# generalized_fast_exact_conformalization__1d7916cc.pdf

Generalized Fast Exact Conformalization

Diyang Li Cornell University diyang01@cs.cornell.edu

Conformal prediction converts nearly any point estimator into a prediction interval under standard assumptions while ensuring valid coverage. However, the extensive computational demands of full conformal prediction are daunting in practice, as it necessitates a comprehensive number of trainings across the entire latent label space. Unfortunately, existing efforts to expedite conformalization often carry strong assumptions and are developed specifically for certain models, or they only offer approximate solution sets. To address this gap, we develop a method for fast exact conformalization of generalized statistical estimation. Our analysis reveals that the structure of the solution path is inherently piecewise smooth, and indicates that utilizing second-order information of difference equations suffices to approximate the entire solution spectrum arbitrarily. We provide a unified view that not only encompasses existing work but also attempts to offer geometric insights. Practically, our framework integrates seamlessly with well-studied numerical solvers. The significant speedups of our algorithm as compared to the existing standard methods are demonstrated across numerous benchmarks.

1 Introduction

In modern algorithmic practice, quantifying uncertainty is crucial for accurate and reliable model predictions. Conformal prediction [1] serves as a powerful statistical tool that leverages the observed data to construct prediction intervals containing the outcome with a predefined probability level. It enjoys model-free coverage guarantee regardless of the underlying distribution of the data. In recent years, conformal prediction has gained increasing attention from the community of machine learning [2 5], data mining [6 8] and computer vision [9, 10]. This growing interest is attributed to the attractive properties that it operates under the assumption of exchangeability, which is a weaker condition than independence and identical distribution, allowing for a wider range of applications in real-world scenarios where data may not meet strict statistical assumptions. Meanwhile, conformal prediction can be combined with almost any existing point estimators, even when the model is potentially misspecified [5].

While exhibits appealing properties, the application of conformal prediction often comes at a high computational cost [11, 12]. Kindly note that in this paper we refer to the full conformal prediction that does not discard training points as opposed to the split conformal prediction, as the latter involves only one single fitting. From a numerical perspective, when constructing a conformal prediction set, one needs to exhaustively search all points (potential candidates) in the label space, where for each point the learning model needs to be refitted and the conformity score needs to be re-calculated. In many scenarios like regression, the number of possible candidates is infinite as the latent label can take an uncountable number of possible values. Conventional conformal prediction works by a grid-search type method to loop over the label space [13, 14], which discretize the interval of interest and subsequently solve a sequence of individual optimization subproblems. To improve such brute-force approach, there have been many efforts in community that devoted to develop better algorithms for computing the prediction set. A natural idea is to generate the set of all solutions

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

Table 1: Representative related work, which are instances of generalized parametric estimations.

Model Reference Exact Loss Regularizer Constrained Path Structure

Least Squares [15] Quadratic \ Piecewise Linear Ridge Regression [16] Quadratic w 2 Piecewise Linear Empirical Risk Minimization [17] Convex Convex Piecewise Smooth Elastic Net [12] Quadratic w 1 Piecewise Linear Generalized Lasso Regression [18] Convex w 1 Piecewise Linear General Formulation (ours) Section 4 PCr PCr Piecewise Smooth

indexed by the latent label candidate using numerical continuation (a.k.a. homotopy) method, and we name this set of optimal solutions as (exact) solution path, as illustrated in Figure 1.

x 1 y1 ... ... x n yn

n x +1 yn+1

Figure 1: Diagram of setting on fast exact conformalization, where label candidate yn+1 should loop over the whole latent label space Y and each possible yn+1 R1 corresponds to one point on the (red) solution path. Conventional practice is to refit the model on each new yn+1 while we employ the path-following algorithm to obtain the whole solution spectrum within 1 execution.

