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We extend the deterministic-control quantum Turing machine (dcq-TM) model to incorporate mixed state inputs and outputs. Moreover, we define dcq-computable states as those that can be accurately approximated by a dcq-TM, and we introduce (conditional) Kolmogorov complexity of quantum states. We show that this notion is machine independent and that the set of dcq-computable states coincides with states having computable classical representations. Furthermore, we prove an algorithmic information version of the no-cloning theorem stating that cloning most quantum states is as difficult as creating them. Finally, we also propose a correlation-aware definition for algorithmic mutual information and shown that it satisfies symmetry of information property.

Offline Meta Reinforcement Learning (OMRL) aims to learn transferable knowledge from offline datasets to enhance the learning process for new target tasks. Context-based Reinforcement Learning (RL) adopts a context encoder to expediently adapt the agent to new tasks by inferring the task representation, and then adjusting the policy based on this inferred representation. In this work, we focus on context-based OMRL, specifically on the challenge of learning task representation for OMRL. We conduct experiments that demonstrate that the context encoder trained on offline datasets might encounter distribution shift between the contexts used for training and testing. To overcome this problem, we present a hard-sampling-based strategy to train a robust task context encoder. Our experimental findings on diverse continuous control tasks reveal that utilizing our approach yields more robust task representations and better testing performance in terms of accumulated returns compared to baseline methods. Our code is available at //github.com/ZJLAB-AMMI/HS-OMRL.

In this work we consider the problem of fitting Random Utility Models (RUMs) to user choices. Given the winner distributions of the subsets of size $k$ of a universe, we obtain a polynomial-time algorithm that finds the RUM that best approximates the given distribution on average. Our algorithm is based on a linear program that we solve using the ellipsoid method. Given that its corresponding separation oracle problem is NP-hard, we devise an approximate separation oracle that can be viewed as a generalization of the weighted feedback arc set problem to hypergraphs. Our theoretical result can also be made practical: we obtain a heuristic that is effective and scales to real-world datasets.

Recently, Armstrong, Guzm\'an, and Sing Long (2021), presented an optimal $O(n^2)$ time algorithm for strict circular seriation (called also the recognition of strict quasi-circular Robinson spaces). In this paper, we give a very simple $O(n\log n)$ time algorithm for computing a compatible circular order for strict circular seriation. When the input space is not known to be strict quasi-circular Robinson, our algorithm is complemented by an $O(n^2)$ time verification of compatibility of the returned order. This algorithm also works for recognition of other types of strict circular Robinson spaces known in the literature. We also prove that the circular Robinson dissimilarities (which are defined by the existence of compatible orders on one of the two arcs between each pair of points) are exactly the pre-circular Robinson dissimilarities (which are defined by a four-point condition).

Reachable sets of nonlinear control systems can in general only be approximated numerically, and these approximations are typically very expensive to compute. In this paper, we explore strategies for choosing the temporal and spatial discretizations of Euler's method for reachable set computation in a non-uniform way to improve the performance of the method.

Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well-structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well-structured sparse matrices, and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

In experimental and observational studies, there is often interest in understanding the mechanism through which an intervention program improves the final outcome. Causal mediation analyses have been developed for this purpose but are primarily considered for the case of perfect treatment compliance, with a few exceptions that require the exclusion restriction assumption. In this article, we consider a semiparametric framework for assessing causal mediation in the presence of treatment noncompliance without the exclusion restriction. We propose a set of assumptions to identify the natural mediation effects for the entire study population and further, for the principal natural mediation effects within subpopulations characterized by the potential compliance behavior. We derive the efficient influence functions for the principal natural mediation effect estimands and motivate a set of multiply robust estimators for inference. The multiply robust estimators remain consistent to their respective estimands under four types of misspecification of the working models and are efficient when all nuisance models are correctly specified. We further introduce a nonparametric extension of the proposed estimators by incorporating machine learners to estimate the nuisance functions. Sensitivity analysis methods are also discussed for addressing key identification assumptions. We demonstrate the proposed methods via simulations and an application to a real data example.

