Recep Can Yavas

Recep Can Yavas, Yuqi Huang, Vincent Y. F. Tan et al.

We develop a general framework for clustering and distribution matching problems with bandit feedback. We consider a $K$-armed bandit model where some subset of $K$ arms is partitioned into $M$ groups. Within each group, the random variable associated to each arm follows the same distribution on a finite alphabet. At each time step, the decision maker pulls an arm and observes its outcome from the random variable associated to that arm. Subsequent arm pulls depend on the history of arm pulls and their outcomes. The decision maker has no knowledge of the distributions of the arms or the underlying partitions. The task is to devise an online algorithm to learn the underlying partition of arms with the least number of arm pulls on average and with an error probability not exceeding a pre-determined value~$δ$. Several existing problems fall under our general framework, including finding $M$ pairs of arms, odd arm identification, and $N$-ary clustering of $K$ arms belong to our general framework. We derive a non-asymptotic lower bound on the average number of arm pulls for any online algorithm with an error probability not exceeding $δ$. Furthermore, we develop a computationally-efficient online algorithm based on the Track-and-Stop method and Frank--Wolfe algorithm, and show that the average number of arm pulls of our algorithm asymptotically matches that of the lower bound. Our refined analysis also uncovers a novel bound on the speed at which the average number of arm pulls of our algorithm converges to the fundamental limit as $δ$ vanishes.

Recep Can Yavas, Vincent Y. F. Tan

We study the best-arm identification problem in sparse linear bandits under the fixed-budget setting. In sparse linear bandits, the unknown feature vector $θ^*$ may be of large dimension $d$, but only a few, say $s \ll d$ of these features have non-zero values. We design a two-phase algorithm, Lasso and Optimal-Design- (Lasso-OD) based linear best-arm identification. The first phase of Lasso-OD leverages the sparsity of the feature vector by applying the thresholded Lasso introduced by Zhou (2009), which estimates the support of $θ^*$ correctly with high probability using rewards from the selected arms and a judicious choice of the design matrix. The second phase of Lasso-OD applies the OD-LinBAI algorithm by Yang and Tan (2022) on that estimated support. We derive a non-asymptotic upper bound on the error probability of Lasso-OD by carefully choosing hyperparameters (such as Lasso's regularization parameter) and balancing the error probabilities of both phases. For fixed sparsity $s$ and budget $T$, the exponent in the error probability of Lasso-OD depends on $s$ but not on the dimension $d$, yielding a significant performance improvement for sparse and high-dimensional linear bandits. Furthermore, we show that Lasso-OD is almost minimax optimal in the exponent. Finally, we provide numerical examples to demonstrate the significant performance improvement over the existing algorithms for non-sparse linear bandits such as OD-LinBAI, BayesGap, Peace, LinearExploration, and GSE.

Recep Can Yavas

We study variable-length feedback (VLF) codes over binary-input memoryless symmetric (BMS) channels using posterior matching with small-enough-difference (SED) partitioning. Prior analyses of SED-based schemes rely on bounded log-likelihood ratio (LLR) increments, restricting their scope to discrete-output channels such as the binary symmetric channel (BSC). We remove this restriction and provide an analysis of posterior matching that covers a broad class of BMS channels, including continuous-output channels such as the binary-input AWGN channel. We derive a novel non-asymptotic achievability bound on the expected decoding time that decomposes into communication, confirmation, and recovery terms with explicit dependence on the channel capacity~$C$, the KL divergence~$C_1$, and the Bhattacharyya parameter of the channel. The proof develops new stopping-time and overshoot bounds for submartingales and random walks with unbounded increments, drawing on tools from renewal theory. On the algorithmic side, we propose a low-complexity encoder that enforces the exact SED partition at every step by grouping messages according to their log-likelihood ratios that are assumed to land on a lattice, and applying a batched correction step that restores the partition balance. The resulting encoder complexity is polynomial in the number of transmitted bits. For continuous-output channels, the lattice structure is enforced through output quantization satisfying an exact induced-lattice constraint; the associated capacity loss is $O(\log B / B^2)$ for a $B$-level quantizer. These results yield a VLF coding scheme for BMS channels that simultaneously achieves strong non-asymptotic performance and practical encoder complexity.