Despite extensive theoretical and empirical efforts, the understanding of conformalization path remains rather deficient. For some simple cases, closed-form characterizations of conformal prediction sets are available, such as knearest neighbors, least squares regression [15], and ridge regression [16] with quadratic loss. The study by [12] presents an exact solution path for conformalized Lasso and elastic net using their statistical property and ℓ1-sparsity analysis. Investigations into more general objectives as discussed in [17, 18] incorporate linear interpolation to approximate the intrinsic piecewise smooth structure, yielding prediction sets lacking of finite-sample calibration. In other terms, [17, 18] offer only an upper bound for the approximation error and fail to control the degree of approximation relative to the optimal solutions. We summarize these relevant prior studies in Table 1. Given that existing exact algorithms are either tailored to a select few models, or turn out to be grid-search type approaches that take a very black-box approach, it prompts the following question:

Can we open the black-box by developing a methodology that better exploits the structure of the path?

We answer this question positively by introducing a differential equation perspective to analyze the ground truth solution path, which enables us to better reveal and exploit the fundamental path structure. This more profound understanding enables us to build more generalized conformalization algorithm and present improved computational guarantees.

1.1 Our contributions

The main contributions brought by this paper are summarized as follows.

Generalizable framework This study aims to extend the application of fast exact conformalization into generalized statistical estimation. We relax the assumptions by considering a cost function that is no longer globally differentiable, but rather piecewise differentiable, while introducing constraints on the weight vector to fit more statistical models. In our analysis, the Clarke subdifferential of the objective is derived, and the set of nonsmooth points of path can essentially be described as a level set of certain smooth functions. By adopting a reparameterization regime, our framework effectively broadens the scope of several existing algorithms in Table 1 and provides a unified view of them.

Theoretical insights We analyze the underlying structure of path through the first-order optimality conditions of the regularized problem, identifying sufficient conditions for a local path to be smooth around a given point and reveal that the structure of the path is inherently piecewise smooth. More precisely, if conditions are met, then the path is locally the projection of a higher-dimensional smooth manifold onto the optimization space, thus offering preliminary geometric intuitions. Our

investigation further suggests that leveraging the second-order information of difference equations can approximate the solution spectrum arbitrarily.

Practical efficiency Practically, our framework is not only straightforward to implement but also computationally efficient. With theoretical analysis, we present explicit expressions for the gradient flows of objective, which is homogenized and well aligned with standard forms used by mainstream numerical libraries for ordinary differential equations (ODE), thereby easing the programming efforts. When crossing potential kinks (or nonsmooth points), the computations are facilitated by boundary conditions pre-set in the numerical solver. Notably, our algorithm eliminates the need for extensive iterations for computing the entire solution spectrum, contrasting sharply with conventional baselines. In experiments, we demonstrate the significant computational speed advantages of our algorithm over existing baselines, without compromising on accuracy.

2 Background

Notation For a set J Rp, we denote by cl(J ) the closure and by int(J ) the interior of J w.r.t. the natural topology on Rp. Define J (i) the i-th element of J and conv(J ) be the convex hull of J , or conv(J ) = {v : v = Pk i=1 ˆθiui, ui J , ˆθi R>0, Pk i=1 ˆθi = 1}. The w(i) is the i-th element of vector w and I( ) is an indicator function. Let z(t) be the derivative dz(t)

dt of function z(t). Let O be the zero matrix and P { } be the event probability. The sgn( ) is the sign function sgn(x) = x |x|(x = 0) or 0 (x = 0) that applied entrywise.

2.1 Model and assumptions

Given the dataset {xi, yi}n i=1, where xi Rp is the sample (covariate) and yi is the i-th label (response) live in label space Y R1. We consider the generalized parametric estimation problem

w arg min w

i=1 Li yi, ηw(xi) +

j=1 λjΩj (w)

s.t. gi (w) = 0, 1 i r, hj (w) 0, 1 j s,

where ηw is the model prediction function, Li is the loss and Ωj is the regularizer. The gi, hj are constraints on w, and parameter λj R>0 controls the degree of regularity. In the following, we will define the piecewise differentiability and state main assumptions that used throughout this work. Definition 1 (PCr Function). Let f : U R be continuous on the open set U Rp and fi : U R, i {1, . . . , k} be a set of r-times continuously differentiable (or Cr) functions for r N { }. If f(x) {fi(x)}i {1,...,k} holds for all x U, then f is an r-times piecewise continuously differentiable (or PCr) function. The {f1, . . . , fk} is a set of selection functions of f.

