In the model of online caching with machine learned advice, introduced by Lykouris and Vassilvitskii, the goal is to solve the caching problem with an online algorithm that has access to next-arrival predictions: when each input element arrives, the algorithm is given a prediction of the next time when the element will reappear. The traditional model for online caching suffers from an Ωplog kq competitive ratio lower bound (on a cache of size k). In contrast, the augmented model admits algorithms which beat this lower bound when the predictions have low error, and asymptotically match the lower bound when the predictions have high error, even if the algorithms are oblivious to the prediction error. In particular, Lykouris and Vassilvitskii showed that there is a prediction-augmented caching algorithm with a competitive ratio of Op1`minp a η{opt, log kqq when the overall ℓ 1 prediction error is bounded by η, and opt is the cost of the optimal offline algorithm.The dependence on k in the competitive ratio is optimal, but the dependence on η{opt may be far from optimal. In this work, we make progress towards closing this gap. Our contributions are twofold. First, we provide an improved algorithm with a competitive ratio of Op1m inppη{optq{k, 1q log kq. Second, we provide a lower bound of Ωplog minppη{optq{pk log kq, kqq.

Partially Observable Markov Decision Processes (POMDPs) are a natural and general model in reinforcement learning that take into account the agent's uncertainty about its current state. In the literature on POMDPs, it is customary to assume access to a planning oracle that computes an optimal policy when the parameters are known, even though the problem is known to be computationally hard. Almost all existing planning algorithms either run in exponential time, lack provable performance guarantees, or require placing strong assumptions on the transition dynamics under every possible policy. In this work, we revisit the planning problem and ask: Are there natural and well-motivated assumptions that make planning easy?Our main result is a quasipolynomial-time algorithm for planning in (one-step) observable POMDPs. Specifically, we assume that well-separated distributions on states lead to wellseparated distributions on observations, and thus the observations are at least somewhat informative in each step. Crucially, this assumption places no restrictions on the transition dynamics of the POMDP; nevertheless, it implies that near-optimal policies admit quasi-succinct descriptions, which is not true in general (under standard hardness assumptions). Our analysis is based on new quantitative bounds for filter stability -i.e. the rate at which an optimal filter for the latent state forgets its initialization. Furthermore, we prove matching hardness for planning in observable POMDPs under the Exponential Time Hypothesis.

Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian N (0, Σ), and we seek an estimator with small excess risk.If the true signal is t-sparse, information-theoretically, it is possible to achieve strong recovery guarantees with only O(t log n) samples. However, computationally efficient algorithms have sample complexity linear in (some variant of) the condition number of Σ. Classical algorithms such as the Lasso can require significantly more samples than necessary even if there is only a single sparse approximate dependency among the covariates.We provide a polynomial-time algorithm that, given Σ, automatically adapts the Lasso to tolerate a small number of approximate dependencies. In particular, we achieve near-optimal sample complexity for constant sparsity and if Σ has few "outlier" eigenvalues. Our algorithm fits into a broader framework of feature adaptation for sparse linear regression with ill-conditioned covariates. With this framework, we additionally provide the first polynomial-factor improvement over brute-force search for constant sparsity t and arbitrary covariance Σ.

For ordered graphs G and H, the ordered Ramsey number r<(G, H) is the smallest n such that every red/blue edge coloring of the complete graph on vertices {1, . . . , n} contains either a blue copy of G or a red copy of H, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the "off-diagonal" ordered Ramsey number r<(M, K3), where M is an ordered matching on n vertices. In particular, Conlon et al. asked what asymptotic bounds (in n) can be obtained for max r<(M, K3), where the maximum is over all ordered matchings M on n vertices. The bestknown upper bound is O(n 2 / log n), whereas the best-known lower bound is Ω((n/ log n) 4/3 ), and Conlon et al. hypothesize that r<(M, K3) = O(n 2− ) for every ordered matching M . We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random matchings with interval chromatic number 2.

Measuring the stability of conclusions derived from Ordinary Least Squares linear regression is critically important, but most metrics either only measure local stability (i.e. against infinitesimal changes in the data), or are only interpretable under statistical assumptions. Recent work proposes a simple, global, finite-sample stability metric: the minimum number of samples that need to be removed so that rerunning the analysis overturns the conclusion [BGM20], specifically meaning that the sign of a particular coefficient of the estimated regressor changes. However, besides the trivial exponential-time algorithm, the only approach for computing this metric is a greedy heuristic that lacks provable guarantees under reasonable, verifiable assumptions; the heuristic provides a loose upper bound on the stability and also cannot certify lower bounds on it.We show that in the low-dimensional regime where the number of covariates is a constant but the number of samples is large, there are efficient algorithms for provably estimating (a fractional version of) this metric. Applying our algorithms to the Boston Housing dataset, we exhibit regression analyses where we can estimate the stability up to a factor of 3 better than the greedy heuristic, and analyses where we can certify stability to dropping even a majority of the samples.