The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This yields a generalized posterior that naturally accommodates structural constraints. As a direct instantiation, we develop MINimalist Thompson Sampling (MINTS). For multi-armed bandits with mean constraints, we establish near-optimal non-asymptotic regret guarantees and sharp almost-sure asymptotic regret characterizations. In particular, MINTS attains the classical Lai--Robbins constant in the unstructured setting and automatically adapts to unimodal structure, achieving the sharp constant determined only by the immediate neighbors of the optimal arm.

Adaptive importance sampling(AIS) uses past samples to update thesampling policyqt.

For eachDk subsets, respectively, we follow the same subsampling procedure used in the singlevariate case. That is, we letq10 = 10% of the first data subset andq20 = 10% of the second data subset.

Online retailing has become increasingly popular over the last decades [17, 28, 52]. The way of product ranking is the crux for online retailers because it determines the consumers' shopping behaviors [17] and thus influences the retailers' revenue [20, 49]. For instance, the probability of consumers' purchasing from a firm or clicking an advertisement is strongly related to the display order[8,3,33].

In addition, we define1 to be a vector with all the entries being 1, andI be the identity matrix. Suppose thatδ > 0andε (0,(1 γ) 1/2]. The remainder of this section is devotedtoprovingTheorem3. VT) to be the policy (resp. The remainder of this section is devotedtoprovingTheorem4.

IMPALA: scalable distributed deep-rl with importance weighted actor-learner architectures.69

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.

When f is convex with respect tox, many efficient algorithms can be powerful tools for solving Problem(1). One well-known algorithm is mirror descent (MD) [5]which is based on Bregman divergence.

C.1 Datasetdescription For WMT'16 English-German experiment, we used the same preprocessed data provided by [31] 1, including the samevalidation(neewsteest2013)andtest (neewsteest2014) splits. The data volume for train, validation and test splits are 4500966, 3000, 3003 sentence pairs respectively. When using LayerDrop we use 50% dropout probability. Similarly,we use beam search with beam size 5and length penalty 1.0 for decoding. First, we show that adding the auxiliary lossLK discretizes the samples and achieve the pruning purpose byenforcingsparsity oftheresulting model.

We study the problem of identifying viewers of arbitrary images based on their eye gaze. Psychological research has derived generative stochastic models of eye movements. In order to exploit this background knowledge within a discriminatively trained classification model, we derive Fisher kernels from different generative models of eye gaze. Experimentally, we find that the performance of the classifier strongly depends on the underlying generative model. Using an SVM with Fisher kernel improves the classification performance over the underlying generative model.