-ie SrSSKrSSKrSSKJr SSKJr SSKJ r SSK J r J r J r JrJrJrJrJrJrJrJr SS KJrJrJrJrJrJr "S S 5r"S S \5r"SS\5r"SS\5r "SS\5r!"SS\5r""SS\5r#"SS\5r$"SS\5r%"SS\5r&"SS\5r'"S S!\5r(\\\ \!\"\#\$\&\'\(S". r)g)#z This module contains loss classes suitable for fitting. It is not part of the public API. Specific losses are used for regression, binary classification or multiclass classification. Nxlogy) check_scalar)_weighted_percentile) CyAbsoluteErrorCyExponentialLossCyHalfBinomialLossCyHalfGammaLossCyHalfMultinomialLossCyHalfPoissonLossCyHalfSquaredErrorCyHalfTweedieLossCyHalfTweedieLossIdentity CyHuberLoss CyPinballLoss) HalfLogitLink IdentityLinkInterval LogitLinkLogLinkMultinomialLogitc\rSrSrSrSrSrSrSSjrSr Sr SS jr SS jr SS jr SS jrSS jrSSjrSSjr\R&S4SjrSrg)BaseLossFa|Base class for a loss function of 1-dimensional targets. Conventions: - y_true.shape = sample_weight.shape = (n_samples,) - y_pred.shape = raw_prediction.shape = (n_samples,) - If is_multiclass is true (multiclass classification), then y_pred.shape = raw_prediction.shape = (n_samples, n_classes) Note that this corresponds to the return value of decision_function. y_true, y_pred, sample_weight and raw_prediction must either be all float64 or all float32. gradient and hessian must be either both float64 or both float32. Note that y_pred = link.inverse(raw_prediction). Specific loss classes can inherit specific link classes to satisfy BaseLink's abstractmethods. Parameters ---------- sample_weight : {None, ndarray} If sample_weight is None, the hessian might be constant. n_classes : {None, int} The number of classes for classification, else None. Attributes ---------- closs: CyLossFunction link : BaseLink interval_y_true : Interval Valid interval for y_true interval_y_pred : Interval Valid Interval for y_pred differentiable : bool Indicates whether or not loss function is differentiable in raw_prediction everywhere. need_update_leaves_values : bool Indicates whether decision trees in gradient boosting need to uptade leave values after having been fit to the (negative) gradients. approx_hessian : bool Indicates whether the hessian is approximated or exact. If, approximated, it should be larger or equal to the exact one. constant_hessian : bool Indicates whether the hessian is one for this loss. is_multiclass : bool Indicates whether n_classes > 2 is allowed. TFNcXlX lSUlSUlX0l[ [ R*[ RSS5UlURRUl g)NF) closslinkapprox_hessianconstant_hessian n_classesrnpinfinterval_y_trueinterval_y_pred)selfrrr"s E/var/www/html/venv/lib/python3.13/site-packages/sklearn/_loss/loss.py__init__BaseLoss.__init__sP  # %"'F#yy88c8URRU5$zUReturn True if y is in the valid range of y_true. Parameters ---------- y : ndarray )r%includesr'ys r(in_y_true_rangeBaseLoss.in_y_true_range##,,Q//r+c8URRU5$)zUReturn True if y is in the valid range of y_pred. Parameters ---------- y : ndarray )r&r.r/s r(in_y_pred_rangeBaseLoss.in_y_pred_ranger3r+cUc[R"U5nURS:Xa$URSS:XaUR S5nUR R UUUUUS9 U$)aCompute the pointwise loss value for each input. Parameters ---------- y_true : C-contiguous array of shape (n_samples,) Observed, true