Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Convergence of stochastic gradient descent for non-smooth problems is a known result. For completeness, wereproduce and adapt ausual proof toour setting. Let us denote byF the class of functions fromX toY we are going to work with. Assumption 1 states that we have a well-specified modelF to estimate the median,i.e. Let us begin by controlling the estimation error.

Landmarks were distributed atdistance 200 from the origin, everyπ3 radians, starting at0. The listener particle had amassof0.1;itsforceactions Episodeslasted50timesteps. The 9-points environment used the same physics engine astriangle and episodes also lasted 50 timesteps. We found that additional time enabled more convergent behaviors: that is, agents hadtolearn toreach alocation andstopformaximal reward,asopposed totraveling atfull speed in a fixed direction. Whenexperimenting with environmental noise, allagents were trained using identical noise models and evaluated with zeronoise.

This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension d of the state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity O(d log d). We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.

We analyze the HyperCube model, an \textit{operator-valued} tensor factorization architecture that discovers group structures and their unitary representations. We provide a rigorous theoretical explanation for this inductive bias by decomposing its objective into a term regulating factor scales ($\mathcal{B}$) and a term enforcing directional alignment ($\mathcal{R} \geq 0$). This decomposition isolates the \textit{collinear manifold} ($\mathcal{R}=0$), to which numerical optimization consistently converges for group isotopes. We prove that this manifold admits feasible solutions exclusively for group isotopes, and that within it, $\mathcal{B}$ exerts a variational pressure toward unitarity. To bridge the gap to the global landscape, we formulate a \textit{Collinearity Dominance Conjecture}, supported by empirical observations. Conditional on this dominance, we prove two key results: (1) the global minimum is achieved by the unitary regular representation for groups, and (2) non-group operations incur a strictly higher objective value, formally quantifying the model's inductive bias toward the associative structure of groups (up to isotopy).

Agnostic learning of Boolean halfspaces is a fundamental problem in computational learning theory, but it is known to be computationally hard even for weak learning. Recent work [CKKMK24] proposed smoothed analysis as a way to bypass such hardness, but existing frameworks rely on additive Gaussian perturbations, making them unsuitable for discrete domains. We introduce a new smoothed agnostic learning framework for Boolean inputs, where perturbations are modeled via random bit flips. This defines a natural discrete analogue of smoothed optimality generalizing the Gaussian case. Under strictly subexponential assumptions on the input distribution, we give an efficient algorithm for learning halfspaces in this model, with runtime and sample complexity approximately n raised to a poly(1/(sigma * epsilon)) factor. Previously, such algorithms were known only with strong structural assumptions for the discrete hypercube, for example, independent coordinates or symmetric distributions. Our result provides the first computationally efficient guarantee for smoothed agnostic learning of halfspaces over the Boolean hypercube, bridging the gap between worst-case intractability and practical learnability in discrete settings.