Satisfaction of the strict saddle property has become a standard assumption in non-convex optimization, and it ensures that many first-order optimization
algorithms will almost always escape saddle points. However, functions exist in machine learning that do not satisfy this property, such as the loss function of a neural network with at least two hidden layers. First-order methods such as gradient descent may converge to non-strict saddle points of such functions, and there do not currently exist any first-order methods that reliably escape non-strict saddle points. To address this need, we demonstrate that regularizing a function with a linear term enforces the strict saddle property, and we provide justification for only regularizing locally, i.e., when the norm of the gradient falls below a certain threshold. We analyze bifurcations that may result from this form of regularization, and then we provide a selection rule for regularizers that depends only on the gradient of an objective function. This rule is shown to guarantee that gradient descent will escape the neighborhoods around a broad class of non-strict saddle points, and this behavior is demonstrated on numerical examples of non-strict saddle points common in the optimization literature.

Linear Regularizers Enforce the Strict Saddle Property

Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing a class of DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coordinate-wise stationary point. We prove that this new optimality condition is always stronger than the standard critical point condition and directional point condition under a mild \textit{locally bounded nonconvexity assumption}. For comparisons, we also include a naive variant of coordinate descent methods based on sequential convex approximation in our study. When the objective function satisfies a \textit{globally bounded nonconvexity assumption} and \textit{Luo-Tseng error bound assumption}, coordinate descent methods achieve \textit{Q-linear} convergence rate. Also, for many applications of interest, we show that the nonconvex one-dimensional subproblem can be computed exactly and efficiently using a breakpoint searching method. Finally, we have conducted extensive experiments on several statistical learning tasks to show the superiority of our approach.

Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergence

This paper presents SVAM (Sequential Variance-Altered MLE), a unified framework for learning generalized linear models under adversarial label corruption in training data. SVAM extends to tasks such as least squares regression, logistic regression, and gamma regression, whereas many existing works on learning with label corruptions focus only on least squares regression. SVAM is based on a novel variance reduction technique that may be of independent interest and works by iteratively solving weighted MLEs over variance-altered versions of the GLM objective. SVAM offers provable model recovery guarantees superior to the state-of-the-art for robust regression even when a constant fraction of training labels are adversarially corrupted. SVAM also empirically outperforms several existing problem-specific techniques for robust regression and classification. Code for SVAM is available at https://github.com/purushottamkar/svam/

Corruption-tolerant Algorithms for Generalized Linear Models

We study risk-sensitive reinforcement learning (RL) based on an entropic risk measure in episodic non-stationary Markov decision processes (MDPs). Both the reward functions and the state transition kernels are unknown and allowed to vary arbitrarily over time with a budget on their cumulative variations. When this variation budget is known a prior, we propose two restart-based algorithms, namely Restart-RSMB and Restart-RSQ, and establish their dynamic regrets. Based on these results, we further present a meta-algorithm that does not require any prior knowledge of the variation budget and can adaptively detect the non-stationarity on the exponential value functions. A dynamic regret lower bound is then established for non-stationary risk-sensitive RL to certify the near-optimality of the proposed algorithms. Our results also show that the risk control and the handling of the non-stationarity can be separately designed in the algorithm if the variation budget is known a prior, while the non-stationary detection mechanism in the adaptive algorithm depends on the risk parameter. This work offers the first non-asymptotic theoretical analyses for the non-stationary risk-sensitive RL in the literature. 

Non-stationary Risk-sensitive Reinforcement Learning: Near-optimal Dynamic Regret, Adaptive Detection, and Separation Design

Modern machine learning models may be susceptible to learning spurious correlations that hold on average but not for the atypical group of samples. To address the problem, previous approaches minimize the empirical worst-group risk. Despite the promise, they often assume that each sample belongs to one and only one group, which does not allow expressing the uncertainty in group labeling. In this paper, we propose a novel framework PG-DRO, which explores the idea of probabilistic group membership for distributionally robust optimization. Key to our framework, we consider soft group membership instead of hard group annotations. The group probabilities can be flexibly generated using either supervised learning or zero-shot approaches. Our framework accommodates samples with group membership ambiguity, offering stronger flexibility and generality than the prior art. We comprehensively evaluate PG-DRO on both image classification and natural language processing benchmarks, establishing superior performance.

