Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex optimization that sequentially minimizes a majorizing surrogate of the objective function in each block coordinate while the other block coordinates are held fixed. We consider a family of BMM algorithms for minimizing nonsmooth nonconvex objectives, where each parameter block is constrained within a subset of a Riemannian manifold. We establish that this algorithm converges asymptotically to the set of stationary points, and attains an $\epsilon$-stationary point within $\widetilde{O}(\epsilon^{-2})$ iterations. In particular, the assumptions for our complexity results are completely Euclidean when the underlying manifold is a product of Euclidean or Stiefel manifolds, although our analysis makes explicit use of the Riemannian geometry. Our general analysis applies to a wide range of algorithms with Riemannian constraints: Riemannian MM, block projected gradient descent, Bures-JKO scheme for Wasserstein variational inference, optimistic likelihood estimation, geodesically constrained subspace tracking, robust PCA, and Riemannian CP-dictionary-learning. We experimentally validate that our algorithm converges faster than standard Euclidean algorithms applied to the Riemannian setting.

The classical static Schrödinger Bridge (SSB) problem, which seeks the most likely stochastic evolution between two marginal probability measures, has been studied extensively in the optimal transport and statistical physics communities, and more recently in machine learning communities in the surge of generative models. The standard approach to solve SSB is to first identify its Kantorovich dual and use Sinkhorn’s algorithm to find the optimal potential functions. While the original SSB is only a strictly convex minimization problem, this approach is known to warrant linear convergence under mild assumptions. In this work, we consider a generalized SSB allowing any strictly increasing divergence functional, far generalizing the entropy functional $x\log x$ in the standard SSB. This problem naturally arises in a wide range of seemingly unrelated problems in entropic optimal transport, random graphs/matrices, and combinatorics. We establish Kantorovich duality and linear convergence of Sinkhorn’s algorithm for the generalized SSB problem under mild conditions. Our results provide a new rigorous foundation for understanding Sinkhorn-type iterative methods in the context of large-scale generalized Schrödinger bridges.

We consider the branch-length estimation problem on a bifurcating tree: a character evolves along the edges of a binary tree according to a two-state symmetric Markov process, and we seek to recover the edge transition probabilities from repeated observations at the leaves. This problem arises in phylogenetics, and is related to latent tree graphical model inference. In general, the log-likelihood function is non-concave and may admit many critical points. Nevertheless, simple coordinate maximization has been known to perform well in practice, defying the complexity of the likelihood landscape. In this work, we provide the first theoretical guarantee as to why this might be the case. We show that deep inside the Kesten-Stigum reconstruction regime, provided with polynomially many $m$ samples (assuming the tree is balanced), there exists a universal parameter regime (independent of the size of the tree) where the log-likelihood function is strongly concave and smooth with high probability. On this high-probability likelihood landscape event, we show that the standard coordinate maximization algorithm converges exponentially fast to the maximum likelihood estimator, which is within $O(1/\sqrt{m})$ from the true parameter, provided a sufficiently close initial point.

We study offline policy optimization for infinite-horizon average-reward Markov decision processes (MDPs) with large or infinite state spaces. Specifically, we propose a pessimistic actor-critic algorithm that uses a computationally efficient linear function class for value function estimation. At the core of our method is a critic that computes a pessimistic estimate of the average reward under the current policy, as well as the corresponding policy gradient, by solving a fixed-point Bellman equation, rather than solving a successive sequence of regression problems as in finite horizon settings. This procedure reduces to solving a second-order cone program, which is computationally tractable. Our theoretical analysis is based on a weak partial data coverage assumption, which requires only that the offline data aligns well with the expected feature vector of a comparator policy. Under this condition, we show that our algorithm achieves the optimal sample complexity of O(\varepsilon^{-2}) for learning a near-optimal policy, up to model misspecification errors.

We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators locally unbiased within the $\Theta(\epsilon)$-ball around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.

Supervised matrix factorization (SMF) is a classical machine learning method that seeks low-dimensional feature extraction and classification tasks at the same time. Training an SMF model involves solving a non-convex and factor-wise constrained optimization problem with at least three blocks of parameters. Due to the high non-convexity and constraints, theoretical understanding of the optimization landscape of SMF has been limited. In this paper, we provide an extensive local landscape analysis for SMF and derive several theoretical and practical applications. Analyzing diagonal blocks of the Hessian naturally leads to a block coordinate descent (BCD) algorithm with adaptive step sizes. We provide global convergence and iteration complexity guarantees for this algorithm. Full Hessian analysis gives minimum $L_{2}$-regularization to guarantee local strong convexity and robustness of parameters. We establish a local estimation guarantee under a statistical SMF model. We also propose a novel GPU-friendly neural implementation of the BCD algorithm and validate our theoretical findings through numerical experiments. Our work contributes to a deeper understanding of SMF optimization, offering insights into the optimization landscape and providing practical solutions to enhance its performance.

