Naima Boutaghane

Laurence Voutquenne -Nazabadioko
email: laurence.nazabadioko@univ-reims.fr

Alain Simon

Dominique Harakat

Kaddour Benlabed

Zahia Kabouche

A new triterpenic diester from the aerial parts of Chrysanthemum macrocarpum

Keywords: Chrysanthemum macrocarpum, Asteraceae, Triterpenic diester, Cyclitol, NMR, Antibacterial activity

A new triterpenic diester, 3,21-dipalmitoyloxy-16β,21α-dihydroxy-β-amyrine(1), along with two natural cyclitols, conduritol C (2) and viburnitol (3), four known triterpenes (4-7), and seven known flavonoids (8-14) were isolated from the aerial parts of Chrysanthemum macrocarpum. Their structures were established on the basis of extensive 1D and 2D NMR ( 1 H, 13 C, COSY, HMBC, HSQC, and ROESY) and ESIMS studies. The chloroform fraction, taraxasterol (4) and β-sitosterol (7) were investigated for their antibacterial activity against Staphylococcus aureus, Enterococcus faecalis, Escherichia coli, Pseudomonas aeruginosa and Klebsiella pneumoniae. The chloroform fraction and taraxasterol (4) showed a weak antibacterial activity and were evaluated for their cytotoxic activity against human colon cancer HT-29 cells and human prostate carcinoma PC3 cells. The results indicated that both the chloroform fraction and taraxasterol (4) inhibited cell proliferation of both PC3 and HT-29 cells.

Introduction

The Chrysanthemum genus (Asteraceae) is found world-wide and some species have been extensively used as food additives or in traditional medicine to treat inflammation, headache, ulcerative colitis, vertigo, eye irritation, hypertension, and respiratory diseases (Jiangsu, 1993;[START_REF] Yang | Experimental study on the immunological and anti-inflammatory activities of Chrysanthemum indicum[END_REF][START_REF] Yoshikawa | Medicinal flowers. II. Inhibitors of nitric oxide production and absolute stereostructures of five new germacrane-type sesquiterpenes, kikkanols D, D monoacetate, E, F, and F monoacetate from the flowers of Chrysanthemum indicum L[END_REF][START_REF] Matsuda | Medicinal flowers VI. Absolute stereostructures of two new flavanone glycosides and a phenylbutanoid glycoside from the flowers of Chrysanthemum indicum L: their inhibitory activities for rat lens aldose reductase[END_REF]. A series of recent studies have suggested that the Chrysanthemum genus possesses a strong anti-cancer and anti-human immunodeficiency virus type 1 (HIV-1) activity. [START_REF] Ukiya | Constituents of Compositae plants III. Anti tumor promoting effects and cytotoxic activity against human cancer cell lines of triterpene diols and triols from edible Chrysanthemum flowers[END_REF][START_REF] Lee | A new anti-HIV flavonoid glucuronide from Chrysanthemum morifolium[END_REF], Anti-inflammatory [START_REF] Cheon | Chrysanthemum indicum Linne extract inhibits the inflammatory response by suppressing NF-kappa B and MAPKs activation in lipopolysaccharide-induced RAW 264.7 macrophages[END_REF], antinociceptive [START_REF] Shi | Mechanisms involved in the antinociception of petroleum ether fraction from the EtOH extract of Chrysanthemum indicum in mice[END_REF] as well as anti-oxidant activities [START_REF] Yan | Experimental studies on the anti-oxidant effects of water extract from Chrysanthemum indicum. ZhongGuo Xian Dai Ying[END_REF] have been also confirmed. The phytochemical profile of Chrysanthemum species has shown the presence of terpenoids [START_REF] Ukiya | Constituents of Compositae Plants. 2. Triterpenediols, triols, and their 3-O-fatty acid esters from edible Chrysanthemum flower extract and their antiinflammatory effects[END_REF], flavonoids and phenolic compounds [START_REF] Harnyl | Identification of the phenolic components of Chrysanthemum flower (Chrysanthemum morifolium Ramat)[END_REF]. Fifteen Chrysanthemum species grow in Algeria (Quezeland Santa, 1963).Chrysanthemum macrocarpum (Sch.Bip.)Coss.&Kralik ex Batt., a synonym of Endopappus macrocarpus (Coss. & Kralik ex Batt. & Trabut) Sch.Bip., is an endemic species used in traditional medicine as a scabicide and to treat intestinal infections. Touareg people also use this species in food for flavouring and as an herbal tea [START_REF] Boukef | Les plantes dans la médecine traditionnelle tunisienne, Collection: agence de cooperation culturelle et technique[END_REF]Bellakhdar, 1997). We have previously analyzed the chemical composition of the essential oil of this species [START_REF] Boutaghane | Composition of the essential oil of Chrysanthemum macrocarpum from Algeria[END_REF]. This paper reports the isolation and structural elucidation of one new triterpene fatty acid diester (1) and two natural cyclitols, in addition to four known triterpenes, and seven known flavonoids from the aerial parts of C. macrocarpum (Fig. 1). The antibacterial activity and cytotoxic activity against HT-29 and PC3 cells of the chloroform soluble fraction and taraxasterol (4) were also evaluated in order to obtain an insight into the beneficial effects of this plant species.

