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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 {\it 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 on $y^*$-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds. 

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 𝜖-stationary point within 𝑂(𝜖−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. 

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 L2-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. 

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 $$O(n^{-2})$$ 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 target locations. We discuss applications of our general framework to decentralized optimization and distributed non-negative matrix factorization. 

Temporal text data, such as news articles or Twitter feeds, often comprises a mixture of long-lasting trends and transient topics. Effective topic modeling strategies should detect both types and clearly locate them in time. We first demonstrate that nonnegative CANDECOMP/PARAFAC decomposition (NCPD) can automatically identify topics of variable persistence. We then introduce sparseness-constrained NCPD (S-NCPD) and its online variant to control the duration of the detected topics more effectively and efficiently, along with theoretical analysis of the proposed algorithms. Through an extensive study on both semi-synthetic and real-world datasets, we find that our S-NCPD and its online variant can identify both short- and long-lasting temporal topics in a quantifiable and controlled manner, which traditional topic modeling methods are unable to achieve. Additionally, the online variant of S-NCPD shows a faster reduction in reconstruction error and results in more coherent topics compared to S-NCPD, thus achieving both computational efficiency and quality of the resulting topics. Our findings indicate that S-NCPD and its online variant are effective tools for detecting and controlling the duration of topics in temporal text data, providing valuable insights into both persistent and transient trends. 

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 calledonline 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 fromDrosophila MelanogasterS2 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. 

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.