, let us consider two specific examples.

As a second example, consider the MaxCut problem for a $d$-regular graph $G$, with adjacency matrix $A_{G}$. This can be addressed by considering the SDP (MC-SDP) with $A=-A_{G}$, and the corresponding non-convex version ($k$-Ncvx-MC-SDP). As shown in the next section, finding a $2{\textup{Rg}}(A)/(n(k-1))$-approximate concave point of ($k$-Ncvx-MC-SDP) yields an $(1-O(1/k))\times 0.878$-approximation of the MaxCut of $G$. For this choice of $A$, we have $\|A\|_{1}=d$, $\|A\|_{2}=d$, and ${\textup{Rg}}(A)=\Theta(nd)$. Therefore, in implementation $(a)$ where all the steps are eigen-step, the number of iterations given by Corollary 1 scales as $T_{H}=\mathcal{O}(nk^{2})$. In implementation $(b)$, we choose $\mu_{G}=\Theta(d)$, and the number of gradient-steps and eigen-steps scale respectively as $T_{G}=\mathcal{O}(n)$ and $T_{H}=\mathcal{O}(nk^{2})$. In terms of floating point operations, the computational costs of one gradient-step and one eigen-step power iteration are the same (which are $\mathcal{O}(ndk)$) as in the example of minimum bisection SDP. The number of iterations in the power method scales as $\mathcal{O}(k\log n)$. Therefore, the two approaches are equivalent. The total number of floating point operations to find a $(1-O(1/k))\times 0.878$ approximate solution of the MaxCut of a $d$-regular graph is upper bounded by $\mathcal{O}(n^{2}dk^{4}\log n)$.

Let us emphasize that the complexity bound in Theorem 2 is not superior to the ones available for some alternative approaches. There is a vast literatures that studies fast SDP solvers [AHK05, AK07, Ste10, GH11]. In particular, [AK07, Ste10] give nearly linear-time algorithms to approximate (MC-SDP). These algorithms are different from the one studied here, and rely on the multiplicative weight update method [AHK12]. Using sketching techniques, their complexity can be further reduced [GH11]. However, in practice, the Burer-Monteiro approach studied here is extremely simple and scales well to large instances [BM03, JMRT16]. Empirically, it appears to have better complexity than what is guaranteed by our theorem. It would be interesting to compare the multiplicative weight update method and the non-convex approach both theoretically and experimentally.

2 Application to MaxCut

We consider the following semidefinite programming relaxation

Denote by $X^{*}$ the solution of this SDP. Goemans and Williamson [GW95] proposed a celebrated rounding scheme using this $X^{*}$, which is guaranteed to find an $\alpha_{*}$-approximate solution to the MaxCut problem (11), where $\alpha_{*}\equiv\min_{\theta\in[0,\pi]}2\theta/(\pi(1-\cos\theta))$, $\alpha_{*}>0.87856$.

The corresponding rank-$k$ non-convex formulation is given by

Applying Theorem 1, we obtain the following result.

For any $k\geq 3$, if $\sigma^{*}$ is a local maximizer of the rank-$k$ non-convex SDP problem (13), then using $\sigma^{*}$ we can find an $\alpha_{*}\times(1-1/(k-1))\geq 0.878\times(1-1/(k-1))$-approximate solution of the MaxCut problem (11). If $\sigma^{*}$ is a $2{\textup{Rg}}(A_{G})/(n(k-1))$-approximate concave point, then using $\sigma^{*}$ we can find an $\alpha_{*}\times(1-2/(k-1))\geq 0.878\times(1-2/(k-1))$-approximate solution of the MaxCut problem.

where $W_{n}\sim{\rm GOE}(n)$, and $\lambda$ is a signal-to-noise ratio. The random matrix model (14) is also known as the ‘spiked model’ [Joh01] or ‘deformed Wigner matrix’ and has attracted significant attention across statistics and probability theory [BAP+05].

The Maximum Likelihood Estimator for recovering the labels $u\in\{\pm 1\}^{n}$ is given by

A natural SDP relaxation of this optimization problem is given –once more– by (MC-SDP).

