5,305 research outputs found
Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded
Decision trees usefully represent sparse, high dimensional and noisy data.
Having learned a function from this data, we may want to thereafter integrate
the function into a larger decision-making problem, e.g., for picking the best
chemical process catalyst. We study a large-scale, industrially-relevant
mixed-integer nonlinear nonconvex optimization problem involving both
gradient-boosted trees and penalty functions mitigating risk. This
mixed-integer optimization problem with convex penalty terms broadly applies to
optimizing pre-trained regression tree models. Decision makers may wish to
optimize discrete models to repurpose legacy predictive models, or they may
wish to optimize a discrete model that particularly well-represents a data set.
We develop several heuristic methods to find feasible solutions, and an exact,
branch-and-bound algorithm leveraging structural properties of the
gradient-boosted trees and penalty functions. We computationally test our
methods on concrete mixture design instance and a chemical catalysis industrial
instance
Minimum and maximum entropy distributions for binary systems with known means and pairwise correlations
Maximum entropy models are increasingly being used to describe the collective
activity of neural populations with measured mean neural activities and
pairwise correlations, but the full space of probability distributions
consistent with these constraints has not been explored. We provide upper and
lower bounds on the entropy for the {\em minimum} entropy distribution over
arbitrarily large collections of binary units with any fixed set of mean values
and pairwise correlations. We also construct specific low-entropy distributions
for several relevant cases. Surprisingly, the minimum entropy solution has
entropy scaling logarithmically with system size for any set of first- and
second-order statistics consistent with arbitrarily large systems. We further
demonstrate that some sets of these low-order statistics can only be realized
by small systems. Our results show how only small amounts of randomness are
needed to mimic low-order statistical properties of highly entropic
distributions, and we discuss some applications for engineered and biological
information transmission systems.Comment: 34 pages, 7 figure
Linear Programming Decoding of Spatially Coupled Codes
For a given family of spatially coupled codes, we prove that the LP threshold
on the BSC of the graph cover ensemble is the same as the LP threshold on the
BSC of the derived spatially coupled ensemble. This result is in contrast with
the fact that the BP threshold of the derived spatially coupled ensemble is
believed to be larger than the BP threshold of the graph cover ensemble as
noted by the work of Kudekar et al. (2011, 2012). To prove this, we establish
some properties related to the dual witness for LP decoding which was
introduced by Feldman et al. (2007) and simplified by Daskalakis et al. (2008).
More precisely, we prove that the existence of a dual witness which was
previously known to be sufficient for LP decoding success is also necessary and
is equivalent to the existence of certain acyclic hyperflows. We also derive a
sublinear (in the block length) upper bound on the weight of any edge in such
hyperflows, both for regular LPDC codes and for spatially coupled codes and we
prove that the bound is asymptotically tight for regular LDPC codes. Moreover,
we show how to trade crossover probability for "LP excess" on all the variable
nodes, for any binary linear code.Comment: 37 pages; Added tightness construction, expanded abstrac
Critical slowing down and hyperuniformity on approach to jamming
Hyperuniformity characterizes a state of matter that is poised at a critical
point at which density or volume-fraction fluctuations are anomalously
suppressed at infinite wavelengths. Recently, much attention has been given to
the link between strict jamming and hyperuniformity in frictionless
hard-particle packings. Doing so requires one to study very large packings,
which can be difficult to jam properly. We modify the rigorous linear
programming method of Donev et al. [J. Comp. Phys. 197, 139 (2004)] in order to
test for jamming in putatively jammed packings of hard-disks in two dimensions.
We find that various standard packing protocols struggle to reliably create
packings that are jammed for even modest system sizes; importantly, these
packings appear to be jammed by conventional tests. We present evidence that
suggests that deviations from hyperuniformity in putative maximally random
jammed (MRJ) packings can in part be explained by a shortcoming in generating
exactly-jammed configurations due to a type of "critical slowing down" as the
necessary rearrangements become difficult to realize by numerical protocols.
Additionally, various protocols are able to produce packings exhibiting
hyperuniformity to different extents, but this is because certain protocols are
better able to approach exactly-jammed configurations. Nonetheless, while one
should not generally expect exact hyperuniformity for disordered packings with
rattlers, we find that when jamming is ensured, our packings are very nearly
hyperuniform, and deviations from hyperuniformity correlate with an inability
to ensure jamming, suggesting that strict jamming and hyperuniformity are
indeed linked. This raises the possibility that the ideal MRJ packings have no
rattlers. Our work provides the impetus for the development of packing
algorithms that produce large disordered strictly jammed packings that are
rattler-free.Comment: 15 pages, 11 figures. Accepted for publication in Phys. Rev.
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