30 research outputs found
Size and depth of monotone neural networks: interpolation and approximation
Monotone functions and data sets arise in a variety of applications. We study
the interpolation problem for monotone data sets: The input is a monotone data
set with points, and the goal is to find a size and depth efficient
monotone neural network, with non negative parameters and threshold units, that
interpolates the data set. We show that there are monotone data sets that
cannot be interpolated by a monotone network of depth . On the other hand,
we prove that for every monotone data set with points in ,
there exists an interpolating monotone network of depth and size .
Our interpolation result implies that every monotone function over
can be approximated arbitrarily well by a depth-4 monotone network, improving
the previous best-known construction of depth . Finally, building on
results from Boolean circuit complexity, we show that the inductive bias of
having positive parameters can lead to a super-polynomial blow-up in the number
of neurons when approximating monotone functions.Comment: 19 page
Community detection and percolation of information in a geometric setting
We make the first steps towards generalizing the theory of stochastic block
models, in the sparse regime, towards a model where the discrete community
structure is replaced by an underlying geometry. We consider a geometric random
graph over a homogeneous metric space where the probability of two vertices to
be connected is an arbitrary function of the distance. We give sufficient
conditions under which the locations can be recovered (up to an isomorphism of
the space) in the sparse regime. Moreover, we define a geometric counterpart of
the model of flow of information on trees, due to Mossel and Peres, in which
one considers a branching random walk on a sphere and the goal is to recover
the location of the root based on the locations of leaves. We give some
sufficient conditions for percolation and for non-percolation of information in
this model.Comment: 21 page
Is this correct? Let's check!
Societal accumulation of knowledge is a complex process. The correctness of
new units of knowledge depends not only on the correctness of new reasoning,
but also on the correctness of old units that the new one builds on. The errors
in such accumulation processes are often remedied by error correction and
detection heuristics.
Motivating examples include the scientific process based on scientific
publications, and software development based on libraries of code.
Natural processes that aim to keep errors under control, such as peer review
in scientific publications, and testing and debugging in software development,
would typically check existing pieces of knowledge -- both for the reasoning
that generated them and the previous facts they rely on. In this work, we
present a simple process that models such accumulation of knowledge and study
the persistence (or lack thereof) of errors.
We consider a simple probabilistic model for the generation of new units of
knowledge based on the preferential attachment growth model, which additionally
allows for errors. Furthermore, the process includes checks aimed at catching
these errors. We investigate when effects of errors persist forever in the
system (with positive probability) and when they get rooted out completely by
the checking process.
The two basic parameters associated with the checking process are the {\em
probability} of conducting a check and the depth of the check. We show that
errors are rooted out if checks are sufficiently frequent and sufficiently
deep. In contrast, shallow or infrequent checks are insufficient to root out
errors.Comment: 29 page
Integrality gaps for random integer programs via discrepancy
We give bounds on the additive gap between the value of a random integer
program with constraints
and that of its linear programming relaxation for a range of distributions on
. Dyer and Frieze (MOR '89) and Borst et al (IPCO '21) respectively
showed that for random packing and Gaussian IPs, where the entries of are
independently distributed according to either uniform or
, that the integrality gap is bounded by with probability at least for . In this
paper, we extend these results to the case where is discretely distributed
(e.g., entries ), and where the columns of are logconcave
distributed. Second, we improve the success probability from constant, for
fixed and , to . Using a connection between
integrality gaps and Branch-and-Bound due to Dey, Dubey and Molinaro (SODA
'21), our gap results imply that Branch-and-Bound is polynomial for these IPs.
Our main technical contribution and the key for achieving the above results,
is a new discrepancy theoretic theorem which gives general conditions for when
a target is equal or very close to a combination of the columns
of a random matrix . Compared to prior results, our theorem handles a much
wider range of distributions on , both continuous and discrete, and achieves
success probability exponentially close to as opposed to constant. We prove
this lemma using a Fourier analytic approach, building on the work of Hoberg
and Rothvoss (SODA '19) and Franks and Saks (RSA '20) who studied similar
questions for combinations
Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
International audienc