Distributionally Robust Optimization with Probabilistic Group

We present regret minimization algorithms for stochastic contextual MDPs under minimum reachability assumption, using an access to an offline least square regression oracle.
    We analyze three different settings: where the dynamics is known, where the dynamics is unknown but independent of the context and the most challenging setting where the dynamics is unknown and context-dependent. For the latter, our algorithm obtains regret bound of
    $\widetilde{O}( (H+{1}/{p_{min}})H|S|^{3/2}\sqrt{|A|T\log(\max\{|\mathcal{G}|,|\mathcal{P}|\}/\delta)})$ with probability $1-\delta$, where $\mathcal{P}$ and $\mathcal{G}$ are finite and realizable function classes used to approximate the dynamics and rewards respectively, $p_{min}$ is the minimum reachability parameter, $S$ is the set of states, $A$ the set of actions, $H$ the horizon, and $T$ the number of episodes.
    To our knowledge, our approach is the first optimistic approach applied to contextual MDPs with general function approximation (i.e., without additional knowledge regarding the function class, such as it being linear and etc.).
    We present a lower bound of $\Omega(\sqrt{T  H |S| |A| \ln(|\mathcal{G}|)/\ln(|A|)})$, on the expected regret which holds even in the case of known dynamics.
    Lastly, we discuss an extension of our results to CMDPs without minimum reachability, that obtains $\widetilde{O}(T^{3/4})$ regret.

Optimism in Face of a Context:Regret Guarantees for Stochastic Contextual MDP

The problem of image aesthetic quality assessment is surprisingly difficult to define precisely. Most early work attempted to estimate the average aesthetic rating of a group of observers, while some recent work has shifted to an approach based on few-shot personalization. In this paper, we connect few-shot personalization, via Immanuel Kant's concept of disinterested judgment, to an argument from feminist aesthetics about the biased tendencies of objective standards for subjective pleasures. To empirically investigate this philosophical debate, we introduce PR-AADB, a relabeling of the existing AADB dataset with labels for pairs of images, and measure how well the existing groundtruth predicts our new pairwise labels. We find, consistent with the feminist critique, that both the existing groundtruth and few-shot personalized predictions represent some users' preferences significantly better than others, but that it is difficult to predict when and for whom the existing groundtruth will be correct. We thus advise against using benchmark datasets to evaluate models for personalized IAQA, and recommend caution when attempting to account for subjective difference using machine learning more generally.

Correct for Whom? Subjectivity and the Evaluation of Personalized Image Aesthetics Assessment Models

How to efficiently explore in reinforcement learning is an open problem. Many exploration algorithms employ the epistemic uncertainty of their own value predictions -- for instance to compute an exploration bonus or upper confidence bound. Unfortunately the required uncertainty is difficult to estimate in general with function approximation.

We propose epistemic value estimation (EVE): a recipe that is compatible with sequential decision making and with neural network function approximators. It equips agents with a tractable posterior over all their parameters from which epistemic value uncertainty can be computed efficiently.

We use the recipe to derive an epistemic Q-Learning agent and observe competitive performance on a series of benchmarks. Experiments confirm that the EVE recipe facilitates efficient exploration in hard exploration tasks.

Exploration via Epistemic Value Estimation

We study critical systems that allocate scarce resources to satisfy basic needs, such as homeless services that provide housing. These systems often support communities disproportionately affected by systemic racial, gender, or other injustices, so it is crucial to design these systems with fairness considerations in mind. To address this problem, we propose a framework for evaluating fairness in contextual resource allocation systems that is inspired by fairness metrics in machine learning. This framework can be applied to evaluate the fairness properties of a historical policy, as well as to impose constraints in the design of new (counterfactual) allocation policies. Our work culminates with a set of incompatibility results that investigate the interplay between the different fairness metrics we propose. Notably, we demonstrate that: 1) fairness in allocation and fairness in outcomes are usually incompatible; 2) policies that prioritize based on a vulnerability score will usually result in unequal outcomes across groups, even if the score is perfectly calibrated; 3) policies using contextual information beyond what is needed to characterize baseline risk and treatment effects can be fairer in their outcomes than those using just baseline risk and treatment effects; and 4) policies using group status in addition to baseline risk and treatment effects are as fair as possible given all available information. Our framework can help guide the discussion among stakeholders in deciding which fairness metrics to impose when allocating scarce resources.