It was proved in [MS16] that the SDP relaxation (MC-SDP) –with a suitable rounding scheme– achieves the information-theoretic threshold $\lambda_{c}=1$ for this problem. In this paper, we prove a similar result for the non-convex problem ($k$-Ncvx-MC-SDP). Namely, we show that for any signal-to-noise ratio $\lambda>1$ there exists a sufficiently large $k$ such that every local maximizer has a non trivial correlation to the ground truth. Below we denote by ${\rm Cr}_{n,k}(A)$ the set of local maximizers of problem ($k$-Ncvx-MC-SDP).

For any $\lambda>1$, there exists a function $k_{*}(\lambda)>0$, such that for any $k>k_{*}(\lambda)$, with high probability, any local maximizer $\sigma$ of the rank-$k$ non-convex SDP ($k$-Ncvx-MC-SDP) problem has non-vanishing correlation with the ground truth parameter. Explicitly, there exists $\varepsilon=\varepsilon(\lambda)>0$ such that

The proof of this theorem is deferred to Section 5.2.

Note that this guarantee is weaker than the one of [MS16], which also presents an explicit rounding scheme to obtain an estimator $\hat{u}\in\{+1,-1\}^{n}$. However, we expect that the techniques of [MS16] should be generalizable to the present setting. A simple rounding scheme takes the sign of principal left singular vector of $\sigma$. We will use this estimator in our numerical experiments in Section 4.

This theorem can be compared with the one of [BBV16] which uses $k=2$ but requires $\lambda>8$. As a side result which improves over [BBV16] for $k=2$, we obtain the following lower bound on the correlation for any $k\geq 2$.

For any $k\geq 2$, the following holds almost surely

The proof is deferred to Section 5.3. Our lower bound converges to $1$ at large $\lambda$, which is the qualitatively correct behavior.

4 Stochastic block model

The planted partition problem (two-groups symmetric stochastic block model), is another well-studied statistical estimation problem that can be reduced to (MC-SDP) [MS16]. We write $G\sim\mathcal{G}(n,p,q)$ if $G$ is a graph over $n$ vertices generated as follows (for simplicity of notation, we assume $n$ even). Let $u\in\{\pm 1\}^{n}$ be a vector of labels that is uniformly random with $u^{\mathsf{T}}\bm{1}=0$. Conditional on this partition, edges are drawn independently with

We consider the case when $p=a/n$ and $q=b/n$ with $a,b={\mathcal{O}}(1)$, and $a>b$, and denote by $d=(a+b)/2$ the average degree. A phase transition occurs as the following signal-to-noise parameter increases

For $\lambda>1$ there exists an efficient estimator that correlates with the true labels with high probability [Mas14, MNS13], whereas no estimator exists below this threshold, regardless of its computational complexity [MNS15].

The Maximum Likelihood Estimator of the vertex labels is given by

where $A_{G}$ is the adjacency matrix of the graph $G$. This optimization problem can again be attacked using the relaxation (MC-SDP), where $A=\overline{A}_{G}\equiv(A_{G}-d/n\cdot\bm{1}\bm{1}^{\mathsf{T}})/\sqrt{d}$ is the scaled and centered adjacency matrix.

where $p_{ij}=a/n$ for $u_{i}=u_{j}$ and $p_{ij}=b/n$ for $u_{i}\neq u_{j}$. In analogy with Theorem 4, we have the following results on the rank-constrained approach to the two-groups stochastic block model.

Consider the rank-$k$ non-convex SDP ($k$-Ncvx-MC-SDP) with $A=\overline{A}_{G}$ the centered, scaled adjacency matrix of graph $G\sim\mathcal{G}(n,a/n,b/n)$. For any $\lambda=\lambda(a,b)>1$, there exists an average degree $d_{*}(\lambda)$ and a rank $k_{*}(\lambda)$, such that for any $d\geq d_{*}(\lambda)$ and $k\geq k_{*}(\lambda)$, with high probability, any local maximizer $\sigma$ has non-vanishing correlation with the true labels. Explicitly, there exists an $\varepsilon=\varepsilon(\lambda)>0$ such that

The proof of this theorem can be found in Section 5.4. As mentioned above, efficient algorithms that estimate the hidden partition better than random guessing for $\lambda>1$ and any $d>1$ have been developed, among others, in [Mas14, MNS13]. However, we expect the optimization approach ($k$-Ncvx-MC-SDP) to share some of the robustness properties of semidefinite programming [MPW16], while scaling well to large instances.

