One challenge of continual learning is "catastrophic forgetting" [Had+20; VT19; Kem+18]: A model However, if contexts similar to A arise repeatedly, this may be undesirable.. In machine learning, many data sets display cyclic or periodic patterns.

Several recent works demonstrate that transformers can implement algorithms like gradient descent. By a careful construction of weights, these works show that multiple layers of transformers are expressive enough to simulate iterations of gradient descent.

The main result is informally presented here.

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm---a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial in $n$ variables of constant degree (as low as three) and outputs a point whose gradient has Euclidean norm at most $2^n$ whenever the polynomial has a critical point, then P=NP. The algorithm is permitted to return an arbitrary point when no critical point exists. We also prove hardness results for approximate computation of critical points under additional structural assumptions, including settings in which existence and uniqueness of a critical point are guaranteed, the function is lower bounded, and approximation is measured in terms of distance to a critical point. Overall, our results stand in contrast to the commonly-held belief that, in nonconvex optimization, approximate computation of critical points is a tractable task.

Fairness and privacy are two vital pillars of trustworthy machine learning. Despite extensive research on these individual topics, the relationship between fairness and privacy has received significantly less attention. In this paper, we utilize the information-theoretic measure Chernoff Information to highlight the data-dependent nature of the relationship among the triad of fairness, privacy, and accuracy. We first define Noisy Chernoff Difference, a tool that allows us to analyze the relationship among the triad simultaneously. We then show that for synthetic data, this value behaves in 3 distinct ways (depending on the distribution of the data). We highlight the data distributions involved in these cases and explore their fairness and privacy implications. Additionally, we show that Noisy Chernoff Difference acts as a proxy for the steepness of the fairness-accuracy curves. Finally, we propose a method for estimating Chernoff Information on data from unknown distributions and utilize this framework to examine the triad dynamic on real datasets. This work builds towards a unified understanding of the fairness-privacy-accuracy relationship and highlights its data-dependent nature.

In Vision-and-Language Navigation (VLN) task, an agent is asked to navigate inside 3D indoor environments following given instructions. Cross-modal alignment is one of the most critical challenges in VLN because the predicted trajectory needs to match the given instruction accurately. In this paper, we address the cross-modal alignment challenge from the perspective of fine-grain. Firstly, to alleviate weak cross-modal alignment supervision from coarse-grained data, we introduce a human-annotated fine-grained VLN dataset, namely Landmark-RxR. Secondly, to further enhance local cross-modal alignment under fine-grained supervision, we investigate the focal-oriented rewards with soft and hard forms, by focusing on the critical points sampled from fine-grained Landmark-RxR. Moreover, to fully evaluate the navigation process, we also propose a re-initialization mechanism that makes metrics insensitive to difficult points, which can cause the agent to deviate from the correct trajectories. Experimental results show that our agent has superior navigation performance on Landmark-RxR, en-RxR and R2R.