Results and discussion

The aerial parts of C. macrocarpum were extracted with 20% aq. MeOH. After MeOH removal, the crude extract in H2O was successively partitioned with CHCl3, EtOAc, and n-BuOH. The CHCl3 soluble fraction was purified by chromatographic separations over silica gel, yielding the new triterpene fatty ester (1) and the known compounds identified as taraxasterol (4) [START_REF] Reynolds | Total assignment of the 13 C spectrum of taraxasteryl acetate by 13 C-13 C connectivity experiments and determination of the stereochemistry of taraxasterol by Xray diffraction[END_REF], stigmasterol (6), β-sitosterol (7) [START_REF] Kojima | Sterol glucosides from Prunella vulgaris[END_REF], and calenduladiol-3-O-myristate (5) [START_REF] Ukiya | Constituents of Compositae Plants. 2. Triterpenediols, triols, and their 3-O-fatty acid esters from edible Chrysanthemum flower extract and their antiinflammatory effects[END_REF] (Fig. 1).

The purification of the EtOAc soluble fraction gave the natural cyclitolsconduritol C

(2) [START_REF] Hiroki | Efficient synthesis of (+), and (-) conduritol C via Palladium (II) catalyzed 1,4-Diacetoxylation in combination with enzymatic hydrolysis[END_REF] and viburnitol (3) [START_REF] Kindl | Untersuchungenüber die biosynthese der cyclite-XIII: Vorkommen and biosynthese von cycliten in Asclepiadaceae[END_REF] along with seven known flavonoids: apigenin (8), 7-O-methylapigenin (9), apigenin-7-O-glucoside (10) (Bennini et al., 1992), kaempferol-3-O-glucoside (11), kaempferol-7-O-glucoside (12) [START_REF] Kim | Identification and in vitro biological activities of flavonols in garlic leaf and shoot: inhibition of soybean lipoxygenase and hyaluronidaseactivities and scavenging of free radicals[END_REF], luteolin-3'-O-glucoside (13) [START_REF] Moiteiro | HPLC quantification of dye flavonoids in Reseda luteola L. from Portugal[END_REF] and 6-C-glucosyl-8-Carabinosylapigenin ( 14) [START_REF] Krafczyk | Phenolic composition of rhubarb[END_REF]. Their structures were elucidated using chemical and spectroscopic methods including 1D and 2D NMR experiments, and ESIMS.