5 SO​(d)SO𝑑{\rm SO}(d) synchronization

In ${\rm SO}(d)$ synchronization we would like to estimate $m$ matrices $R_{1},\dots,R_{m}$ in the special orthogonal group

The Maximum Likelihood Estimator for recovering the group elements $R_{i}\in{\rm SO}(d)$ solves the problem of the form

In analogy with the MaxCut SDP, we obtain the following Grothendieck-type inequality.

For an $\varepsilon$-approximate concave point $\sigma\in{\mathcal{M}}_{o,d,k}$ of the rank-$k$ non-convex Orthogonal-Cut SDP problem ($k$-Ncvx-OC-SDP), we have

The proof of this theorem is a generalization of the proof of Theorem 1, and is deferred to Section 5.5.

Proof of Theorem 1

In this section we present the proof of Theorem 1, while deferring other proofs to Section 5. Notice that the present proof is simpler and provides a tighter bound with respect to the one of [Mon16]. Before passing to the actual proof, we make a few remarks about the geometry of optimization on ${\mathcal{M}}_{k}$.

We will write $\Lambda=\Lambda(\sigma)=\text{ddiag}\left(A\sigma\sigma^{\mathsf{T}}\right)$ and often drop the dependence on $\sigma$ for simplicity. At $\sigma\in{\mathcal{M}}_{k}$, let $\nabla^{2}{f}(\sigma)$ and ${\textup{Hess}}{f}(\sigma)$ be respectively the Euclidean and the Riemannian Hessian of ${f}$. The Riemannian Hessian is a symmetric operator on the tangent space and is given by projecting the directional derivative of the gradient vector field (we use $D$ to denote the directional derivative):

In particular, we will use the following identity

2 Proof of Theorem 1

Let $\sigma$ be an $\varepsilon$-approximate concave point of ${f}(\sigma)$ on ${\mathcal{M}}_{k}$. Using the definition and Equation (20), we have (for $\Lambda=\text{ddiag}(A\sigma\sigma^{\mathsf{T}})$)

where the expectation is taken over the random matrix $G$.

The left hand side of the last equation gives

Note that ${\textup{Tr}}(\Lambda)={f}(\sigma)$. Crucially, if we let $\bar{X}_{ij}=\langle v_{i},v_{j}\rangle(\langle\sigma_{i},\sigma_{j}\rangle)^{2}$, we have $\bar{X}_{ii}=1$ and $\bar{X}\succeq 0$. Thus we have ${\textup{\rm SDP}}(-A)\geq\langle-A,\bar{X}\rangle$. Therefore, we have

Rearranging the terms gives the conclusion.

Numerical illustration

In this section we carry out some numerical experiments to illustrate our results. We also find interesting phenomena which are not captured by our analysis.

Although Theorem 2 provides a complexity bound for the Riemannian trust-region method (RTR), we observe that (projected) gradient ascent also converges very fast. That is, gradient ascent rapidly increases the objective function, is not trapped at a saddle point, and converges to a local maximizer eventually. In Figure 2, we take $A\sim{\rm GOE}(1000)$, and use projected gradient ascent to solve the optimization problem ($k$-Ncvx-MC-SDP) with a random initialization and fixed step size. Figure 2a shows that the objective function increases rapidly and converges within a small interval from the local maximum (which is upper bounded by the value ${\textup{\rm SDP}}(A)$). Also the gap between the value obtained by this procedure and the value ${\textup{\rm SDP}}(A)$ decreases rapidly with $k$. Figure 2b shows that the Riemannian gradient decreases very rapidly, but presents some non-monotonicity. We believe these bumps occur when the iterates are close to saddle points.

In Figure 3, we examine some geometric properties of the rank-$k$ non-convex SDP. As above, we explore the landscape of this problem by projected gradient ascent. In Figure 3a, we plot the curvature $\lambda_{\max}({\textup{Hess}}{f}(\sigma))$ versus the gap from the SDP value $({\textup{\rm SDP}}(A)-{f}(\sigma))/n\times 2$ along the iterations. When ${f}(\sigma)$ is far from ${\textup{\rm SDP}}(A)$, there is a linear relationship between these two quantities, which is consistent with Theorem 1. In Figure 3b, we plot the gap between ${\textup{\rm SDP}}(A)$ and ${f}(\sigma^{*})$ for a local maximizer $\sigma^{*}\in{\mathcal{M}}_{k}$ that is produced by projected gradient ascent, for different values of $k$. These data are averaged over $10$ realizations of the random matrix $A$. This gap converges to zero as $k$ gets large, and is upper bounded by the curve ${\textup{Rg}}(A)/(15(k-1))$. This coincides with Theorem 1, which predicts that this gap must be smaller than ${\textup{Rg}}(A)/(k-1)$. Note however that –in this case– Theorem 1 is overly pessimistic, and the gap appears to decrease very rapidly with $k$.

