Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly.

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is _not_ geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability $1$. For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability $1$, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the end state of a stochastic Riemannian algorithm can only be a local minimizer.

This paper focuses on the short and sparse blind deconvolu-tion problem, where the one unknown signal is short and the other one is sparsely and randomly supported.

We study personalized federated learning for multivariate responses where client models are heterogeneous yet share variable-level structure. Existing entry-wise penalties ignore cross-response dependence, while matrix-wise fusion over-couples clients. We propose a Sparse Row-wise Fusion (SROF) regularizer that clusters row vectors across clients and induces within-row sparsity, and we develop RowFed, a communication-efficient federated algorithm that embeds SROF into a linearized ADMM framework with privacy-preserving partial participation. Theoretically, we establish an oracle property for SROF-achieving correct variable-level group recovery with asymptotic normality-and prove convergence of RowFed to a stationary solution. Under random client participation, the iterate gap contracts at a rate that improves with participation probability. Empirically, simulations in heterogeneous regimes show that RowFed consistently lowers estimation and prediction error and strengthens variable-level cluster recovery over NonFed, FedAvg, and a personalized matrix-fusion baseline. A real-data study further corroborates these gains while preserving interpretability. Together, our results position row-wise fusion as an effective and transparent paradigm for large-scale personalized federated multivariate learning, bridging the gap between entry-wise and matrix-wise formulations.

Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We develop a nonconvex reformulation designed to exploit this staged structure. Our reformulation has only simple bound constraints, enabling solution via projected gradient methods and their accelerated variants.