We initiate the study of generalized AC0 circuits comprised of negations and arbitrary unbounded fan-in gates that only need to be constant over inputs of Hamming weight $\ge k$, which we denote GC0$(k)$. The gate set of this class includes biased LTFs like the $k$-$OR$ (output $1$ iff $\ge k$ bits are 1) and $k$-$AND$ (output $0$ iff $\ge k$ bits are 0), and thus can be seen as an interpolation between AC0 and TC0. We establish a tight multi-switching lemma for GC0$(k)$ circuits, which bounds the probability that several depth-2 GC0$(k)$ circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-$d$ size-$s$ AC0 circuits lifts to depth-$d$ size-$s^{.99}$ GC0$(.01\log s)$ circuits with no loss in parameters (other than hidden constants). Our result has the following applications: 1.Size-$2^{\Omega(n^{1/d})}$ depth-$d$ GC0$(\Omega(n^{1/d}))$ circuits do not correlate with parity (extending a result of H{\aa}stad (SICOMP, 2014)). 2. Size-$n^{\Omega(\log n)}$ GC0$(\Omega(\log^2 n))$ circuits with $n^{.249}$ arbitrary threshold gates or $n^{.499}$ arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). 3. There is a seed length $O((\log m)^{d-1}\log(m/\varepsilon)\log\log(m))$ pseudorandom generator against size-$m$ depth-$d$ GC0$(\log m)$ circuits, matching the AC0 lower bound of H{\aa}stad stad up to a $\log\log m$ factor (extending a result of Lyu (CCC, 2022)). 4. Size-$m$ GC0$(\log m)$ circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).

Randomized controlled trials (RCTs) are increasingly prevalent in education research, and are often regarded as a gold standard of causal inference. Two main virtues of randomized experiments are that they (1) do not suffer from confounding, thereby allowing for an unbiased estimate of an intervention's causal impact, and (2) allow for design-based inference, meaning that the physical act of randomization largely justifies the statistical assumptions made. However, RCT sample sizes are often small, leading to low precision; in many cases RCT estimates may be too imprecise to guide policy or inform science. Observational studies, by contrast, have strengths and weaknesses complementary to those of RCTs. Observational studies typically offer much larger sample sizes, but may suffer confounding. In many contexts, experimental and observational data exist side by side, allowing the possibility of integrating "big observational data" with "small but high-quality experimental data" to get the best of both. Such approaches hold particular promise in the field of education, where RCT sample sizes are often small due to cost constraints, but automatic collection of observational data, such as in computerized educational technology applications, or in state longitudinal data systems (SLDS) with administrative data on hundreds of thousand of students, has made rich, high-dimensional observational data widely available. We outline an approach that allows one to employ machine learning algorithms to learn from the observational data, and use the resulting models to improve precision in randomized experiments. Importantly, there is no requirement that the machine learning models are "correct" in any sense, and the final experimental results are guaranteed to be exactly unbiased. Thus, there is no danger of confounding biases in the observational data leaking into the experiment.

Lexicase selection is a widely used parent selection algorithm in genetic programming, known for its success in various task domains such as program synthesis, symbolic regression, and machine learning. Due to its non-parametric and recursive nature, calculating the probability of each individual being selected by lexicase selection has been proven to be an NP-hard problem, which discourages deeper theoretical understanding and practical improvements to the algorithm. In this work, we introduce probabilistic lexicase selection (plexicase selection), a novel parent selection algorithm that efficiently approximates the probability distribution of lexicase selection. Our method not only demonstrates superior problem-solving capabilities as a semantic-aware selection method, but also benefits from having a probabilistic representation of the selection process for enhanced efficiency and flexibility. Experiments are conducted in two prevalent domains in genetic programming: program synthesis and symbolic regression, using standard benchmarks including PSB and SRBench. The empirical results show that plexicase selection achieves state-of-the-art problem-solving performance that is competitive to the lexicase selection, and significantly outperforms lexicase selection in computation efficiency.