target values. raw_prediction : C-contiguous array of shape (n_samples,) or array of shape (n_samples, n_classes) Raw prediction values (in link space). sample_weight : None or C-contiguous array of shape (n_samples,) Sample weights. loss_out : None or C-contiguous array of shape (n_samples,) A location into which the result is stored. If None, a new array might be created. n_threads : int, default=1 Might use openmp thread parallelism. Returns ------- loss : array of shape (n_samples,) Element-wise loss function. rry_trueraw_prediction sample_weightloss_out n_threads)r# empty_likendimshapesqueezerloss)r'r9r:r;r<r=s r(rB BaseLoss.lossss<  }}V,H   ! #(<(rFr?r@rAr loss_gradient)r'r9r:r;r<rGr=s r(rHBaseLoss.loss_gradientsL  #==0!}}^< ==7I7IJ  !==~~NL   ! #(<(r?r@rArgradient)r'r9r:r;rGr=s r(rKBaseLoss.gradient s>  ==8L   ! #(<(r?r@rArgradient_hessian)r'r9r:r;rGrNr=s r(rOBaseLoss.gradient_hessian=sN  "!}}^<  mmN; !}}[9  -- 5K   ! #(<(%--a0K ##)'%# $ ((r+c N[R"URUUSSUS9US9$)aCompute the weighted average loss. Parameters ---------- y_true : C-contiguous array of shape (n_samples,) Observed, true target values. raw_prediction : C-contiguous array of shape (n_samples,) or array of shape (n_samples, n_classes) Raw prediction values (in link space). sample_weight : None or C-contiguous array of shape (n_samples,) Sample weights. n_threads : int, default=1 Might use openmp thread parallelism. Returns ------- loss : float Mean or averaged loss function. Nr8weights)r#averagerB)r'r9r:r;r=s r(__call__BaseLoss.__call__s:(zz II-"#  "  r+c[R"XSS9nS[R"UR5R-nUR R [R*:XaSnOKUR R(aUR R nOUR R U-nUR R[R:XaSnOKUR R(aUR RnOUR RU- nUcUcURRU5$URR[R"X5U55$)aCompute raw_prediction of an intercept-only model. This can be used as initial estimates of predictions, i.e. before the first iteration in fit. Parameters ---------- y_true : array-like of shape (n_samples,) Observed, true target values. sample_weight : None or array of shape (n_samples,) Sample weights. Returns ------- raw_prediction : numpy scalar or array of shape (n_classes,) Raw predictions of an intercept-only model. rrSaxis N) r#rTfinforFepsr&lowr$ low_inclusivehighhigh_inclusiverclip)r'r9r;y_predr\a_mina_maxs r(fit_intercept_onlyBaseLoss.fit_intercept_onlys(FB288FLL)---    # #w .E  ! ! / /((,,E((,,s2E    $ $ .E  ! ! 0 0((--E((--3E =U]99>>&) )99>>"''&"?@ @r+c.[R"U5$)z`Calculate term dropped in loss. With this term added, the loss of perfect predictions is zero. )r# zeros_liker'r9r;s r(constant_to_optimal_zero!BaseLoss.constant_to_optimal_zeros }}V$$r+FcXU[R[R4;a[SUS35eUR(aXR 4nOU4n[R "XBUS9nUR(a[R"SUS9nXV4$[R "XBUS9nXV4$)aInitialize arrays for gradients and hessians. Unless hessians are constant, arrays are initialized with undefined values. Parameters ---------- n_samples : int The number of samples, usually passed to `fit()`. dtype : {np.float64, np.float32}, default=np.float64 The dtype of the arrays gradient and hessian. order : {'C', 'F'}, default='F' Order of the arrays gradient and hessian. The default 'F' makes the arrays contiguous along samples. Returns ------- gradient : C-contiguous array of