7 research outputs found
Leading strategies in competitive on-line prediction
We start from a simple asymptotic result for the problem of on-line
regression with the quadratic loss function: the class of continuous
limited-memory prediction strategies admits a "leading prediction strategy",
which not only asymptotically performs at least as well as any continuous
limited-memory strategy but also satisfies the property that the excess loss of
any continuous limited-memory strategy is determined by how closely it imitates
the leading strategy. More specifically, for any class of prediction strategies
constituting a reproducing kernel Hilbert space we construct a leading
strategy, in the sense that the loss of any prediction strategy whose norm is
not too large is determined by how closely it imitates the leading strategy.
This result is extended to the loss functions given by Bregman divergences and
by strictly proper scoring rules.Comment: 20 pages; a conference version is to appear in the ALT'2006
proceeding
Competitive on-line learning with a convex loss function
We consider the problem of sequential decision making under uncertainty in
which the loss caused by a decision depends on the following binary
observation. In competitive on-line learning, the goal is to design decision
algorithms that are almost as good as the best decision rules in a wide
benchmark class, without making any assumptions about the way the observations
are generated. However, standard algorithms in this area can only deal with
finite-dimensional (often countable) benchmark classes. In this paper we give
similar results for decision rules ranging over an arbitrary reproducing kernel
Hilbert space. For example, it is shown that for a wide class of loss functions
(including the standard square, absolute, and log loss functions) the average
loss of the master algorithm, over the first observations, does not exceed
the average loss of the best decision rule with a bounded norm plus
. Our proof technique is very different from the standard ones and
is based on recent results about defensive forecasting. Given the probabilities
produced by a defensive forecasting algorithm, which are known to be well
calibrated and to have good resolution in the long run, we use the expected
loss minimization principle to find a suitable decision.Comment: 26 page
Calibration and Internal no-Regret with Partial Monitoring
Calibrated strategies can be obtained by performing strategies that have no
internal regret in some auxiliary game. Such strategies can be constructed
explicitly with the use of Blackwell's approachability theorem, in an other
auxiliary game. We establish the converse: a strategy that approaches a convex
-set can be derived from the construction of a calibrated strategy. We
develop these tools in the framework of a game with partial monitoring, where
players do not observe the actions of their opponents but receive random
signals, to define a notion of internal regret and construct strategies that
have no such regret
On-line regression competitive with reproducing kernel Hilbert spaces
We consider the problem of on-line prediction of real-valued labels, assumed
bounded in absolute value by a known constant, of new objects from known
labeled objects. The prediction algorithm's performance is measured by the
squared deviation of the predictions from the actual labels. No stochastic
assumptions are made about the way the labels and objects are generated.
Instead, we are given a benchmark class of prediction rules some of which are
hoped to produce good predictions. We show that for a wide range of
infinite-dimensional benchmark classes one can construct a prediction algorithm
whose cumulative loss over the first N examples does not exceed the cumulative
loss of any prediction rule in the class plus O(sqrt(N)); the main differences
from the known results are that we do not impose any upper bound on the norm of
the considered prediction rules and that we achieve an optimal leading term in
the excess loss of our algorithm. If the benchmark class is "universal" (dense
in the class of continuous functions on each compact set), this provides an
on-line non-stochastic analogue of universally consistent prediction in
non-parametric statistics. We use two proof techniques: one is based on the
Aggregating Algorithm and the other on the recently developed method of
defensive forecasting.Comment: 37 pages, 1 figur
Stronger Calibration Lower Bounds via Sidestepping
We consider an online binary prediction setting where a forecaster observes a
sequence of bits one by one. Before each bit is revealed, the forecaster
predicts the probability that the bit is . The forecaster is called
well-calibrated if for each , among the bits for which the
forecaster predicts probability , the actual number of ones, , is
indeed equal to . The calibration error, defined as , quantifies the extent to which the forecaster deviates from being
well-calibrated. It has long been known that an calibration error
is achievable even when the bits are chosen adversarially, and possibly based
on the previous predictions. However, little is known on the lower bound side,
except an bound that follows from the trivial example of
independent fair coin flips.
In this paper, we prove an bound on the calibration
error, which is the first super- lower bound for this setting to the
best of our knowledge. The technical contributions of our work include two
lower bound techniques, early stopping and sidestepping, which circumvent the
obstacles that have previously hindered strong calibration lower bounds. We
also propose an abstraction of the prediction setting, termed the
Sign-Preservation game, which may be of independent interest. This game has a
much smaller state space than the full prediction setting and allows simpler
analyses. The lower bound follows from a general reduction
theorem that translates lower bounds on the game value of Sign-Preservation
into lower bounds on the calibration error.Comment: To appear in STOC 202
Predictions as statements and decisions
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