Structure elucidation of compounds 1-3

Compound 1 was obtained as a viscous liquid. The HRESIMS (positive ion mode) exhibited a quasimolecular ion peak at m/z 957.8634 [M+Na] + (calc. for C62H110O5Na, 957.8653) suggesting a molecular formula of C62H110O5. The positive ESIMS spectrum gave the same pseudomolecular ion peak at m/z 957 [M+Na] + with a ion fragment observed at m/z 701 [M+Na-C16H32O2] + in agreement with the loss of a palmitoyloxy side chain. The presence of two acyloxy groups in compound 1 was supported by the presence in the 13 C NMR spectrum of two carbonyl signals at δC 173.3 and 173.7 and in the 1 H NMR spectra of signals due to two methylene groups bearing a carbonyl group at δH 2.32 (q t d, J = 7.1, 1.0 Hz, H-2') and 2.36 (q t d, J = 7.7, 2.7 Hz, H-2''), and for two terminal methyl groups at δH 0.90 (6H,t,J = 7.4 Hz,. The 1 H NMR spectra of 1 showed the presence of eight methyl singlets at δ 0.81, 0.88, 0.89, 0.91, 1.00, 1.01, 1.02 and 1.27, an olefinic proton at δ 5.33 (t, J = 3.5Hz, H-12), three methine protons bearing an oxygen at δ 4.43 (dd,J = 11.7,4.7 Hz,4.53 (dd,J = 9.3,5.5 Hz, H-3) and 4.73 (brt, J = 3.4 Hz, H-21) (Table 1). The basic Δ 12 oleanene skeleton [START_REF] Mahato | 13 C NMR spectra of pentacylic triterpenoids. A compilation and some salient features[END_REF] Compound 2 was obtained as colourless crystals and afforded a [M+Na] + quasimolecular ion peak at m/z 169 suggesting the molecular formula of C6H10O4Na. In the 13 C NMR spectrum, six carbon signals were observed at δC 66.6, 71.2, 72.4, 72.5, 126.6 and 132.4, and the corresponding 1 H signals were observed at δH3. 95, 3.64, 4.20, 3.43, 5.83 and 5.75, respectively (Table 2). The 1 H-1 H COSY spectrum of 2 showed a cyclic spin system with correlations observed between the olefinic protons at δH 5.75 and δH5.83 and both protons H-4 (δH4.20) and H-1 (δH3.95). H-4 and H-1 were also correlated with H-3 (δH3.43) and H-2 (δH3.64), respectively, and H-2 with H-3, suggesting a cyclohexenetetraol. In the HMBC spectrum H-4 and H-1correlated with C-5 and C-6 indicating that the double bond was located in the C-5 (6) positions. The small coupling constant of H-4 (t, J = 4.5 Hz) indicate an α-configuration of the hydroxyl group. The large coupling constants up to 7 Hz for protons H-1, H-2 and H-3 indicate a β-configuration for the three hydroxyl groups in these positions. Therefore, 2 was deduced as being cyclohex-5-ene 1β,2β,3β,4α-tetraol, also named conduritol C previously obtained by synthesis [START_REF] Balci | Conduritols and related compounds[END_REF]. To our knowledge, this is the first time that this compound has been isolated in a natural product.

Compound 3 displayed a quasi-molecular ion peak at m/z 187 [M+Na] + indicating a molecular weight of 164 which is 18 mass units greater than that of 2 and suggesting a molecular formula of C6H12O5. Careful comparison of the NMR spectroscopic data of 3 with that of 2 (Table 2) showed in the 1 H and 13 C NMR signals of 3, the absence of the two olefinic carbon signals at δC126.6 and 132.4 and the occurrence of hydroxyl group (δH 4.02/δC 69.6) and methylene carbon at δC 37.1. Full examination of the 2D NMR spectroscopic data further confirmed the structure of compound 3 as cyclohexa-1β,2α,3β,4α,5α-pentaol also named viburnitol (Kindl, H., et al., 1969).

Antibacterial and cytotoxic activities

The antibacterial activity of the CHCl3 soluble fraction, EtOAc soluble fraction, n-BuOH soluble fraction, and compounds 4 and 7 was performed using the disc diffusion method against two Gram-positive (S. aureus and E. faecalis) and three Gram-negative (E. coli, P. aeruginosa, K.pneumoniae) bacteria (Table 3). The chloroform fraction and 4 and 7 were further analyzed by TLC-direct bioautography on silica gel GF254 to determine the number of active compounds. As observed with the disc diffusion method, the assayfor bioautography demonstrated weak inhibition zones of chloroform fraction and taraxasterol (4) against the growth of Gram-negative (E. coli, P. aeruginosa, K.pneumoniae) bacteria. The clear zones were located separately on TLC suggesting that more than one compounds at different Rf values involved in antibacterial effects. Nevertheless, no inhibition zone was observed with compound 7 at the concentration tested. Minimum inhibitory concentration (MICs) determination was performed by the serial dilution method (Table 3), ampicillin was used as a positive control in these tests. Only the CHCl3 fractionof C. macrocarpum notably inhibited the growth of all the tested bacteria (Gram-positive and Gram negative), and was shown to possess the strongest antibacterial activity against Gram negative with the largest inhibition zones (30 and 24 mm for P. aeruginosa and E. coli, respectively) and lowest MICs 0.5 to 8 µg/ml.Compound 4 also showed relatively modest activity, most particularly against Gram negative bacteria K. pneumonia (32 µg/ml), E. Coli (32 µg/ml) and P. aeruginosa (16 µg/ml).Previous findings have shown that taraxasterol exhibited antibacterial properties [START_REF] Singh | Anti-inflammatory and antimicrobial activities of triterpenoids from Strobilanthes callosus Nees[END_REF][START_REF] Singh | Estimation of triterpenoids from Heliotropium marifolium Koen ex. Retz. in vivo and in vitro. I. Antimicrobial screening[END_REF][START_REF] Villarreal | Cytotoxic and antimicrobial screening of selected terpenoids from Asteraceae species[END_REF][START_REF] Dai | Taraxastane-type triterpenoids from Saussurea petrovii[END_REF]. The antibacterial activity of the chloroform fraction could, in part, be associated with the triterpenes such as taraxasterol (4), the main component of the chloroform fraction. In fact, it was also possible that the components in lower percentage might be involved in some type of synergism with the other active compounds. Our results support the use of C. macrocarpum in traditional medicine for diseases caused by infective agents.

