Dear all, the next GSSI Math Colloquium will be held on Thursday
January 28 at3pm (Italian time).
The speaker is Anders
Hansen, with a lecture connecting computational mathematics
with deep learning and AI. More details below.
Anders Hansen is Associate Professor at University of Cambridge,
where he leads the Applied Functional and Harmonic Analysis group,
and Full Professor of Mathematics at the University of Oslo.
Please feel free to distribute this announcement as you see fit.
Looking forward to seeing you all on Thursday!
Paolo Antonelli, Stefano Marchesani, Francesco Tudisco and Francesco
Viola
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Title: On the foundations of computational mathematics, Smale's 18th
problem and the potential limits of AI
Abstract:
There is a profound optimism on the impact of deep learning (DL) and
AI in the sciences with Geoffrey Hinton concluding that 'They should
stop training radiologists now'. However, DL has an Achilles heel:
it is universally unstable so that small changes in the initial data
can lead to large errors in the final result. This has been
documented in a wide variety of applications. Paradoxically, the
existence of stable neural networks for these applications is
guaranteed by the celebrated Universal Approximation Theorem,
however, the stable neural networks are never computed by the
current training approaches. We will address this problem and the
potential limitations of AI from a foundations point of view.
Indeed, the current situation in AI is comparable to the situation
in mathematics in the early 20th century, when David Hilbert’s
optimism (typically reflected in his 10th problem) suggested no
limitations to what mathematics could prove and no restrictions on
what computers could compute. Hilbert’s optimism was turned upside
down by Goedel and Turing, who established limitations on what
mathematics can prove and which problems computers can solve
(however, without limiting the impact of mathematics and computer
science).
We predict a similar outcome for modern AI and DL, where the
limitations of AI (the main topic of Smale’s 18th problem) will be
established through the foundations of computational mathematics. We
sketch the beginning of such a program by demonstrating how there
exist neural networks approximating classical mappings in scientific
computing, however, no algorithm (even randomised) can compute such
a network to even 1-digit accuracy (with probability better than
1/2). We will also show how instability is inherit in the
methodology of DL demonstrating that there is no easy remedy, given
the current methodology. Finally, we will demonstrate basic examples
in inverse problems where there exists (untrained) neural networks
that can easily compute a solution to the problem, however, the
current DL techniques will need 10^80 data points in the training
set to get even 1% success rate.
—
Francesco Tudisco
Assistant Professor
School of Mathematics
GSSI Gran Sasso Science Institute
Web: https://ftudisco.gitlab.io