arXiv subjects

Yuval Peres

Publications and source records attributed to Yuval Peres.

At least 145 records · Page 8Linked to original sources

arXiv2014

Online Learning with Composite Loss Functions

We study a new class of online learning problems where each of the online algorithm's actions is assigned an adversarial value, and the loss of the algorithm at each step is a known and deterministic function of the values assigned to its recent actions. This class includes problems where the algorithm's loss is the minimum over the recent adversarial values, the maximum over the recent values, or a linear combination of the recent values. We analyze the minimax regret of this class of problems when the algorithm receives bandit feedback, and prove that when the minimum or maximum functions are used, the minimax regret is $\tilde Ω(T^{2/3})$ (so called hard online learning problems), and when a linear function is used, the minimax regret is $\tilde O(\sqrt{T})$ (so called easy learning problems). Previously, the only online learning problem that was known to be provably hard was the multi-armed bandit with switching costs.

cs.LG↗

arXiv2014

Permuted Random Walk Exits Typically in Linear Time

Given a permutation sigma of the integers {-n,-n+1,...,n} we consider the Markov chain X_{sigma}, which jumps from k to sigma (k\pm 1) equally likely if k\neq -n,n. We prove that the expected hitting time of {-n,n} starting from any point is Theta(n) with high probability when sigma is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.

math.PR↗

arXiv2014

Wald for non-stopping times: The rewards of impatient prophets

Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative random variables. Wald's identity states that the random sum $S_T:=X_1+\cdots+X_T$ has expectation $E(T)) E(X_1)$ provided $T$ is a stopping time. We prove here that for any $1<α\leq 2$, if $T$ is an arbitrary nonnegative random variable, then $S_T$ has finite expectation provided that $X_1$ has finite $α$-moment and $T$ has finite $1/(α-1)$-moment. We also prove a variant in which $T$ is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d.\ sequence $X_i$ violating them, there is a $T$ satisfying the given condition for which $S_T$ (and, in fact, $X_T$) has infinite expectation. An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.

math.PR↗

arXiv2014

Martingale defocusing and transience of a self-interacting random walk

Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.

math.PR↗

arXiv2013

Escape rates for rotor walk in Z^d

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z^d for the initial configuration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order n in dimensions d>=3, and of order n/log(n) in dimension 2.

math.PR↗

arXiv2013

Bandits with Switching Costs: T^{2/3} Regret

We study the adversarial multi-armed bandit problem in a setting where the player incurs a unit cost each time he switches actions. We prove that the player's $T$-round minimax regret in this setting is $\widetildeΘ(T^{2/3})$, thereby closing a fundamental gap in our understanding of learning with bandit feedback. In the corresponding full-information version of the problem, the minimax regret is known to grow at a much slower rate of $Θ(\sqrt{T})$. The difference between these two rates provides the \emph{first} indication that learning with bandit feedback can be significantly harder than learning with full-information feedback (previous results only showed a different dependence on the number of actions, but not on $T$.) In addition to characterizing the inherent difficulty of the multi-armed bandit problem with switching costs, our results also resolve several other open problems in online learning. One direct implication is that learning with bandit feedback against bounded-memory adaptive adversaries has a minimax regret of $\widetildeΘ(T^{2/3})$. Another implication is that the minimax regret of online learning in adversarial Markov decision processes (MDPs) is $\widetildeΘ(T^{2/3})$. The key to all of our results is a new randomized construction of a multi-scale random walk, which is of independent interest and likely to prove useful in additional settings.

cs.LG↗

arXiv2013

Dimension of Fractional Brownian motion with variable drift

Let $X$ be a fractional Brownian motion in $\mathbb{R}^d$. For any Borel function $f:[0,1] \to \mathbb{R}^d$, we express the Hausdorff dimension of the image and the graph of $X+f$ in terms of $f$. This is new even for the case of Brownian motion and continuous $f$, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of $f$ is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dimension of the graph of $X+f$ in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of $f$ and that of $X$. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of $X+f$ can disagree.

math.PR↗

arXiv2013

Harmonic maps on amenable groups and a diffusive lower bound for random walks

We prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs. Similar estimates hold for finite graphs, up to the relaxation time of the walk. Our approach uses nonconstant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok-Korevaar-Schoen theorem on the existence of such harmonic maps by constructing them from the heat flow on a Følner set.

