Algorithm evaluation and comparison are fundamental questions in machine learning and statistics -- how well does an algorithm perform at a given modeling task, and which algorithm performs best? Many methods have been developed to assess algorithm performance, often based around cross-validation type strategies, retraining the algorithm of interest on different subsets of the data and assessing its performance on the held-out data points. Despite the broad use of such procedures, the theoretical properties of these methods are not yet fully understood. In this work, we explore some fundamental limits for answering these questions with limited amounts of data. In particular, we make a distinction between two questions: how good is an algorithm $A$ at the problem of learning from a training set of size $n$, versus, how good is a particular fitted model produced by running $A$ on a particular training data set of size $n$? Our main results prove that, for any test that treats the algorithm $A$ as a ``black box'' (i.e., we can only study the behavior of $A$ empirically), there is a fundamental limit on our ability to carry out inference on the performance of $A$, unless the number of available data points $N$ is many times larger than the sample size $n$ of interest. (On the other hand, evaluating the performance of a particular fitted model is easy as long as a holdout data set is available -- that is, as long as $N-n$ is not too small.) We also ask whether an assumption of algorithmic stability might be sufficient to circumvent this hardness result. Surprisingly, we find that this is not the case: the same hardness result still holds for the problem of evaluating the performance of $A$, aside from a high-stability regime where fitted models are essentially nonrandom. Finally, we also establish similar hardness results for the problem of comparing multiple algorithms.
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Despite the considerable success of Bregman proximal-type algorithms, such as mirror descent, in machine learning, a critical question remains: Can existing stationarity measures, often based on Bregman divergence, reliably distinguish between stationary and non-stationary points? In this paper, we present a groundbreaking finding: All existing stationarity measures necessarily imply the existence of spurious stationary points. We further establish an algorithmic independent hardness result: Bregman proximal-type algorithms are unable to escape from a spurious stationary point in finite steps when the initial point is unfavorable, even for convex problems. Our hardness result points out the inherent distinction between Euclidean and Bregman geometries, and introduces both fundamental theoretical and numerical challenges to both machine learning and optimization communities.
This paper leverages the statistics of extreme values to predict the worst-case convergence times of machine learning algorithms. Timing is a critical non-functional property of ML systems, and providing the worst-case converge times is essential to guarantee the availability of ML and its services. However, timing properties such as worst-case convergence times (WCCT) are difficult to verify since (1) they are not encoded in the syntax or semantics of underlying programming languages of AI, (2) their evaluations depend on both algorithmic implementations and underlying systems, and (3) their measurements involve uncertainty and noise. Therefore, prevalent formal methods and statistical models fail to provide rich information on the amounts and likelihood of WCCT. Our key observation is that the timing information we seek represents the extreme tail of execution times. Therefore, extreme value theory (EVT), a statistical discipline that focuses on understanding and predicting the distribution of extreme values in the tail of outcomes, provides an ideal framework to model and analyze WCCT in the training and inference phases of ML paradigm. Building upon the mathematical tools from EVT, we propose a practical framework to predict the worst-case timing properties of ML. Over a set of linear ML training algorithms, we show that EVT achieves a better accuracy for predicting WCCTs than relevant statistical methods such as the Bayesian factor. On the set of larger machine learning training algorithms and deep neural network inference, we show the feasibility and usefulness of EVT models to accurately predict WCCTs, their expected return periods, and their likelihood.
While reinforcement learning (RL) algorithms have been successfully applied across numerous sequential decision-making problems, their generalization to unforeseen testing environments remains a significant concern. In this paper, we study the problem of out-of-distribution (OOD) detection in RL, which focuses on identifying situations at test time that RL agents have not encountered in their training environments. We first propose a clarification of terminology for OOD detection in RL, which aligns it with the literature from other machine learning domains. We then present new benchmark scenarios for OOD detection, which introduce anomalies with temporal autocorrelation into different components of the agent-environment loop. We argue that such scenarios have been understudied in the current literature, despite their relevance to real-world situations. Confirming our theoretical predictions, our experimental results suggest that state-of-the-art OOD detectors are not able to identify such anomalies. To address this problem, we propose a novel method for OOD detection, which we call DEXTER (Detection via Extraction of Time Series Representations). By treating environment observations as time series data, DEXTER extracts salient time series features, and then leverages an ensemble of isolation forest algorithms to detect anomalies. We find that DEXTER can reliably identify anomalies across benchmark scenarios, exhibiting superior performance compared to both state-of-the-art OOD detectors and high-dimensional changepoint detectors adopted from statistics.
