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.

This problem involves identifying a density function that accurately represents the distribution of a given dataset.

We study theoretical properties of a broad class of regularized algorithms with vector-valued output.

Wepropose twoinnovativealgorithms, DP-GLMtron and DP-TAGLMtron, that outperform the conventional DPSGD. Inlight ofthevast quantities of personal and sensitiveinformation involved, traditional methods of ensuring privacy are encountering significant challenges.

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".