We study a mean change point testing problem for high-dimensional data, with exponentially- or polynomially-decaying tails. In each case, depending on the $\ell_0$-norm of the mean change vector, we separately consider dense and sparse regimes. We characterise the boundary between the dense and sparse regimes under the above two tail conditions for the first time in the change point literature and propose novel testing procedures that attain optimal rates in each of the four regimes up to a poly-iterated logarithmic factor. Our results quantify the costs of heavy-tailedness on the fundamental difficulty of change point testing problems for high-dimensional data by comparing to the previous results under Gaussian assumptions. To be specific, when the error vectors follow sub-Weibull distributions, a CUSUM-type statistic is shown to achieve a minimax testing rate up to $\sqrt{\log\log(8n)}$. When the error distributions have polynomially-decaying tails, admitting bounded $\alpha$-th moments for some $\alpha \geq 4$, we introduce a median-of-means-type test statistic that achieves a near-optimal testing rate in both dense and sparse regimes. In particular, in the sparse regime, we further propose a computationally-efficient test to achieve the exact optimality. Surprisingly, our investigation in the even more challenging case of $2 \leq \alpha < 4$, unveils a new phenomenon that the minimax testing rate has no sparse regime, i.e. testing sparse changes is information-theoretically as hard as testing dense changes. This phenomenon implies a phase transition of the minimax testing rates at $\alpha = 4$.
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Quantum machine learning has become an area of growing interest but has certain theoretical and hardware-specific limitations. Notably, the problem of vanishing gradients, or barren plateaus, renders the training impossible for circuits with high qubit counts, imposing a limit on the number of qubits that data scientists can use for solving problems. Independently, angle-embedded supervised quantum neural networks were shown to produce truncated Fourier series with a degree directly dependent on two factors: the depth of the encoding and the number of parallel qubits the encoding applied to. The degree of the Fourier series limits the model expressivity. This work introduces two new architectures whose Fourier degrees grow exponentially: the sequential and parallel exponential quantum machine learning architectures. This is done by efficiently using the available Hilbert space when encoding, increasing the expressivity of the quantum encoding. Therefore, the exponential growth allows staying at the low-qubit limit to create highly expressive circuits avoiding barren plateaus. Practically, parallel exponential architecture was shown to outperform the existing linear architectures by reducing their final mean square error value by up to 44.7% in a one-dimensional test problem. Furthermore, the feasibility of this technique was also shown on a trapped ion quantum processing unit.
Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control (SC) methods have quickly become one of the leading methods for estimating causal effects in observational studies in settings with panel data. Formal discussions often motivate SC methods by the assumption that the potential outcomes were generated by a factor model. Here we study SC methods from a design-based perspective, assuming a model for the selection of the treated unit(s) and period(s). We show that the standard SC estimator is generally biased under random assignment. We propose a Modified Unbiased Synthetic Control (MUSC) estimator that guarantees unbiasedness under random assignment and derive its exact, randomization-based, finite-sample variance. We also propose an unbiased estimator for this variance. We document in settings with real data that under random assignment, SC-type estimators can have root mean-squared errors that are substantially lower than that of other common estimators. We show that such an improvement is weakly guaranteed if the treated period is similar to the other periods, for example, if the treated period was randomly selected. While our results only directly apply in settings where treatment is assigned randomly, we believe that they can complement model-based approaches even for observational studies.
Information about action costs is critical for real-world AI planning applications. Rather than rely solely on declarative action models, recent approaches also use black-box external action cost estimators, often learned from data, that are applied during the planning phase. These, however, can be computationally expensive, and produce uncertain values. In this paper we suggest a generalization of deterministic planning with action costs that allows selecting between multiple estimators for action cost, to balance computation time against bounded estimation uncertainty. This enables a much richer -- and correspondingly more realistic -- problem representation. Importantly, it allows planners to bound plan accuracy, thereby increasing reliability, while reducing unnecessary computational burden, which is critical for scaling to large problems. We introduce a search algorithm, generalizing $A^*$, that solves such planning problems, and additional algorithmic extensions. In addition to theoretical guarantees, extensive experiments show considerable savings in runtime compared to alternatives.
Effectively measuring and modeling the reliability of a trained model is essential to the real-world deployment of monocular depth estimation (MDE) models. However, the intrinsic ill-posedness and ordinal-sensitive nature of MDE pose major challenges to the estimation of uncertainty degree of the trained models. On the one hand, utilizing current uncertainty modeling methods may increase memory consumption and are usually time-consuming. On the other hand, measuring the uncertainty based on model accuracy can also be problematic, where uncertainty reliability and prediction accuracy are not well decoupled. In this paper, we propose to model the uncertainty of MDE models from the perspective of the inherent probability distributions originating from the depth probability volume and its extensions, and to assess it more fairly with more comprehensive metrics. By simply introducing additional training regularization terms, our model, with surprisingly simple formations and without requiring extra modules or multiple inferences, can provide uncertainty estimations with state-of-the-art reliability, and can be further improved when combined with ensemble or sampling methods. A series of experiments demonstrate the effectiveness of our methods.