Now we turn to study the MaxCut problem. Note that Theorem 3 gives a guarantee for the approximation ratio for the cut induced by any local maximizer of the rank-$k$ non-convex SDP ($k$-Ncvx-MC-SDP). In Figure 4, we take the graph to be an Erdős-Rényi graph with $n=1000$ and average degree $d=50$. We plot the cut value found by rounding the maximizer of the rank-$k$ non-convex SDP, for $k$ from $2$ to $10$, and also for $k=n$ which corresponds to the (MC-SDP). Surprisingly, the cut value found by solving rank-$k$ non-convex problem is typically bigger than the cut value found by solving the original SDP. This provides a further reason to adopt the non-convex approach ($k$-Ncvx-MC-SDP). It appears to provide a significantly tight relaxation for random instances.

Other proofs

Note that problem (13) is equivalent to problem ($k$-Ncvx-MC-SDP) with matrix $A=-A_{G}$. Applying Theorem 1, and noting that the elements of $A_{G}$ are non-negative, we for any local maximizer $\sigma^{*}$ of the problem (13), and any $X^{*}$ optimal solution of the SDP (12),

Applying the randomized rounding scheme of [GW95], we sample a vector $u\sim{\sf N}(\bm{0},{\mathbf{I}}_{k})$, and define $v\in\{\pm 1\}^{n}$ by $v_{i}=\text{sign}(\langle\sigma_{i}^{*},u\rangle)$, then we obtain

Therefore, for any local maximizer $\sigma^{*}$, it gives an $\alpha_{*}\times(1-1/(k-1))$-approximate solution of the MaxCut problem.

If $\sigma^{*}$ is an $\varepsilon=2{\textup{Rg}}(A_{G})/(n(k-1))$-approximate concave point, using Theorem 1 and the same argument, we can prove that it gives an $\alpha_{*}\times(1-2/(k-1))$-approximate solution of the MaxCut problem.

2 Proof of Theorem 4

Let $A(\lambda)=\lambda/n\cdot uu^{\mathsf{T}}+W_{n}$. For any local maximum $\sigma\in{\rm Cr}_{n,k}$ of the rank-$k$ non-convex MaxCut SDP problem, according to Theorem 1, we have

Using the convergence of the SDP value as proved in [MS16, Theorem 5], for any $\lambda>1$, there exists $\Delta(\lambda)>0$ such that, for any $\delta>0$, the following holds with high probability

Since $\Delta(\lambda)>0$ for $\lambda>1$, there exists a $k_{*}(\lambda)$ such that the above expression is greater than $\varepsilon$ for sufficiently small $\varepsilon$ and $\delta$, which concludes the proof.

3 Proof of Theorem 5

We decompose the proof into two parts. In part $(a)$, we prove that almost surely

using only the second order optimality condition. In part $(b)$, we incorporate the first order optimality condition and prove that as $\lambda\geq 12k$, we have almost surely

Plugging in the expression of $A$, we obtain

Letting $\bar{X}_{ij}=u_{i}u_{j}\langle\sigma_{i},\sigma_{j}\rangle^{2}$, we have

Recall that $\text{rank}(\sigma\sigma^{\mathsf{T}})=k$, and ${\textup{Tr}}(\sigma\sigma^{\mathsf{T}})=n$. Thus, we get the lower bound

Also note that $\bar{X}$ is a feasible point of (MC-SDP). Therefore,

where we used the fact that for a GOE matrix $W_{n}$, we have $\lim_{n\rightarrow\infty}\|W_{n}\|_{{\rm op}}=2$ almost surely [AGZ10].

Part (b)𝑏(b)

In part $(a)$ we only used the second order optimality condition. In this part of the proof, we will incorporate the first order optimality condition. Note that as $\lambda<12k$, the bound in part $(a)$ is better. So in this part, we only consider the case when $\lambda\geq 12k$.

We decompose the proof into the following steps.

Step 1 Upper bound on $\langle\bm{1},v_{j}\rangle^{2}/n^{2}$, for $j=2,\ldots,k$, using the first order optimality condition.

