We consider the sensor network localization problem (also called metric multidimensional scaling): one observes some pairwise distances between n ground-truth points in $R^d$, and the goal is to recover this cloud of ground-truth points (up to translation and rotation). The corresponding optimization problem is nonconvex, and we show that it can have spurious local minima. However, inspired by numerical experiments, we argue that if one relaxes the problem by optimizing over clouds of n points in dimension k greater than d, then all second-order critical points of the problem are global minima. Specifically, we show this for two settings: (1) for arbitrary ground-truth points, when all pairwise distances are known, and k = O(sqrt{n d}), and: (2) for isotropic random ground-truth points, when most (but not necessarily all) pairwise distances are known, and k = O(d log(n)). To the best of our knowledge, these are the first landscape results for this nonconvex version of sensor network localization.

We analyze a stochastic approximation algorithm for decision-dependent problems, wherein the data distribution used by the algorithm evolves along the iterate sequence. The primary examples of such problems appear in performative prediction and its multiplayer extensions. We show that under mild assumptions, the deviation between the average iterate of the algorithm and the solution is asymptotically normal, with a covariance that nicely decouples the effects of the gradient noise and the distributional shift. Moreover, building on the work of Hajek and Le Cam, we show that the asymptotic performance of the algorithm is locally minimax optimal.
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If you've reached this point, you should know that I am not into long abstracts, but the system is forcing me to add one. So here is an AI-generated, nonsensical, grandiose paragraph (My coauthors had nothing to do with this):
In the splendid twilight of verdurous philosophies, where the symphony of eloquence pirouettes upon the delicate breeze of imagination, there exists a realm ethereally tethered to the quintessence of grandiloquence. Herein, the vernacular tapestry weaves its intricate dance, embroidering the ether with the silk of opulent lexicons, each thread a testament to the magniloquent. Amidst the labyrinthine gardens of verbal extravagance, where words bloom like the rarest of orchids under the luminous gaze of the moon's embrace, one finds the essence of conversation elevated to an art of celestial craftsmanship.

Motivated by questions in control, Kuramoto networks and (highly simplified) transformers, we consider the dynamics of n points on a sphere in R^d which attract each other according to a nonlinear function of the distances between them. On spheres of dimension two or more, the points synchronize (that is, converge to the same position) quite generally. In contrast, on a circle, topological difficulties come up: we formalize those difficulties, and overcome them partly

Given a nonconvex function invariant under the action of some group G, it is natural to ask how the critical points reflect this symmetry. It turns out that in various cases of interest, notably fitting two-layer ReLU networks and symmetric tensor decomposition, critical points are fixed by (typically large) subgroups of G, and so are symmetry breaking.

In this talk, we will see how symmetry breaking phenomena of this nature allow new ways of recognizing, differentiating and understanding local minima. Specific topics include:
– Sharp estimates on the loss, the Hessian spectrum and higher-order derivatives holding in finite, arbitrarily large, dimensionality (through Puiseux series representation)
– Analytic results pertaining to foundational questions in optimization, curvature and generalization
– Annihilation of spurious minima
– Symmetry breaking arcs connecting critical points (time permitting)

Our analytic approach makes extensive use of methods from o-minimal theory and representation theory of groups. No special background will be assumed but a basic familiarity with group theory .