Fairness in Contextual Resource Allocation Systems: Metrics and Incompatibility Results

Constrained Reinforcement Learning (CRL) burgeons broad interest in recent years, which pursues maximizing long-term returns while constraining costs. Although CRL can be cast as a multi-objective optimization problem, it is still facing the key challenge that gradient-based Pareto optimization methods tend to stick to known Pareto-optimal solutions even when they yield poor returns (e.g., the safest self-driving car that never moves) or violate the constraints (e.g., the record-breaking racer that crashes the car). In this paper, we propose Gradient-adaptive Constrained Policy Optimization (GCPO for short), a novel Pareto optimization method for CRL with two adaptive gradient recalibration techniques. First, to find Pareto-optimal solutions with balanced performance over all targets, we propose gradient rebalancing which forces the agent to improve more on under-optimized objectives at every policy iteration. Second, to guarantee that the cost constraints are satisfied, we propose gradient perturbation that can temporarily sacrifice the returns for costs. Experiments on the SafetyGym benchmarks show that our method consistently outperforms previous CRL methods in reward while satisfying the constraints.

Gradient-Adaptive Pareto Optimization for Constrained Reinforcement Learning

Task-Free Continual Learning (TFCL) represents a challenging scenario for lifelong learning because it does not access any task information. The Dynamic Expansion Model (DEM) has shown promising results in this scenario due to its scalability and generalisation power. However, existing work from DEM focuses only on addressing forgetting and ignores model compression, which is hardly deployed in practical systems. In this work, we aim to simultaneously address network forgetting and model compression by developing the Lifelong Compression Mixture Model (LGMM) equipped with the maximum mean discrepancy (MMD) based expansion criterion for model expansion. A diversity-aware sample selection approach is proposed to selectively store a variety of samples to promote information diversity among the components of the LGMM, which allows more knowledge to be captured with an appropriate model size. Since we can not access the past data and task information in TFCL, it is challenging to apply the existing compression technologies in LCMM. Inspired by this, we propose a data-free component discarding mechanism that evaluates a knowledge relation graph matrix describing the relevance between each pair of components. A greedy selection procedure is proposed to statistically select  overlapping components and remove them from the LGMM. The proposed discarding mechanism can be performed during or after the training. Experiments on different datasets show that LGMM achieves the best performance for TFCL.

Lifelong Compression Mixture Model via Knowledge Relationship Graph

This paper introduces and studies zero-base generalized few-shot learning (zero-base GFSL), which is an extreme yet practical version of few-shot learning problem. Motivated by the cases where base data is not available due to privacy or ethical issues, the goal of zero-base GFSL is to newly incorporate the knowledge of few samples of novel classes into a pretrained model without any samples of base classes. According to our analysis, we discover the fact that both mean and variance of the weight distribution of novel classes are not properly established, compared to those of base classes. The existing GFSL methods attempt to make the weight norms balanced, which we find help only the variance part, but discard the importance of mean of weights particularly for novel classes, leading to the limited performance in the GFSL problem even with base data. In this paper, we overcome this limitation by proposing a simple yet effective normalization method that can effectively control both mean and variance of the weight distribution of novel classes without using any base samples and thereby achieve a satisfactory performance on both novel and base classes. Our experimental results somewhat surprisingly show that the proposed zero-base GFSL method that does not utilize any base samples even outperforms the existing GFSL methods that make the best use of base data.  Our implementation is available at: https://github.com/bigdata-inha/Zero-Base-GFSL.

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Linear Regularizers Enforce the Strict Saddle Property

Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergence

Corruption-tolerant Algorithms for Generalized Linear Models

Non-stationary Risk-sensitive Reinforcement Learning: Near-optimal Dynamic Regret, Adaptive Detection, and Separation Design

Distributionally Robust Optimization with Probabilistic Group

Optimism in Face of a Context:Regret Guarantees for Stochastic Contextual MDP

Correct for Whom? Subjectivity and the Evaluation of Personalized Image Aesthetics Assessment Models

Exploration via Epistemic Value Estimation

Fairness in Contextual Resource Allocation Systems: Metrics and Incompatibility Results

Gradient-Adaptive Pareto Optimization for Constrained Reinforcement Learning

Lifelong Compression Mixture Model via Knowledge Relationship Graph

Better Generalized Few-Shot Learning Even Without Base Data

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