30,669 research outputs found
PAC-Bayesian Bound for the Conditional Value at Risk
Conditional Value at Risk (CVaR) is a family of "coherent risk measures"
which generalize the traditional mathematical expectation. Widely used in
mathematical finance, it is garnering increasing interest in machine learning,
e.g., as an alternate approach to regularization, and as a means for ensuring
fairness. This paper presents a generalization bound for learning algorithms
that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type
and is guaranteed to be small when the empirical CVaR is small. We achieve this
by reducing the problem of estimating CVaR to that of merely estimating an
expectation. This then enables us, as a by-product, to obtain concentration
inequalities for CVaR even when the random variable in question is unbounded
Optimal Best-Arm Identification Methods for Tail-Risk Measures
Conditional value-at-risk (CVaR) and value-at-risk (VaR) are popular
tail-risk measures in finance and insurance industries as well as in highly
reliable, safety-critical uncertain environments where often the underlying
probability distributions are heavy-tailed. We use the multi-armed bandit
best-arm identification framework and consider the problem of identifying the
arm from amongst finitely many that has the smallest CVaR, VaR, or weighted sum
of CVaR and mean. The latter captures the risk-return trade-off common in
finance. Our main contribution is an optimal -correct algorithm that
acts on general arms, including heavy-tailed distributions, and matches the
lower bound on the expected number of samples needed, asymptotically (as
approaches ). The algorithm requires solving a non-convex
optimization problem in the space of probability measures, that requires
delicate analysis. En-route, we develop new non-asymptotic empirical
likelihood-based concentration inequalities for tail-risk measures which are
tighter than those for popular truncation-based empirical estimators.Comment: 55 pages, 4 figure
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Can Feedback Traders Rock the Markets? A Logistic Tale of Persistence and Chaos
This paper introduces a nonlinear feedback trading model at high frequency. All price adjustment is endogenous, driven by asset return and volatility in the previous trading period. There is no stochastic uncertainty or asymmetric information. The dynamics of expected returns display stable or unstable behaviorâincluding the possibility of turbulence and chaosâas a function of market liquidity (inverse price impact) and the concentration of investor beliefs, which is proportional to the intensity of positive feedback. The results highlight the complementary role of investor diversity and market liquidity in maintaining financial stability
Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods
We obtain a Bernstein-type inequality for sums of Banach-valued random
variables satisfying a weak dependence assumption of general type and under
certain smoothness assumptions of the underlying Banach norm. We use this
inequality in order to investigate in the asymptotical regime the error upper
bounds for the broad family of spectral regularization methods for reproducing
kernel decision rules, when trained on a sample coming from a mixing
process.Comment: 39 page
Learning the dependence structure of rare events: a non-asymptotic study
Assessing the probability of occurrence of extreme events is a crucial issue
in various fields like finance, insurance, telecommunication or environmental
sciences. In a multivariate framework, the tail dependence is characterized by
the so-called stable tail dependence function (STDF). Learning this structure
is the keystone of multivariate extremes. Although extensive studies have
proved consistency and asymptotic normality for the empirical version of the
STDF, non-asymptotic bounds are still missing. The main purpose of this paper
is to fill this gap. Taking advantage of adapted VC-type concentration
inequalities, upper bounds are derived with expected rate of convergence in
O(k^-1/2). The concentration tools involved in this analysis rely on a more
general study of maximal deviations in low probability regions, and thus
directly apply to the classification of extreme data
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