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Daniil Ryabko

Publications and source records attributed to Daniil Ryabko.

At least 37 records · Page 2Linked to original sources
arXiv2013

Clustering processes

The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for the case of a known number of clusters, consistency can be achieved under the only assumption that the joint distribution of the data is stationary ergodic (no parametric or Markovian assumptions, no assumptions of independence, neither between nor within the samples). If the number of clusters is unknown, consistency can be achieved under appropriate assumptions on the mixing rates of the processes. (again, no parametric or independence assumptions). In both cases we give examples of simple (at most quadratic in each argument) algorithms which are consistent.

cs.LG↗
arXiv2013

Optimal Regret Bounds for Selecting the State Representation in Reinforcement Learning

We consider an agent interacting with an environment in a single stream of actions, observations, and rewards, with no reset. This process is not assumed to be a Markov Decision Process (MDP). Rather, the agent has several representations (mapping histories of past interactions to a discrete state space) of the environment with unknown dynamics, only some of which result in an MDP. The goal is to minimize the average regret criterion against an agent who knows an MDP representation giving the highest optimal reward, and acts optimally in it. Recent regret bounds for this setting are of order $O(T^{2/3})$ with an additive term constant yet exponential in some characteristics of the optimal MDP. We propose an algorithm whose regret after $T$ time steps is $O(\sqrt{T})$, with all constants reasonably small. This is optimal in $T$ since $O(\sqrt{T})$ is the optimal regret in the setting of learning in a (single discrete) MDP.

cs.LG↗
arXiv2013

A consistent clustering-based approach to estimating the number of change-points in highly dependent time-series

The problem of change-point estimation is considered under a general framework where the data are generated by unknown stationary ergodic process distributions. In this context, the consistent estimation of the number of change-points is provably impossible. However, it is shown that a consistent clustering method may be used to estimate the number of change points, under the additional constraint that the correct number of process distributions that generate the data is provided. This additional parameter has a natural interpretation in many real-world applications. An algorithm is proposed that estimates the number of change-points and locates the changes. The proposed algorithm is shown to be asymptotically consistent; its empirical evaluations are provided.

stat.ML↗
arXiv2013

Online Regret Bounds for Undiscounted Continuous Reinforcement Learning

We derive sublinear regret bounds for undiscounted reinforcement learning in continuous state space. The proposed algorithm combines state aggregation with the use of upper confidence bounds for implementing optimism in the face of uncertainty. Beside the existence of an optimal policy which satisfies the Poisson equation, the only assumptions made are Holder continuity of rewards and transition probabilities.

cs.LG↗
arXiv2013

Selecting the State-Representation in Reinforcement Learning

The problem of selecting the right state-representation in a reinforcement learning problem is considered. Several models (functions mapping past observations to a finite set) of the observations are given, and it is known that for at least one of these models the resulting state dynamics are indeed Markovian. Without knowing neither which of the models is the correct one, nor what are the probabilistic characteristics of the resulting MDP, it is required to obtain as much reward as the optimal policy for the correct model (or for the best of the correct models, if there are several). We propose an algorithm that achieves that, with a regret of order T^{2/3} where T is the horizon time.

cs.LG↗
arXiv2012

Regret Bounds for Restless Markov Bandits

We consider the restless Markov bandit problem, in which the state of each arm evolves according to a Markov process independently of the learner's actions. We suggest an algorithm that after $T$ steps achieves $\tilde{O}(\sqrt{T})$ regret with respect to the best policy that knows the distributions of all arms. No assumptions on the Markov chains are made except that they are irreducible. In addition, we show that index-based policies are necessarily suboptimal for the considered problem.

cs.LG↗
arXiv2012

Confidence Sets in Time-Series Filtering

The problem of filtering of finite-alphabet stationary ergodic time series is considered. A method for constructing a confidence set for the (unknown) signal is proposed, such that the resulting set has the following properties: First, it includes the unknown signal with probability $γ$, where $γ$ is a parameter supplied to the filter. Second, the size of the confidence sets grows exponentially with the rate that is asymptotically equal to the conditional entropy of the signal given the data. Moreover, it is shown that this rate is optimal.

cs.IT↗
arXiv2012

Nonparametric Statistical Inference for Ergodic Processes

In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.

cs.IT↗
arXiv2011

Constructing Perfect Steganographic Systems

We propose steganographic systems for the case when covertexts (containers) are generated by a finite-memory source with possibly unknown statistics. The probability distributions of covertexts with and without hidden information are the same; this means that the proposed stegosystems are perfectly secure, i.e. an observer cannot determine whether hidden information is being transmitted. The speed of transmission of hidden information can be made arbitrary close to the theoretical limit - the Shannon entropy of the source of covertexts. An interesting feature of the suggested stegosystems is that they do not require any (secret or public) key. At the same time, we outline some principled computational limitations on steganography. We show that there are such sources of covertexts, that any stegosystem that has linear (in the length of the covertext) speed of transmission of hidden text must have an exponential Kolmogorov complexity. This shows, in particular, that some assumptions on the sources of covertext are necessary.

cs.CR↗
arXiv2010

Clustering processes

The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for the case of a known number of clusters, consistency can be achieved under the only assumption that the joint distribution of the data is stationary ergodic (no parametric or Markovian assumptions, no assumptions of independence, neither between nor within the samples). If the number of clusters is unknown, consistency can be achieved under appropriate assumptions on the mixing rates of the processes. (again, no parametric or independence assumptions). In both cases we give examples of simple (at most quadratic in each argument) algorithms which are consistent.

