Let's be Honest: An Optimal No-Regret Framework for Zero-Sum Games

We revisit the problem of solving two-player zero-sum games in the decentralized setting. We propose a simple algorithmic framework that simultaneously achieves the best rates for honest regret as well as adversarial regret, and in addition resolves the open problem of removing the logarithmic terms in convergence to the value of the game. We achieve this goal in three steps. First, we provide a novel analysis of the optimistic mirror descent (OMD), showing that it can be modified to guarantee fast convergence for both honest regret and value of the game, when the players are playing collaboratively. Second, we propose a new algorithm, dubbed as robust optimistic mirror descent (ROMD), which attains optimal adversarial regret without knowing the time horizon beforehand. Finally, we propose a simple signaling scheme, which enables us to bridge OMD and ROMD to achieve the best of both worlds. Numerical examples are presented to support our theoretical claims and show that our non-adaptive ROMD algorithm can be competitive to OMD with adaptive step-size selection.Comment: Proceedings of the 35th International Conference on Machine Learnin

A Sphere-Packing Error Exponent for Mismatched Decoding

We derive a sphere-packing error exponent for coded transmission over discrete memoryless channels with a fixed decoding metric. By studying the error probability of the code over an auxiliary channel, we find a lower bound to the probability of error of mismatched decoding. The bound is shown to decay exponentially for coding rates smaller than a new upper bound to the mismatch capacity. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to the mismatch capacity

A Single-Letter Upper Bound to the Mismatch Capacity

We derive a single-letter upper bound to the mismatched-decoding capacity for discrete memoryless channels. The bound is expressed as the mutual information of a transformation of the channel, such that a maximum-likelihood decoding error on the translated channel implies a mismatched-decoding error in the original channel. In particular, a strong converse is shown to hold for this upper-bound: if the rate exceeds the upper-bound, the probability of error tends to 1 exponentially when the block-length tends to infinity. We also show that the underlying optimization problem is a convex-concave problem and that an efficient iterative algorithm converges to the optimal solution. In addition, we show that, unlike achievable rates in the literature, the multiletter version of the bound does not improve. A number of examples are discussed throughout the paper.European Research Council under Grant 725411, and by the Spanish Ministry of Economy and Competitiveness under Grant TEC2016-78434-C3-1-R

Effectiveness of Trauma-Focused Cognitive-Behavioral Therapy on the Grief Symptoms and Behavioral Problems of Bereaved Children (One-Month Follow-Up)

The purpose of the present study was to investigate the effectiveness of trauma-focused cognitive-behavioral therapy on the bereavement symptoms and behavioral problems of grieving children. In this investigation, a single-case experimental design with incongruent multiple baseline designs was used. The statistical population of the present research was comprised of bereaved children in the city of Karaj who have lost a parent within the past six months. The investigation was conducted between January and May of 2021. The research sample consisted of three minors aged 10 to 11 who were selected using a method of purposive sampling based on the inclusion criteria. During two and a half months, the experimental group received ten 75-minute sessions of trauma-focused cognitive-behavioral therapy. This study utilized the Children’s Grief Questionnaire (CGQ) and the Child Cehavior Checklist (CBCL) as questionnaires. The study’s data were analyzed using the statistical software SPSS-22, visual representation, the dynamic change index, and the improvement percentage formula. The degree of improvement in the variable of grief was 34, 27.53, and 29 for the first, second, and third subjects, respectively, and the calculated dynamic change index was 3.18, 3.56, and 3.14. Moreover, the degree of improvement in the variable of internalized behavioral problems was 46.66, 31.50, and 38.02 for the first, second, and third subjects, respectively, and the calculated dynamic change index was 2.07, 2.96, and 2.14. In addition, the degree of externalized behavioral disorders was 46.98, 40.82, and 45.92 for the first, second, and third subjects, respectively, and the calculated dynamic change index was 2.34, 2.02, and 1.98. After treatment, the amount of dynamic change index was greater than Z (1.96); the results of this study suggest that cognitive-behavioral therapy is an effective method for reducing the bereavement and behavioral problems of grieving children

Let’s be honest: An optimal no-regret framework for zero-sum games

We revisit the problem of solving two-player zero- sum games in the decentralized setting. We pro- pose a simple algorithmic framework that simulta- neously achieves the best rates for honest regret as well as adversarial regret, and in addition resolves the open problem of removing the logarithmic terms in convergence to the value of the game. We achieve this goal in three steps. First, we provide a novel analysis of the optimistic mirror descent (OMD), showing that it can be modified to guarantee fast convergence for both honest re- gret and value of the game, when the players are playing collaboratively. Second, we propose a new algorithm, dubbed as robust optimistic mir- ror descent (ROMD), which attains optimal ad- versarial regret without knowing the time horizon beforehand. Finally, we propose a simple signal- ing scheme, which enables us to bridge OMD and ROMD to achieve the best of both worlds. Numerical examples are presented to support our theoretical claims and show that our non-adaptive ROMD algorithm can be competitive to OMD with adaptive step-size selection

