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We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.

In a sports competition, a team might lose a powerful incentive to exert full effort if its final rank does not depend on the outcome of the matches still to be played. Therefore, the organiser should reduce the probability of such a situation to the extent possible. Our paper provides a classification scheme to identify these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless games. A statistical model is estimated to simulate the UEFA Champions League groups and compare the candidate schedules used in the 2021/22 season according to the competitiveness of the matches played in the last round(s). The option followed in four of the eight groups is found to be optimal under a wide set of parameters. Minimising the number of strongly stakeless matches is verified to be a likely goal in the computer draw of the fixture that remains hidden from the public.

Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.

Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.

In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the $p(x)-$Laplacian operator. Our implementation is based in the {\em decomposition--coordination} method that allows us, via an iterative process, to solve in each step a linear differential equation and a nonlinear algebraic equation. Our code is implemented in {\sc MatLab} in 2 dimensions and turns out to be extremely efficient from the computational point of view.

Policy gradient (PG) estimation becomes a challenge when we are not allowed to sample with the target policy but only have access to a dataset generated by some unknown behavior policy. Conventional methods for off-policy PG estimation often suffer from either significant bias or exponentially large variance. In this paper, we propose the double Fitted PG estimation (FPG) algorithm. FPG can work with an arbitrary policy parameterization, assuming access to a Bellman-complete value function class. In the case of linear value function approximation, we provide a tight finite-sample upper bound on policy gradient estimation error, that is governed by the amount of distribution mismatch measured in feature space. We also establish the asymptotic normality of FPG estimation error with a precise covariance characterization, which is further shown to be statistically optimal with a matching Cramer-Rao lower bound. Empirically, we evaluate the performance of FPG on both policy gradient estimation and policy optimization, using either softmax tabular or ReLU policy networks. Under various metrics, our results show that FPG significantly outperforms existing off-policy PG estimation methods based on importance sampling and variance reduction techniques.

Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.

Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.

Modern neural network training relies heavily on data augmentation for improved generalization. After the initial success of label-preserving augmentations, there has been a recent surge of interest in label-perturbing approaches, which combine features and labels across training samples to smooth the learned decision surface. In this paper, we propose a new augmentation method that leverages the first and second moments extracted and re-injected by feature normalization. We replace the moments of the learned features of one training image by those of another, and also interpolate the target labels. As our approach is fast, operates entirely in feature space, and mixes different signals than prior methods, one can effectively combine it with existing augmentation methods. We demonstrate its efficacy across benchmark data sets in computer vision, speech, and natural language processing, where it consistently improves the generalization performance of highly competitive baseline networks.

We introduce the first system towards the novel task of answering complex multisentence recommendation questions in the tourism domain. Our solution uses a pipeline of two modules: question understanding and answering. For question understanding, we define an SQL-like query language that captures the semantic intent of a question; it supports operators like subset, negation, preference and similarity, which are often found in recommendation questions. We train and compare traditional CRFs as well as bidirectional LSTM-based models for converting a question to its semantic representation. We extend these models to a semisupervised setting with partially labeled sequences gathered through crowdsourcing. We find that our best model performs semi-supervised training of BiDiLSTM+CRF with hand-designed features and CCM(Chang et al., 2007) constraints. Finally, in an end to end QA system, our answering component converts our question representation into queries fired on underlying knowledge sources. Our experiments on two different answer corpora demonstrate that our system can significantly outperform baselines with up to 20 pt higher accuracy and 17 pt higher recall.