Procedural content generation (PCG) is a growing field, with numerous applications in the video game industry and great potential to help create better games at a fraction of the cost of manual creation. However, much of the work in PCG is focused on generating relatively straightforward levels in simple games, as it is challenging to design an optimisable objective function for complex settings. This limits the applicability of PCG to more complex and modern titles, hindering its adoption in industry. Our work aims to address this limitation by introducing a compositional level generation method that recursively composes simple low-level generators to construct large and complex creations. This approach allows for easily-optimisable objectives and the ability to design a complex structure in an interpretable way by referencing lower-level components. We empirically demonstrate that our method outperforms a non-compositional baseline by more accurately satisfying a designer's functional requirements in several tasks. Finally, we provide a qualitative showcase (in Minecraft) illustrating the large and complex, but still coherent, structures that were generated using simple base generators.
In many industrial applications, obtaining labeled observations is not straightforward as it often requires the intervention of human experts or the use of expensive testing equipment. In these circumstances, active learning can be highly beneficial in suggesting the most informative data points to be used when fitting a model. Reducing the number of observations needed for model development alleviates both the computational burden required for training and the operational expenses related to labeling. Online active learning, in particular, is useful in high-volume production processes where the decision about the acquisition of the label for a data point needs to be taken within an extremely short time frame. However, despite the recent efforts to develop online active learning strategies, the behavior of these methods in the presence of outliers has not been thoroughly examined. In this work, we investigate the performance of online active linear regression in contaminated data streams. Our study shows that the currently available query strategies are prone to sample outliers, whose inclusion in the training set eventually degrades the predictive performance of the models. To address this issue, we propose a solution that bounds the search area of a conditional D-optimal algorithm and uses a robust estimator. Our approach strikes a balance between exploring unseen regions of the input space and protecting against outliers. Through numerical simulations, we show that the proposed method is effective in improving the performance of online active learning in the presence of outliers, thus expanding the potential applications of this powerful tool.
In this paper, we focus on the high-dimensional double sparse structure, where the parameter of interest simultaneously encourages group-wise sparsity and element-wise sparsity in each group. By combining the Gilbert-Varshamov bound and its variants, we develop a novel lower bound technique for the metric entropy of the parameter space, specifically tailored for the double sparse structure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lower bounds on the estimation error using an information-theoretic approach, leveraging our proposed lower bound technique and Fano's inequality. To complement the lower bounds, we establish matching upper bounds through a direct analysis of constrained least-squares estimators and utilize results from empirical processes. A significant finding of our study is the discovery of a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$. Furthermore, we extend the theoretical results to the double sparse regression model and determine its minimax rate for estimation error. To tackle double sparse linear regression, we develop the DSIHT (Double Sparse Iterative Hard Thresholding) algorithm, demonstrating its optimality in the minimax sense. Finally, we demonstrate the superiority of our method through numerical experiments.
We study the problem of exact support recovery for high-dimensional sparse linear regression under independent Gaussian design when the signals are weak, rare, and possibly heterogeneous. Under a suitable scaling of the sample size and signal sparsity, we fix the minimum signal magnitude at the information-theoretic optimal rate and investigate the asymptotic selection accuracy of best subset selection (BSS) and marginal screening (MS) procedures. We show that despite the ideal setup, somewhat surprisingly, marginal screening can fail to achieve exact recovery with probability converging to one in the presence of heterogeneous signals, whereas BSS enjoys model consistency whenever the minimum signal strength is above the information-theoretic threshold. To mitigate the computational intractability of BSS, we also propose an efficient two-stage algorithmic framework called ETS (Estimate Then Screen) comprised of an estimation step and gradient coordinate screening step, and under the same scaling assumption on sample size and sparsity, we show that ETS achieves model consistency under the same information-theoretic optimal requirement on the minimum signal strength as BSS. Finally, we present a simulation study comparing ETS with LASSO and marginal screening. The numerical results agree with our asymptotic theory even for realistic values of the sample size, dimension and sparsity.
The projection predictive variable selection is a decision-theoretically justified Bayesian variable selection approach achieving an outstanding trade-off between predictive performance and sparsity. Its projection problem is not easy to solve in general because it is based on the Kullback-Leibler divergence from a restricted posterior predictive distribution of the so-called reference model to the parameter-conditional predictive distribution of a candidate model. Previous work showed how this projection problem can be solved for response families employed in generalized linear models and how an approximate latent-space approach can be used for many other response families. Here, we present an exact projection method for all response families with discrete and finite support, called the augmented-data projection. A simulation study for an ordinal response family shows that the proposed method performs better than or similarly to the previously proposed approximate latent-space projection. The cost of the slightly better performance of the augmented-data projection is a substantial increase in runtime. Thus, in such cases, we recommend the latent projection in the early phase of a model-building workflow and the augmented-data projection for final results. The ordinal response family from our simulation study is supported by both projection methods, but we also include a real-world cancer subtyping example with a nominal response family, a case that is not supported by the latent projection.
This paper focuses on spatial time-optimal motion planning, a generalization of the exact time-optimal path following problem that allows the system to plan within a predefined space. In contrast to state-of-the-art methods, we drop the assumption that a collision-free geometric reference is given. Instead, we present a two-stage motion planning method that solely relies on a goal location and a geometric representation of the environment to compute a time-optimal trajectory that is compliant with system dynamics and constraints. To do so, the proposed scheme first computes an obstacle-free Pythagorean Hodograph parametric spline, and second solves a spatially reformulated minimum-time optimization problem. The spline obtained in the first stage is not a geometric reference, but an extension of the environment representation, and thus, time-optimality of the solution is guaranteed. The efficacy of the proposed approach is benchmarked by a known planar example and validated in a more complex spatial system, illustrating its versatility and applicability.
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