shape (n_samples,) or array of shape (n_samples, n_classes) Empty array (allocated but not initialized) to be used as argument gradient_out. hessian : C-contiguous array of shape (n_samples,), array of shape (n_samples, n_classes) or shape (1,) Empty (allocated but not initialized) array to be used as argument hessian_out. If constant_hessian is True (e.g. `HalfSquaredError`), the array is initialized to ``1``. zCValid options for 'dtype' are np.float32 and np.float64. Got dtype=z instead.)r@rForder)r)r@rF) r#float32float64 ValueError is_multiclassr"emptyr!ones)r' n_samplesrFrnr@rKhessians r(init_gradient_and_hessian"BaseLoss.init_gradient_and_hessians8 RZZ0 0"G9.    /ELE88%EB  ggD6G  hhUuEG  r+)r rr!r&r%rr"N)NNrNNNrNr)__name__ __module__ __qualname____firstlineno____doc__differentiableneed_update_leaves_valuesrrr)r1r5rBrHrKrOrUrerjr#rprw__static_attributes__r+r(rrFs/tN %M900 +b=&F /j@)D >(AT%:<31!r+rc0^\rSrSrSrSU4SjjrSrU=r$)HalfSquaredErroriaHalf squared error with identity link, for regression. Domain: y_true and y_pred all real numbers Link: y_pred = raw_prediction For a given sample x_i, half squared error is defined as:: loss(x_i) = 0.5 * (y_true_i - raw_prediction_i)**2 The factor of 0.5 simplifies the computation of gradients and results in a unit hessian (and is consistent with what is done in LightGBM). It is also half the Normal distribution deviance. cT>[TU][5[5S9 USLUlg)Nrr)superr)rrr!r'r; __class__s r(r)HalfSquaredError.__init__s( 13,.I - 5r+)r!ryr|r}r~rrr)r __classcell__rs@r(rrs"66r+rcB^\rSrSrSrSrSrSU4SjjrSSjrSr U=r $) AbsoluteErroriaAbsolute error with identity link, for regression. Domain: y_true and y_pred all real numbers Link: y_pred = raw_prediction For a given sample x_i, the absolute error is defined as:: loss(x_i) = |y_true_i - raw_prediction_i| Note that the exact hessian = 0 almost everywhere (except at one point, therefore differentiable = False). Optimization routines like in HGBT, however, need a hessian > 0. Therefore, we assign 1. FTcb>[TU][5[5S9 SUlUSLUlg)NrT)rr)r rr r!rs r(r)AbsoluteError.__init__3s/ 0|~F" - 5r+cJUc[R"USS9$[XS5$)}Compute raw_prediction of an intercept-only model. This is the weighted median of the target, i.e. over the samples axis=0. rrY2)r#medianrris r(re AbsoluteError.fit_intercept_only8s(  99V!, ,'rB Br+r r!ry r|r}r~rrrrr)rerrrs@r(rrs&"N $6 C Cr+rcB^\rSrSrSrSrSrSU4SjjrS SjrSr U=r $) PinballLossiDa3Quantile loss aka pinball loss, for regression. Domain: y_true and y_pred all real numbers quantile in (0, 1) Link: y_pred = raw_prediction For a given sample x_i, the pinball loss is defined as:: loss(x_i) = rho_{quantile}(y_true_i - raw_prediction_i) rho_{quantile}(u) = u * (quantile - 1_{u<0}) = -u *(1 - quantile) if u < 0 u * quantile if u >= 0 Note: 2 * PinballLoss(quantile=0.5) equals AbsoluteError(). Note that the exact hessian = 0 almost everywhere (except at one point, therefore differentiable = False). Optimization routines like in HGBT, however, need a hessian > 0. Therefore, we assign 1. Additional Attributes --------------------- quantile : float The quantile level of the quantile to be estimated. Must be in range (0, 1). FTc >[US[RSSSS9 [TU][ [ U5S9[5S9 SUlUSLUl g) Nquantilerrneither target_typemin_valmax_valinclude_boundaries)rrT) rnumbersRealrr)rfloatrr r!)r'r;rrs r(r)PinballLoss.