406 research outputs found
Sparsity regret bounds for individual sequences in online linear regression
We consider the problem of online linear regression on arbitrary deterministic sequences when the ambient dimension d can be much larger than the number of time rounds T. We introduce the notion of sparsity regret bound, which is a deterministic online counterpart of recent risk bounds derived in the stochastic setting under a sparsity scenario. We prove such regret bounds for an online-learning algorithm called SeqSEW and based on exponential weighting and data-driven truncation. In a second part we apply a parameter-free version of this algorithm to the stochastic setting (regression model with random design). This yields risk bounds of the same flavor as in Dalalyan and Tsybakov (2011) but which solve two questions left open therein. In particular our risk bounds are adaptive (up to a logarithmic factor) to the unknown variance of the noise if the latter is Gaussian. We also address the regression model with fixed design
Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case
We demonstrate that, in the classical non-stochastic regret minimization
problem with decisions, gains and losses to be respectively maximized or
minimized are fundamentally different. Indeed, by considering the additional
sparsity assumption (at each stage, at most decisions incur a nonzero
outcome), we derive optimal regret bounds of different orders. Specifically,
with gains, we obtain an optimal regret guarantee after stages of order
, so the classical dependency in the dimension is replaced by
the sparsity size. With losses, we provide matching upper and lower bounds of
order , which is decreasing in . Eventually, we also
study the bandit setting, and obtain an upper bound of order when outcomes are losses. This bound is proven to be optimal up to the
logarithmic factor
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
Fast rates in learning with dependent observations
In this paper we tackle the problem of fast rates in time series forecasting
from a statistical learning perspective. In a serie of papers (e.g. Meir 2000,
Modha and Masry 1998, Alquier and Wintenberger 2012) it is shown that the main
tools used in learning theory with iid observations can be extended to the
prediction of time series. The main message of these papers is that, given a
family of predictors, we are able to build a new predictor that predicts the
series as well as the best predictor in the family, up to a remainder of order
. It is known that this rate cannot be improved in general. In this
paper, we show that in the particular case of the least square loss, and under
a strong assumption on the time series (phi-mixing) the remainder is actually
of order . Thus, the optimal rate for iid variables, see e.g. Tsybakov
2003, and individual sequences, see \cite{lugosi} is, for the first time,
achieved for uniformly mixing processes. We also show that our method is
optimal for aggregating sparse linear combinations of predictors
Sparse Stochastic Bandits
In the classical multi-armed bandit problem, d arms are available to the
decision maker who pulls them sequentially in order to maximize his cumulative
reward. Guarantees can be obtained on a relative quantity called regret, which
scales linearly with d (or with sqrt(d) in the minimax sense). We here consider
the sparse case of this classical problem in the sense that only a small number
of arms, namely s < d, have a positive expected reward. We are able to leverage
this additional assumption to provide an algorithm whose regret scales with s
instead of d. Moreover, we prove that this algorithm is optimal by providing a
matching lower bound - at least for a wide and pertinent range of parameters
that we determine - and by evaluating its performance on simulated data
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