The universal approximation Theorem 3.1 immediately implies another universal approximation Thus y (t) solves the ODE (2.6), with initial condition y (0) = y (0) = 0 . Reconstruction of a continuous signal from its sine transform. Step 0: (Equicontinuity) We recall the following fact from topology. F (τ):= null f (τ), for τ 0, f ( τ), for τ 0. Since F is odd, the Fourier transform of F is given by We provide the details below. The next step in the proof of the fundamental Lemma 3.5 needs the following preliminary result in By (B.3), this implies that It follows from Lemma 3.4 that for any input By the sine transform reconstruction Lemma B.1, there exists It follows from Lemma 3.6, that there exists Indeed, Lemma 3.7 shows that time-delays of any given input signal can be approximated with any Step 1: By the Fundamental Lemma 3.5, there exist It follows from Lemma 3.6, that there exists an oscillator Step 3: Finally, by Lemma 3.8, there exists an oscillator network,

Thislinkbetween the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error,that depends onthespectrum oftheRKHS kernel.

We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.

In this work, we identify anovel set of conditions that ensure convergence with probability 1 ofQ-learning with linear function approximation, by proposing a twotime-scalevariationthereof.

We provide a more extension discussion for the context of this work. Firstly, when closed-form expressions for the optimizer of a function are unavailable, solving optimization problems requires iterative schemes such as gradient ascent [31]. Their convergence to global extrema is predicated on concavity and the tractability of computing ascent directions. When the objective takes the form of an expected value of a function parameterized by a random variable, stochastic approximations are required [36, 24]. The PG Theorem mentioned above gives a specific form for obtaining ascent directions with respect to a parameterized family of stationary policies via trajectories in a Markov decision process, when the objective is the expected cumulative return [44], which gives rise to the REINFORCE algorithm.