Mismatched Decoding: Capacity and Error Exponent Upper Bounds

Mismatched decoding models the coded communication problem in scenarios where a sub-optimal decoder is used. These situations arise when optimal maximum-likelihood decoding cannot be used: i) the channel transition is unknown and imperfectly estimated or, ii) when, for complexity reasons, the channel likelihood is too difficult to compute and an alternative decoding metric is needed. In addition, important problems in information theory such as zero-error capacity or erasures-only capacity can be cast as instances of mismatched decoding. In the mismatched decoding problem, the optimal maximum-likelihood decoder is replaced by a maximum metric decoder, where the metric is not necessarily the channel likelihood. This thesis studies the problem of channel coding with mismatched decoding. Single-letter characterization of mismatch capacity is a long standing open problem in information theory. Lower bounds for the mismatch capacity have been studied extensively using random coding techniques. On the other hand, upperbounds for the mismatch capacity were not explored until a few years ago. In this thesis we study a novel technique for tackling this problem based on analysing the probability of error of a given codebook using an auxiliary channel and a hypothetical decoder. Careful design of the auxiliary channel leads to a lower bound on the probability of error of the codebook in the original mismatched decoding problem. The first version of our upper bound based on the mentioned technique studies the case where we design the auxiliary channel with the following property. When a codeword is sent over the auxiliary channel and the hypothetical decoder makes an error then the mismatched decoder makes an error for sure. The results is this part are mainly derived using the method of types and graph theory. Moreover, the resulting bound is shown to be computable using convex optimization techniques. A simple algorithm is introduced and is shown to converge to the derived bound. A multiletter version of the bound is studied and is shown to yield no improvement over the single-letter bound. An implication of the bound for binary-input channels is derived which provides a simple sufficient condition for the mismatch capacity being strictly less than the matched capacity. Finally, using some examples the application of the bound is illustrated. A generalized version of the mentioned method is used to derive an upper bound to the reliability function. In contrast to the previous part, when a codeword is sent over the auxiliary channel and the hypothetical decoder makes an error, the mismatched decoder makes an error with a positive probability. Moreover, this probability is conditioned to be bounded away from zero by the inverse of a polynomial function of the block length. This is a strictly weaker design assumption for the auxiliary channel and the hypothetical decoder compared to the previous one. Nevertheless, the rate that the reliability functions becomes equal to zero is shown to be an upper bound to the mismatch capacity. To prove the validity of this upper bound on the reliability function the method of types and graph theory are mainly used. The optimization problem for computing this upper bound is shown to be non-convex. This upper bound is compared to the existing upper bounds on the mismatch capacity and is shown to improve over the recent ones. Using several numerical examples the application of the bound is illustrated

An Upper Bound to the Mismatch Capacity

We derive a single-letter upper bound to the mismatched-decoding capacity for discrete memoryless channels. The bound is expressed as the mutual information of a transformation of the channel, such that a maximum-likelihood decoding error on the translated channel implies a mismatched-decoding error in the original channel. We show this bound recovers the binary-input binary-output mismatch capacity which is known to either be the channel capacity or zero. In addition, a strong converse is shown for this upper bound: if the rate exceeds the upper-bound, the probability of error tends to 1 exponentially when the block-length tends to infinity

A Single-Letter Upper Bound to the Mismatch Capacity

We derive a single-letter upper bound to the mismatched-decoding capacity for discrete memoryless channels. The bound is expressed as the mutual information of a transformation of the channel, such that a maximum-likelihood decoding error on the translated channel implies a mismatched-decoding error in the original channel. In particular, it is shown that if the rate exceeds the upper-bound, the probability of error tends to one exponentially when the block-length tends to infinity. We also show that the underlying optimization problem is a convex-concave problem and that an efficient iterative algorithm converges to the optimal solution. In addition, we show that, unlike achievable rates in the literature, the multiletter version of the bound cannot not improve. A number of examples are discussed throughout the paper

A Sphere-Packing Error Exponent for Mismatched Decoding

We derive a sphere-packing error exponent for coded transmission over discrete memoryless channels with a fixed decoding metric. By studying the error probability of the code over an auxiliary channel, we find a lower bound to the probability of error of mismatched decoding. The bound is shown to decay exponentially for coding rates smaller than a new upper bound to the mismatch capacity which is established in this paper. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to the mismatch capacity

A sphere-packing exponent for mismatched decoding

Comunicació presentada a 2021 IEEE International Symposium on Information Theory (ISIT), celebrat del 12 al 20 de juliol de 2021 de manera virtual.We derive a sphere-packing error exponent for mismatched decoding over discrete memoryless channels. We find a lower bound to the probability of error of mismatched decoding that decays exponentially for coding rates smaller than a new upper bound to the mismatch capacity. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to the mismatch capacity.This work was supported in part by the European Research Council under Grant 725411