Neural Information Processing Systems http://nips.cc/

The proliferation of saddle points, rather than poor local minima, is increasingly understood to be a primary obstacle in large-scale non-convex optimization for machine learning. Variable elimination algorithms, like Variable Projection (VarPro), have long been observed to exhibit superior convergence and robustness in practice, yet a principled understanding of why they so effectively navigate these complex energy landscapes has remained elusive. In this work, we provide a rigorous geometric explanation by comparing the optimization landscapes of the original and reduced formulations. Through a rigorous analysis based on Hessian inertia and the Schur complement, we prove that variable elimination fundamentally reshapes the critical point structure of the objective function, revealing that local maxima in the reduced landscape are created from, and correspond directly to, saddle points in the original formulation. Our findings are illustrated on the canonical problem of non-convex matrix factorization, visualized directly on two-parameter neural networks, and finally validated in training deep Residual Networks, where our approach yields dramatic improvements in stability and convergence to superior minima. This work goes beyond explaining an existing method; it establishes landscape simplification via saddle point transformation as a powerful principle that can guide the design of a new generation of more robust and efficient optimization algorithms.

The move-acceptance hyper-heuristic was recently shown to be able to leave local optima with astonishing efficiency (Lissovoi et al., Artificial Intelligence (2023)). In this work, we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic Cliff$_d$ and Jump$_m$ function classes. (i) Instead of randomly choosing between the only-improving and any-move acceptance operator, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on Jump$_m$ functions with gap parameter $m$ from $Ω(n^{2m-1})$ to $O(n^{m+1})$. (ii) We then replace the all-moves acceptance operator with the operator that only accepts worsenings. Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to $O(n^3 \log n)$ independent of the gap size. In general, we prove a remarkably good runtime of $O(n^{k+1} \log n)$ for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT$_k$, which contains a large number of functions having $k$ successive local optima, and which contains the commonly studied Jump$_m$ and Cliff$_d$ functions for $k=2$.

Interactive visualization of embedding projections is a useful technique for understanding data and evaluating machine learning models. Labeling data within these visualizations is critical for interpretation, as labels provide an overview of the projection and guide user navigation. However, most methods for producing labels require clustering the points, which can be computationally expensive as the number of points grows. In this paper, we describe an efficient clustering approach using kernel density estimation in the projected 2D space instead of points. This algorithm can produce high-quality cluster regions from a 2D density map in a few hundred milliseconds, orders of magnitude faster than current approaches. We contribute the design of the algorithm, benchmarks, and applications that demonstrate the utility of the algorithm, including labeling and summarization.

Clustering is often a challenging problem because of the inherent ambiguity in what the "correct" clustering should be. Even when the number of clusters $K$ is known, this ambiguity often still exists, particularly when there is variation in density among different clusters, and clusters have multiple relatively separated regions of high density. In this paper we propose an information-theoretic characterization of when a $K$-clustering is ambiguous, and design an algorithm that recovers the clustering whenever it is unambiguous. This characterization formalizes the situation when two high density regions within a cluster are separable enough that they look more like two distinct clusters than two truly distinct clusters in the clustering. The algorithm first identifies $K$ partial clusters (or "seeds") using a density-based approach, and then adds unclustered points to the initial $K$ partial clusters in a greedy manner to form a complete clustering. We implement and test a version of the algorithm that is modified to effectively handle overlapping clusters, and observe that it requires little parameter selection and displays improved performance on many datasets compared to widely used algorithms for non-convex cluster recovery.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to have the geometric property that "all local optima are (approximately) global optima", and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult. In this paper, we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised leaning, especially in learning latent variable models.

We develop hard clustering based on likelihood rather than distance and prove convergence. We also provide simulations and real data examples.