When working with PCr-functions in a local sense, it is useful to only consider the selection functions that have an impact on the local behavior around a given point. Definition 2 (Essentially Active Set). Let f : U R be a PCr function on the open set U Rp with a set of selection functions {f1, . . . , fk}. Denote IK = {1, . . . , k}. Then given a x1 U, Ia f (x1) {i IK : f(x1) = fi(x1)} is called the active set at x1, and the Ie f(x1) {i IK : x1 cl ( int ( {x2 U : f(x2) = fi(x2)}))} is the essentially active set at x1.

Assumption 1. We assume that Li and Ωj in (1) are PCr functions each with a set of selection functions S

k Ie Li{Dk Li} and S

k Ie Ωj {Dk Ωj}, respectively.

Assumption 2. We assume that Ωj and Li are non-differentiable at w with multiple active selection functions, where j {1, . . . , m}, i {1, . . . , n + 1}. We further assume that Ia Ωj(w) Ie Ωj(w), Ia Li(w) Ie Li(w) holds for all w considered and all Ωj, Li in the following.

Assumption 2 ensures that all selection functions we consider are actually relevant for the representation of Ωj, Li. i.e., it does not matter if we consider the active or the essentially active set in the underlying optimizing space, which allows for an easier representation of Dk Ωj and Dk Li.

2.2 Conformal prediction

Definition 3 (Symmetrical Algorithm). A deterministic algorithm A : (x1, . . . , xn) A is symmetric if for any permutation τ of {1, . . . , n}, A(x1, . . . , xn) a.s. = A(xτ(1), . . . , xτ(n)).

Definition 4 (Conformity Score). The conformity score function A, symmetric in its first n inputs, is defined as A( x1, . . . , xn; xn+1): R(p+1) (n+1) R, where xi (xi, yi).

Conformal prediction starts from a conventional model fitting stage, followed by the evaluation of conformity score and the construction of prediction set. The score function A serves as a measure of deviation or conformity, assessing the extent to which the new input xn+1 aligns with the previously fitted model. A higher conformity score indicates a better match between xn+1 and the model [1, 14]. For a new instance xn+1 where the prediction region is desired, the conformalization method operates by assigning a p-value to each latent yn+1 Y, formalized as

ˆpyn+1 = 1 1 n + 1

i=1 I (Ai An+1) , (2)

where Ai A( x1, .., xi 1, xi+1, .., xn+1; xi). Specifically, in density estimation, A is defined as ηw (xn+1), where ηw is the density function estimated from the augmented dataset. For regression tasks, A might be set as |yn+1 ηw (xn+1)|, with ηw being the regression function trained by the dataset including n + 1 samples. Under the assumption of exchangeability among the pairs {xi, yi}n+1 i=1 , the ˆpyn+1 returned by (2) has been demonstrated to be statistically valid [13, 14]. To generate the prediction set Γ( ), one thresholds these p-values at a prescribed error level α (0, 1), resulting in Γ(xn+1) = yn+1 : ˆpyn+1 α . (3)

Theorem 1. [19] Suppose that {xi, yi}n+1 i=1 are exchangeable and the fitting algorithm A is symmetric. Conformal prediction applied on {xi, yi}n i=1 {xn+1} outputs a set Γ( ) such that

P y n+1 Γ(xn+1) 1 α, (4)

where y n+1 is the ground truth (n + 1)-th label.

Theorem 1 (a.k.a. coverage guarantee) requires only exchangeability of input data and symmetry of the conformity score function, which are met by nearly all prevalent model fitting algorithms. In alignment with the analysis taken in prior research, we treat yn+1 := yn+1(z) as a function of scalar variable z. We utilize it to facilitate the traversal of yn+1 across the entire label space Y, and compute the homotopy solution path {w (z) : zmin z zmax} (also shown in Figure 1). We assume that yn+1( ): [zmin, zmax] Y is continuously differentiable in terms of z for simplicity.

3 Main results

We present our main results for fast exact conformalization. The discussions here focus on positive z,1 but the derivation extends easily to include negative values as well, which will be discussed later.

