Risk Modeling

Distributional reinforcement learning algorithms have attempted to utilize estimated uncertainty for exploration, such as optimism in the face of uncertainty. However, using the estimated variance for optimistic exploration may cause biased data collection and hinder convergence or performance. In this paper, we present a novel distributional reinforcement learning that selects actions by randomizing risk criterion without losing the risk-neutral objective. We provide a perturbed distributional Bellman optimality operator by distorting the risk measure. Also,we prove the convergence and optimality of the proposed method with the weaker contraction property. Our theoretical results support that the proposed method does not fall into biased exploration and is guaranteed to converge to an optimal return. Finally, we empirically show that our method outperforms other existing distribution-based algorithms in various environments including Atari 55 games.

Offline reinforcement learning (RL) is suitable for safety-critical domains where online exploration is not feasible. In such domains, decision-making should take into consideration the risk of catastrophic outcomes. In other words, decision-making should be *risk-averse*. An additional challenge of offline RL is avoiding *distributional shift*, i.e. ensuring that state-action pairs visited by the policy remain near those in the dataset. Previous offline RL algorithms that consider risk combine offline RL techniques (to avoid distributional shift), with risk-sensitive RL algorithms (to achieve risk-aversion). In this work, we propose risk-aversion as a mechanism to jointly address *both* of these issues. We propose a model-based approach, and use an ensemble of models to estimate epistemic uncertainty, in addition to aleatoric uncertainty. We train a policy that is risk-averse, and avoids high uncertainty actions. Risk-aversion to epistemic uncertainty prevents distributional shift, as areas not covered by the dataset have high epistemic uncertainty. Risk-aversion to aleatoric uncertainty discourages actions that are risky due to environment stochasticity. Thus, by considering epistemic uncertainty via a model ensemble and introducing risk-aversion, our algorithm (1R2R) avoids distributional shift in addition to achieving risk-aversion to aleatoric risk. Our experiments show that 1R2R achieves strong performance on deterministic benchmarks, and outperforms existing approaches for risk-sensitive objectives in stochastic domains.

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers. Using this framework, we analyze the uniform-mass binning (UMB) recalibration method and establish a finite-sample risk upper bound of order $\tilde{O}(B/n + 1/B^2)$ where $B$ is the number of bins and $n$ is the sample size. By balancing calibration and sharpness, we further determine that the optimal number of bins for UMB scales with $n^{1/3}$, resulting in a risk bound of approximately $O(n^{-2/3})$. Additionally, we tackle the challenge of label shift by proposing a two-stage approach that adjusts the recalibration function using limited labeled data from the target domain. Our results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. We validate our theoretical findings through numerical simulations, which confirm the tightness of the proposed bounds, the optimal number of bins, and the effectiveness of label shift adaptation.

We provide the first polynomial-time convergence guarantees for the probabilistic flow ODE implementation (together with a corrector step) of score-based generative modeling. Our analysis is carried out in the wake of recent results obtaining such guarantees for the SDE-based implementation (i.e., denoising diffusion probabilistic modeling or DDPM), but requires the development of novel techniques for studying deterministic dynamics without contractivity. Through the use of a specially chosen corrector step based on the underdamped Langevin diffusion, we obtain better dimension dependence than prior works on DDPM ($O(\sqrt d)$ vs. $O(d)$, assuming smoothness of the data distribution), highlighting potential advantages of the ODE framework.

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

Generalized Additive Models (GAMs) are a family of flexible and interpretable models with old roots in statistics. GAMs are often used with pairwise interactions to improve model accuracy while still retaining flexibility and interpretability but lead to computational challenges as we are dealing with order of $p^2$ terms. It is desirable to restrict the number of components (i.e., encourage sparsity) for easier interpretability, and better computational and statistical properties. Earlier approaches, considering sparse pairwise interactions, have limited scalability, especially when imposing additional structural interpretability constraints. We propose a flexible GRAND-SLAMIN framework that can learn GAMs with interactions under sparsity and additional structural constraints in a differentiable end-to-end fashion. We customize first-order gradient-based optimization to perform sparse backpropagation to exploit sparsity in additive effects for any differentiable loss function in a GPU-compatible manner. Additionally, we establish novel non-asymptotic prediction bounds for our estimators with tree-based shape functions. Numerical experiments on real-world datasets show that our toolkit performs favorably in terms of performance, variable selection and scalability when compared with popular toolkits to fit GAMs with interactions. Our work expands the landscape of interpretable modeling while maintaining prediction accuracy competitive with non-interpretable black-box models. Our code is available at https://github.com/mazumder-lab/grandslamin.

We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.