We introduce Multistage Conditional Compositional Optimization (MCCO) as a new paradigm for decision-making under uncertainty that combines aspects of multistage stochastic programming and conditional stochastic optimization. MCCO minimizes a nest of conditional expectations and nonlinear cost functions. It has numerous applications and arises, for example, in optimal stopping, linear-quadratic regulator problems, distributionally robust contextual bandits, as well as in problems involving dynamic risk measures. The naïve nested sampling approach for MCCO suffers from the curse of dimensionality familiar from scenario tree-based multistage stochastic programming, that is, its scenario complexity grows exponentially with the number of nests. We develop new multilevel Monte Carlo techniques for MCCO whose scenario complexity grows only polynomially with the desired accuracy.

Estimation of heterogeneous long-term treatment effects (HLTEs) is widely used for personalized decision-making in marketing, economics, and medicine, where short-term randomized experiments are often combined with long-term observational data. However, HLTE estimation is challenging due to limited overlap in treatment or in observing long-term outcomes for certain subpopulations, which can lead to unstable HLTE estimates with large finite-sample variance. To address this challenge, we introduce the LT-O-learners (Long-Term Orthogonal Learners), a set of novel orthogonal learners for HLTE estimation. The learners are designed for the canonical HLTE setting that combines a short-term randomized dataset $\mathcal{D}_1$ with a long-term historical dataset $\mathcal{D}_2$. The key idea of our LT-O-Learners is to retarget the learning objective by introducing custom overlap weights that downweight samples with low overlap in treatment or in long-term observation. We show that the retargeted loss is equivalent to the weighted oracle loss and satisfies Neyman-orthogonality, which means our learners are robust to errors in the nuisance estimation. We further provide a general error bound for the LT-O-Learners and give the conditions under which quasi-oracle rate can be achieved. Finally, our LT-O-learners are model-agnostic and can thus be instantiated with arbitrary machine learning models. We conduct empirical evaluations on synthetic and semi-synthetic benchmarks to confirm the theoretical properties of our LT-O-Learners, especially the robustness in low-overlap settings. To the best of our knowledge, ours are the first orthogonal learners for HLTE estimation that are robust to low overlap that is common in long-term outcomes.

By identifying an instrumental variable correlated with the variableX butunrelated tothe confounders, researchers can isolate the exogenous variation inX and estimate acausal relationship betweenX andY.

As afollow-up work, [26]further released the rank-one restriction on the action feature matrices, andtheyintroduced analgorithm LowGLOC based ontheonline-to-confidenceset conversion [2]for generalized low-rank matrix bandits with O( p (d1+d2)3rT)regret bound.

In this paper, we study the finite-time convergence of nonlinear SA with multiplecoupledsequences.

More specifically,inour main result (Theorem 3.2) we identify a set of sufficient conditions on the initialization and the network topology under which theglobal convergence ofgradient descent isobtained.

Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks.

Inthisworkwepresenta scalable quasi-Bayesian procedure for IV regression, building upon the recently developed kernelized IV models.

Table1: Linearnetwork Layer# Name Layer Inshape Outshape 1 Flatten() (3,32,32) 3072 2 fc1 nn.Linear(3072, 200) 3072 200 3 fc2 nn.Linear(200, 1) 200 1 Fully-connected Network We conduct further experiments on several different fully-connected networks with 4 hidden layers with various activation functions. Our subset is smaller because of the computation limitation when calculating the Gram matrix. Experiments show that the properties along GD trajectory(e.g. We consider simple linear networks, fully-connected networks, convolutional networks in this appendix. The following Figure 4 illustrates the positive correlation between thesharpness andtheA-norm, andtherelationship between theloss D(t) 2 and R(t) 2 alongthetrajectory.