Alhad Sethi

Alhad Sethi, Kavali Sofia Sagar, Shubhada Agrawal et al.

We study one-sided and $α$-correct sequential hypothesis testing for data generated by an ergodic Markov chain. The null hypothesis is that the unknown transition matrix belongs to a prescribed set $P$ of stochastic matrices, and the alternative corresponds to a disjoint set $Q$. We establish a tight non-asymptotic instance-dependent lower bound on the expected stopping time of any valid sequential test under the alternative. Our novel analysis improves the existing lower bounds, which are either asymptotic or provably sub-optimal in this setting. Our lower bound incorporates both the stationary distribution and the transition structure induced by the unknown Markov chain. We further propose an optimal test whose expected stopping time matches this lower bound asymptotically as $α\to 0$. We illustrate the usefulness of our framework through applications to sequential detection of model misspecification in Markov Chain Monte Carlo and to testing structural properties, such as the linearity of transition dynamics, in Markov decision processes. Our findings yield a sharp and general characterization of optimal sequential testing procedures under Markovian dependence.

Manuj Mukherjee, Sagnik Chatterjee, Alhad Sethi

Nitinawarat and Narayan proposed a perfect secret key generation scheme for the so-called \emph{pairwise independent network (PIN) model} by exploiting the combinatorial properties of the underlying graph, namely the spanning tree packing rate. This work considers a generalization of the PIN model where the underlying graph is replaced with a hypergraph, and makes progress towards designing similar perfect secret key generation schemes by exploiting the combinatorial properties of the hypergraph. Our contributions are two-fold. We first provide a capacity achieving scheme for a complete $t$-uniform hypergraph on $m$ vertices by leveraging a packing of the complete $t$-uniform hypergraphs by what we refer to as star hypergraphs, and designing a scheme that gives $\binom{m-2}{t-2}$ bits of perfect secret key per star graph. Our second contribution is a 2-bit perfect secret key generation scheme for 3-uniform star hypergraphs whose projections are cycles. This scheme is then extended to a perfect secret key generation scheme for generic 3-uniform hypergraphs by exploiting star graph packing of 3-uniform hypergraphs and Hamiltonian packings of graphs. The scheme is then shown to be capacity achieving for certain classes of hypergraphs.

Sagnik Chatterjee, Manuj Mukherjee, Alhad Sethi

In this work, we upper bound the generalization error of batch learning algorithms trained on samples drawn from a mixing stochastic process (i.e., a dependent data source) both in expectation and with high probability. Unlike previous results by Mohri et al. (2010) and Fu et al. (2023), our work does not require any stability assumptions on the batch learner, which allows us to derive upper bounds for any batch learning algorithm trained on dependent data. This is made possible due to our use of the Online-to-Batch ( OTB ) conversion framework, which allows us to shift the burden of stability from the batch learner to an artificially constructed online learner. We show that our bounds are equal to the bounds in the i.i.d. setting up to a term that depends on the decay rate of the underlying mixing stochastic process. Central to our analysis is a new notion of algorithmic stability for online learning algorithms based on Wasserstein distances of order one. Furthermore, we prove that the EWA algorithm, a textbook family of online learning algorithms, satisfies our new notion of stability. Following this, we instantiate our bounds using the EWA algorithm.