Deep Reinforcement Learning (RL) has demonstrated success in solving complex sequential decision-making problems by integrating neural networks with the RL framework. However, training deep RL models poses several challenges, such as the need for extensive hyperparameter tuning and high computational costs. Transfer learning has emerged as a promising strategy to address these challenges by enabling the reuse of knowledge from previously learned tasks for new, related tasks. This avoids the need for retraining models entirely from scratch. A commonly used approach for transfer learning in RL is to leverage the internal representations learned by the neural network during training. Specifically, the activations from the last hidden layer can be viewed as refined state representations that encapsulate the essential features of the input. In this work, we investigate whether these representations can be used as input for training simpler models, such as linear function approximators, on new tasks. We observe that the representations learned by standard deep RL models can be highly correlated, which limits their effectiveness when used with linear function approximation. To mitigate this problem, we propose a novel deep Q-learning approach that introduces a regularization term to reduce positive correlations between feature representation of states. By leveraging these reduced correlated features, we enable more effective use of linear function approximation in transfer learning. Through experiments and ablation studies on standard RL benchmarks and MinAtar games, we demonstrate the efficacy of our approach in improving transfer learning performance and thereby reducing computational overhead.

Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert-Varshamov lower bound on the rate of optimal codes in$L_{1}$metric. Several different code spaces are analyzed, including the simplex and the hypercube in${\mathbb {Z}}^{n}$, all of which are inspired by concrete data storage and transmission models such as the permutation channel, the repetition channel, the adjacent transposition (bit-shift) channel, the multilevel flash memory channel, etc.

Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert–Varshamov (GV) bound for the sticky insertion and the constrained-synthesis channel.

We revisit the well-known Gilbert-Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that GV bound can be determined via the solution of some optimization problem. Later, Marcus and Roth (1992) modified the optimization problem and improved the GV bound in many instances. In this work, we provide explicit numerical procedures to solve these two optimization problems and hence, compute the bounds. We then show the procedures can be further simplified when we plot the respective curves.

Transmit a codeword x, that belongs to an (ℓ − 1)deletion-correcting code of length n, over a t-deletion channel for some 1 ≤ ℓ ≤ t < n. Levenshtein, in 2001, proposed the problem of determining N(n,ℓ,t) + 1, the minimum number of distinct channel outputs required to uniquely reconstruct x. Prior to this work, N(n,ℓ,t) is known only when ℓ ∈ {1,2}. Here, we provide an asymptotically exact solution for all values of ℓ and t. Specifically, we show that $N(n,\ell ,t) = \binom{{2\ell }}{\ell}/(t - \ell )!{n^{t - \ell }} - O\left( {{n^t}^{ - \ell - 1}} \right)$ and in the special instance where ℓ = t, we show that $N(n,\ell ,\ell ) = \binom{{2\ell }}{\ell}$. We also provide a conjecture on the exact value of N(n,ℓ,t) for all values of n, ℓ, and t.

Abstract In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed $$\varepsilon > 0$$ ε > 0 , approximating the reoptimization problem with respect to a given $$(1+\varepsilon )$$ ( 1 + ε ) -approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

In reoptimization problems, one is given an optimal solution to a problem instance and a local modification of the instance. The goal is to obtain a solution for the modified instance. The additional information about the instance provided by the given solution plays a central role: we aim to use that information in order to obtain better solutions than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed epsilon > 0, approximating the reoptimization problem with respect to a given (1+epsilon)-approximation is as hard as approximating the Steiner tree problem itself (whereas with a given optimal solution to the original problem it is known that one can obtain considerably improved results). Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.