We analyze inexact Riemannian gradient descent (RGD) where Riemannian gradients and retractions are inexactly (and cheaply) computed. Our focus is on understanding when inexact RGD converges and what is the complexity in the general nonconvex and constrained setting. We answer these questions in a general framework of tangential Block Majorization-Minimization (tBMM). We establish that tBMM converges to an $\epsilon$-stationary point within $O(\epsilon^{-2})$ iterations. Under a mild assumption, the results still hold when the subproblem is solved inexactly in each iteration provided the total optimality gap is bounded. Our general analysis applies to a wide range of classical algorithms with Riemannian constraints including inexact RGD and proximal gradient method on Stiefel manifolds. We numerically validate that tBMM shows improved performance over existing methods when applied to various problems, including nonnegative tensor decomposition with Riemannian constraints, regularized nonnegative matrix factorization, and low-rank matrix recovery problems.

For obtaining optimal first-order convergence guarantees for stochastic optimization, it is necessary to use a recurrent data sampling algorithm that samples every data point with sufficient frequency. Most commonly used data sampling algorithms (e.g., i.i.d., MCMC, random reshuffling) are indeed recurrent under mild assumptions. In this work, we show that for a particular class of stochastic optimization algorithms, we do not need any further property (e.g., independence, exponential mixing, and reshuffling) beyond recurrence in data sampling to guarantee optimal rate of first-order convergence. Namely, using regularized versions of Minimization by Incremental Surrogate Optimization (MISO), we show that for non-convex and possibly non-smooth objective functions with constraints, the expected optimality gap converges at an optimal rate $O(n^{-1/2})$ under general recurrent sampling schemes. Furthermore, the implied constant depends explicitly on the ’speed of recurrence’, measured by the expected amount of time to visit a data point, either averaged (’target time’) or supremized (’hitting time’) over the starting locations. We discuss applications of our general framework to decentralized optimization and distributed non-negative matrix factorization.

Stochastic majorization-minimization (SMM) is a class of stochastic optimization algorithms that proceed by sampling new data points and minimizing a recursive average of surrogate functions of an objective function. The surrogates are required to be strongly convex and the existing convergence rate analysis for the general non-convex setting was not available. In this paper, we propose an extension of SMM where surrogates are allowed to be only weakly convex or block multi-convex, and the averaged surrogates are approximately minimized with proximal regularization or block-minimized within diminishing radii, respectively. For the general nonconvex constrained setting with non-i.i.d. data samples, we show that the first-order optimality gap of the proposed algorithm decays at the rate $\widetilde{O}(n^{-1/4})$ for the empirical loss and $\widetilde{O}(n^{-1/8})$ for the expected loss, where $n$ denotes the number of data samples processed. Under some additional assumption, the latter convergence rate can be improved to $\widetilde{O}(n^{-1/4})$. As a corollary, we obtain the first convergence rate bounds for various optimization methods under general nonconvex non-i.i.d. data setting: Double-averaging projected gradient descent and its generalizations, proximal point empirical risk minimization, and online matrix/tensor decomposition algorithms. We also provide experimental validation of our results.

Dictionary learning (DL), implemented via matrix factorization (MF), is commonly used in computational biology to tackle ubiquitous clustering problems. The method is favored due to its conceptual simplicity and relatively low computational complexity. However, DL algorithms produce results that lack interpretability in terms of real biological data. Additionally, they are not optimized for graph-structured data and hence often fail to handle them in a scalable manner. In order to address these limitations, we propose a novel DL algorithm called online convex network dictionary learning (online cvxNDL). Unlike classical DL algorithms, online cvxNDL is implemented via MF and designed to handle extremely large datasets by virtue of its online nature. Importantly, it enables the interpretation of dictionary elements, which serve as cluster representatives, through convex combinations of real measurements. Moreover, the algorithm can be applied to data with a network structure by incorporating specialized subnetwork sampling techniques. To demonstrate the utility of our approach, we apply cvxNDL on 3D-genome RNAPII ChIA-Drop data with the goal of identifying important long-range interaction patterns (long-range dictionary elements). ChIA-Drop probes higher-order interactions, and produces data in the form of hypergraphs whose nodes represent genomic fragments. The hyperedges represent observed physical contacts. Our hypergraph model analysis has the objective of creating an interpretable dictionary of long-range interaction patterns that accurately represent global chromatin physical contact maps. Through the use of dictionary information, one can also associate the contact maps with RNA transcripts and infer cellular functions. To accomplish the task at hand, we focus on RNAPII-enriched ChIA-Drop data from Drosophila Melanogaster S2 cell lines. Our results offer two key insights. First, we demonstrate that online cvxNDL retains the accuracy of classical DL (MF) methods while simultaneously ensuring unique interpretability and scalability. Second, we identify distinct collections of proximal and distal interaction patterns involving chromatin elements shared by related processes across different chromosomes, as well as patterns unique to specific chromosomes. To associate the dictionary elements with biological properties of the corresponding chromatin regions, we employ Gene Ontology (GO) enrichment analysis and perform multiple RNA coexpression studies.

