On the Finite-Time Complexity and Practical Computation of Approximate Stationarity Concepts of Lipschitz Functions

We report a practical finite-time algorithmic scheme to compute approximately stationary points for nonconvex nonsmooth Lipschitz functions. In particular, we are interested in two kinds of approximate stationarity notions for nonconvex nonsmooth problems, i.e., Goldstein approximate stationarity (GAS) and near-approximate stationarity (NAS). For GAS, our scheme removes the unrealistic subgradient selection oracle assumption in (Zhang et al., 2020, Assumption 1) and computes GAS with the same finite-time complexity. For NAS, Davis & Drusvyatskiy (2019) showed that $\rho$-weakly convex functions admit finite-time computation, while Tian & So (2021) provided the matching impossibility results of dimension-free finite-time complexity for first-order methods. Complement to these developments, in this paper, we isolate a new class of functions that could be Clarke irregular (and thus not weakly convex anymore) and show that our new algorithmic scheme can compute NAS points for functions in that class within finite time. To demonstrate the wide applicability of our new theoretical framework, we show that $\rho$-margin SVM, $1$-layer, and $2$-layer ReLU neural networks, all being Clarke irregular, satisfy our new conditions.Comment: 20 pages, 3 figures, ICML 202

ViCor: Bridging Visual Understanding and Commonsense Reasoning with Large Language Models

In our work, we explore the synergistic capabilities of pre-trained vision-and-language models (VLMs) and large language models (LLMs) for visual commonsense reasoning (VCR). We categorize the problem of VCR into visual commonsense understanding (VCU) and visual commonsense inference (VCI). For VCU, which involves perceiving the literal visual content, pre-trained VLMs exhibit strong cross-dataset generalization. On the other hand, in VCI, where the goal is to infer conclusions beyond image content, VLMs face difficulties. We find that a baseline where VLMs provide perception results (image captions) to LLMs leads to improved performance on VCI. However, we identify a challenge with VLMs' passive perception, which often misses crucial context information, leading to incorrect or uncertain reasoning by LLMs. To mitigate this issue, we suggest a collaborative approach where LLMs, when uncertain about their reasoning, actively direct VLMs to concentrate on and gather relevant visual elements to support potential commonsense inferences. In our method, named ViCor, pre-trained LLMs serve as problem classifiers to analyze the problem category, VLM commanders to leverage VLMs differently based on the problem classification, and visual commonsense reasoners to answer the question. VLMs will perform visual recognition and understanding. We evaluate our framework on two VCR benchmark datasets and outperform all other methods that do not require in-domain supervised fine-tuning

The Benefits of Being Distributional: Small-Loss Bounds for Reinforcement Learning

While distributional reinforcement learning (RL) has demonstrated empirical success, the question of when and why it is beneficial has remained unanswered. In this work, we provide one explanation for the benefits of distributional RL through the lens of small-loss bounds, which scale with the instance-dependent optimal cost. If the optimal cost is small, our bounds are stronger than those from non-distributional approaches. As warmup, we show that learning the cost distribution leads to small-loss regret bounds in contextual bandits (CB), and we find that distributional CB empirically outperforms the state-of-the-art on three challenging tasks. For online RL, we propose a distributional version-space algorithm that constructs confidence sets using maximum likelihood estimation, and we prove that it achieves small-loss regret in the tabular MDPs and enjoys small-loss PAC bounds in latent variable models. Building on similar insights, we propose a distributional offline RL algorithm based on the pessimism principle and prove that it enjoys small-loss PAC bounds, which exhibit a novel robustness property. For both online and offline RL, our results provide the first theoretical benefits of learning distributions even when we only need the mean for making decisions

An Adaptive Incremental Gradient Method With Support for Non-Euclidean Norms

Stochastic variance reduced methods have shown strong performance in solving finite-sum problems. However, these methods usually require the users to manually tune the step-size, which is time-consuming or even infeasible for some large-scale optimization tasks. To overcome the problem, we propose and analyze several novel adaptive variants of the popular SAGA algorithm. Eventually, we design a variant of Barzilai-Borwein step-size which is tailored for the incremental gradient method to ensure memory efficiency and fast convergence. We establish its convergence guarantees under general settings that allow non-Euclidean norms in the definition of smoothness and the composite objectives, which cover a broad range of applications in machine learning. We improve the analysis of SAGA to support non-Euclidean norms, which fills the void of existing work. Numerical experiments on standard datasets demonstrate a competitive performance of the proposed algorithm compared with existing variance-reduced methods and their adaptive variants

Efficient Private SCO for Heavy-Tailed Data via Clipping

We consider stochastic convex optimization for heavy-tailed data with the guarantee of being differentially private (DP). Prior work on this problem is restricted to the gradient descent (GD) method, which is inefficient for large-scale problems. In this paper, we resolve this issue and derive the first high-probability bounds for the private stochastic method with clipping. For general convex problems, we derive excess population risks \Tilde{O}\left(\frac{d^{1/7}\sqrt{\ln\frac{(n \epsilon)^2}{\beta d}}}{(n\epsilon)^{2/7}}\right) and \Tilde{O}\left(\frac{d^{1/7}\ln\frac{(n\epsilon)^2}{\beta d}}{(n\epsilon)^{2/7}}\right) under bounded or unbounded domain assumption, respectively (here $n$ is the sample size, $d$ is the dimension of the data, $\beta$ is the confidence level and $\epsilon$ is the private level). Then, we extend our analysis to the strongly convex case and non-smooth case (which works for generalized smooth objectives with H$\ddot{\text{o}}$lder-continuous gradients). We establish new excess risk bounds without bounded domain assumption. The results above achieve lower excess risks and gradient complexities than existing methods in their corresponding cases. Numerical experiments are conducted to justify the theoretical improvement