We show that the subgradient method converges only to local minimizers when applied to generic Lipschitz continuous and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, the argument we present is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on a certain distinguished submanifold that captures the nonsmooth activity of the function. In the process, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier and extend stochastic processes techniques of Pemantle.

The common practice for training commonsense models has gone from-human-to-corpus-to-machine: humans author commonsense knowledge graphs in order to train commonsense models. In this work, we investigate an alternative, from-machine-to-corpus-to-machine: general language models author these commonsense knowledge graphs to train commonsense models. Our study leads to a new framework, Symbolic Knowledge Distillation. As with prior art in Knowledge Distillation (Hinton et al., 2015), our approach uses larger models to teach smaller models. A key difference is that we distill knowledge symbolically-as text-in addition to the neural model. We also distill only one aspect-the commonsense of a general language model teacher, allowing the student to be a different type, a commonsense model. Altogether, we show that careful prompt engineering and a separately trained critic model allow us to selectively distill high-quality causal commonsense from GPT-3, a general language model. Empirical results demonstrate that, for the first time, a human-authored commonsense knowledge graph is surpassed by our automatically distilled variant in all three criteria: quantity, quality, and diversity. In addition, it results in a neural commonsense model that surpasses the teacher model's commonsense capabilities despite its 100x smaller size. We apply this to the ATOMIC resource, and share our new symbolic knowledge graph and commonsense models.

Most existing dialogue systems fail to respond properly to potentially unsafe user utterances by either ignoring or passively agreeing with them. To address this issue, we introduce ProsocialDialog, the first large-scale multi-turn dialogue dataset to teach conversational agents to respond to problematic content following social norms. Covering diverse unethical, problematic, biased, and toxic situations, ProsocialDialog contains responses that encourage prosocial behavior, grounded in commonsense social rules (i.e., rules-of-thumb, RoTs). Created via a human-AI collaborative framework, ProsocialDialog consists of 58K dialogues, with 331K utterances, 160K unique RoTs, and 497K dialogue safety labels accompanied by free-form rationales. With this dataset, we introduce a dialogue safety detection module, Canary, capable of generating RoTs given conversational context, and a socially-informed dialogue agent, Prost. Empirical results show that Prost generates more socially acceptable dialogues compared to other state-of-the-art language and dialogue models in both in-domain and out-of-domain settings. Additionally, Canary effectively guides conversational agents and off-the-shelf language models to generate significantly more prosocial responses. Our work highlights the promise and importance of creating and steering conversational AI to be socially responsible.

We study the asymmetric matrix factorization problem under a natural nonconvex formulation with arbitrary overparametrization. The model-free setting is considered, with minimal assumption on the rank or singular values of the observed matrix, where the global optima provably overfit. We show that vanilla gradient descent with small random initialization sequentially recovers the principal components of the observed matrix. Consequently, when equipped with proper early stopping, gradient descent produces the best low-rank approximation of the observed matrix without explicit regularization. We provide a sharp characterization of the relationship between the approximation error, iteration complexity, initialization size and stepsize. Our complexity bound is almost dimension-free and depends logarithmically on the approximation error, with significantly more lenient requirements on the stepsize and initialization compared to prior work. Our theoretical results provide accurate prediction for the behavior gradient descent, showing good agreement with numerical experiments.

As AI systems become increasingly powerful and pervasive, there are growing concerns about machines' morality or a lack thereof. Yet, teaching morality to machines is a formidable task, as morality remains among the most intensely debated questions in humanity, let alone for AI. Existing AI systems deployed to millions of users, however, are already making decisions loaded with moral implications, which poses a seemingly impossible challenge: teaching machines moral sense, while humanity continues to grapple with it. To explore this challenge, we introduce Delphi, an experimental framework based on deep neural networks trained directly to reason about descriptive ethical judgments, e.g., "helping a friend" is generally good, while "helping a friend spread fake news" is not. Empirical results shed novel insights on the promises and limits of machine ethics; Delphi demonstrates strong generalization capabilities in the face of novel ethical situations, while off-the-shelf neural network models exhibit markedly poor judgment including unjust biases, confirming the need for explicitly teaching machines moral sense. Yet, Delphi is not perfect, exhibiting susceptibility to pervasive biases and inconsistencies. Despite that, we demonstrate positive use cases of imperfect Delphi, including using it as a component model within other imperfect AI systems. Importantly, we interpret the operationalization of Delphi in light of prominent ethical theories, which leads us to important future research questions.

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust $\ell_1$ loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank. On the other hand, we show that the RDPP condition holds under generic settings, such as Gaussian measurements under independent or adversarial sparse corruptions, where the result could be of independent interest. Both the exact recovery and the convergence rate of the proposed subgradient method are numerically verified in the overspecified regime. Moreover, our experiment further shows that our particular design of diminishing stepsize effectively prevents overfitting for robust recovery under overparameterized models, such as robust matrix sensing and learning robust deep image prior. This regularization effect is worth further investigation.