527 research outputs found
Second-order Quantile Methods for Experts and Combinatorial Games
We aim to design strategies for sequential decision making that adjust to the
difficulty of the learning problem. We study this question both in the setting
of prediction with expert advice, and for more general combinatorial decision
tasks. We are not satisfied with just guaranteeing minimax regret rates, but we
want our algorithms to perform significantly better on easy data. Two popular
ways to formalize such adaptivity are second-order regret bounds and quantile
bounds. The underlying notions of 'easy data', which may be paraphrased as "the
learning problem has small variance" and "multiple decisions are useful", are
synergetic. But even though there are sophisticated algorithms that exploit one
of the two, no existing algorithm is able to adapt to both.
In this paper we outline a new method for obtaining such adaptive algorithms,
based on a potential function that aggregates a range of learning rates (which
are essential tuning parameters). By choosing the right prior we construct
efficient algorithms and show that they reap both benefits by proving the first
bounds that are both second-order and incorporate quantiles
Adaptation to Easy Data in Prediction with Limited Advice
We derive an online learning algorithm with improved regret guarantees for
`easy' loss sequences. We consider two types of `easiness': (a) stochastic loss
sequences and (b) adversarial loss sequences with small effective range of the
losses. While a number of algorithms have been proposed for exploiting small
effective range in the full information setting, Gerchinovitz and Lattimore
[2016] have shown the impossibility of regret scaling with the effective range
of the losses in the bandit setting. We show that just one additional
observation per round is sufficient to circumvent the impossibility result. The
proposed Second Order Difference Adjustments (SODA) algorithm requires no prior
knowledge of the effective range of the losses, , and achieves an
expected regret guarantee, where is the time horizon and is the number
of actions. The scaling with the effective loss range is achieved under
significantly weaker assumptions than those made by Cesa-Bianchi and Shamir
[2018] in an earlier attempt to circumvent the impossibility result. We also
provide a regret lower bound of , which almost
matches the upper bound. In addition, we show that in the stochastic setting
SODA achieves an pseudo-regret bound that holds simultaneously
with the adversarial regret guarantee. In other words, SODA is safe against an
unrestricted oblivious adversary and provides improved regret guarantees for at
least two different types of `easiness' simultaneously.Comment: Fixed a mistake in the proof and statement of Theorem
Combining Adversarial Guarantees and Stochastic Fast Rates in Online Learning
We consider online learning algorithms that guarantee worst-case regret rates
in adversarial environments (so they can be deployed safely and will perform
robustly), yet adapt optimally to favorable stochastic environments (so they
will perform well in a variety of settings of practical importance). We
quantify the friendliness of stochastic environments by means of the well-known
Bernstein (a.k.a. generalized Tsybakov margin) condition. For two recent
algorithms (Squint for the Hedge setting and MetaGrad for online convex
optimization) we show that the particular form of their data-dependent
individual-sequence regret guarantees implies that they adapt automatically to
the Bernstein parameters of the stochastic environment. We prove that these
algorithms attain fast rates in their respective settings both in expectation
and with high probability
High-Dimensional Prediction for Sequential Decision Making
We study the problem of making predictions of an adversarially chosen
high-dimensional state that are unbiased subject to an arbitrary collection of
conditioning events, with the goal of tailoring these events to downstream
decision makers. We give efficient algorithms for solving this problem, as well
as a number of applications that stem from choosing an appropriate set of
conditioning events.
For example, we can efficiently make predictions targeted at polynomially
many decision makers, giving each of them optimal swap regret if they
best-respond to our predictions. We generalize this to online combinatorial
optimization, where the decision makers have a very large action space, to give
the first algorithms offering polynomially many decision makers no regret on
polynomially many subsequences that may depend on their actions and the
context. We apply these results to get efficient no-subsequence-regret
algorithms in extensive-form games (EFGs), yielding a new family of regret
guarantees for EFGs that generalizes some existing EFG regret notions, e.g.
regret to informed causal deviations, and is generally incomparable to other
known such notions.
Next, we develop a novel transparent alternative to conformal prediction for
building valid online adversarial multiclass prediction sets. We produce class
scores that downstream algorithms can use for producing valid-coverage
prediction sets, as if these scores were the true conditional class
probabilities. We show this implies strong conditional validity guarantees
including set-size-conditional and multigroup-fair coverage for polynomially
many downstream prediction sets. Moreover, our class scores can be guaranteed
to have improved loss, cross-entropy loss, and generally any Bregman
loss, compared to any collection of benchmark models, yielding a
high-dimensional real-valued version of omniprediction.Comment: Added references, Arxiv abstract edite
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