cs.LG↗
arXiv2009

On Finding Predictors for Arbitrary Families of Processes

The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $μ$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $μ$ belongs to an arbitrary but known class $C$ of stochastic process measures. We are interested in predictors $ρ$ whose conditional probabilities converge (in some sense) to the "true" $μ$-conditional probabilities if any $μ\in C$ is chosen to generate the sequence. The contribution of this work is in characterizing the families $C$ for which such predictors exist, and in providing a specific and simple form in which to look for a solution. We show that if any predictor works, then there exists a Bayesian predictor, whose prior is discrete, and which works too. We also find several sufficient and necessary conditions for the existence of a predictor, in terms of topological characterizations of the family $C$, as well as in terms of local behaviour of the measures in $C$, which in some cases lead to procedures for constructing such predictors. It should be emphasized that the framework is completely general: the stochastic processes considered are not required to be i.i.d., stationary, or to belong to any parametric or countable family.

cs.LG↗
arXiv2009

An impossibility result for process discrimination

Two series of binary observations $x_1,x_1,...$ and $y_1,y_2,...$ are presented: at each time $n\in\N$ we are given $x_n$ and $y_n$. It is assumed that the sequences are generated independently of each other by two B-processes. We are interested in the question of whether the sequences represent a typical realization of two different processes or of the same one. We demonstrate that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts earlier positive results on B-processes, in particular those showing that there are consistent $\bar d$-distance estimates for this class of processes.

math.PR↗
arXiv2009

Characterizing predictable classes of processes

The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $μ$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $μ$ belongs to an arbitrary class $\C$ of stochastic processes. We are interested in predictors $ρ$ whose conditional probabilities converge to the "true" $μ$-conditional probabilities if any $μ\in\C$ is chosen to generate the data. We show that if such a predictor exists, then a predictor can also be obtained as a convex combination of a countably many elements of $\C$. In other words, it can be obtained as a Bayesian predictor whose prior is concentrated on a countable set. This result is established for two very different measures of performance of prediction, one of which is very strong, namely, total variation, and the other is very weak, namely, prediction in expected average Kullback-Leibler divergence.

cs.AI↗
arXiv2009

Using Kolmogorov Complexity for Understanding Some Limitations on Steganography

Recently perfectly secure steganographic systems have been described for a wide class of sources of covertexts. The speed of transmission of secret information for these stegosystems is proportional to the length of the covertext. In this work we show that there are sources of covertexts for which such stegosystems do not exist. The key observation is that if the set of possible covertexts has a maximal Kolmogorov complexity, then a high-speed perfect stegosystem has to have complexity of the same order.

cs.CC↗
arXiv2008

On the Possibility of Learning in Reactive Environments with Arbitrary Dependence

We address the problem of reinforcement learning in which observations may exhibit an arbitrary form of stochastic dependence on past observations and actions, i.e. environments more general than (PO)MDPs. The task for an agent is to attain the best possible asymptotic reward where the true generating environment is unknown but belongs to a known countable family of environments. We find some sufficient conditions on the class of environments under which an agent exists which attains the best asymptotic reward for any environment in the class. We analyze how tight these conditions are and how they relate to different probabilistic assumptions known in reinforcement learning and related fields, such as Markov Decision Processes and mixing conditions.

cs.LG↗
arXiv2007

Using Data Compressors to Construct Rank Tests

Nonparametric rank tests for homogeneity and component independence are proposed, which are based on data compressors. For homogeneity testing the idea is to compress the binary string obtained by ordering the two joint samples and writing 0 if the element is from the first sample and 1 if it is from the second sample and breaking ties by randomization (extension to the case of multiple samples is straightforward). $H_0$ should be rejected if the string is compressed (to a certain degree) and accepted otherwise. We show that such a test obtained from an ideal data compressor is valid against all alternatives. Component independence is reduced to homogeneity testing by constructing two samples, one of which is the first half of the original and the other is the second half with one of the components randomly permuted.

cs.DS↗
arXiv2006

Provably Secure Universal Steganographic Systems

We propose a simple universal (that is, distribution--free) steganographic system in which covertexts with and without hidden texts are statistically indistinguishable. The stegosystem can be applied to any source generating i.i.d. covertexts with unknown distribution, and the hidden text is transmitted exactly, with zero probability of error. Moreover, the proposed steganographic system has two important properties. First, the rate of transmission of hidden information approaches the Shannon entropy of the covertext source as the size of blocks used for hidden text encoding tends to infinity. Second, if the size of the alphabet of the covertext source and its minentropy tend to infinity then the number of bits of hidden text per letter of covertext tends to $\log(n!)/n$ where $n$ is the (fixed) size of blocks used for hidden text encoding. The proposed stegosystem uses randomization.

cs.CR↗
arXiv2006

On Sequence Prediction for Arbitrary Measures

Suppose we are given two probability measures on the set of one-way infinite finite-alphabet sequences and consider the question when one of the measures predicts the other, that is, when conditional probabilities converge (in a certain sense) when one of the measures is chosen to generate the sequence. This question may be considered a refinement of the problem of sequence prediction in its most general formulation: for a given class of probability measures, does there exist a measure which predicts all of the measures in the class? To address this problem, we find some conditions on local absolute continuity which are sufficient for prediction and which generalize several different notions which are known to be sufficient for prediction. We also formulate some open questions to outline a direction for finding the conditions on classes of measures for which prediction is possible.

cs.LG↗