17 research outputs found
Upper Tail Estimates with Combinatorial Proofs
We study generalisations of a simple, combinatorial proof of a Chernoff bound
similar to the one by Impagliazzo and Kabanets (RANDOM, 2010).
In particular, we prove a randomized version of the hitting property of
expander random walks and apply it to obtain a concentration bound for expander
random walks which is essentially optimal for small deviations and a large
number of steps. At the same time, we present a simpler proof that still yields
a "right" bound settling a question asked by Impagliazzo and Kabanets.
Next, we obtain a simple upper tail bound for polynomials with input
variables in which are not necessarily independent, but obey a certain
condition inspired by Impagliazzo and Kabanets. The resulting bound is used by
Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number
of calls in a black-box construction of a pseudorandom generator from a one-way
function.
We then show that the same technique yields the upper tail bound for the
number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph,
matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math,
2002).Comment: Full version of the paper from STACS 201
A Matrix Expander Chernoff Bound
We prove a Chernoff-type bound for sums of matrix-valued random variables
sampled via a random walk on an expander, confirming a conjecture due to
Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the
Golden-Thompson inequality which improves in some ways the inequality of
Sutter, Berta, and Tomamichel, and may be of independent interest, as well as
an adaptation of an argument for the scalar case due to Healy. Secondarily, we
also provide a generic reduction showing that any concentration inequality for
vector-valued martingales implies a concentration inequality for the
corresponding expander walk, with a weakening of parameters proportional to the
squared mixing time.Comment: Fixed a minor bug in the proof of Theorem 3.
A PCP Characterization of AM
We introduce a 2-round stochastic constraint-satisfaction problem, and show
that its approximation version is complete for (the promise version of) the
complexity class AM. This gives a `PCP characterization' of AM analogous to the
PCP Theorem for NP. Similar characterizations have been given for higher levels
of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the
result for AM might be of particular significance for attempts to derandomize
this class.
To test this notion, we pose some `Randomized Optimization Hypotheses'
related to our stochastic CSPs that (in light of our result) would imply
collapse results for AM. Unfortunately, the hypotheses appear over-strong, and
we present evidence against them. In the process we show that, if some language
in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there
exist hard-on-average optimization problems of a particularly elegant form.
All our proofs use a powerful form of PCPs known as Probabilistically
Checkable Proofs of Proximity, and demonstrate their versatility. We also use
known results on randomness-efficient soundness- and hardness-amplification. In
particular, we make essential use of the Impagliazzo-Wigderson generator; our
analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page
Deterministic Coupon Collection and Better Strong Dispersers
Hashing is one of the main techniques in data processing and algorithm design for very large data sets. While random hash functions satisfy most desirable properties, it is often too expensive to store a fully random hash function. Motivated by this, much attention has been given to designing small families of hash functions suitable for various applications. In this work, we study the question of designing space-efficient hash families H = {h:[U] -> [N]} with the natural property of \u27covering\u27: H is said to be covering if any set of Omega(N log N) distinct items from the universe (the "coupon-collector limit") are hashed to cover all N bins by most hash functions in H. We give an explicit covering family H of size poly(N) (which is optimal), so that hash functions in H can be specified efficiently by O(log N) bits.
We build covering hash functions by drawing a connection to "dispersers", which are quite well-studied and have a variety of applications themselves. We in fact need strong dispersers and we give new constructions of strong dispersers which may be of independent interest. Specifically, we construct strong dispersers with optimal entropy loss in the high min-entropy, but very small error (poly(n)/2^n for n bit sources) regimes. We also provide a strong disperser construction with constant error but for any min-entropy. Our constructions achieve these by using part of the source to replace seed from previous non-strong constructions in surprising ways. In doing so, we take two of the few constructions of dispersers with parameters better than known extractors and make them strong
Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified
We prove the first Chernoff-Hoeffding bounds for general nonreversible
finite-state Markov chains based on the standard L_1 (variation distance)
mixing-time of the chain. Specifically, consider an ergodic Markov chain M and
a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu =
E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step
random walk (v_1,...,v_t) on M starting from the stationary distribution pi has
expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the
L_1 mixing-time of M. We show that the probability of X deviating from its mean
by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is
at most exp(-Omega(delta^2 mu t / T)) for 0 <= delta <= 1, and exp(-Omega(delta
mu t / T)) for delta > 1. In fact, the bounds hold even if the weight functions
f_i's for i in [t] are distinct, provided that all of them have the same mean
mu.
We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on
the spectral expansion lambda of M, which is the square root of the second
largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the
time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t|
>= delta mu t ] is at most exp(-Omega(delta^2 (1-lambda) mu t)) for 0 <= delta
1.
Both of our results extend to continuous-time Markov chains, and to the case
where the walk starts from an arbitrary distribution x, at a price of a
multiplicative factor depending on the distribution x in the concentration
bounds
Learning to Order for Inventory Systems with Lost Sales and Uncertain Supplies
We consider a stochastic lost-sales inventory control system with a lead time
over a planning horizon . Supply is uncertain, and is a function of the
order quantity (due to random yield/capacity, etc). We aim to minimize the
-period cost, a problem that is known to be computationally intractable even
under known distributions of demand and supply. In this paper, we assume that
both the demand and supply distributions are unknown and develop a
computationally efficient online learning algorithm. We show that our algorithm
achieves a regret (i.e. the performance gap between the cost of our algorithm
and that of an optimal policy over periods) of when
. We do so by 1) showing our algorithm cost is higher by at most
for any compared to an optimal constant-order policy
under complete information (a well-known and widely-used algorithm) and 2)
leveraging its known performance guarantee from the existing literature. To the
best of our knowledge, a finite-sample (and polynomial in )
regret bound when benchmarked against an optimal policy is not known before in
the online inventory control literature. A key challenge in this learning
problem is that both demand and supply data can be censored; hence only
truncated values are observable. We circumvent this challenge by showing that
the data generated under an order quantity allows us to simulate the
performance of not only but also for all , a key
observation to obtain sufficient information even under data censoring. By
establishing a high probability coupling argument, we are able to evaluate and
compare the performance of different order policies at their steady state
within a finite time horizon. Since the problem lacks convexity, we develop an
active elimination method that adaptively rules out suboptimal solutions