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Applications of Reinforcement Learning (RL), in which agents learn to make a sequence of decisions despite lacking complete information about the latent states of the controlled system, that is, they act under partial observability of the states, are ubiquitous. Partially observable RL can be notoriously difficult -- well-known information-theoretic results show that learning partially observable Markov decision processes (POMDPs) requires an exponential number of samples in the worst case. Yet, this does not rule out the existence of large subclasses of POMDPs over which learning is tractable. In this paper we identify such a subclass, which we call weakly revealing POMDPs. This family rules out the pathological instances of POMDPs where observations are uninformative to a degree that makes learning hard. We prove that for weakly revealing POMDPs, a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to guarantee polynomial sample complexity. To the best of our knowledge, this is the first provably sample-efficient result for learning from interactions in overcomplete POMDPs, where the number of latent states can be larger than the number of observations.

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST, where $\tilde{\Omega}$ hides a $1/(\text{polylog } n)$ factor.

Maximal Independent Set (MIS) is one of the central and most well-studied problems in distributed computing. Even after four decades of intensive research, the best-known (randomized) MIS algorithms take $O(\log{n})$ worst-case rounds on general graphs (where $n$ is the number of nodes), while the best-known lower bound is $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$ rounds. Breaking past the $O(\log{n})$ worst-case bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is that we show that MIS can be computed in (worst-case) awake complexity of $O(\log \log n)$ rounds that is (essentially) exponentially better compared to the (traditional) round complexity lower bound of $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$. Specifically, we present the following results. (1) We present a randomized distributed (Monte Carlo) algorithm for MIS that with high probability computes an MIS and has $O(\log\log{n})$-rounds awake complexity. This algorithm has (traditional) {\em round complexity} that is $O(poly(n))$. Our bounds hold in the $CONGEST(O(polylog n))$ model where only $O(polylog n)$ (specifically $O(\log^3 n)$) bits are allowed to be sent per edge per round. (2) We also show that we can drastically reduce the round complexity at the cost of a slight increase in awake complexity by presenting a randomized MIS algorithm with $O(\log \log n \log^* n )$ awake complexity and $O(\log^3 n \log \log n \log^*n)$ round complexity in the $CONGEST(O(polylog n))$ model.

Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\frac{\Delta}{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.

The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of length $n$ such that the dissimilarity between any elements is an integer between zero and $c$, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is $O(c n^2)$, which can be translated to $O(n^2)$ for constant dissimilarity functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.

We study dynamic algorithms for the problem of maximizing a monotone submodular function over a stream of $n$ insertions and deletions. We show that any algorithm that maintains a $(0.5+\epsilon)$-approximate solution under a cardinality constraint, for any constant $\epsilon>0$, must have an amortized query complexity that is $\mathit{polynomial}$ in $n$. Moreover, a linear amortized query complexity is needed in order to maintain a $0.584$-approximate solution. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve $(0.5-\epsilon)$-approximation with a $\mathsf{poly}\log(n)$ amortized query complexity. On the positive side, when the stream is insertion-only, we present efficient algorithms for the problem under a cardinality constraint and under a matroid constraint with approximation guarantee $1-1/e-\epsilon$ and amortized query complexities $\smash{O(\log (k/\epsilon)/\epsilon^2)}$ and $\smash{k^{\tilde{O}(1/\epsilon^2)}\log n}$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.

Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, i.e., a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size, and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for Non-Gaussian Component Analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like $\tan(z)$, $z/(\exp(z)-1)$, $\sec(z)$, etc., particularly, reciprocals, quotients and compositions of holonomic functions, are generally not holonomic. Therefore their power series are inaccessible by the holonomic framework. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of $\tan(z)$ and $\sec(z)$ the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations to characterize several non-holonomic power series and prove non-trivial identities. We discuss this algorithm and its implementation for Maple 2022.

There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs $(\mathcal{A},\mathcal{T})$, such that if the distribution on examples in the data passes the tester $\mathcal{T}$ then one can safely trust the output of the agnostic learner $\mathcal{A}$ on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of $n^{\tilde{O}(1/\epsilon^4)}$. This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the $L_1$ and EMD distance measures. A key step in the analysis is a novel characterization of concentration and anti-concentration properties of a distribution whose low-degree moments approximately match those of a Gaussian. We also use tools from polynomial approximation theory. In contrast, we show strong lower bounds on the combined run-times of tester-learner pairs for the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Through these lower bounds we exhibit natural problems where there is a dramatic gap between standard agnostic learning run-time and the run-time of the best tester-learner pair.