We present the first optimal rates for infinite-dimensional vector-valued
ridge regression on a continuous scale of norms that interpolate between $L_2$
and the hypothesis space, which we consider as a vector-valued reproducing
kernel Hilbert space. These rates allow to treat the misspecified case in which
the true regression function is not contained in the hypothesis space. We
combine standard assumptions on the capacity of the hypothesis space with a
novel tensor product construction of vector-valued interpolation spaces in
order to characterize the smoothness of the regression function. Our upper
bound not only attains the same rate as real-valued kernel ridge regression,
but also removes the assumption that the target regression function is bounded.
For the lower bound, we reduce the problem to the scalar setting using a
projection argument. We show that these rates are optimal in most cases and
independent of the dimension of the output space. We illustrate our results for
the special case of vector-valued Sobolev spaces.

我们提出了一个连续标准的无限维向量值岭回归问题的第一个最优率，在 $L_2$ 和假设空间之间插值的范数上，我们认为这是一个向量值再生核希尔伯特空间。这些率允许处理真实回归函数不包含在假设空间中的不正确情况。我们将假设空间的容量标准假设与一种新的向量值插值空间的张量积构造相结合，以刻画回归函数的平滑程度。我们的上界不仅达到与实值核岭回归相同的速率，而且消除了目标回归函数有界的假设。对于下界，我们使用投影论证将问题简化为标量情况。我们证明这些速率在大多数情况下是最优的，并且与输出空间的维度无关。我们以向量值 Sobolev 空间的特殊情况来说明我们的结果。

针对向量 - 值正则化最小二乘算法的最佳 Sobolev 范数速率

Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized  Least-Squares Algorithm

We consider online reinforcement learning (RL) in episodic Markov decision
processes (MDPs) under the linear $q^\pi$-realizability assumption, where it is
assumed that the action-values of all policies can be expressed as linear
functions of state-action features. This class is known to be more general than
linear MDPs, where the transition kernel and the reward function are assumed to
be linear functions of the feature vectors. As our first contribution, we show
that the difference between the two classes is the presence of states in
linearly $q^\pi$-realizable MDPs where for any policy, all the actions have
approximately equal values, and skipping over these states by following an
arbitrarily fixed policy in those states transforms the problem to a linear
MDP. Based on this observation, we derive a novel (computationally inefficient)
learning algorithm for linearly $q^\pi$-realizable MDPs that simultaneously
learns what states should be skipped over and runs another learning algorithm
on the linear MDP hidden in the problem. The method returns an
$\epsilon$-optimal policy after $\text{polylog}(H, d)/\epsilon^2$ interactions
with the MDP, where $H$ is the time horizon and $d$ is the dimension of the
feature vectors, giving the first polynomial-sample-complexity online RL
algorithm for this setting. The results are proved for the misspecified case,
where the sample complexity is shown to degrade gracefully with the
misspecification error.

在线强化学习中的线性可实现的马尔可夫决策过程 (MDP)，提出了一种计算效率较低的学习算法，通过跳过特定状态转化为线性 MDP，并证明了该算法在这种情况下具有多项式样本复杂度。