Convergence analysis of Markov chain Monte Carlo methods in high-dimensional statistical applications is increasingly recognized. In this paper, we develop general mixing time bounds for Metropolis-Hastings algorithms on discrete spaces by building upon and refining some recent theoretical advancements in Bayesian model selection problems. We establish sufficient conditions for a class of informed Metropolis-Hastings algorithms to attain relaxation times that are independent of the problem dimension. These conditions are grounded in high-dimensional statistical theory and allow for possibly multimodal posterior distributions. We obtain our results through two independent techniques: the multicommodity flow method and single-element drift condition analysis; we find that the latter yields a tighter mixing time bound. Our results and proof techniques are readily applicable to a broad spectrum of statistical problems with discrete parameter spaces.
Adversarial attacks on machine learning algorithms have been a key deterrent to the adoption of AI in many real-world use cases. They significantly undermine the ability of high-performance neural networks by forcing misclassifications. These attacks introduce minute and structured perturbations or alterations in the test samples, imperceptible to human annotators in general, but trained neural networks and other models are sensitive to it. Historically, adversarial attacks have been first identified and studied in the domain of image processing. In this paper, we study adversarial examples in the field of natural language processing, specifically text classification tasks. We investigate the reasons for adversarial vulnerability, particularly in relation to the inherent dimensionality of the model. Our key finding is that there is a very strong correlation between the embedding dimensionality of the adversarial samples and their effectiveness on models tuned with input samples with same embedding dimension. We utilize this sensitivity to design an adversarial defense mechanism. We use ensemble models of varying inherent dimensionality to thwart the attacks. This is tested on multiple datasets for its efficacy in providing robustness. We also study the problem of measuring adversarial perturbation using different distance metrics. For all of the aforementioned studies, we have run tests on multiple models with varying dimensionality and used a word-vector level adversarial attack to substantiate the findings.
One of the current challenges in physically-based simulations, and, more specifically, fluid simulations, is to produce visually appealing results at interactive rates, capable of being used in multiple forms of media. In recent times, a lot of effort has been made with regards to this with the use of multi-core architectures, as many of the computations involved in the algorithms for these simulations are very well suited for these architectures. Although there is a considerable amount of works regarding acceleration techniques in this field, there is yet room to further explore and analyze some of them. To investigate this problem, we surveyed the topic of fluid simulations and some of the recent contributions towards this field. Additionally, we implemented two versions of a fluid simulation algorithm, one on the CPU and the other on the GPU using NVIDIA's CUDA framework, with the intent of gaining a better understanding of the effort needed to move these simulations to a multi-core architecture and the performance gains that we get with it.
Judgment aggregation is a framework to aggregate individual opinions on multiple, logically connected issues into a collective outcome. These opinions are cast by judges, which can be for example referees, experts, advisors or jurors, depending on the application and context. It is open to manipulative attacks such as \textsc{Manipulation} where judges cast their judgments strategically. Previous works have shown that most computational problems corresponding to these manipulative attacks are \NP-hard. This desired computational barrier, however, often relies on formulas that are either of unbounded size or of complex structure. We revisit the computational complexity for various \textsc{Manipulation} and \textsc{Bribery} problems in premise-based judgment aggregation, now focusing on simple and realistic formulas. We restrict all formulas to be clauses that are (positive) monotone, Horn-clauses, or have bounded length. For basic variants of \textsc{Manipulation}, we show that these restrictions make several variants, which were in general known to be \NP-hard, polynomial-time solvable. Moreover, we provide a P vs.\ NP dichotomy for a large class of clause restrictions (generalizing monotone and Horn clauses) by showing a close relationship between variants of \textsc{Manipulation} and variants of \textsc{Satisfiability}. For Hamming distance based \textsc{Manipulation}, we show that \NP-hardness even holds for positive monotone clauses of length three, but the problem becomes polynomial-time solvable for positive monotone clauses of length two. For \textsc{Bribery}, we show that \NP-hardness even holds for positive monotone clauses of length two, but it becomes polynomial-time solvable for the same clause set if there is a constant budget.
We study the Out-of-Distribution (OOD) generalization in machine learning and propose a general framework that provides information-theoretic generalization bounds. Our framework interpolates freely between Integral Probability Metric (IPM) and $f$-divergence, which naturally recovers some known results (including Wasserstein- and KL-bounds), as well as yields new generalization bounds. Moreover, we show that our framework admits an optimal transport interpretation. When evaluated in two concrete examples, the proposed bounds either strictly improve upon existing bounds in some cases or recover the best among existing OOD generalization bounds.
Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
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