3.1 Surrogate function

Lemma 1. Let f : U R be a PCr-function on the open set U Rp and let Cr-functions {f1, . . . , fk} be a set of selection functions of f. Then for any x U, there exists an open neighborhood U U of x on which f is also a continuous selection of {fi : i Ie f(x)}.

Drawing insights from Lemma 1, we minimize the surrogate function of (1) as

min w Rp Ez (w) :=

i=1 Li (yi, ηw(xi)) +

j=1 λjΩj(w)+ρ

i=1 |gi(w)|+ρ

j=1 max {0, hj(w)} , (5)

1For technicality reasons, we enforce z > 0 even the limit w (0+) limz 0+ w (z) might be well-defined.

where ρ R>0 and yn+1 = yn+1 (z). This definition of Ez (w) is meaningful regardless of whether the contributing functions are convex. Denote model estimation terms P

j λjΩj as LM(w|z), and the minimizer of (5) as w (z). It is interesting to compare Ez (w) to the Lagrangian function

Lz(w) := LM (w|z) +

i=1 λigi(w) +

j=1 µjhj(w), (6)

which captures the behavior of LM( ) near the optimum. At a constrained minimum w , the Lagrangian satisfies the stationarity condition L(w ) = 0; its inequality multipliers µj are nonnegative and satisfy the complementary slackness µjhj(w ) = 0. In the penalized (5), one usually takes

ρ > max{| λ1|, . . . , | λr|, µ1, . . . , µs}, (7)

which creates the favorable circumstances: (i) Lz(w) Ez(w) for all w, (ii) Lz(w) LM(w|z) = Ez(w) for all feasible w, (iii) Lz(w ) = LM(w |z) = Ez(w ) with profound consequences.

Theorem 2. Our surrogate function Ez(w) is increasing in ρ. Furthermore, Ez(w) is strictly convex for one ρ > 0 if and only if it is strictly convex for all ρ > 0. Likewise, it is coercive for one ρ > 0 if and only if it is coercive for all ρ > 0. Finally, if LM( ) is strictly convex (or coercive), then Ez(w) for all ρ are strictly convex (or coercive).

Given Theorem 2, several classical results [20] state that minimizing Ez(w) is effective in minimizing LM( ) subject to the constraints if we choose ρ by (7).

3.2 Conformal path characterization

Definition 5 (Clarke Subdifferential). For continuous function f : U R on the open set U Rp, let Θ U be the set of points in which f is not differentiable. The Clarke subdifferential of f at x U is defined as

f(x) conv({φ Rp : {xi} i=1 Rp\Θ with lim i xi = x, lim i f(xi) = φ}).

Specifically, if f is continuously differentiable in x, f(x) = { f(x)}. While PCr-functions are generally non-smooth, we can use the Clarke subdifferential to obtain first-order optimality of (5),

k Ia Li(w )

ˆθk Li(w ) Dk Li(yi, ηw (xi))+ X

k Ia Ln+1(w )

ˆθk Ln+1(w ) Dk Ln+1(yn+1(z), ηw (xn+1)) D (w ),

i=1 ˆθgi gi(w )+ρ

j=1 ˆθhj hj(w )+

k Ia Ωj (w ) λj ˆθk Ωj(w ) Dk Ωj (w )+D (w ) = 0, (8)

where ˆθk Ωj, ˆθk Li is the k-th auxiliary parameter for convex hull of each Ωj, Li. The (8) is accompanied by the active sets conditions and the subdifferentials conditions, rewritten in detail as

Dk Li(w ) Dli Li(w ) = 0, k Ia Li(w )\{li}, i Ie Li Dk Ωj(w ) Drj Ωj(w ) = 0, k Ia Ωj(w )\{rj}, j Ie Ωj X

k Ia Li(w )

ˆθk Li(w ) 1 = 0, ˆθk Li(w ) 0, 1 i n + 1

k Ia Ωj(w )

ˆθk Ωj(w ) 1 = 0, ˆθk Ωj(w ) 0, 1 j m

where rj, li is randomly selected from Ia Ωj, Ia Li and being fixed, with coefficients satisfying ˆθgi

{ 1} gi(w) < 0 [ 1, 1] gi(w) = 0 {1} gi(w) > 0 , ˆθhj

{0} hj(w) < 0 [0, 1] hj(w) = 0 {1} hj(w) > 0 . In this work, we specialize to the case where the

constraint functions gi (1 i r) and hj (1 j s) are affine, i.e., the gradients gi(w), hj(w) are constant.2 We define gi and hj as constraint residuals gi(w) := v i w di, hj(w) := ω j w ej.

