Our contributions are: (1) We develop a generalized framework to optimize non-differentiable ranking-based functions byextending theerror-drivenoptimization ofAPLoss.(2)Weprovethat

Forl = 2, i. e., for 3 - cycles, thisrelationship is X ikX kj= X ij. Figure 1: Averagematchingerrorunderthe LBCmodel ( left) and LACmodel ( right).

In Step 9, PS{ j( j i,t)}kj=1 lie collaboration St. Inparticulari 2 H, define Yi,t = 1 m X

Two other important techniques are mixed precision training [36] and in-place activated BatchNorm [53]. Mixed precision training involves training using both 32-bit and 16-bit IEEE floating point numbers depending onthenumerical sensitivityofdifferent layers [36].

First we provide Lemma 8, which will be used in the proof of Proposition 2 and 4. Lemma 8. Given n non-negative feature vectors fi =[fi0,fi1,...,fim], where i=1,...,n, there exists n matrices Qi with shape nm m and n vector ˆfi =QifTi, s.t. Proposition 2. A factor graph G =(V,C,E) with variable log potentialsθi(xi) and factor log potentials ϕc(xc) can be converted to a factor graph G0 with the same variable potentials and the decomposed log-potentials ϕic(xi,zc) using a one-layer FGNN. Without loss of generality, we assume that logφc(xc)>1. Then for each i the item θic(xi,zc) in (9) have kn+1 entries, and each entry is either a scaled entry of the vectorgc or arbitrary negative number less than maxxcθc(xc). Thusifweorganize θic(xi,zc) asalength-kn+1 vector fic, thenwedefinea kn+1 kn matrix Qci, where if and only if thelth entry of fic is set to the mth entry of gc multiplied by 12 1/|s(c)|, the entry of Qci in lth row, mth column will be set to 1/|s(c)|; all the other entries of Qci is set to some negative number smaller than maxxcθc(xc).

Bayesian Optimization specifically aims toincrease sample efficiencyfor hard optimization algorithms, and consequently can help achieve better solutions without incurring large societal costs. In the 2-objective case, instead of padding the box decomposition, the Pareto frontier under each posterior sample can be padded instead by repeating a point on the Pareto Frontier such that the padded Pareto frontier under every posterior sample has exactlymaxt|Pt| points. Since the sequentialNEHVIis equivalent to theqNEHVIwith q = 1, we prove Theorem 1 for the generalq > 1 case. Recall from Section C.2, that using the method of common random numbers tofixthebasesamples, theIEPandCBDformulations areequivalent. Note that the box decomposition of the non-dominated space{S1,...,SKt} and the number of rectangles in the box decomposition depend onζt.

In this paper, we study dataset distillation (DD), from a novel perspective and introduce adataset factorizationapproach, termedHaBa, which is a plug-andplay strategy portable to any existing DD baseline.