At thet-th backward propagation step, we can derive the gradient il(wt)toupdatei-th module inM. The number in the bracket represents the batch size. We see that when the batch size is small (i.e.,32), the gradientvariancesaresimilar. N and K indicate the number of FPN levels and region proposals fed into the detection head. To evaluate this assumption, as shown in Figure 1, we have three observations. As illustrated by the second figure in Figure 1, the gradient misalignment phenomenon between detection head and backbone has been reduced.

Manynon-parametric regression methods areproposed to solve the regression problem by assuming thatf falls into certain function classes, including polynomial splines Stone (1994), local polynomials Cleveland (1979); Stone (1977), the spectral algorithmsCaponnetto(2006);CaponnettoandDeVito(2007);CaponnettoandYao(2010),etc.

In Appendix E, we detail our experimental settings and exhibit additional experimental results.

We present a computationally efficient algorithm for learning in this framework that simultaneously achieves near-optimal regret bounds in both stochastic and adversarial environments. The bound against oblivious adversaries is O( αT), where T is the time horizon andα is the independence number of the feedback graph.

W = 2for = 10 2and = 10 4, omitted architectures.

LetAbean nHermitian matrixandletBbea(n 1) (n 1)matrixwhich is constructed by deleting thei-th row andi-th column ofA. Denote thatΦ = [ϕ(x1),...,ϕ(xn)] Rn D, where D is the dimension of feature spaceH. Performing rank-n singular value decomposition (SVD) onΦ, we have Φ = HΣV, where H Rn n, Σ Rn n is a diagonal matrix whose diagonal elements are the singular values of Φ,andV RD n. F(α) in Eq.(21) is proven differentiable and thep-th component of the gradient is F(α) αp = Then, a reduced gradient descent algorithm [26] is adopted to optimize Eq.(21). The three deep neural networks are pre-trained on the ImageNet[5].

This appendix contains additional material for the paper "Optimistic tree search strategies forblack-box combinatorial optimization".

As such we see that this variance depends on the structure of the densityρX with the variance of (I +λL) 1δX, and the labelling noise with the variance of(Y |X).

We summarize our methods for computing/estimating the gradient of the log determinant arising inmaximum likelihood training ofrectangular flows. Algorithm 2showstheexactmethod, where jvp(f,z,)denotes computingJ[f](z) usingforward-mode AD,and i Rd isthei-thstandard basis vector, i.e. a one-hot vector with a1 on its i-th coordinate. Note that / θlogdetAθ is computed using backpropagation. Thefor loop is easily parallelized in practice.

Ournew bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enoughn).