Complexity of Minimizing Projected-Gradient-Dominated Functions with Stochastic First-order Oracles

This work investigates the performance limits of projected stochastic first-order methods for minimizing functions under the $(\alpha,\tau,\mathcal{X})$-projected-gradient-dominance property, that asserts the sub-optimality gap $F(\mathbf{x})-\min_{\mathbf{x}'\in \mathcal{X}}F(\mathbf{x}')$ is upper-bounded by $\tau\cdot\|\mathcal{G}_{\eta,\mathcal{X}}(\mathbf{x})\|^{\alpha}$ for some $\alpha\in[1,2)$ and $\tau>0$ and $\mathcal{G}_{\eta,\mathcal{X}}(\mathbf{x})$ is the projected-gradient mapping with $\eta>0$ as a parameter. For non-convex functions, we show that the complexity lower bound of querying a batch smooth first-order stochastic oracle to obtain an $\epsilon$-global-optimum point is $\Omega(\epsilon^{-{2}/{\alpha}})$. Furthermore, we show that a projected variance-reduced first-order algorithm can obtain the upper complexity bound of $\mathcal{O}(\epsilon^{-{2}/{\alpha}})$, matching the lower bound. For convex functions, we establish a complexity lower bound of $\Omega(\log(1/\epsilon)\cdot\epsilon^{-{2}/{\alpha}})$ for minimizing functions under a local version of gradient-dominance property, which also matches the upper complexity bound of accelerated stochastic subgradient methods.

Causal Effect Identification in LiNGAM Models with Latent Confounders

We study the generic identifiability of causal effects in linear non-Gaussian acyclic models (LiNGAM) with latent variables. We consider the problem in two main settings: When the causal graph is known a priori, and when it is unknown. In both settings, we provide a complete graphical characterization of the identifiable direct or total causal effects among observed variables. Moreover, we propose efficient algorithms to certify the graphical conditions. Finally, we propose an adaptation of the reconstruction independent component analysis (RICA) algorithm that estimates the causal effects from the observational data given the causal graph. Experimental results show the effectiveness of the proposed method in estimating the causal effects.

Soft Preference Optimization: Aligning Language Models to Expert Distributions

We propose Soft Preference Optimization (SPO), a method for aligning generative models, such as Large Language Models (LLMs), with human preferences, without the need for a reward model. SPO optimizes model outputs directly over a preference dataset through a natural loss function that integrates preference loss with a regularization term across the model's entire output distribution rather than limiting it to the preference dataset. Although SPO does not require the assumption of an existing underlying reward model, we demonstrate that, under the Bradley-Terry (BT) model assumption, it converges to a softmax of scaled rewards, with the distribution's "softness" adjustable via the softmax exponent, an algorithm parameter. We showcase SPO's methodology, its theoretical foundation, and its comparative advantages in simplicity, computational efficiency, and alignment precision.

MetaOptimize: A Framework for Optimizing Step Sizes and Other Meta-parameters

This paper addresses the challenge of optimizing meta-parameters (i.e., hyperparameters) in machine learning algorithms, a critical factor influencing training efficiency and model performance. Moving away from the computationally expensive traditional meta-parameter search methods, we introduce MetaOptimize framework that dynamically adjusts meta-parameters, particularly step sizes (also known as learning rates), during training. More specifically, MetaOptimize can wrap around any first-order optimization algorithm, tuning step sizes on the fly to minimize a specific form of regret that accounts for long-term effect of step sizes on training, through a discounted sum of future losses. We also introduce low complexity variants of MetaOptimize that, in conjunction with its adaptability to multiple optimization algorithms, demonstrate performance competitive to those of best hand-crafted learning rate schedules across various machine learning applications.

