A major bottleneck in scientific discovery consists of narrowing an exponentially large set of objects, such as proteins or molecules, to a small set of promising candidates with desirable properties. While this process can rely on expert knowledge, recent methods leverage reinforcement learning (RL) guided by a proxy reward function to enable this filtering. By employing various forms of entropy regularization, these methods aim to learn samplers that generate diverse candidates that are highly rated by the proxy function. In this work, we make two main contributions. First, we show that these methods are liable to generate overly diverse, suboptimal candidates in large search spaces. To address this issue, we introduce a novel unified operator that combines several regularized RL operators into a general framework that better targets peakier sampling distributions. Secondly, we offer a novel, robust RL perspective of this filtering process. The regularization can be interpreted as robustness to a compositional form of uncertainty in the proxy function (i.e., the true evaluation of a candidate differs from the proxy's evaluation). Our analysis leads us to a novel, easy-to-use algorithm we name trajectory general mellowmax (TGM): we show it identifies higher quality, diverse candidates than baselines in both synthetic and real-world tasks. Code: https://github.com/marcojira/tgm.

We propose REpresentation-Aware Distributionally Robust Estimation (READ), a novel framework for Wasserstein distributionally robust learning that accounts for predictive representations when guarding against distributional shifts. Unlike classical approaches that treat all feature perturbations equally, READ embeds a multidimensional alignment parameter into the transport cost, allowing the model to differentially discourage perturbations along directions associated with informative representations. This yields robustness to feature variation while preserving invariant structure. Our first contribution is a theoretical foundation: we show that seminorm regularizations for linear regression and binary classification arise as Wasserstein distributionally robust objectives, thereby providing tractable reformulations of READ and unifying a broad class of regularized estimators under the DRO lens. Second, we adopt a principled procedure for selecting the Wasserstein radius using the techniques of robust Wasserstein profile inference. This further enables the construction of valid, representation-aware confidence regions for model parameters with distinct geometric features. Finally, we analyze the geometry of READ estimators as the alignment parameters vary and propose an optimization algorithm to estimate the projection of the global optimum onto this solution surface. This procedure selects among equally robust estimators while optimally constructing a representation structure. We conclude by demonstrating the effectiveness of our framework through extensive simulations and a real-world study, providing a powerful robust estimation grounded in learning representation.

In high-dimensional sparse regression, the \textsc{Lasso} estimator offers excellent theoretical guarantees but is well-known to produce biased estimates. To address this, \cite{Javanmard2014} introduced a method to ``debias" the \textsc{Lasso} estimates for a random sub-Gaussian sensing matrix $\boldsymbol{A}$. Their approach relies on computing an ``approximate inverse" $\boldsymbol{M}$ of the matrix $\boldsymbol{A}^\top \boldsymbol{A}/n$ by solving a convex optimization problem. This matrix $\boldsymbol{M}$ plays a critical role in mitigating bias and allowing for construction of confidence intervals using the debiased \textsc{Lasso} estimates. However the computation of $\boldsymbol{M}$ is expensive in practice as it requires iterative optimization. In the presented work, we re-parameterize the optimization problem to compute a ``debiasing matrix" $\boldsymbol{W} := \boldsymbol{AM}^{\top}$ directly, rather than the approximate inverse $\boldsymbol{M}$. This reformulation retains the theoretical guarantees of the debiased \textsc{Lasso} estimates, as they depend on the \emph{product} $\boldsymbol{AM}^{\top}$ rather than on $\boldsymbol{M}$ alone. Notably, we provide a simple, computationally efficient, closed-form solution for $\boldsymbol{W}$ under similar conditions for the sensing matrix $\boldsymbol{A}$ used in the original debiasing formulation, with an additional condition that the elements of every row of $\boldsymbol{A}$ have uncorrelated entries. Also, the optimization problem based on $\boldsymbol{W}$ guarantees a unique optimal solution, unlike the original formulation based on $\boldsymbol{M}$. We verify our main result with numerical simulations.