We keep track of the following index sets determined by signs of constraint residuals:

NE = {i : gi(w) = v i w di < 0}, ZE = {i : gi(w) = v i w di = 0}, PE = {i : gi(w) = v i w di > 0},

NI = {j : hj(w) = ω j w ej < 0}, ZI = {j : hj(w) = ω j w ej = 0}, PI = {j : hj(w) = ω j w ej > 0}. (10)

To characterize the path, we further introduce a reparameterization in terms of an auxiliary variable t 0 (thought of as time), whereby for a given T > 0 we introduce functions z( ) : [0, T] [zmin, zmax] and ξ( ) : [zmin, zmax] R such that: ξ( ) is Lipschitz, z( ) is differentiable on (0, T), and we have z = ξ(z(t)) for all t (0, T). In a slight abuse of notation, we define the path w.r.t. t as n w (t) w (z(t)) : t [0, T] o . To enhance structural clarity, we denote n Z = |ZE ZI| and represent the certain matrix inversion as H(w |z) U Z UZ On Z n Z

1 = P(w |z) Q(w |z) Q (w |z) R

where the rows of matrix UZ are the constant differentials, v i for i ZE and ω j for j ZI, and

k Ia Li(w )

ˆθk Li(w ) 2Dk Li(yi, ηw (xi)) +

k Ia Ωj (w ) λj ˆθk Ωj(w ) 2Dk Ωj (w )

k Ia Ln+1(w )

ˆθk Ln+1(w ) 2Dk Ln+1(yn+1(z) , ηw (xn+1)) .

Now we are fully equipped to unveil the structure of homotopy path, as shown in next subsection.

3.3 Piecewise smooth structure & Unified view

To describe the structure, we define the kink (non-smooth point) as the point that ˆθk Ωj, ˆθk Li hit the restriction bound in (9), or set in (10) is violated so the entire structure changes. Theorem 3. Given an optimum (w 0, z0) at t0, and assume that t0 is not a kink, then there exists an open neighborhood of t0 such that w (t) is a C1 function of t and satisfies the following autonomous system

w (t) Υ(w , z) = X

k Ia Ln+1(w )

ˆθk Ln+1(w )ξ(z) P(w |z) h z Dk Ln+1(yn+1(z) , ηw (xn+1)) i ,

(13) where P(w |z) is obtained from the definition in (11), and the ˆθLi, ˆθΩj can be solved from (8).

The (13) offers an explicit gradient flow of the solution path in underlying optimization space. Solving this ODE numerically would recover all the optimal parametric solutions, which provide the conformity scores and one can obtain the required p-values (2) that used for exact conformal prediction via (3). Theorem 4. The optimality solution (w (t), z(t)) has a unique trajectory for t (0, T). The w (t) is continuous if selections Dk Li( ), Dk Ωj( ) are convex. Furthermore, if gradients of constraints { gi(w) : gi(w) = 0} { hj(w) : hj(w) = 0} are affinely independent at the solution w (z) over an open neighborhood of z, then the coefficient paths ˆθgi, ˆθhj are unique and continuous at z. Theorem 5. On a optimality path with set configuration (10), the coefficients for constraints satisfies

r Z ˆθZE(w ) ˆθZI(w )

ρD (w ) + u Z

2In principle a similar algorithm can be developed for the general convex constraint where the hj are relaxed to convex, but that is beyond the scope of current paper.

where Q(w ) is from (11), and

i NE vi + X

i PE vi + X

Although Theorem 3, 4, and 5 are highly technical and may difficult to grasp on first glance, they lay the groundwork for practical application, as illustrated later in Section 4. Specifically, Theorem 4 ensures continuity, which is fundamental for the numerical ODE solvers. Theorem 5, on the other hand, provides a rule for handling constraints.

