864 research outputs found
On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters
Kalman filter is a key tool for time-series forecasting and analysis. We show
that the dependence of a prediction of Kalman filter on the past is decaying
exponentially, whenever the process noise is non-degenerate. Therefore, Kalman
filter may be approximated by regression on a few recent observations.
Surprisingly, we also show that having some process noise is essential for the
exponential decay. With no process noise, it may happen that the forecast
depends on all of the past uniformly, which makes forecasting more difficult.
Based on this insight, we devise an on-line algorithm for improper learning
of a linear dynamical system (LDS), which considers only a few most recent
observations. We use our decay results to provide the first regret bounds
w.r.t. to Kalman filters within learning an LDS. That is, we compare the
results of our algorithm to the best, in hindsight, Kalman filter for a given
signal. Also, the algorithm is practical: its per-update run-time is linear in
the regression depth
Unconstrained Dynamic Regret via Sparse Coding
Motivated by the challenge of nonstationarity in sequential decision making,
we study Online Convex Optimization (OCO) under the coupling of two problem
structures: the domain is unbounded, and the comparator sequence
is arbitrarily time-varying. As no algorithm can guarantee low
regret simultaneously against all comparator sequences, handling this setting
requires moving from minimax optimality to comparator adaptivity. That is,
sensible regret bounds should depend on certain complexity measures of the
comparator relative to one's prior knowledge.
This paper achieves a new type of these adaptive regret bounds via a sparse
coding framework. The complexity of the comparator is measured by its energy
and its sparsity on a user-specified dictionary, which offers considerable
versatility. Equipped with a wavelet dictionary for example, our framework
improves the state-of-the-art bound (Jacobsen & Cutkosky, 2022) by adapting to
both () the magnitude of the comparator average , rather than the maximum ; and ()
the comparator variability , rather than the
uncentered sum . Furthermore, our analysis is simpler due
to decoupling function approximation from regret minimization.Comment: Split the two results from the previous version. Expanded the results
on Haar wavelets. Improved writin
Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States
Consider an online convex optimization problem where the loss functions are
self-concordant barriers, smooth relative to a convex function , and
possibly non-Lipschitz. We analyze the regret of online mirror descent with
. Then, based on the result, we prove the following in a unified manner.
Denote by the time horizon and the parameter dimension. 1. For online
portfolio selection, the regret of , a variant of
exponentiated gradient due to Helmbold et al., is when . This improves on the original regret bound for . 2. For online portfolio
selection, the regret of online mirror descent with the logarithmic barrier is
. The regret bound is the same as that of Soft-Bayes due
to Orseau et al. up to logarithmic terms. 3. For online learning quantum states
with the logarithmic loss, the regret of online mirror descent with the
log-determinant function is also . Its per-iteration
time is shorter than all existing algorithms we know.Comment: 19 pages, 1 figur
Improved Online Conformal Prediction via Strongly Adaptive Online Learning
We study the problem of uncertainty quantification via prediction sets, in an
online setting where the data distribution may vary arbitrarily over time.
Recent work develops online conformal prediction techniques that leverage
regret minimization algorithms from the online learning literature to learn
prediction sets with approximately valid coverage and small regret. However,
standard regret minimization could be insufficient for handling changing
environments, where performance guarantees may be desired not only over the
full time horizon but also in all (sub-)intervals of time. We develop new
online conformal prediction methods that minimize the strongly adaptive regret,
which measures the worst-case regret over all intervals of a fixed length. We
prove that our methods achieve near-optimal strongly adaptive regret for all
interval lengths simultaneously, and approximately valid coverage. Experiments
show that our methods consistently obtain better coverage and smaller
prediction sets than existing methods on real-world tasks, such as time series
forecasting and image classification under distribution shift
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