The first order optimality condition gives $A\sigma=\text{ddiag}(A\sigma\sigma^{\mathsf{T}})\,\sigma$, which implies that

for any $i\neq j$, where we denoted $u\circ v$ the entry-wise product of $u$ and $v$. Replacing $A$ by its expression gives

We take the norm of this expression and, recalling that $\langle v_{i},v_{j}\rangle=0$, we obtain

Notice that $\|v_{j}\|_{\infty}\leq 1,\forall j\in[k]$, hence

Without loss of generality, let us assume that $\|v_{1}\|_{2}\geq\|v_{j}\|_{2}$ for $j\geq 2$ which implies

for $j=2,\ldots,k$, where we use the fact that for a GOE matrix $W_{n}$, we have $\lim_{n\rightarrow\infty}\|W_{n}\|_{{\rm op}}=2$ almost surely.

Step 2 Lower bound on $\langle\bm{1},v_{1}\rangle^{2}/n^{2}$.

We combine equation (26) and (29) to get almost surely

Since we assumed that $\lambda\geq 12k$ and $k\geq 2$, we obtain, almost surely,

The second inequality above is loose but it is sufficient for our purposes.

Step 3 Upper bound on $\|v_{a}\|_{2}^{2}$ for $a\in\{2,\ldots,k\}$.

In Equation (28), let us take $i=1$ and $j=a\in\{2,\ldots,k\}$, we have

Combining equation (31) and (30) results in the following upper bound for $\lambda\geq 12k$,

holding almost surely for any $a\in\{2,\ldots,k\}$.

Using the second order stationarity condition with this choice of $\xi_{i}$, we have

Consider the first term $B_{1}$. It is easy to see that the second order stationary condition implies $(\Lambda-A)_{ii}\geq 0$. Thus, we have

Next consider the second term $B_{2}$. We have

where the last inequality is because $|v_{a,i}|\leq 1$ so that $\|v_{a}\circ v_{a}\|_{2}\leq\|v_{a}\|_{2}$.

Here is the justification of the above fact. For $X$ in the elliptope, we have $X_{ii}=1$ and $X\succeq 0$. For any $Z$ satisfying $Z\succeq 0$ and ${\textup{Tr}}(Z)\leq 1$, $X\circ Z$ also satisfies $X\circ Z\succeq 0$ and ${\textup{Tr}}(X\circ Z)\leq 1$. Therefore, using the variational representation of the operator norm, we have

Noting that $f(\sigma)=\lambda/n\cdot\|\sigma^{\mathsf{T}}\bm{1}\|_{2}^{2}+\langle\sigma,W_{n}\sigma\rangle$ and $\langle\bm{1},A\bm{1}\rangle=n\lambda+\langle\bm{1},W_{n}\bm{1}\rangle$, we rewrite Equation (34) as following

Plug in the lower bound of $B_{1}$, $B_{2}$, $B_{3}$, we have almost surely

Here we used Equation (32), $\lambda\geq 12k\geq 24$, and the fact that for a GOE matrix $W_{n}$, we have $\lim_{n\rightarrow\infty}\|W_{n}\|_{{\rm op}}=2$ almost surely.

4 Proof of Theorem 6

The proof is similar to the proof of Theorem 4, where the GOE matrix $W_{n}$ is replaced by the noise matrix $E$.

Applying Theorem 1 with the matrix $\overline{A}_{G}(\lambda)$, similar to Equation (24), we have

According to [MS16, Theorem 8], the gap between the SDPs with the two different noise matrices is bounded with high probability by a function of the average degree $d$

According to [MS16, Theorem 5], for any $\delta>0$ and $\lambda>1$, there exists a function $\Delta(\lambda)>0$ such that with high probability, we have

Combining the above results, we have for any $\delta>0$, with high probability

For a sufficiently small $\varepsilon>0$, taking $\delta$ sufficiently small, and taking successively $d$ and $k$ sufficiently large, the above expression will be greater than $\varepsilon$, which concludes the proof. ∎

5 Proof of Theorem 7

We decompose the proof into three parts. In the first part, we do the calculation for a general non-convex problem. In the second part, we focus on the non-convex problem ($k$-Ncvx-OC-SDP). In the third part, we prove a claim we made in the second part.

