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The study of statistical estimation without distributional assumptions on data values, but with knowledge of data collection methods was recently introduced by Chen, Valiant and Valiant (NeurIPS 2020). In this framework, the goal is to design estimators that minimize the worst-case expected error. Here the expectation is over a known, randomized data collection process from some population, and the data values corresponding to each element of the population are assumed to be worst-case. Chen, Valiant and Valiant show that, when data values are $\ell_{\infty}$-normalized, there is a polynomial time algorithm to compute an estimator for the mean with worst-case expected error that is within a factor $\frac{\pi}{2}$ of the optimum within the natural class of semilinear estimators. However, their algorithm is based on optimizing a somewhat complex concave objective function over a constrained set of positive semidefinite matrices, and thus does not come with explicit runtime guarantees beyond being polynomial time in the input. In this paper we design provably efficient algorithms for approximating the optimal semilinear estimator based on online convex optimization. In the setting where data values are $\ell_{\infty}$-normalized, our algorithm achieves a $\frac{\pi}{2}$-approximation by iteratively solving a sequence of standard SDPs. When data values are $\ell_2$-normalized, our algorithm iteratively computes the top eigenvector of a sequence of matrices, and does not lose any multiplicative approximation factor. We complement these positive results by stating a simple combinatorial condition which, if satisfied by a data collection process, implies that any (not necessarily semilinear) estimator for the mean has constant worst-case expected error.

ByteScheduler partitions and rearranges tensor transmissions to improve the communication efficiency of distributed Deep Neural Network (DNN) training. The configuration of hyper-parameters (i.e., the partition size and the credit size) is critical to the effectiveness of partitioning and rearrangement. Currently, ByteScheduler adopts Bayesian Optimization (BO) to find the optimal configuration for the hyper-parameters beforehand. In practice, however, various runtime factors (e.g., worker node status and network conditions) change over time, making the statically-determined one-shot configuration result suboptimal for real-world DNN training. To address this problem, we present a real-time configuration method (called AutoByte) that automatically and timely searches the optimal hyper-parameters as the training systems dynamically change. AutoByte extends the ByteScheduler framework with a meta-network, which takes the system's runtime statistics as its input and outputs predictions for speedups under specific configurations. Evaluation results on various DNN models show that AutoByte can dynamically tune the hyper-parameters with low resource usage, and deliver up to 33.2\% higher performance than the best static configuration in ByteScheduler.

We study the problem of online tree exploration by a deterministic mobile agent. Our main objective is to establish what features of the model of the mobile agent and the environment allow linear exploration time. We study agents that, upon entering to a node, do not receive as input the edge via which they entered. In such a model, deterministic memoryless exploration is infeasible, hence the agent needs to be allowed to use some memory. The memory can be located at the agent or at each node. The existing lower bounds show that if the memory is either only at the agent or only at the nodes, then the exploration needs superlinear time. We show that tree exploration in dual-memory model, with constant memory at the agent and logarithmic at each node is possible in linear time when one of two additional features is present: fixed initial state of the memory at each node (so called clean memory) or a single movable token. We present two algorithms working in linear time for arbitrary trees in these two models. On the other hand, in our lower bound we show that if the agent has a single bit of memory and one bit is present at each node, then exploration may require quadratic time on paths, if the initial memory at nodes could be set arbitrarily (so called dirty memory). This shows that having clean node memory or a token allows linear exploration of trees in the model with two types of memory, but having neither of those features may lead to quadratic exploration time even on a simple path.

This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator). We first consider $\gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $\mathcal{S}$ and action space $\mathcal{A}$. Despite a number of prior works tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, all prior results suffer from a severe sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^2}$. The current paper overcomes this barrier by certifying the minimax optimality of two algorithms -- a perturbed model-based algorithm and a conservative model-based algorithm -- as soon as the sample size exceeds the order of $\frac{|\mathcal{S}||\mathcal{A}|}{1-\gamma}$ (modulo some log factor). Moving beyond infinite-horizon MDPs, we further study time-inhomogeneous finite-horizon MDPs, and prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level. To the best of our knowledge, this work delivers the first minimax-optimal guarantees that accommodate the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically infeasible).

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in $[0,1]^d$ of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most $B$ in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to "learn" characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints.

The curse of dimensionality is a widely known issue in reinforcement learning (RL). In the tabular setting where the state space $\mathcal{S}$ and the action space $\mathcal{A}$ are both finite, to obtain a nearly optimal policy with sampling access to a generative model, the minimax optimal sample complexity scales linearly with $|\mathcal{S}|\times|\mathcal{A}|$, which can be prohibitively large when $\mathcal{S}$ or $\mathcal{A}$ is large. This paper considers a Markov decision process (MDP) that admits a set of state-action features, which can linearly express (or approximate) its probability transition kernel. We show that a model-based approach (resp.$~$Q-learning) provably learns an $\varepsilon$-optimal policy (resp.$~$Q-function) with high probability as soon as the sample size exceeds the order of $\frac{K}{(1-\gamma)^{3}\varepsilon^{2}}$ (resp.$~$$\frac{K}{(1-\gamma)^{4}\varepsilon^{2}}$), up to some logarithmic factor. Here $K$ is the feature dimension and $\gamma\in(0,1)$ is the discount factor of the MDP. Both sample complexity bounds are provably tight, and our result for the model-based approach matches the minimax lower bound. Our results show that for arbitrarily large-scale MDP, both the model-based approach and Q-learning are sample-efficient when $K$ is relatively small, and hence the title of this paper.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.