Theorem 6. Suppose that Dk Li( ) is µ-strongly convex for µ 0, Dk Ωj( ) is σ-strongly convex for σ > 0, and P(w |z), z Dk Ln+1( ), z Dk Ln+1( ) are all locally ℓ-Lipschitz continuous. Suppose further that ξ( ) is Lipschitz continuous on [zmin, zmax] and satisfies |ξ(z)| C for all z [zmin, zmax]. Then, it holds that Υ( , ) defined in (13) is uniformly ℓΥ-Lipschitz continuous with ℓΥ = Cℓ2 + 2 Cℓ (n+1)µ+Pm j=1 λjσ for any z [zmin, zmax] when the active selections IL, IΩare fixed.

The essence of Theorem 6 lies in its suggestion that the dynamics of w (t) is piecewise continuous, i.e., w (t) maintains smoothness between two adjacent kinks. By considering specific choices of ξ( ) and yn+1(z), our system (13) generalizes some previously studied methodologies in fast conformalization. First, consider the scenario with an equally spaced discretization of the interval [0, T], namely tk = k h for some fixed step-size h > 0. Thus, the sequence zk := z(tk) is approximately given by zk+1 zk + h ξ(zk). Intuitively, the choice of ξ( ) controls the dynamic of z( ) and generalizes some previously considered sequences {zk} for the problem (5). For example, letting ξ(z) := 1 we recover the arithmetic sequence in [12] and letting ξ(z) := z we recover the geometric sequence in [17].

4 Fast conformalization algorithm

The complete algorithm on the fast exact conformalization is outlined in the plate referred as Algorithm 1.

Algorithm 1 Fast Exact Conformalization Algorithm Input: Training data {xi, yi}n i=1, new covariate xn+1, range [zmin, zmax], initial solution w 0, regularization strength {λj}m j=1, miscoverage level α (0, 1). 1: \\ Full Path Generation 2: z 0, set NE, PE, PI and all IL, IΩby w 0. 3: while 0 z zmax do 4: while partitions NE, PE, PI, IL, IΩare met do 5: Calculate H(w |z), r Z as in (12), (14). 6: Solve ODE system (13). 7: end while 8: Update NE, PE, PI, IL, IΩby index violator(s). 9: end while 10: z 0. 11: while zmin z 0 do 12: Repeat the above procedure analogously for negative values of z, obtaining {w (z) : zmin z 0}. 13: end while 14: \\ Conformal Set Generation 15: for i = 1 to n + 1 do 16: Calculate conformity score path Ai for i-th sample. 17: end for 18: Calculate the path of ˆpyn+1 by (2). 19: Γ(xn+1) yn+1 : ˆpyn+1 α . Output: Conformal prediction set Γ(xn+1).

Table 2: Numerical results for average empirical coverage, average length of the conformal prediction set, and the total number of kinks observed. One standard error is given in the parenthesis following the average number.

Dataset Model (Parameter) Grid1/Grid2 SCP Exact # Kinks Coverage Length Coverage Length Coverage Length

fried NLS(\) 0.81(0.003) 5.82(0.53) 0.81(0.007) 5.55(0.94) 0.80(0.009) 5.30(0.72) 6 cadata NLS(\) 0.82(0.007) 5.54(0.12) 0.82(0.007) 5.63(0.94) 0.80(0.003) 5.35(0.02) 2 delta NLS(\) 0.81(0.008) 5.63(0.16) 0.81(0.002) 5.79(0.53) 0.80(0.002) 5.26(0.27) 5 cadata GFM(λ1 =0.03) 0.82(0.001) 11.64(0.13) 0.81(0.004) 11.38(0.36) 0.80(0.003) 11.34(0.27) 7 cadata GFM(λ1 =0.02) 0.83(0.008) 10.72(0.91) 0.81(0.004) 10.39(0.97) 0.80(0.001) 10.13(0.21) 5 elevator GFM(λ1 =0.03) 0.83(0.003) 11.09(0.58) 0.81(0.007) 11.34(0.97) 0.80(0.002) 10.68(0.81) 9 fried IGR(λ1 =0.1, λ2 =0.02) 0.81(0.002) 19.14(0.78) 0.81(0.001) 19.69(0.96) 0.81(0.002) 19.05(0.19) 12 cadata IGR(λ1 =0.1, λ2 =0.02) 0.82(0.003) 19.99(0.33) 0.80(0.002) 19.16(0.29) 0.80(0.002) 19.14(0.82) 13 elevator IGR(λ1 =0.2, λ2 =0.05) 0.82(0.003) 19.99(0.85) 0.81(0.001) 19.53(0.53) 0.81(0.005) 19.48(0.75) 7

