We study the problem of globally recovering a dictionary from a set of signals via $\ell_1$-minimization. We assume that the signals are generated as i.i.d. random linear combinations of the $K$ atoms from a complete reference dictionary $D^*\in \mathbb R^{K\times K}$, where the linear combination coefficients are from either a Bernoulli type model or exact sparse model. First, we obtain a necessary and sufficient norm condition for the reference dictionary $D^*$ to be a sharp local minimum of the expected $\ell_1$ objective function. Our result substantially extends that of Wu and Yu (2015) and allows the combination coefficient to be non-negative. Secondly, we obtain an explicit bound on the region within which the objective value of the reference dictionary is minimal. Thirdly, we show that the reference dictionary is the unique sharp local minimum, thus establishing the first known global property of $\ell_1$-minimization dictionary learning. Motivated by the theoretical results, we introduce a perturbation-based test to determine whether a dictionary is a sharp local minimum of the objective function. In addition, we also propose a new dictionary learning algorithm based on Block Coordinate Descent, called DL-BCD, which is guaranteed to have monotonic convergence. Simulation studies show that DL-BCD has competitive performance in terms of recovery rate compared to many state-of-the-art dictionary learning algorithms.

Federated Learning is the current state of the art in supporting secure multi-party ML: data is maintained on the owner's device and is aggregated through a secure protocol. However, this process assumes a trusted centralized infrastructure for coordination and clients must trust that the central service does not maliciously omit client contributions or use the byproducts of client data. As a response, we propose Biscotti: a fully decentralized P2P approach to multi-party ML, which uses blockchain and crypto primitives to coordinate a privacy-preserving ML process between peering clients. Our evaluation demonstrates that Biscotti is scalable, fault tolerant, and defends against known attacks. For example, Biscotti is able to protect the performance of the global model at scale even when 45% of adversaries are trying to poison the model. The implementation can be found at: https://github.com/DistributedML/Biscotti

A central capability of intelligent systems is the ability to continuously build upon previous experiences to speed up and enhance learning of new tasks. Two distinct research paradigms have studied this question. Meta-learning views this problem as learning a prior over model parameters that is amenable for fast adaptation on a new task, but typically assumes the set of tasks are available together as a batch. In contrast, online (regret based) learning considers a sequential setting in which problems are revealed one after the other, but conventionally train only a single model without any task-specific adaptation. This work introduces an online meta-learning setting, which merges ideas from both the aforementioned paradigms to better capture the spirit and practice of continual lifelong learning. We propose the follow the meta leader algorithm which extends the MAML algorithm to this setting. Theoretically, this work provides an $\mathcal{O}(\log T)$ regret guarantee with only one additional higher order smoothness assumption in comparison to the standard online setting. Our experimental evaluation on three different large-scale tasks suggest that the proposed algorithm significantly outperforms alternatives based on traditional online learning approaches.

Comparing different algorithms is hard. For almost any pair of algorithms and measure of algorithm performance like running time or solution quality, each algorithm will perform better than the other on some inputs.a For example, the insertion sort algorithm is faster than merge sort on already-sorted arrays but slower on many other inputs. When two algorithms have incomparable performance, how can we deem one of them "better than" the other? Worst-case analysis is a specific modeling choice in the analysis of algorithms, where the overall performance of an algorithm is summarized by its worst performance on any input of a given size. The "better" algorithm is then the one with superior worst-case performance. Merge sort, with its worst-case asymptotic running time of Θ(n log n) for arrays of length n, is better in this sense than insertion sort, which has a worst-case running time of Θ(n2). While crude, worst-case analysis can be tremendously useful, and it is the dominant paradigm for algorithm analysis in theoretical computer science. A good worst-case guarantee is the best-case scenario for an algorithm, certifying its general-purpose utility and absolving its users from understanding which inputs are relevant to their applications. Remarkably, for many fundamental computational problems, there are algorithms with excellent worst-case performance guarantees. The lion's share of an undergraduate algorithms course comprises algorithms that run in linear or near-linear time in the worst case.

We propose a new method for local distance metric learning based on sample similarity as side information. These local metrics, which utilize conical combinations of metric weight matrices, are learned from the pooled spatial characteristics of the data, as well as the similarity profiles between the pairs of samples, whose distances are measured. The main objective of our framework is to yield metrics, such that the resulting distances between similar samples are small and distances between dissimilar samples are above a certain threshold. For learning and inference purposes, we describe a transductive, as well as an inductive algorithm; the former approach naturally befits our framework, while the latter one is provided in the interest of faster learning. Experimental results on a collection of classification problems imply that the new methods may exhibit notable performance advantages over alternative metric learning approaches that have recently appeared in the literature.