__init__es]   (   x9  # - 5r+cUc-[R"USURR-SS9$[ XSURR-5$)rdrr)r# percentilerrrris r(rePinballLoss.fit_intercept_onlyusM  ==tzz/B/B)BK K'sTZZ-@-@'@ r+r)N?ryrrs@r(rrDs$:N $6  r+rcB^\rSrSrSrSrSrSU4SjjrS SjrSr U=r $) HuberLossiaHuber loss, for regression. Domain: y_true and y_pred all real numbers quantile in (0, 1) Link: y_pred = raw_prediction For a given sample x_i, the Huber loss is defined as:: loss(x_i) = 1/2 * abserr**2 if abserr <= delta delta * (abserr - delta/2) if abserr > delta abserr = |y_true_i - raw_prediction_i| delta = quantile(abserr, self.quantile) Note: HuberLoss(quantile=1) equals HalfSquaredError and HuberLoss(quantile=0) equals delta * (AbsoluteError() - delta/2). Additional Attributes --------------------- quantile : float The quantile level which defines the breaking point `delta` to distinguish between absolute error and squared error. Must be in range (0, 1). Reference --------- .. [1] Friedman, J.H. (2001). :doi:`Greedy function approximation: A gradient boosting machine <10.1214/aos/1013203451>`. Annals of Statistics, 29, 1189-1232. FTc >[US[RSSSS9 X l[TU][ [U5S9[5S9 SUl S Ul g) Nrrrrr)deltarTF) rrrrrr)rrrr r!)r'r;rrrs r(r)HuberLoss.__init__s]   (  !  E%L1  # %r+c0Uc[R"USSS9nO [XS5nX- n[R"U5[R"UR R [R"U55-nU[R"XRS9-$)rrrrrR) r#rrsignminimumrrabsrT)r'r9r;rdiffterms r(reHuberLoss.fit_intercept_onlysr  ]]62A6F)&DFwwt}rzz$***:*:BFF4LII 4???r+)r r!r)Ng?rryrrs@r(rrs'BN $&"@@r+rc:^\rSrSrSrSU4SjjrSSjrSrU=r$)HalfPoissonLossiaOHalf Poisson deviance loss with log-link, for regression. Domain: y_true in non-negative real numbers y_pred in positive real numbers Link: y_pred = exp(raw_prediction) For a given sample x_i, half the Poisson deviance is defined as:: loss(x_i) = y_true_i * log(y_true_i/exp(raw_prediction_i)) - y_true_i + exp(raw_prediction_i) Half the Poisson deviance is actually the negative log-likelihood up to constant terms (not involving raw_prediction) and simplifies the computation of the gradients. We also skip the constant term `y_true_i * log(y_true_i) - y_true_i`. c>[TU][5[5S9 [ S[ R SS5Ulg)NrrTF)rr)rrrr#r$r%rs r(r)HalfPoissonLoss.__init__s2 02C'2664?r+c0[X5U- nUbX2-nU$ryrr'r9r;rs r(rj(HalfPoissonLoss.constant_to_optimal_zeros$V$v-  $  !D r+r%ry r|r}r~rrr)rjrrrs@r(rrs(@r+rc:^\rSrSrSrSU4SjjrSSjrSrU=r$) HalfGammaLossia&Half Gamma deviance loss with log-link, for regression. Domain: y_true and y_pred in positive real numbers Link: y_pred = exp(raw_prediction) For a given sample x_i, half Gamma deviance loss is defined as:: loss(x_i) = log(exp(raw_prediction_i)/y_true_i) + y_true/exp(raw_prediction_i) - 1 Half the Gamma deviance is actually proportional to the negative log- likelihood up to constant terms (not involving raw_prediction) and simplifies the computation of the gradients. We also skip the constant term `-log(y_true_i) - 1`. c>[TU][5[5S9 [ S[ R SS5Ulg)NrrF)rr)r rrr#r$r%rs r(r)HalfGammaLoss.