222,131 research outputs found
On the power of foundation models
With infinitely many high-quality data points, infinite computational power,
an infinitely large foundation model with a perfect training algorithm and
guaranteed zero generalization error on the pretext task, can the model be used
for everything? This question cannot be answered by the existing theory of
representation, optimization or generalization, because the issues they mainly
investigate are assumed to be nonexistent here. In this paper, we show that
category theory provides powerful machinery to answer this question. We have
proved three results. The first one limits the power of prompt-based learning,
saying that the model can solve a downstream task with prompts if and only if
the task is representable. The second one says fine tuning does not have this
limit, as a foundation model with the minimum required power (up to symmetry)
can theoretically solve downstream tasks with fine tuning and enough resources.
Our final result can be seen as a new type of generalization theorem, showing
that the foundation model can generate unseen objects from the target category
(e.g., images) using the structural information from the source category (e.g.,
texts). Along the way, we provide a categorical framework for supervised and
self-supervised learning, which might be of independent interest
The Markowitz Category
We give an algebraic definition of a Markowitz market and classify markets up
to isomorphism. Given this classification, the theory of portfolio optimization
in Markowitz markets without short selling constraints becomes trivial.
Conversely, this classification shows that, up to isomorphism, there is little
that can be said about a Markowitz market that is not already detected by the
theory of portfolio optimization. In particular, if one seeks to develop a
simplified low-dimensional model of a large financial market using
mean--variance analysis alone, the resulting model can be at most
two-dimensional.Comment: 1 figur
The Design of the Fifth Answer Set Programming Competition
Answer Set Programming (ASP) is a well-established paradigm of declarative
programming that has been developed in the field of logic programming and
nonmonotonic reasoning. Advances in ASP solving technology are customarily
assessed in competition events, as it happens for other closely-related
problem-solving technologies like SAT/SMT, QBF, Planning and Scheduling. ASP
Competitions are (usually) biennial events; however, the Fifth ASP Competition
departs from tradition, in order to join the FLoC Olympic Games at the Vienna
Summer of Logic 2014, which is expected to be the largest event in the history
of logic. This edition of the ASP Competition series is jointly organized by
the University of Calabria (Italy), the Aalto University (Finland), and the
University of Genova (Italy), and is affiliated with the 30th International
Conference on Logic Programming (ICLP 2014). It features a completely
re-designed setup, with novelties involving the design of tracks, the scoring
schema, and the adherence to a fixed modeling language in order to push the
adoption of the ASP-Core-2 standard. Benchmark domains are taken from past
editions, and best system packages submitted in 2013 are compared with new
versions and solvers.
To appear in Theory and Practice of Logic Programming (TPLP).Comment: 10 page
Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers
In this work, using Moreau envelopes, we define a complete metric for the set
of proper lower semicontinuous convex functions. Under this metric, the
convergence of each sequence of convex functions is epi-convergence. We show
that the set of strongly convex functions is dense but it is only of the first
category. On the other hand, it is shown that the set of convex functions with
strong minima is of the second category
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