Let ${\mathbf{B}}=[B_{1},\ldots,B_{s}]$ and ${\mathbf{c}}=(c_{1},\ldots,c_{s})$. We denote ${\textup{\rm SDP}}(A,{\mathbf{B}},{\mathbf{c}})$ the maximum of the above SDP problem:

We assume ${\textup{\rm SDP}}(A,{\mathbf{B}},{\mathbf{c}})<\infty$.

For a fixed integer $k$, the Burer-Monteiro approach considers the following non-convex problem:

with $\lambda_{i}=\sum_{j=1}^{s}M_{ij}{\textup{Tr}}(B_{j}A\sigma\sigma^{\mathsf{T}})$. We will write $\Lambda=\Lambda(\sigma)=\sum_{i=1}^{s}\lambda_{i}B_{i}$. The Riemannian Hessian ${\textup{Hess}}{f}(\sigma)$ applied on the direction $U\in T_{\sigma}{\mathcal{M}}_{k}^{{\mathbf{B}},{\mathbf{c}}}$ gives

Therefore, according to the definition of the $\varepsilon$-approximate concave point $\sigma\in\mathcal{M}_{k}^{{\mathbf{B}},{\mathbf{c}}}$, we have

where the expectation is taken over the random mapping $G$. Expanding the left hand side gives

The second term in the last equation gives

Part 2

Now let’s consider the case of the rank-$k$ non-convex Orthogonal-Cut SDP problem ($k$-Ncvx-OC-SDP). There are $s=d(d+1)/2\times m$ constraints corresponding to the set $\{(B_{i},c_{i}):i\in[s]\}=\{(E_{ii},1):i\in[n]\}\bigcup\cup_{t=1}^{m}\{((E_{ij}+E_{ji})/\sqrt{2},0):(t-1)d+1\leq i<j\leq td\}$, where $E_{ij}=e_{i}e_{j}^{\mathsf{T}}$. We will denote ${\mathcal{M}}_{o,d,k}$ the optimization manifold:

It is straightforward to verify that for any $\sigma\in{\mathcal{M}}_{o,d,k}$, we have $(\langle B_{i}\sigma,B_{j}\sigma\rangle)_{ij=1}^{s}={\mathbf{I}}_{s}$. Thus, we have $M={\mathbf{I}}_{s}$. In the following calculation, we write $X=\sigma\sigma^{\mathsf{T}}$. Recall that $X^{*}$ is a global maximizer of problem (OC-SDP), and $X^{*}=VV^{\mathsf{T}}$.

Now, let us calculate each term in Equation (39), for the specific problem ($k$-Ncvx-OC-SDP). For the second term in Equation (39), we derived Equation (40). One can check with some calculations that for any $\sigma\in{\mathcal{M}}_{o,d,k}$, we have

For the fourth term in Equation (39), we derived Equation (42). Following the calculation in Equation (42), we have

For the third term in Equation (39), we derived Equation (41). Following the calculation in Equation (41), we have

where we define $\bar{X}=2/(d+1)\cdot(\sum_{kl=1}^{s}\langle B_{k},E_{ij}B_{l}X\rangle\langle X^{*}B_{k},B_{l}X\rangle)_{i,j=1}^{n}$. Here, we claim that $\bar{X}$ is a feasible point of the Orthogonal-Cut SDP problem (OC-SDP). We will prove this claim in part $3$.

For any feasible point $X$ of the Orthogonal-Cut SDP problem (OC-SDP), we have ${f}(\sigma)=\langle\Lambda,X\rangle$. Therefore, from Equation (39), we obtain

Letting $k_{d}=2k/(d+1)$, rearranging the above inequality, we have

which finally gives the desired inequality

Part 3

Now, let us check that $\bar{X}$ is a feasible point of the Orthogonal-Cut SDP problem (OC-SDP). The reason is given by the following Fact $(a)$ and $(b)$.

Fact $(b)$ The $(i,i)$’th block of $\bar{X}$ equals ${\mathbf{I}}_{d}$. To show this, we assume $d\geq 2$, and due to the symmetry, we just need to check $\bar{X}_{11}=1$ and $\bar{X}_{12}=0$. We denote $J_{ij}=E_{ii}\delta_{ij}+(E_{ij}+E_{ji})/\sqrt{2}\cdot(1-\delta_{ij})$, and we rewrite $\bar{X}_{ij}$ as

where $\Gamma_{a}=\{(k,s,l,t):1+(a-1)d\leq k\leq s\leq ad,1+(a-1)d\leq l\leq t\leq ad\}$. We have the following series of simplification

The third equality used the fact that $X$ and $X^{*}$ are feasible point so that their $(i,i)$’th block are ${\mathbf{I}}_{d}$. Similarly, we have

The last equality is because $J_{ks}J_{ks}$ is always a diagonal matrix.