__init__s1 0wyA'2665%@r+cH[R"U5*S- nUbX2-nU$r{)r#logrs r(rj&HalfGammaLoss.constant_to_optimal_zeros)v"  $  !D r+rryrrs@r(rrs&Ar+rc:^\rSrSrSrSU4SjjrSSjrSrU=r$)HalfTweedieLossi aHalf Tweedie deviance loss with log-link, for regression. Domain: y_true in real numbers for power <= 0 y_true in non-negative real numbers for 0 < power < 2 y_true in positive real numbers for 2 <= power y_pred in positive real numbers power in real numbers Link: y_pred = exp(raw_prediction) For a given sample x_i, half Tweedie deviance loss with p=power is defined as:: loss(x_i) = max(y_true_i, 0)**(2-p) / (1-p) / (2-p) - y_true_i * exp(raw_prediction_i)**(1-p) / (1-p) + exp(raw_prediction_i)**(2-p) / (2-p) Taking the limits for p=0, 1, 2 gives HalfSquaredError with a log link, HalfPoissonLoss and HalfGammaLoss. We also skip constant terms, but those are different for p=0, 1, 2. Therefore, the loss is not continuous in `power`. Note furthermore that although no Tweedie distribution exists for 0 < power < 1, it still gives a strictly consistent scoring function for the expectation. c>[TU][[U5S9[ 5S9 UR R S::a1[[R*[RSS5Ul gUR R S:a"[S[RSS5Ul g[S[RSS5Ul gN)powerrrFrT) rr)rrrrrrr#r$r%r'r;rrs r(r)HalfTweedieLoss.__init__*s #%,7   ::  q #+RVVGRVVUE#JD ZZ   !#+ArvvtU#CD #+Arvvue#DD r+cURRS:Xa[5RXS9$URRS:Xa[ 5RXS9$URRS:Xa[ 5RXS9$URRn[ R"[ R"US5SU- 5SU- - SU- - nUbXB-nU$)Nr)r9r;rr)rrrrjrrr#maximum)r'r9r;prs r(rj(HalfTweedieLoss.constant_to_optimal_zero6s ::  q #%>>? ZZ   ""$==> ZZ   " ?;;<    A88BJJvq11q59QUCq1uMD(%Kr+rNg?ryrrs@r(rr s< Er+rc0^\rSrSrSrSU4SjjrSrU=r$)HalfTweedieLossIdentityiKa"Half Tweedie deviance loss with identity link, for regression. Domain: y_true in real numbers for power <= 0 y_true in non-negative real numbers for 0 < power < 2 y_true in positive real numbers for 2 <= power y_pred in positive real numbers for power != 0 y_pred in real numbers for power = 0 power in real numbers Link: y_pred = raw_prediction For a given sample x_i, half Tweedie deviance loss with p=power is defined as:: loss(x_i) = max(y_true_i, 0)**(2-p) / (1-p) / (2-p) - y_true_i * raw_prediction_i**(1-p) / (1-p) + raw_prediction_i**(2-p) / (2-p) Note that the minimum value of this loss is 0. Note furthermore that although no Tweedie distribution exists for 0 < power < 1, it still gives a strictly consistent scoring function for the expectation. cz>[TU][[U5S9[ 5S9 UR R S::a1[[R*[RSS5Ul O]UR R S:a"[S[RSS5Ul O![S[RSS5Ul UR R S:Xa1[[R*[RSS5Ul g[S[RSS5Ul gr) rr)rrrrrrr#r$r%r&rs r(r) HalfTweedieLossIdentity.__init__gs +%,?   ::  q #+RVVGRVVUE#JD ZZ   !#+ArvvtU#CD #+Arvvue#DD ::  q #+RVVGRVVUE#JD #+Arvvue#DD r+r&r%rrrs@r(rrKs6EEr+rc@^\rSrSrSrSU4SjjrSSjrSrSrU=r $)HalfBinomialLossiya Half Binomial deviance loss with logit link, for binary classification. This is also know as binary cross entropy, log-loss and logistic loss. Domain: y_true in [0, 1], i.e. regression on the unit interval y_pred in (0, 1), i.e. boundaries excluded Link: y_pred = expit(raw_prediction) For a given sample x_i, half Binomial deviance is defined as the negative log-likelihood of the Binomial/Bernoulli distribution