Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Stochastic subspace methods have gained popularity as gradient descent based techniques for large scale optimisation problems, especially in distributed settings. In this paper, we introduce the technique of "persistence of memory" to greatly extend and improve the random subspace methods. To this end, we leverage a vector that is only weakly correlated with the gradient in order to provide a guiding structure to the generative process of the random subspace along which the descent is going to take place. This guidance vector may be fixed for a large number of iterations, only to be refreshed at wide intervals (on whose size we can provide guarantees in terms of problem parameters). In important machine learning settings, such as optimisation problems embodying sparsity or a minibatch structure, we show that the guidance vector can be obtained in an effective and computationally inexpensive manner by leveraging the structured properties of the problem. En route, we establish to our knowledge the first theoretical analysis of classical SSD methods for sparse functions. In a local neighbourhood of the optimum, we demonstrate an alignment phenomenon of our gradient estimates with a low-lying eigenvector of the Hessian, allowing a once-for-all computation of the guidance vector which renders the method computationally favourable even in scenarios with unstructured objectives.

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis--Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.

Stochastic estimators are fundamental to large-scale optimization, where population quantities must be inferred from noisy oracle observations. Although influential methods such as momentum, SPIDER, STORM, and PAGE have been highly successful, their analyses are largely estimator-specific and expectation-based, obscuring the structural tradeoffs that determine reliability. In this paper, we develop a unified framework for stochastic variance-reduced estimation based on a recursion with three components: memory retention, reset probability, and a correction term for iterate movement. This framework recovers several classical estimators, motivates new second-order variants, and yields a bias-variance decomposition of estimation error. Our main result is a unified high-probability bound proved using a new dimension-free vector-valued Freedman inequality, valid for smooth normed spaces involving random sums of vector martingales. The result applies in both Euclidean and non-Euclidean settings, including the analysis of mirror-descent-based methods in Banach spaces. As applications, we obtain high-probability oracle complexities for unconstrained optimization with mirror descent, establishing the logarithmic dependence on the confidence level. We also derive the first $\tilde{\mathcal{O}}(\varepsilon^{-3})$ oracle-complexity bounds for stochastic optimization with expectation constraints, improving upon the existing $\tilde{\mathcal{O}}(\varepsilon^{-4})$ complexity by leveraging variance-reduced estimation for the first time in this setting.