arXiv subjects

Debabrota Basu

Publications and source records attributed to Debabrota Basu.

60 records · Page 4Linked to original sources

arXiv2020

Differential Privacy at Risk: Bridging Randomness and Privacy Budget

The calibration of noise for a privacy-preserving mechanism depends on the sensitivity of the query and the prescribed privacy level. A data steward must make the non-trivial choice of a privacy level that balances the requirements of users and the monetary constraints of the business entity. We analyse roles of the sources of randomness, namely the explicit randomness induced by the noise distribution and the implicit randomness induced by the data-generation distribution, that are involved in the design of a privacy-preserving mechanism. The finer analysis enables us to provide stronger privacy guarantees with quantifiable risks. Thus, we propose privacy at risk that is a probabilistic calibration of privacy-preserving mechanisms. We provide a composition theorem that leverages privacy at risk. We instantiate the probabilistic calibration for the Laplace mechanism by providing analytical results. We also propose a cost model that bridges the gap between the privacy level and the compensation budget estimated by a GDPR compliant business entity. The convexity of the proposed cost model leads to a unique fine-tuning of privacy level that minimises the compensation budget. We show its effectiveness by illustrating a realistic scenario that avoids overestimation of the compensation budget by using privacy at risk for the Laplace mechanism. We quantitatively show that composition using the cost optimal privacy at risk provides stronger privacy guarantee than the classical advanced composition.

cs.CR↗

arXiv2020

Construction and Random Generation of Hypergraphs with Prescribed Degree and Dimension Sequences

We propose algorithms for construction and random generation of hypergraphs without loops and with prescribed degree and dimension sequences. The objective is to provide a starting point for as well as an alternative to Markov chain Monte Carlo approaches. Our algorithms leverage the transposition of properties and algorithms devised for matrices constituted of zeros and ones with prescribed row- and column-sums to hypergraphs. The construction algorithm extends the applicability of Markov chain Monte Carlo approaches when the initial hypergraph is not provided. The random generation algorithm allows the development of a self-normalised importance sampling estimator for hypergraph properties such as the average clustering coefficient.We prove the correctness of the proposed algorithms. We also prove that the random generation algorithm generates any hypergraph following the prescribed degree and dimension sequences with a non-zero probability. We empirically and comparatively evaluate the effectiveness and efficiency of the random generation algorithm. Experiments show that the random generation algorithm provides stable and accurate estimates of average clustering coefficient, and also demonstrates a better effective sample size in comparison with the Markov chain Monte Carlo approaches.

cs.DS↗

arXiv2019

Near-optimal Optimistic Reinforcement Learning using Empirical Bernstein Inequalities

We study model-based reinforcement learning in an unknown finite communicating Markov decision process. We propose a simple algorithm that leverages a variance based confidence interval. We show that the proposed algorithm, UCRL-V, achieves the optimal regret $\tilde{\mathcal{O}}(\sqrt{DSAT})$ up to logarithmic factors, and so our work closes a gap with the lower bound without additional assumptions on the MDP. We perform experiments in a variety of environments that validates the theoretical bounds as well as prove UCRL-V to be better than the state-of-the-art algorithms.

cs.LG↗

arXiv2019

Near-optimal Bayesian Solution For Unknown Discrete Markov Decision Process

We tackle the problem of acting in an unknown finite and discrete Markov Decision Process (MDP) for which the expected shortest path from any state to any other state is bounded by a finite number $D$. An MDP consists of $S$ states and $A$ possible actions per state. Upon choosing an action $a_t$ at state $s_t$, one receives a real value reward $r_t$, then one transits to a next state $s_{t+1}$. The reward $r_t$ is generated from a fixed reward distribution depending only on $(s_t, a_t)$ and similarly, the next state $s_{t+1}$ is generated from a fixed transition distribution depending only on $(s_t, a_t)$. The objective is to maximize the accumulated rewards after $T$ interactions. In this paper, we consider the case where the reward distributions, the transitions, $T$ and $D$ are all unknown. We derive the first polynomial time Bayesian algorithm, BUCRL{} that achieves up to logarithm factors, a regret (i.e the difference between the accumulated rewards of the optimal policy and our algorithm) of the optimal order $\tilde{\mathcal{O}}(\sqrt{DSAT})$. Importantly, our result holds with high probability for the worst-case (frequentist) regret and not the weaker notion of Bayesian regret. We perform experiments in a variety of environments that demonstrate the superiority of our algorithm over previous techniques. Our work also illustrates several results that will be of independent interest. In particular, we derive a sharper upper bound for the KL-divergence of Bernoulli random variables. We also derive sharper upper and lower bounds for Beta and Binomial quantiles. All the bound are very simple and only use elementary functions.

cs.LG↗

arXiv2019

BelMan: Bayesian Bandits on the Belief--Reward Manifold

We propose a generic, Bayesian, information geometric approach to the exploration--exploitation trade-off in multi-armed bandit problems. Our approach, BelMan, uniformly supports pure exploration, exploration--exploitation, and two-phase bandit problems. The knowledge on bandit arms and their reward distributions is summarised by the barycentre of the joint distributions of beliefs and rewards of the arms, the \emph{pseudobelief-reward}, within the beliefs-rewards manifold. BelMan alternates \emph{information projection} and \emph{reverse information projection}, i.e., projection of the pseudobelief-reward onto beliefs-rewards to choose the arm to play, and projection of the resulting beliefs-rewards onto the pseudobelief-reward. It introduces a mechanism that infuses an exploitative bias by means of a \emph{focal distribution}, i.e., a reward distribution that gradually concentrates on higher rewards. Comparative performance evaluation with state-of-the-art algorithms shows that BelMan is not only competitive but can also outperform other approaches in specific setups, for instance involving many arms and continuous rewards.

cs.LG↗

arXiv2019

Topological Data Analysis with $ε$-net Induced Lazy Witness Complex

Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of topological data analysis was curbed by the computational cost of constructing such simplicial representations. The lazy witness complex is a computationally feasible approximation of the underlying topological structure of a point cloud. It is built in reference to a subset of points, called landmarks, rather than considering all the points as in the Čech and Vietoris-Rips complexes. The choice and the number of landmarks dictate the effectiveness and efficiency of the approximation. We adopt the notion of $ε$-cover to define $ε$-net. We prove that $ε$-net, as a choice of landmarks, is an $ε$-approximate representation of the point cloud and the induced lazy witness complex is a $3$-approximation of the induced Vietoris-Rips complex. Furthermore, we propose three algorithms to construct $ε$-net landmarks. We establish the relationship of these algorithms with the existing landmark selection algorithms. We empirically validate our theoretical claims. We empirically and comparatively evaluate the effectiveness, efficiency, and stability of the proposed algorithms on synthetic and real datasets.

cs.CG↗