and can be expressed as:: loss(x_i) = log(1 + exp(raw_pred_i)) - y_true_i * raw_pred_i See The Elements of Statistical Learning, by Hastie, Tibshirani, Friedman, section 4.4.1 (about logistic regression). Note that the formulation works for classification, y = {0, 1}, as well as logistic regression, y = [0, 1]. If you add `constant_to_optimal_zero` to the loss, you get half the Bernoulli/binomial deviance. More details: Inserting the predicted probability y_pred = expit(raw_prediction) in the loss gives the well known:: loss(x_i) = - y_true_i * log(y_pred_i) - (1 - y_true_i) * log(1 - y_pred_i) cj>[TU][5[5SS9 [ SSSS5UlgNrrrr"rrT)rr)r rrr%rs r(r)HalfBinomialLoss.__init__s8 $&  (1dD9r+cP[X5[SU- SU- 5-nUbX2-nU$r{rrs r(rj)HalfBinomialLoss.constant_to_optimal_zeros3V$uQZV'DD  $  !D r+c4URS:Xa$URSS:XaURS5n[R"URSS4UR S9nUR RU5USS2S4'SUSS2S4- USS2S4'U$zPredict probabilities. Parameters ---------- raw_prediction : array of shape (n_samples,) or (n_samples, 1) Raw prediction values (in link space). Returns ------- proba : array of shape (n_samples, 2) Element-wise class probabilities. rrrrENr?r@rAr#rsrFrinverser'r:probas r( predict_probaHalfBinomialLoss.predict_proba   ! #(<(:r+rc\^\rSrSrSrSrS U4SjjrSrS SjrSr S Sjr S r U=r $) HalfMultinomialLossia4Categorical cross-entropy loss, for multiclass classification. Domain: y_true in {0, 1, 2, 3, .., n_classes - 1} y_pred has n_classes elements, each element in (0, 1) Link: y_pred = softmax(raw_prediction) Note: We assume y_true to be already label encoded. The inverse link is softmax. But the full link function is the symmetric multinomial logit function. For a given sample x_i, the categorical cross-entropy loss is defined as the negative log-likelihood of the multinomial distribution, it generalizes the binary cross-entropy to more than 2 classes:: loss_i = log(sum(exp(raw_pred_{i, k}), k=0..n_classes-1)) - sum(y_true_{i, k} * raw_pred_{i, k}, k=0..n_classes-1) See [1]. Note that for the hessian, we calculate only the diagonal part in the classes: If the full hessian for classes k and l and sample i is H_i_k_l, we calculate H_i_k_k, i.e. k=l. Reference --------- .. [1] :arxiv:`Simon, Noah, J. Friedman and T. Hastie. "A Blockwise Descent Algorithm for Group-penalized Multiresponse and Multinomial Regression". <1311.6529>` Tc>[TU][5[5US9 [ S[ R SS5Ul[ SSSS5Ulg)NrrTFr) rr)r rrr#r$r%r&)r'r;r"rs r(r)HalfMultinomialLoss.__init__sP ')!#  (2664?'1eU;r+cURRU5=(a, [R"UR [ 5U:H5$r-)r%r.r#allastypeintr/s r(r1#HalfMultinomialLoss.in_y_true_ranges6##,,Q/NBFF188C=A;M4NNr+c[R"URURS9n[R"UR5R n[ UR5H<n[R"X:HUSS9X5'[R"X5USU- 5X5'M> URRUSSS245RS5$)zCompute raw_prediction of an intercept-only model. This is the softmax of the weighted average of the target, i.e. over the samples axis=0. rErrXrN) r#zerosr"rFr[r\rangerTrarreshape)r'r9r;outr\ks r(re&HalfMultinomialLoss.fit_intercept_onlys hht~~V\\:hhv||$((t~~&AZZ ]KCFWWSVS!c'2CF'yy~~c$'l+33B77r+c8URRU5$)zPredict probabilities. Parameters ---------- raw_prediction : array of shape (n_samples, n_classes) Raw prediction values (in link space). Returns ------- proba : array