2097 4194 6291 8388 10485 12582 14679 16776 Size of dataset (delta)

Running Time of NLS (s)

Grid1 Grid2 Exact

900 1800 2700 3600 4500 5400 6300 7200 Size of dataset (fried)

Running Time of GFM (s)

Grid1( 1 = 0.01) Grid1( 1 = 0.03)

Grid2( 1 = 0.01)

Grid2( 1 = 0.03)

Exact( 1 = 0.01) Exact( 1 = 0.03)

4500 9000 13500 18000 22500 27000 31500 36000 Size of dataset (fried)

Running Time of IGR (s)

Grid1( 1, 2 = 0.1, 0.02)

Grid1( 1, 2 = 0.2, 0.03)

Grid2( 1, 2 = 0.1, 0.02)

Grid2( 1, 2 = 0.2, 0.03)

Exact( 1, 2 = 0.1, 0.02)

Exact( 1, 2 = 0.2, 0.03)

Figure 2: The running time under different sizes of datasets.

4.1 On solving (11)

To avoid the direct computation of matrix inversion (11), our practical algorithm involves sweep operator [22, 23]. Suppose A is an p p symmetric matrix, sweeping on the k-th diagonal entry Akk = 0 of A results in a matrix b A with entries

Akk , ˆAik = Aik

Akk (i = k) , ˆAkj = Akj

Akk (j = k) , ˆAij = Aij Aik Akj

Akk (i, j = k) . (15)

Since the sweeping (15) preserves symmetry, all operations can be performed solely on either the lower or upper-triangular part of A to ease the computational burden [24]. To begin with, we initiate

with a sweeping tableau as h H(w |z) * UZ O

i , and further sweeping of diagonal entries of block H

yields M h M11 * M21 M22

i . Then we reinitialize our new tableau in the form of h M22 M21 * M11

and further sweeping of diagonal entries of block M22 makes h R Q (w |z) * P(w |z)

i . Compared to direct

inversion, it also decreases a O(p2 + n2 Z) storage space.

4.2 Efficiency

Algorithm 1 shows a very favourable behavior empirically, and converges remarkably faster than the standard grid-search type algorithm. We argue that this observation is actually quite natural. Indeed, our algorithm can follow the ground truth solution path, and the numerical integration process is fully deterministic, which avoids large fluctuations between the iteration steps like stochastic gradient descent. Therefore, the solving process of Algorithm 1 exhibits a more stable character being completely deterministic and has no extensive loops, which explains the much faster convergences observed in practice. In contrast, the standard grid-search type method would cost N-times the original batch iterative algorithms, where N is the number of grid points.

5 Numerical experiments

We provide experimental results on real-world benchmarks to validate our derived algorithm. All experiments presented in this study were conducted on a workstation running the Ubuntu 18.04 operating system, equipped with Intel Xeon Gold 5218R CPU 64 and 60.9 GB of RAM. We integrate a system of ordinary differential equations using lsoda from the FORTRAN library, where an

interface for Sci Py is available using the odepack. The concrete parameter settings of ODE solver are shown in the Table 3, wherein the numerical solver exploit the Runge-Kutta method of order 4 or 5. The parameterizers are set to yn+1(z) = 4z, ξ(z) = 1, respectively.

Table 3: List of the key parameters in the numerical solver.

Parameters Descriptions Values rtol allowed relative error in the solution 1e-6 atol allowed absolute error in the solution 1.49e-8 tcrit vector of critical points set by known / explicit kinks h0 initial step size for the integration 0.02*(zmax zmin) hmax maximum absolute step size allowed 0.1*(zmax zmin) hmin minimum absolute step size allowed 1e-7 mxstep maximum number of steps allowed for each point 400 mxordn maximum order to be allowed for the non-stiff method 12 mxords maximum order to be allowed for the stiff method. 5 ixpr extra printing at method switches True mxhnil maximum number of messages printed 15 tfirst the required order of the first two arguments True full_output return a dictionary of optional outputs True

Model For evaluation, here we employ 3 specific forms of (1), i.e., Nonnegative Least Squares (NLS) [25], Graph-guided Fused Model (GFM) [26], and Inverse Gaussian Regression (IGR) [27].

