Partitioning a set of elements into subsets of a priori unknown sizes is essential in many applications. These subset sizes are rarely explicitly learned - be it the cluster sizes in clustering applications or the number of shared versus independent generative latent factors in weakly-supervised learning. Probability distributions over correct combinations of subset sizes are non-differentiable due to hard constraints, which prohibit gradient-based optimization. In this work, we propose the differentiable hypergeometric distribution. The hypergeometric distribution models the probability of different group sizes based on their relative importance. We introduce reparameterizable gradients to learn the importance between groups and highlight the advantage of explicitly learning the size of subsets in two typical applications: weakly-supervised learning and clustering. In both applications, we outperform previous approaches, which rely on suboptimal heuristics to model the unknown size of groups.
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After just a few hundred training updates, a standard probabilistic model for language generation has likely not yet learnt many semantic or syntactic rules of natural language, making it difficult to estimate the probability distribution over next tokens. Yet around this point, these models have identified a simple, loss-minimising behaviour: to output the unigram distribution of the target training corpus. The use of such a heuristic raises the question: Can we initialise our models with this behaviour and save precious compute resources and model capacity? Here we show that we can effectively endow standard neural language generation models with a separate module that reflects unigram frequency statistics as prior knowledge, simply by initialising the bias term in a model's final linear layer with the log-unigram distribution. We use neural machine translation as a test bed for this simple technique and observe that it: (i) improves learning efficiency; (ii) achieves better overall performance; and perhaps most importantly (iii) appears to disentangle strong frequency effects by encouraging the model to specialise in non-frequency-related aspects of language.
Autoencoders have been extensively used in the development of recent anomaly detection techniques. The premise of their application is based on the notion that after training the autoencoder on normal training data, anomalous inputs will exhibit a significant reconstruction error. Consequently, this enables a clear differentiation between normal and anomalous samples. In practice, however, it is observed that autoencoders can generalize beyond the normal class and achieve a small reconstruction error on some of the anomalous samples. To improve the performance, various techniques propose additional components and more sophisticated training procedures. In this work, we propose a remarkably straightforward alternative: instead of adding neural network components, involved computations, and cumbersome training, we complement the reconstruction loss with a computationally light term that regulates the norm of representations in the latent space. The simplicity of our approach minimizes the requirement for hyperparameter tuning and customization for new applications which, paired with its permissive data modality constraint, enhances the potential for successful adoption across a broad range of applications. We test the method on various visual and tabular benchmarks and demonstrate that the technique matches and frequently outperforms alternatives. We also provide a theoretical analysis and numerical simulations that help demonstrate the underlying process that unfolds during training and how it can help with anomaly detection. This mitigates the black-box nature of autoencoder-based anomaly detection algorithms and offers an avenue for further investigation of advantages, fail cases, and potential new directions.
Given a dataset of $n$ i.i.d. samples from an unknown distribution $P$, we consider the problem of generating a sample from a distribution that is close to $P$ in total variation distance, under the constraint of differential privacy (DP). We study the problem when $P$ is a multi-dimensional Gaussian distribution, under different assumptions on the information available to the DP mechanism: known covariance, unknown bounded covariance, and unknown unbounded covariance. We present new DP sampling algorithms, and show that they achieve near-optimal sample complexity in the first two settings. Moreover, when $P$ is a product distribution on the binary hypercube, we obtain a pure-DP algorithm whereas only an approximate-DP algorithm (with slightly worse sample complexity) was previously known.
The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of ${|T\rangle}^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. For the "approximate" rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than $\tilde \Omega(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde \Omega (n^2)$ for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over $\mathbb{F}_2$, resolving a question in [Wil18]. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with $T$ gates by [LKS18].