Learning Unknown Intervention Targets in Structural Causal Models from Heterogeneous Data

We study the problem of identifying the unknown intervention targets in structural causal models where we have access to heterogeneous data collected from multiple environments. The unknown intervention targets are the set of endogenous variables whose corresponding exogenous noises change across the environments. We propose a two-phase approach which in the first phase recovers the exogenous noises corresponding to unknown intervention targets whose distributions have changed across environments. In the second phase, the recovered noises are matched with the corresponding endogenous variables. For the recovery phase, we provide sufficient conditions for learning these exogenous noises up to some component-wise invertible transformation. For the matching phase, under the causal sufficiency assumption, we show that the proposed method uniquely identifies the intervention targets. In the presence of latent confounders, the intervention targets among the observed variables cannot be determined uniquely. We provide a candidate intervention target set which is a superset of the true intervention targets. Our approach improves upon the state of the art as the returned candidate set is always a subset of the target set returned by previous work. Moreover, we do not require restrictive assumptions such as linearity of the causal model or performing invariance tests to learn whether a distribution is changing across environments which could be highly sample inefficient. Our experimental results show the effectiveness of our proposed algorithm in practice.

Efficiently Escaping Saddle Points for Non-Convex Policy Optimization

Policy gradient (PG) is widely used in reinforcement learning due to its scalability and good performance. In recent years, several variance-reduced PG methods have been proposed with a theoretical guarantee of converging to an approximate first-order stationary point (FOSP) with the sample complexity of $O(\epsilon^{-3})$. However, FOSPs could be bad local optima or saddle points. Moreover, these algorithms often use importance sampling (IS) weights which could impair the statistical effectiveness of variance reduction. In this paper, we propose a variance-reduced second-order method that uses second-order information in the form of Hessian vector products (HVP) and converges to an approximate second-order stationary point (SOSP) with sample complexity of $\tilde{O}(\epsilon^{-3})$. This rate improves the best-known sample complexity for achieving approximate SOSPs by a factor of $O(\epsilon^{-0.5})$. Moreover, the proposed variance reduction technique bypasses IS weights by using HVP terms. Our experimental results show that the proposed algorithm outperforms the state of the art and is more robust to changes in random seeds.

Fast Causal Orientation Learning in Directed Acyclic Graphs

Causal relationships among a set of variables are commonly represented by a directed acyclic graph. The orientations of some edges in the causal DAG can be discovered from observational/interventional data. Further edges can be oriented by iteratively applying so-called Meek rules. Inferring edges' orientations from some previously oriented edges, which we call Causal Orientation Learning (COL), is a common problem in various causal discovery tasks. In these tasks, it is often required to solve multiple COL problems and therefore applying Meek rules could be time-consuming. Motivated by Meek rules, we introduce Meek functions that can be utilized in solving COL problems. In particular, we show that these functions have some desirable properties, enabling us to speed up the process of applying Meek rules. In particular, we propose a dynamic programming (DP) based method to apply Meek functions. Moreover, based on the proposed DP method, we present a lower bound on the number of edges that can be oriented as a result of intervention. We also propose a method to check whether some oriented edges belong to a causal DAG. Experimental results show that the proposed methods can outperform previous work in several causal discovery tasks in terms of running-time.

Stochastic Second-Order Methods Provably Beat SGD For Gradient-Dominated Functions

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that SCRN improves the best-known sample complexity of stochastic gradient descent in achieving $\epsilon$-global optimum by a factor of $\mathcal{O}(\epsilon^{-1/2})$. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the sample complexity of SCRN can be improved by a factor of ${\mathcal{O}}(\epsilon^{-1/2})$ using a variance reduction method with time-varying batch sizes. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.

A Unified Experiment Design Approach for Cyclic and Acyclic Causal Models

We study experiment design for the unique identification of the causal graph of a system where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design. Unlike the case of acyclic graphs, learning the skeleton of the causal graph from observational distribution may not be possible. Furthermore, intervening on a variable does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for the unique identification of the causal graph in the worst case.

Adaptive Momentum-Based Policy Gradient with Second-Order Information

The variance reduced gradient estimators for policy gradient methods has been one of the main focus of research in the reinforcement learning in recent years as they allow acceleration of the estimation process. We propose a variance reduced policy gradient method, called SGDHess-PG, which incorporates second-order information into stochastic gradient descent (SGD) using momentum with an adaptive learning rate. SGDHess-PG algorithm can achieve $\epsilon$-approximate first-order stationary point with $\tilde{O}(\epsilon^{-3})$ number of trajectories, while using a batch size of $O(1)$ at each iteration. Unlike most previous work, our proposed algorithm does not require importance sampling techniques which can compromise the advantage of variance reduction process. Our extensive experimental results show the effectiveness of the proposed algorithm on various control tasks and its advantage over the state of the art in practice.