We focus on analyzing the classical stochastic projected gradient methods under a general dependent data sampling scheme for constrained smooth nonconvex optimization. We show the worst-case rate of convergence $\tilde{O}(t^{-1/4})$ and complexity $\tilde{O}(\varepsilon^{-4})$ for achieving an $\varepsilon$-near stationary point in terms of the norm of the gradient of Moreau envelope and gradient mapping. While classical convergence guarantee requires i.i.d. data sampling from the target distribution, we only require a mild mixing condition of the conditional distribution, which holds for a wide class of Markov chain sampling algorithms. This improves the existing complexity for the constrained smooth nonconvex optimization with dependent data from $\tilde{O}(\varepsilon^{-8})$ to $\tilde{O}(\varepsilon^{-4})$ with a significantly simpler analysis. We illustrate the generality of our approach by deriving convergence results with dependent data for stochastic proximal gradient methods, adaptive stochastic gradient algorithm AdaGrad and stochastic gradient algorithm with heavy ball momentum. As an application, we obtain first online nonnegative matrix factorization algorithms for dependent data based on stochastic projected gradient methods with adaptive step sizes and optimal rate of convergence.

We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for ‘Wasserstein CP-dictionary learning’, which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.

Supervised matrix factorization (SMF) is a classical machine learning method that simultaneously seeks feature extraction and classification tasks, which are not necessarily a priori aligned objectives. Our goal is to use SMF to learn low-rank latent factors that offer interpretable, data-reconstructive, and class-discriminative features, addressing challenges posed by high-dimensional data. Training SMF model involves solving a nonconvex and possibly constrained optimization with at least three blocks of parameters. Known algorithms are either heuristic or provide weak convergence guarantees for special cases. In this paper, we provide a novel framework that `lifts' SMF as a low-rank matrix estimation problem in a combined factor space and propose an efficient algorithm that provably converges exponentially fast to a global minimizer of the objective with arbitrary initialization under mild assumptions. Our framework applies to a wide range of SMF-type problems for multi-class classification with auxiliary features. To showcase an application, we demonstrate that our algorithm successfully identified well-known cancer-associated gene groups for various cancers.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph into a large network. We propose two complementary MCMC algorithms for sampling random graph homomorphisms and establish bounds on their mixing times and the concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neighborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also \commHL{demonstrate the performance of} our framework on the tasks of network clustering and subgraph classification on the Facebook100 dataset and on Word Adjacency Networks of a set of classic novels.

Online Tensor Factorization (OTF) is a fundamental tool in learning low-dimensional interpretable features from streaming multi-modal data. While various algorithmic and theoretical aspects of OTF have been investigated recently, a general convergence guarantee to stationary points of the objective function without any incoherence or sparsity assumptions is still lacking even for the i.i.d. case. In this work, we introduce a novel algorithm that learns a CANDECOMP/PARAFAC (CP) basis from a given stream of tensor-valued data under general constraints, including nonnegativity constraints that induce interpretability of the learned CP basis. We prove that our algorithm converges almost surely to the set of stationary points of the objective function under the hypothesis that the sequence of data tensors is generated by an underlying Markov chain. Our setting covers the classical i.i.d. case as well as a wide range of application contexts including data streams generated by independent or MCMC sampling. Our result closes a gap between OTF and Online Matrix Factorization in global convergence analysis for CP-decompositions. Experimentally, we show that our algorithm converges much faster than standard algorithms for nonnegative tensor factorization tasks on both synthetic and real-world data. Also, we demonstrate the utility of our algorithm on a diverse set of examples from image, video, and time-series data, illustrating how one may learn qualitatively different CP-dictionaries from the same tensor data by exploiting the tensor structure in multiple ways.