Motivated by alternating learning dynamics in two-player games, a recent work by Cevher et al.(2024) shows that $o(\sqrt{T})$ alternating regret is possible for any $T$-round adversarial Online Linear Optimization (OLO) problem, and left as an open question whether the same is true for general Online Convex Optimization (OCO). We answer this question in the affirmative by showing that the continuous Hedge algorithm achieves $\tilde{\mathcal{O}}(d^{\frac{2}{3}}T^{\frac{1}{3}})$ alternating regret for any adversarial $d$-dimensional OCO problems. We show that this implies an alternating learning dynamic that finds a Nash equilibrium for any convex-concave zero-sum games or a coarse correlated equilibrium for any convex two-player general-sum games at a rate of $\tilde{\mathcal{O}}(d^{\frac{2}{3}}/T^{\frac{2}{3}})$. To further improve the time complexity and/or the dimension dependence, we propose another simple algorithm, Follow-the-Regularized-Leader with a regularizer whose convex conjugate is 3rd-order smooth, for OCO with smooth and self-concordant loss functions (such as linear or quadratic losses). We instantiate our algorithm with different regularizers and show that, for example, when the decision set is the $\ell_2$ ball, our algorithm achieves $\tilde{\mathcal{O}}(T^{\frac{2}{5}})$ alternating regret with no dimension dependence (and a better $\tilde{\mathcal{O}}(T^{\frac{1}{3}})$ bound for quadratic losses). We complement our results by showing some algorithm-specific alternating regret lower bounds, including a somewhat surprising $\Omega(\sqrt{T})$ lower bound for a Regret Matching variant that is widely used in alternating learning dynamics.

Transfer learning is a popular strategy to leverage external knowledge and improve statistical efficiency, particularly with a limited target sample. We propose a novel knowledge-guided Wasserstein Distributionally Robust Optimization (KG-WDRO) framework that adaptively incorporates multiple sources of external knowledge to overcome the conservativeness of vanilla WDRO, which often results in overly pessimistic shrinkage toward zero. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

In this work we introduce a new optimisation method called SAGA in the spirit of SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support for composite objectives where a proximal operator is used on the regulariser. Unlike SDCA, SAGA supports non-strongly convex problems directly, and is adaptive to any inherent strong convexity of the problem. We give experimental results showing the effectiveness of our method.

The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the unregularized barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension $d > 1$. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time $O(m \log{m})$ and linear space complexity $O(m)$ primal-dual algorithm, the Wasserstein-Descent $\dot{\mathbb{H}}^1$-Ascent (WDHA) algorithm, for computing the exact barycenter when the input probability density functions are discretized on an $m$-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., $1024 \times 1024$ images) 2D synthetic and real data.

In this work we introduce a new optimisation method called SAGA in the spirit of SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support for composite objectives where a proximal operator is used on the regulariser. Unlike SDCA, SAGA supports non-strongly convex problems directly, and is adaptive to any inherent strong convexity of the problem. We give experimental results showing the effectiveness of our method.

We derive generic information-theoretic and PAC-Bayesian generalization bounds involving an arbitrary convex comparator function, which measures the discrepancy between the training and population loss. The bounds hold under the assumption that the cumulant-generating function (CGF) of the comparator is upper-bounded by the corresponding CGF within a family of bounding distributions. We show that the tightest possible bound is obtained with the comparator being the convex conjugate of the CGF of the bounding distribution, also known as the Cram\'er function. This conclusion applies more broadly to generalization bounds with a similar structure. This confirms the near-optimality of known bounds for bounded and sub-Gaussian losses and leads to novel bounds under other bounding distributions.

Abstract--We introduce a new methodology dubbed "safe The rest of the paper is organized as follows. Our peeling This paper focuses on the resolution of the so-called "l IV and its performance is regularized least-squares" problem given by illustrated in Sec. V. Proofs of the results presented in the p Unfortunately, this problem also turns out to be has to be understood component-wise, e.g., x [l,u] means NP-hard [4, Th. 1]. Moreover, η() denotes the indicator function flurry of contributions proposing tractable procedures able to which equals to 0 if the condition in argument is fulfilled and recover approximate solutions of (1-P). This observation, combined with some recent III. Due to space limitation, we only review the elements of A standard approach is to use a Branch-and-Bound (BnB) interest to introduce the proposed peeling method. We refer the algorithm that solves (1-P), see [6-11]. In this paper, we propose a new strategy, dubbed "safe peeling", to accelerate the exact resolution of (1-P). In a A. Pruning nutshell, our contribution is a computationally simple test applied at each node of the BnB decision tree to identify some The crux of BnB methods consists in identifying and discarding intervals of R In this section, we introduce our proposed peeling procedure. This assumption will One ubiquitous approach in the literature [7, 9, 13] to find be relaxed later on in Sec. A proof of this result is available in App. A. A consequence On the one hand, item i) implies that the pruning test (4) of preserving the pruning decision is that taking the new involves the following quantity (rather than p This additional constraint usually takes the form " M x This impairment pertains to a large class of mixed-integer problems and with M > 0 and is known as "Big-M" constraint, see [7, Sec.