Decision-makers often have access to a machine-learned prediction about demand, referred to as advice, which can potentially be utilized in online decision-making processes for resource allocation. However, exploiting such advice poses challenges due to its potential inaccuracy. To address this issue, we propose a framework that enhances online resource allocation decisions with potentially unreliable machine-learned (ML) advice. We assume here that this advice is represented by a general convex uncertainty set for the demand vector. We introduce a parameterized class of Pareto optimal online resource allocation algorithms that strike a balance between consistent and robust ratios. The consistent ratio measures the algorithm's performance (compared to the optimal hindsight solution) when the ML advice is accurate, while the robust ratio captures performance under an adversarial demand process when the advice is inaccurate. Specifically, in a C-Pareto optimal setting, we maximize the robust ratio while ensuring that the consistent ratio is at least C. Our proposed C-Pareto optimal algorithm is an adaptive protection level algorithm, which extends the classical fixed protection level algorithm introduced in Littlewood (2005) and Ball and Queyranne (2009). Solving a complex non-convex continuous optimization problem characterizes the adaptive protection level algorithm. To complement our algorithms, we present a simple method for computing the maximum achievable consistent ratio, which serves as an estimate for the maximum value of the ML advice. Additionally, we present numerical studies to evaluate the performance of our algorithm in comparison to benchmark algorithms. The results demonstrate that by adjusting the parameter C, our algorithms effectively strike a balance between worst-case and average performance, outperforming the benchmark algorithms.
In recent years, promising statistical modeling approaches to tensor data analysis have been rapidly developed. Traditional multivariate analysis tools, such as multivariate regression and discriminant analysis, are generalized from modeling random vectors and matrices to higher-order random tensors. One of the biggest challenges to statistical tensor models is the non-Gaussian nature of many real-world data. Unfortunately, existing approaches are either restricted to normality or implicitly using least squares type objective functions that are computationally efficient but sensitive to data contamination. Motivated by this, we adopt a simple tensor t-distribution that is, unlike the commonly used matrix t-distributions, compatible with tensor operators and reshaping of the data. We study the tensor response regression with tensor t-error, and develop penalized likelihood-based estimation and a novel one-step estimation. We study the asymptotic relative efficiency of various estimators and establish the one-step estimator's oracle properties and near-optimal asymptotic efficiency. We further propose a high-dimensional modification to the one-step estimation procedure and show that it attains the minimax optimal rate in estimation. Numerical studies show the excellent performance of the one-step estimator.
In the scenario of class-incremental learning (CIL), deep neural networks have to adapt their model parameters to non-stationary data distributions, e.g., the emergence of new classes over time. However, CIL models are challenged by the well-known catastrophic forgetting phenomenon. Typical methods such as rehearsal-based ones rely on storing exemplars of old classes to mitigate catastrophic forgetting, which limits real-world applications considering memory resources and privacy issues. In this paper, we propose a novel rehearsal-free CIL approach that learns continually via the synergy between two Complementary Learning Subnetworks. Our approach involves jointly optimizing a plastic CNN feature extractor and an analytical feed-forward classifier. The inaccessibility of historical data is tackled by holistically controlling the parameters of a well-trained model, ensuring that the decision boundary learned fits new classes while retaining recognition of previously learned classes. Specifically, the trainable CNN feature extractor provides task-dependent knowledge separately without interference; and the final classifier integrates task-specific knowledge incrementally for decision-making without forgetting. In each CIL session, it accommodates new tasks by attaching a tiny set of declarative parameters to its backbone, in which only one matrix per task or one vector per class is kept for knowledge retention. Extensive experiments on a variety of task sequences show that our method achieves competitive results against state-of-the-art methods, especially in accuracy gain, memory cost, training efficiency, and task-order robustness. Furthermore, to make the non-growing backbone (i.e., a model with limited network capacity) suffice to train on more incoming tasks, a graceful forgetting implementation on previously learned trivial tasks is empirically investigated.
Learning with noisy labels (LNL) is challenging as the model tends to memorize noisy labels, which can lead to overfitting. Many LNL methods detect clean samples by maximizing the similarity between samples in each category, which does not make any assumptions about likely noise sources. However, we often have some knowledge about the potential source(s) of noisy labels. For example, an image mislabeled as a cheetah is more likely a leopard than a hippopotamus due to their visual similarity. Thus, we introduce a new task called Learning with Noisy Labels and noise source distribution Knowledge (LNL+K), which assumes we have some knowledge about likely source(s) of label noise that we can take advantage of. By making this presumption, methods are better equipped to distinguish hard negatives between categories from label noise. In addition, this enables us to explore datasets where the noise may represent the majority of samples, a setting that breaks a critical premise of most methods developed for the LNL task. We explore several baseline LNL+K approaches that integrate noise source knowledge into state-of-the-art LNL methods across three diverse datasets and three types of noise, where we report a 5-15% boost in performance compared with the unadapted methods. Critically, we find that LNL methods do not generalize well in every setting, highlighting the importance of directly exploring our LNL+K task.
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.
While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.
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