math.PR↗

arXiv2013

Mixing time for the Ising model: a uniform lower bound for all graphs

Consider Glauber dynamics for the Ising model on a graph of $n$ vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least $n\log n/f(Δ)$, where $Δ$ is the maximum degree and $f(Δ) = Θ(Δ\log^2 Δ)$. Their result applies to more general spin systems, and in that generality, they showed that some dependence on $Δ$ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any $n$-vertex graph is at least $(1/4+o(1))n \log n$.

math.PR↗

arXiv2013

Markov type and threshold embeddings

For two metric spaces X and Y, say that X {threshold-embeds} into Y if there exist a number K > 0 and a family of Lipschitz maps $f_τ : X \to Y : τ> 0 \}$ such that for every $x,y \in X$, \[ d_X(x,y) \geq τ=> d_Y(f_τ(x),f_τ(y)) \geq \|φ_τ\|_{\Lip} τ/K \] where $\|f_τ\|_{\Lip}$ denotes the Lipschitz constant of $f_τ$. We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. This suggests some non-linear analogs of Kwapien's theorem. For instance, a subset $X \subseteq L_1$ threshold-embeds into Hilbert space if and only if X has Markov type 2.

math.MG↗

arXiv2013

Localization for controlled random walks and martingales

We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that P(S_n^u=0) decays at polynomial rate n^{-α} where α>0 can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.

math.PR↗

arXiv2013

Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate μ while at the same time a random walker moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side length n, we prove that in the subcritical regime, the mixing times for both the full system and the random walker are n^2/μ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice Z^d holds for this model as well.

math.PR↗

arXiv2013

Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures

Let X_1 ,..., X_n be a collection of binary valued random variables and let f : {0,1}^n -> R be a Lipschitz function. Under a negative dependence hypothesis known as the {\em strong Rayleigh} condition, we show that f - E f satisfies a concentration inequality generalizing the classical Gaussian concentration inequality for sums of independent Bernoullis: P (S_n - E S_n > a) < exp (-2 a^2 / n). The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and spanning tree measures. For instance, the number of vertices of odd degree in a uniform random spanning tree of a graph satisfies a Gaussian concentration inequality with n replaced by |V|, the number of vertices. We also prove a continuous version for concentration of Lipschitz functionals of a determinantal point process.

math.PR↗

arXiv2013

Total variation cutoff in a tree

We construct a family of trees on which a lazy simple random walk exhibits total variation cutoff. The main idea behind the construction is that hitting times of large sets should be concentrated around their means. For this sequence of trees we compute the mixing time, the relaxation time and the cutoff window.

math.PR↗

arXiv2013

Chip-Firing and Rotor-Routing on Directed Graphs

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

math.CO↗

arXiv2013

Mixing times are hitting times of large sets

We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. We relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states x and large sets A of the hitting time of A starting from x. We also prove that the first time when averaging over two consecutive time steps is close to stationarity is equivalent to the mixing time of the lazy version of the chain.

math.PR↗

arXiv2013

Sensitivity of mixing times

In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded factors in a certain way.

math.PR↗

arXiv2012

Decayed MCMC Filtering

Filtering---estimating the state of a partially observable Markov process from a sequence of observations---is one of the most widely studied problems in control theory, AI, and computational statistics. Exact computation of the posterior distribution is generally intractable for large discrete systems and for nonlinear continuous systems, so a good deal of effort has gone into developing robust approximation algorithms. This paper describes a simple stochastic approximation algorithm for filtering called {em decayed MCMC}. The algorithm applies Markov chain Monte Carlo sampling to the space of state trajectories using a proposal distribution that favours flips of more recent state variables. The formal analysis of the algorithm involves a generalization of standard coupling arguments for MCMC convergence. We prove that for any ergodic underlying Markov process, the convergence time of decayed MCMC with inverse-polynomial decay remains bounded as the length of the observation sequence grows. We show experimentally that decayed MCMC is at least competitive with other approximation algorithms such as particle filtering.

cs.AI↗