Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

How to design efficient algorithms is a central problem for reinforcement learning (RL).

Wewill also writeζθ(x) = fθ(x) f?(x)to denote the fitting error. We use Gaussian initialization: if` {1,2,...,L}, the weights are initialized as

In this section, we present some additional experiments. Empirical setup Most of the experimental setups are the same as those in Section 6, except that now we use 5 parties instead of 3 parties. There are 90 dimensions for a single data in YearPredictionMSD dataset, and we let each party hold 18 dimensions. Empirical results We plot the training loss instead of the testing loss since we are comparing differentobjectivefunctions. A.4 Experimentsonotherdatasets In this section, we present the experiment results on another dataset.

Another line of research on theoretical 13 studies with partially observability considers the environment with rich observations [26, 12, 13].

In this setting, unlike basic setting, objective and constraints are not linear. We focus on a single state-action pairs,a, stage h, and objectivem. Similarly, in constrained settings, its estimated resource consumptions are underestimates of the true resource consumptions. B.5 BoundingtheBellmanerror We now provide an upper bound on the Bellman error which arises in the RHS of the regret decomposition(Proposition3.3). When neither failure events occur (probability 1 2δ), Proposition 3.3 upper bounds either of reward or consumption regret by In this section, we prove the main guarantee for the convex-concave setting.

In this paper, we discuss the expressivity and trainability of narrow neural networks. Appendix H introduces the following contents.