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SUMMARY:Low-rank matrix optimization landscapes with overparametrization
DTSTART:20260504T103000
DTEND:20260504T120000
DTSTAMP:20260531T070214Z
UID:71516c18c5cf14b925260278e0b967d1039e02c0453665f92a2f154e
CATEGORIES:Conferences - Seminars
DESCRIPTION:Andrew McRae\nI will discuss the global nonconvex landscape of
  low-rank matrix optimization. In particular\, we consider the landscape u
 nder restricted strong convexity and restricted smoothness assumptions. We
  focus on the effect of rank overparametrization\, that is\, optimizing ov
 er matrices of rank strictly larger than that of the global optimum. Our m
 ain contribution is twofold:\n\nFirst\, we give a positive landscape resul
 t for smooth factored formulations\, showing that\, under certain conditio
 ns on the rank parameters and restricted strong convexity/smoothness const
 ants\, every second-order critical point is globally optimal. This general
 izes and unifies previous state-of-the-art results\; in particular\, our r
 esult allows for nuclear-norm regularization and applies to asymmetric mat
 rix problems without requiring balancing of the factorization. A more stat
 istical version of this result gives\, under classical statistical assumpt
 ions\, optimal recovery of low-rank matrices from noisy linear measurement
 s.\n\nSecond\, we construct a family of counterexamples showing that our p
 ositive result is optimal in terms of all problem parameters. In particula
 r\, contrary to popular belief\, rank overparameterization does not always
  improve the optimization landscape. Although our examples are adversarial
 \, empirical evidence suggests that this phenomenon extends to standard st
 atistical matrix sensing settings.\n\nJoint work with Richard Y. Zhang (UI
 UC).\n 
LOCATION:CM 1 517 https://plan.epfl.ch/?room==CM%201%20517
STATUS:CONFIRMED
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