44 research outputs found
Exact Topology and Parameter Estimation in Distribution Grids with Minimal Observability
Limited presence of nodal and line meters in distribution grids hinders their
optimal operation and participation in real-time markets. In particular lack of
real-time information on the grid topology and infrequently calibrated line
parameters (impedances) adversely affect the accuracy of any operational power
flow control. This paper suggests a novel algorithm for learning the topology
of distribution grid and estimating impedances of the operational lines with
minimal observational requirements - it provably reconstructs topology and
impedances using voltage and injection measured only at the terminal (end-user)
nodes of the distribution grid. All other (intermediate) nodes in the network
may be unobserved/hidden. Furthermore no additional input (e.g., number of grid
nodes, historical information on injections at hidden nodes) is needed for the
learning to succeed. Performance of the algorithm is illustrated in numerical
experiments on the IEEE and custom power distribution models
Minimum Weight Perfect Matching via Blossom Belief Propagation
Max-product Belief Propagation (BP) is a popular message-passing algorithm
for computing a Maximum-A-Posteriori (MAP) assignment over a distribution
represented by a Graphical Model (GM). It has been shown that BP can solve a
number of combinatorial optimization problems including minimum weight
matching, shortest path, network flow and vertex cover under the following
common assumption: the respective Linear Programming (LP) relaxation is tight,
i.e., no integrality gap is present. However, when LP shows an integrality gap,
no model has been known which can be solved systematically via sequential
applications of BP. In this paper, we develop the first such algorithm, coined
Blossom-BP, for solving the minimum weight matching problem over arbitrary
graphs. Each step of the sequential algorithm requires applying BP over a
modified graph constructed by contractions and expansions of blossoms, i.e.,
odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs,
where n is the number of vertices in the original graph. In essence, the
Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom
algorithm by jumping at once over many sub-steps with a single BP. Moreover,
our result provides an interpretation of the Edmonds' algorithm as a sequence
of LPs
Generalization Bounds for Stochastic Gradient Descent via Localized -Covers
In this paper, we propose a new covering technique localized for the
trajectories of SGD. This localization provides an algorithm-specific
complexity measured by the covering number, which can have
dimension-independent cardinality in contrast to standard uniform covering
arguments that result in exponential dimension dependency. Based on this
localized construction, we show that if the objective function is a finite
perturbation of a piecewise strongly convex and smooth function with
pieces, i.e. non-convex and non-smooth in general, the generalization error can
be upper bounded by , where is the number of
data samples. In particular, this rate is independent of dimension and does not
require early stopping and decaying step size. Finally, we employ these results
in various contexts and derive generalization bounds for multi-index linear
models, multi-class support vector machines, and -means clustering for both
hard and soft label setups, improving the known state-of-the-art rates