of shape (n_samples, n_classes) Element-wise class probabilities. )rr)r'r:s r(r!HalfMultinomialLoss.predict_probasyy  00r+c  UcGUc-[R"U5n[R"U5nO0[R"U5nOUc[R"U5nURRUUUUUUS9 XE4$)aCompute gradient and class probabilities fow raw_prediction. Parameters ---------- y_true : C-contiguous array of shape (n_samples,) Observed, true target values. raw_prediction : array of shape (n_samples, n_classes) Raw prediction values (in link space). sample_weight : None or C-contiguous array of shape (n_samples,) Sample weights. gradient_out : None or array of shape (n_samples, n_classes) A location into which the gradient is stored. If None, a new array might be created. proba_out : None or array of shape (n_samples, n_classes) A location into which the class probabilities are stored. If None, a new array might be created. n_threads : int, default=1 Might use openmp thread parallelism. Returns ------- gradient : array of shape (n_samples, n_classes) Element-wise gradients. proba : array of shape (n_samples, n_classes) Element-wise class probabilities. )r9r:r;rG proba_outr=)r#r>rgradient_proba)r'r9r:r;rGrr=s r(r"HalfMultinomialLoss.gradient_probasH   !}}^< MM.9 !}}Y7   l3I !!)'% " &&r+r)Nryrz) r|r}r~rrrrr)r1rerrrrrs@r(rrs= DM<O 8 1&5'5'r+rc@^\rSrSrSrSU4SjjrSSjrSrSrU=r $)ExponentialLossiIaExponential loss with (half) logit link, for binary classification. This is also know as boosting loss. Domain: y_true in [0, 1], i.e. regression on the unit interval y_pred in (0, 1), i.e. boundaries excluded Link: y_pred = expit(2 * raw_prediction) For a given sample x_i, the exponential loss is defined as:: loss(x_i) = y_true_i * exp(-raw_pred_i)) + (1 - y_true_i) * exp(raw_pred_i) See: - J. Friedman, T. Hastie, R. Tibshirani. "Additive logistic regression: a statistical view of boosting (With discussion and a rejoinder by the authors)." Ann. Statist. 28 (2) 337 - 407, April 2000. https://doi.org/10.1214/aos/1016218223 - A. Buja, W. Stuetzle, Y. Shen. (2005). "Loss Functions for Binary Class Probability Estimation and Classification: Structure and Applications." Note that the formulation works for classification, y = {0, 1}, as well as "exponential logistic" regression, y = [0, 1]. Note that this is a proper scoring rule, but without it's canonical link. More details: Inserting the predicted probability y_pred = expit(2 * raw_prediction) in the loss gives:: loss(x_i) = y_true_i * sqrt((1 - y_pred_i) / y_pred_i) + (1 - y_true_i) * sqrt(y_pred_i / (1 - y_pred_i)) cj>[TU][5[5SS9 [ SSSS5Ulgr)rr)r rrr%rs r(r)ExponentialLoss.__init__ms8 #%  (1dD9r+cRS[R"USU- -5-nUbX2-nU$)Nr)r#sqrtrs r(rj(ExponentialLoss.constant_to_optimal_zerous1BGGFa&j122  $  !D r+c4URS:Xa$URSS:XaURS5n[R"URSS4UR S9nUR RU5USS2S4'SUSS2S4- USS2S4'U$rrrs r(rExponentialLoss.predict_proba|rr+rryrrs@r(rrIs!F:r+r) squared_errorabsolute_error pinball_loss huber_loss poisson_loss gamma_loss tweedie_loss binomial_lossmultinomial_lossexponential_loss)*rrnumpyr# scipy.specialrutilsr utils.statsr_lossr r r r r rrrrrrrrrrrrrrrrrrrrrrrrr_LOSSESrr+r(r s* .    8z!z!B6x6.#CH#CL<(<~F@F@Rh@H>=h=@+Eh+E\BxBJH'(H'VFhFT&###%+' r+