10 research outputs found
Greedy Algorithms for Optimal Distribution Approximation
The approximation of a discrete probability distribution by an
-type distribution is considered. The approximation error is
measured by the informational divergence
, which is an appropriate measure, e.g.,
in the context of data compression. Properties of the optimal approximation are
derived and bounds on the approximation error are presented, which are
asymptotically tight. It is shown that -type approximations that minimize
either , or
, or the variational distance
can all be found by using specific
instances of the same general greedy algorithm.Comment: 5 page
Channels With Cost Constraints: Strong Converse and Dispersion
This paper shows the strong converse and the dispersion of memoryless channels with cost constraints and performs a refined analysis of the third-order term in the asymptotic expansion of the maximum achievable channel coding rate, showing that it is equal to (1/2)((log n)/n) in most cases of interest. The analysis is based on a nonasymptotic converse bound expressed in terms of the distribution of a random variable termed the mathsf b -tilted information density, which plays a role similar to that of the mathsf d -tilted information in lossy source coding. We also analyze the fundamental limits of lossy joint-source-channel coding over channels with cost constraints
Bandit Social Learning: Exploration under Myopic Behavior
We study social learning dynamics where the agents collectively follow a
simple multi-armed bandit protocol. Agents arrive sequentially, choose arms and
receive associated rewards. Each agent observes the full history (arms and
rewards) of the previous agents, and there are no private signals. While
collectively the agents face exploration-exploitation tradeoff, each agent acts
myopically, without regards to exploration. Motivating scenarios concern
reviews and ratings on online platforms.
We allow a wide range of myopic behaviors that are consistent with
(parameterized) confidence intervals, including the "unbiased" behavior as well
as various behaviorial biases. While extreme versions of these behaviors
correspond to well-known bandit algorithms, we prove that more moderate
versions lead to stark exploration failures, and consequently to regret rates
that are linear in the number of agents. We provide matching upper bounds on
regret by analyzing "moderately optimistic" agents.
As a special case of independent interest, we obtain a general result on
failure of the greedy algorithm in multi-armed bandits. This is the first such
result in the literature, to the best of our knowledg
Robust Voting Rules from Algorithmic Robust Statistics
Maximum likelihood estimation furnishes powerful insights into voting theory,
and the design of voting rules. However the MLE can usually be badly corrupted
by a single outlying sample. This means that a single voter or a group of
colluding voters can vote strategically and drastically affect the outcome.
Motivated by recent progress in algorithmic robust statistics, we revisit the
fundamental problem of estimating the central ranking in a Mallows model, but
ask for an estimator that is provably robust, unlike the MLE.
Our main result is an efficiently computable estimator that achieves nearly
optimal robustness guarantees. In particular the robustness guarantees are
dimension-independent in the sense that our overall accuracy does not depend on
the number of alternatives being ranked. As an immediate consequence, we show
that while the landmark Gibbard-Satterthwaite theorem tells us a strong
impossiblity result about designing strategy-proof voting rules, there are
quantitatively strong ways to protect against large coalitions if we assume
that the remaining voters voters are honest and their preferences are sampled
from a Mallows model. Our work also makes technical contributions to
algorithmic robust statistics by designing new spectral filtering techniques
that can exploit the intricate combinatorial dependencies in the Mallows model
Sharper Bounds in Lattice-Based Cryptography using the Rényi Divergence
The Rényi divergence is a measure of divergence between distributions. It has recently found several applications in lattice-based cryptography. The contribution of this paper is twofold.
First, we give theoretic results which renders it more efficient and easier to use. This is done by providing two lemmas, which give tight bounds in very common situations { for distributions that are tailcut or have a bounded relative error. We then connect the Rényi divergence to the max-log distance. This allows the Rényi divergence to indirectly benefit from all the advantages of a distance.
Second, we apply our new results to five practical usecases. It allows us to claim 256 bits of security for a floating-point precision of 53 bits, in cases that until now either required more than 150 bits of precision or were limited to 100 bits of security: rejection sampling, trapdoor sampling (61 bits in this case) and a new sampler by Micciancio and Walter. We also propose a new and compact approach for table-based sampling, and squeeze the standard deviation of trapdoor samplers by a factor that provides a gain of 30 bits of security in practice