Therefore, we proved that $\bar{X}$ is a feasible point of the Orthogonal-Cut SDP problem (OC-SDP). ∎

6 Proof of Theorem 2

(Gradient-step) Fix $\mu_{G}\leq 2\|A\|_{1}$. For any point $\sigma\in\mathcal{M}_{k}$ such that $\|{\textup{grad}}{f}(\sigma)\|_{F}\geq\mu_{G}$, taking searching direction $u={\textup{grad}}{f}(\sigma)/\|{\textup{grad}}{f}(\sigma)\|_{F}$ and step size $\eta=\mu_{G}/(20\|A\|_{1})$, we have

The second order expansion of ${f}(\sigma(t))$ around with $t\leq 1$ gives

The second inequality used the bound on the second order derivative in Lemma 5 in Appendix A.1. Now we take $t=\mu_{G}/(20\|A\|_{1})$. Since $\mu_{G}\leq 2\|A\|_{1}$, we have $t\leq 1$. Plugging this $t$ into the above equation completes the proof. ∎

(Eigen-step) For any point $\sigma\in\mathcal{M}_{k}$, and $u\in T_{\sigma}\mathcal{M}_{k}$ satisfying $\|u\|_{F}=1$, $\langle u,{\textup{grad}}{f}(\sigma)\rangle\geq 0$, and $\lambda_{H}=\lambda_{H}(\sigma,u)={\textup{Hess}}{f}(\sigma)[u,u]>0$, choosing $\eta=\lambda_{H}/(100\|A\|_{1})$, we have

The third order expansion of ${f}(\sigma(t))$ around for $t\leq 1$ gives

The second inequality used the bound on the third order derivative in Lemma 6 in Appendix A.1. Now we take $t=\lambda_{H}/(100\|A\|_{1})$. Note that we always have $\lambda_{H}(\sigma,u)\leq\|{\textup{Hess}}{f}(\sigma)\|_{2}\leq\|A-\Lambda\|_{2}\leq 2\|A\|_{1}$, and therefore we have $t\leq 2\|A\|_{1}/(100\|A\|_{1})\leq 1$. Plugging this $t$ into the above equation completes the proof. ∎

The last lower bound on the increment of objective function for eigen-step used the loose bound in Lemma 6. Using Lemma 7, we can give an improved bound for the eigen-step when the norm of the gradient is small. In particular we take $\mu_{G}=\|A\|_{2}$.

(Improved bound for eigen-step) For any point $\sigma\in\mathcal{M}_{k}$ with $\|{\textup{grad}}{f}(\sigma)\|_{F}\leq\mu_{G}=\|A\|_{2}$, and $u\in T_{\sigma}\mathcal{M}_{k}$ satisfying $\|u\|_{F}=1$, $\langle u,{\textup{grad}}{f}(\sigma)\rangle\geq 0$ and $\lambda_{H}=\lambda_{H}(\sigma,u)={\textup{Hess}}f(\sigma)[u,u]>0$, choosing $\eta=\min\left(\sqrt{\lambda_{H}/(216\|A\|_{1})},\lambda_{H}/(12\|A\|_{2})\right)$, we have

The third order expansion of ${f}(\sigma(t))$ around for $t\leq 1$ gives

The first inequality used the improved bound on the third order derivative of Lemma 7 in Appendix A.1, which imply in particular $\|{\textup{grad}}f(\sigma)\|_{F}\leq\|A\|_{2}$. Taking $t=\min\left(\sqrt{\lambda_{H}/(216\|A\|_{1})},\lambda_{H}/(12\|A\|_{2})\right)<1$ completes the proof. ∎

We are now at a good position to prove Theorem 2.