42 84 126 168 210 252 294 336 Number of grids (fried dataset)

Running Time of GFM (s)

Grid1( 1 = 0.03)

Grid1( 1 = 0.05)

Grid2( 1 = 0.03)

Grid2( 1 = 0.05)

Exact( 1 = 0.03)

Exact( 1 = 0.05)

Figure 3: The training time against grid numbers.

Conformalization A conformal prediction set with target coverage level 0.8 (α = 0.2) is calculated for each sample in testing set using each of the 4 methods, i.e., the standard grid-point evaluation method (Grid1) [1], grid-point method with warm-restart strategy (Grid2) [28], the split conformal prediction method (SCP) [4], and our exact conformalization method in Algorithm 1 (Exact). We use the conformity score function Ai = |yi ηw (xi)|. Conventionally, the interval [ymin n+1, ymax n+1] (part of the input) can be chosen simply as [y[1], y[n]], where y[1] y[2] y[n] are the order statistics of the response variable. In experiments we set the search range even more conservatively, enlarging the sample range by 50% of length [ymin n+1, ymax n+1] := y[1] 0.25(y[n] y[1]), y[n] + 0.25(y[n] y[1]) .

0 1 2 3 0.0

Probability

0.027 0.002 0.0

Running time (log(s))

100 200 300 400 500 k

Figure 4: The histogram of kink numbers and the running time of Algorithm 1 against k, where k lives in yn+1 := k z(k N).

Dataset Our experiments were conducted using real-world datasets. We employ real-world datasets from Open ML [29] and UCI repository [30] in simulations. We randomly partition the dataset into training set, testing set, and calibration set (used in SCP) with 70%, 10%, and 20% of the total samples. To facilitate optimization, we have standardized the entire original dataset by removing the mean and scaling to unit variance for the features, and adjusting the mean of all labels to 0.

Setup Our central claim is twofold, encompassing both accuracy and efficiency. We first report the average empirical coverage, average length of the prediction set in Table 2. Regarding running efficiency, we present average training time per dataset in Figure 2 while varying the scale of training set. In Figure 3, we compare the training times when different grid numbers are used in Grid1 and Grid2. We further plot the histogram of kink numbers, and the running time against various yn+1( ) in Figure 4.

Results & Analysis From Table 2 we observe that all these methods provide valid and nearly perfect coverage. The grid and exact method give similar lengths, where the slight difference is due

to the rounding between neighboring grid points. The SCP produces wider intervals due to a less efficient use of data. Given Figure 2, our exact method is much faster than the baselines, with same solid performance. We can also learn from the Figure 3 that the time of Algorithm 1 still compares favorably against the grid-search type method, even when the grid is sparse. Figure 4 found that there exists tiny number of kinks in majority runnings, which offers hope for the future expansion of our algorithm, and indicates that the choice of yn+1 can make a difference in efficiency, as it will determine the solving interval and the total query times of gradient.

6 Conclusion

In this work, we present a unified framework and an elaborate algorithm with statistical analysis for fast exact conformalization regarding generalized parametric estimation. We illustrate the strong and competitive performance of proposed methods in a series of benchmarks.

In future work, a potential direction is to consider scenarios where labels are multidimensional, such as multi-task learning, in which the label space Y would be indexed by multiple independent parameters (z1, . . . , z K). Under such conditions, the homotopy solution path would extend to a solution surface, and our ODE system could be reformulated as a corresponding system of partial differential equations. Additionally, compared to some previous work, Algorithm 1 offers increased speed but at the cost of higher memory requirements. It necessitates storing all training samples throughout the process for gradient queries. We believe it would be interesting to use recent advancements like the Kroneckerfactored approximate method to potentially enhance the memory scalability.

Acknowledgement

I would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the presentation of this paper.

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