Given a sequence of convex functions $f_0, f_1, \ldots, f_T$, we study the problem of sampling from the Gibbs distribution $\pi_t \propto e^{-\sum_{k=0}^t f_k}$ for each epoch $t$ in an online manner. This problem occurs in applications to machine learning, Bayesian statistics, and optimization where one constantly acquires new data, and must continuously update the distribution. Our main result is an algorithm that generates independent samples from a distribution that is a fixed $\varepsilon$ TV-distance from $\pi_t$ for every $t$ and, under mild assumptions on the functions, makes poly$\log(T)$ gradient evaluations per epoch. All previous results for this problem imply a bound on the number of gradient or function evaluations which is at least linear in $T$. While we assume the functions have bounded second moment, we do not assume strong convexity. In particular, we show that our assumptions hold for online Bayesian logistic regression, when the data satisfy natural regularity properties. In simulations, our algorithm achieves accuracy comparable to that of a Markov chain specialized to logistic regression. Our main result also implies the first algorithm to sample from a $d$-dimensional log-concave distribution $\pi_T \propto e^{-\sum_{k=0}^T f_k}$ where the $f_k$'s are not assumed to be strongly convex and the total number of gradient evaluations is roughly $T\log(T)+\mathrm{poly}(d),$ as opposed to $T\cdot \mathrm{poly}(d)$ implied by prior works. Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain that has a carefully designed variance reduction step built-in with fixed constant batch size. Technically, lack of strong convexity is a significant barrier to the analysis, and, here, our main contribution is a martingale exit time argument showing the chain is constrained to a ball of radius roughly poly$\log(T)$ for the duration of the algorithm.

We present a Bayesian hierarchical multi-view mixture model termed Symphony that simultaneously learns clusters of cells representing cell types and their underlying gene regulatory networks by integrating data from two views: single-cell gene expression data and paired epigenetic data, which is informative of gene-gene interactions. This model improves interpretation of clusters as cell types with similar expression patterns as well as regulatory networks driving expression, by explaining gene-gene covariances with the biological machinery regulating gene expression. We show the theoretical advantages of the multi-view learning approach and present a Variational EM inference procedure. We demonstrate superior performance on both synthetic data and real genomic data with subtypes of peripheral blood cells compared to other methods.

The softmax function on top of a final linear layer is the de facto method to output probability distributions in neural networks. In many applications such as language models or text generation, this model has to produce distributions over large output vocabularies. Recently, this has been shown to have limited representational capacity due to its connection with the rank bottleneck in matrix factorization. However, little is known about the limitations of linear-softmax for quantities of practical interest such as cross entropy or mode estimation, a direction that we theoretically and empirically explore here. As an efficient and effective solution to alleviate this issue, we propose to learn parametric monotonic functions on top of the logits. We theoretically investigate the rank increasing capabilities of such monotonic functions. Empirically, our method improves in two different quality metrics over the traditional softmax-linear layer in synthetic and real language model experiments, adding little time or memory overhead, while being comparable to the more computationally expensive mixture of softmaxes.

Perinatal stroke (PS) is a serious condition that, if undetected and thus untreated, often leads to life-long disability, in particular Cerebral Palsy (CP). In clinical settings, Prechtl's General Movement Assessment (GMA) can be used to classify infant movements using a Gestalt approach, identifying infants at high risk of developing PS. Training and maintenance of assessment skills are essential and expensive for the correct use of GMA, yet many practitioners lack these skills, preventing larger-scale screening and leading to significant risks of missing opportunities for early detection and intervention for affected infants. We present an automated approach to GMA, based on body-worn accelerometers and a novel sensor data analysis method-Discriminative Pattern Discovery (DPD)-that is designed to cope with scenarios where only coarse annotations of data are available for model training. We demonstrate the effectiveness of our approach in a study with 34 newborns (21 typically developing infants and 13 PS infants with abnormal movements). Our method is able to correctly recognise the trials with abnormal movements with at least the accuracy that is required by newly trained human annotators (75%), which is encouraging towards our ultimate goal of an automated PS screening system that can be used population-wide.

We explore the use of graph networks to deal with irregular-geometry detectors in the context of particle reconstruction. Thanks to their representation-learning capabilities, graph networks can exploit the full detector granularity, while natively managing the event sparsity and arbitrarily complex detector geometries. We introduce two distance-weighted graph network architectures, dubbed GarNet and GravNet layers, and apply them to a typical particle reconstruction task. The performance of the new architectures is evaluated on a data set of simulated particle interactions on a toy model of a highly granular calorimeter, loosely inspired by the endcap calorimeter to be installed in the CMS detector for the High-Luminosity LHC phase. We study the clustering of energy depositions, which is the basis for calorimetric particle reconstruction, and provide a quantitative comparison to alternative approaches. The proposed algorithms outperform existing methods or reach competitive performance with lower computing-resource consumption. Being geometry-agnostic, the new architectures are not restricted to calorimetry and can be easily adapted to other use cases, such as tracking in silicon detectors.