Denote ${f}^{*}={\textup{\rm SDP}}(A)-1/(k-1)\cdot({\textup{\rm SDP}}(A)+{\textup{\rm SDP}}(-A))$ and $g(\sigma)={f}^{*}-{f}(\sigma)$. Let $T$ be the number of iterations and $\{\sigma^{0},\sigma^{1},\ldots,\sigma^{T}\}\subset\mathcal{M}_{k}$ the iterates returned by our RTR algorithm from an arbitrary initialization $\sigma^{0}\in\mathcal{M}_{k}$. We are only interested in the convergence rate as $g(\sigma)>0$, namely the convergence rate below the gap. Since our algorithm is an ascent algorithm, without loss of generality, we assume $g(\sigma^{0}),\ldots,g(\sigma^{T})>0$ (otherwise the theorem will hold automatically).

At each point $\sigma\in\mathcal{M}_{k}$, Theorem 1 gives the following lower bound on the highest curvature

We will use this information to bound the algorithm’s convergence rate.

Case 1. First, we consider the case when all the RTR steps are eigen-steps. In each iteration, the algorithm constructs an update direction $u^{t}$ with curvature $\lambda_{H}(\sigma^{t},u^{t})\geq\lambda_{H,\max}(\sigma^{t})/2$. According to Lemma 3, we have

which implies $g(\sigma^{t+1})\leq g(\sigma^{t})$. Thus, we have

Therefore, we obtain the convergence rate $g(\sigma^{T})\leq 400\|A\|_{1}n\sqrt{n/T}$. This implies that

as soon as $T\geq 64\cdot 10^{4}\cdot n\|A\|_{1}^{2}/\varepsilon^{2}$.

Case 2. Then, we consider the case where we set $\mu_{G}=\|A\|_{2}$, and we use the gradient step as $\|{\textup{grad}}{f}(\sigma)\|_{F}>\mu_{G}$, and use the eigen-step as $\|{\textup{grad}}{f}(\sigma)\|_{F}\leq\mu_{G}$. First let us bound the number of gradient steps. According to Lemma 1, we have

Hence, we deduce the upper bound $T_{G}\leq 40\cdot\|A\|_{1}{\textup{Rg}}(A)/\|A\|_{2}^{2}$.

Then let us bound the number of eigen-steps. Let us denote $\mathcal{I}$ and $\mathcal{J}\subset\{0,1,\ldots,T-1\}$ the subsets of indices corresponding to eigensteps with respectively $\lambda_{H}\geq 3\|A\|_{2}^{2}/(2\|A\|_{1})$ and $\lambda_{H}<3\|A\|_{2}^{2}/(2\|A\|_{1})$. According to Lemma 3, we have for all $t\in\mathcal{J}$

Summing the contributions of the above two equations gives the convergence rate

Acknowledgements

A.M. was partially supported by the NSF grant CCF-1319979. S.M. was supported by Office of Technology Licensing Stanford Graduate Fellowship.

References

Appendix A Some technical steps

In this section, we give an upper bound to the second and third derivatives of $({f}\circ\sigma)(t)=\langle\sigma(t),A\sigma(t)\rangle$ (these notations are defined below). These bounds are important in bounding the complexity of the Riemannian trust-region method in solving the non-convex SDP problem.

To calculate the first three derivatives of $\sigma(t)$, we expand each row of $\sigma(t+r)$ up to third order in $r$:

By matching each expansion coefficient to the corresponding derivative, we obtain the desired result. ∎

For $({f}\circ\sigma)(t)$ as defined above

We explicitly calculate the second derivative

Noticing that $\|u(t)\|_{F}\leq\|u(0)\|_{F}=1$, we can use the bounds derived in Appendix A.2 to obtain the following inequality

For $({f}\circ\sigma)(t)$ as defined above

We explicitly calculate the third derivative

The inequality is obtained by upper bounding each term using the bounds derived in Appendix A.2. ∎

The above bound on the third derivative of order $\|A\|_{1}$ as $t\to 0$. The next lemma proves a bound of order $\|A\|_{2}+\|{\textup{grad}}{f}(\sigma(0))\|_{F}$ as $t\to 0$. If $\|{\textup{grad}}{f}(\sigma(0))\|_{F}$ is small, this improves the above bound.

For $({f}\circ\sigma)(t)$ as defined above, an improved bound on its third derivative gives

From the proof in the previous lemma, we have

We next bound more carefully $g(t)=\langle\sigma(t),AD(t)u(t)\rangle$. Simple calculation gives us

According to the bounds in Appendix A.2, we have

According to the Taylor expansion of $g(t)$ around and $t\geq 0$ at first order, we have

A.2 All the bounds