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Convolutional image classifiers can achieve high predictive accuracy, but quantifying their uncertainty remains an unresolved challenge, hindering their deployment in consequential settings. Existing uncertainty quantification techniques, such as Platt scaling, attempt to calibrate the network's probability estimates, but they do not have formal guarantees. We present an algorithm that modifies any classifier to output a predictive set containing the true label with a user-specified probability, such as 90%. The algorithm is simple and fast like Platt scaling, but provides a formal finite-sample coverage guarantee for every model and dataset. Our method modifies an existing conformal prediction algorithm to give more stable predictive sets by regularizing the small scores of unlikely classes after Platt scaling. In experiments on both Imagenet and Imagenet-V2 with ResNet-152 and other classifiers, our scheme outperforms existing approaches, achieving coverage with sets that are often factors of 5 to 10 smaller than a stand-alone Platt scaling baseline.

One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.

We study the problem of efficiently summarizing a short video into several keyframes, leveraging recent progress in fast graph sampling. Specifically, we first construct a similarity path graph (SPG) $\mathcal{G}$, represented by graph Laplacian matrix $\mathbf{L}$, where the similarities between adjacent frames are encoded as positive edge weights. We show that maximizing the smallest eigenvalue $\lambda_{\min}(\mathbf{B})$ of a coefficient matrix $\mathbf{B} = \text{diag}(\mathbf{a}) + \mu \mathbf{L}$, where $\mathbf{a}$ is the binary keyframe selection vector, is equivalent to minimizing a worst-case signal reconstruction error. We prove that, after partitioning $\mathcal{G}$ into $Q$ sub-graphs $\{\mathcal{G}^q\}^Q_{q=1}$, the smallest Gershgorin circle theorem (GCT) lower bound of $Q$ corresponding coefficient matrices -- $\min_q \lambda^-_{\min}(\mathbf{B}^q)$ -- is a lower bound for $\lambda_{\min}(\mathbf{B})$. This inspires a fast graph sampling algorithm to iteratively partition $\mathcal{G}$ into $Q$ sub-graphs using $Q$ samples (keyframes), while maximizing $\lambda^-_{\min}(\mathbf{B}^q)$ for each sub-graph $\mathcal{G}^q$. Experimental results show that our algorithm achieves comparable video summarization performance as state-of-the-art methods, at a substantially reduced complexity.

A user generates n independent and identically distributed data random variables with a probability mass function that must be guarded from a querier. The querier must recover, with a prescribed accuracy, a given function of the data from each of n independent and identically distributed query responses upon eliciting them from the user. The user chooses the data probability mass function and devises the random query responses to maximize distribution privacy as gauged by the (Kullback-Leibler) divergence between the former and the querier's best estimate of it based on the n query responses. Considering an arbitrary function, a basic achievable lower bound for distribution privacy is provided that does not depend on n and corresponds to worst-case privacy. Worst-case privacy equals the logsum cardinalities of inverse atoms under the given function, with the number of summands decreasing as the querier recovers the function with improving accuracy. Next, upper (converse) and lower (achievability) bounds for distribution privacy, dependent on n, are developed. The former improves upon worst-case privacy and the latter does so under suitable assumptions; both converge to it as n grows. The converse and achievability proofs identify explicit strategies for the user and the querier.

We give algorithms for sampling several structured logconcave families to high accuracy. We further develop a reduction framework, inspired by proximal point methods in convex optimization, which bootstraps samplers for regularized densities to improve dependences on problem conditioning. A key ingredient in our framework is the notion of a "restricted Gaussian oracle" (RGO) for $g: \mathbb{R}^d \rightarrow \mathbb{R}$, which is a sampler for distributions whose negative log-likelihood sums a quadratic and $g$. By combining our reduction framework with our new samplers, we obtain the following bounds for sampling structured distributions to total variation distance $\epsilon$. For composite densities $\exp(-f(x) - g(x))$, where $f$ has condition number $\kappa$ and convex (but possibly non-smooth) $g$ admits an RGO, we obtain a mixing time of $O(\kappa d \log^3\frac{\kappa d}{\epsilon})$, matching the state-of-the-art non-composite bound; no composite samplers with better mixing than general-purpose logconcave samplers were previously known. For logconcave finite sums $\exp(-F(x))$, where $F(x) = \frac{1}{n}\sum_{i \in [n]} f_i(x)$ has condition number $\kappa$, we give a sampler querying $\widetilde{O}(n + \kappa\max(d, \sqrt{nd}))$ gradient oracles to $\{f_i\}_{i \in [n]}$; no high-accuracy samplers with nontrivial gradient query complexity were previously known. For densities with condition number $\kappa$, we give an algorithm obtaining mixing time $O(\kappa d \log^2\frac{\kappa d}{\epsilon})$, improving the prior state-of-the-art by a logarithmic factor with a significantly simpler analysis; we also show a zeroth-order algorithm attains the same query complexity.

Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree $d$ for an arbitrary constant number of layers $p$ and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree $d$, in the limit $d \to \infty$, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree $d$, and finally to diluted spin-models with $D$-body interactions ($D \geq 3$). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.

Can every connected graph burn in $\lceil \sqrt{n} \rceil $ steps? While this conjecture remains open, we prove that it is asymptotically true when the graph is much larger than its \emph{growth}, which is the maximal distance of a vertex to a well-chosen path in the graph. In fact, we prove that the conjecture for graphs of bounded growth boils down to a finite number of cases. Through an improved (but still weaker) bound for all trees, we argue that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best lower bound was $23$.

Influence maximization is the task of selecting a small number of seed nodes in a social network to maximize the spread of the influence from these seeds, and it has been widely investigated in the past two decades. In the canonical setting, the whole social network as well as its diffusion parameters is given as input. In this paper, we consider the more realistic sampling setting where the network is unknown and we only have a set of passively observed cascades that record the set of activated nodes at each diffusion step. We study the task of influence maximization from these cascade samples (IMS), and present constant approximation algorithms for this task under mild conditions on the seed set distribution. To achieve the optimization goal, we also provide a novel solution to the network inference problem, that is, learning diffusion parameters and the network structure from the cascade data. Comparing with prior solutions, our network inference algorithm requires weaker assumptions and does not rely on maximum-likelihood estimation and convex programming. Our IMS algorithms enhance the learning-and-then-optimization approach by allowing a constant approximation ratio even when the diffusion parameters are hard to learn, and we do not need any assumption related to the network structure or diffusion parameters.

Deep neural networks have achieved great successes on the image captioning task. However, most of the existing models depend heavily on paired image-sentence datasets, which are very expensive to acquire. In this paper, we make the first attempt to train an image captioning model in an unsupervised manner. Instead of relying on manually labeled image-sentence pairs, our proposed model merely requires an image set, a sentence corpus, and an existing visual concept detector. The sentence corpus is used to teach the captioning model how to generate plausible sentences. Meanwhile, the knowledge in the visual concept detector is distilled into the captioning model to guide the model to recognize the visual concepts in an image. In order to further encourage the generated captions to be semantically consistent with the image, the image and caption are projected into a common latent space so that they can be used to reconstruct each other. Given that the existing sentence corpora are mainly designed for linguistic research and thus with little reference to image contents, we crawl a large-scale image description corpus of 2 million natural sentences to facilitate the unsupervised image captioning scenario. Experimental results show that our proposed model is able to produce quite promising results without using any labeled training pairs.

This paper strives to find amidst a set of sentences the one best describing the content of a given image or video. Different from existing works, which rely on a joint subspace for their image and video caption retrieval, we propose to do so in a visual space exclusively. Apart from this conceptual novelty, we contribute \emph{Word2VisualVec}, a deep neural network architecture that learns to predict a visual feature representation from textual input. Example captions are encoded into a textual embedding based on multi-scale sentence vectorization and further transferred into a deep visual feature of choice via a simple multi-layer perceptron. We further generalize Word2VisualVec for video caption retrieval, by predicting from text both 3-D convolutional neural network features as well as a visual-audio representation. Experiments on Flickr8k, Flickr30k, the Microsoft Video Description dataset and the very recent NIST TrecVid challenge for video caption retrieval detail Word2VisualVec's properties, its benefit over textual embeddings, the potential for multimodal query composition and its state-of-the-art results.