To determine the cytotoxicity of the chloroform soluble fraction of C. macrocarpum and taraxasterol (4), cell proliferation was evaluated by MTT assay on two cell lines, the human HT-29 colorectal cancer cell line and the human PC3 prostate cancer cell line.

The chloroform fraction and taraxasterol inhibited cell proliferation of both HT-29 and PC3 cancer cells in a dose-and time-dependent manner. The IC50 were calculated on PC3 and HT-29 cells at 48h as being 69.3 and 103.1 µg/ml, respectively, for chloroform fraction and 37.1 µM and 89.7 µM, respectively, for taraxasterol (4) (Table 4).

Taraxasterol seems to be responsible of the activity of the extract. It is known to possess antiproliferative activity against various cancer cells [START_REF] Villarreal | Cytotoxic and antimicrobial screening of selected terpenoids from Asteraceae species[END_REF][START_REF] Dai | Taraxastane-type triterpenoids from Saussurea petrovii[END_REF].

In conclusion, a new triterpenic diester, 3,21-dipalmitoyloxy-16β,21α-dihydroxy-βamyrine(1), and two natural cyclitols, conduritol C (2) and viburnitol(3), together with four known triterpenes, and seven known flavonoids were isolated from the aerial parts of Chrysanthemum macrocarpum. This study clearly show that the chloroform fraction presents a good activity against Gram-negative bacteria (P. aeruginosa, K. pneumonia, E. coli) with MICs values of 0.5, 4 and 8 µg/ml, respectively, better than ampicilline used as control. At these concentrations, no cytotoxic activity was observed on PC3 and HT-29 cells. It can be concluded that this plant may serve as a very good source of potential antibacterial agent and justify their used in traditional medicine as scabicides and for the treatment of intestinal infections.

Experimental

General experimental procedures

Plant material

The aerial parts of Chrysanthemum macrocarpum(Sch.Bip.) Coss.&Kralik ex Batt.were collected in the flowering season from Ghardaia (Algerian septentrionalsahara) in April 2005. The botanical determination was performed by Pr Gérard De Bélair from Annaba-University (Algeria). A voucher specimen has been deposited in the botanic Museum of Angers, France under reference MBAng 2005-8.

Extraction and isolation

Dried and powdered aerial parts (1380 g) of C. macrocarpum were macerated in 20% aq. MeOH (10 l). The hydromethanolic extract was concentrated to dryness (under low pressure). The residue was suspended in H2O and successively partitioned with CHCl3, EtOAc, and n-BuOH (3 300 ml, each), respectively, affording a CHCl3 soluble fraction (20 g), an EtOAc-soluble fraction (4 g), and a n-BuOH-soluble fraction (20 g). A part (10 g) of the CHCl3 soluble fraction was fractionated by silica gel CC, using a gradient of cyclohexane-EtOAc (1:0 to 0:1) to give 20 fractions. The fraction 6 (964 mg) eluted with cyclohexane-EtOAc (95:5) was fractionated by silica gel CC eluted with a gradient of cyclohexane-CHCl3 (1:0 to8:2).Fractions 71-99 (290 mg), eluted with cyclohexane-CHCl3 (9:1) were subjected to silica gel CC using 100% toluene as solvent, and then fractions 62-75 (141 mg) were further purified by TLC preparative in toluene-acetone (98:2) to give compound 1 (6.5 mg). The fraction 9 (680 mg) was submitted to silica gel CC using a gradient of toluene-acetone (1:0 to8:2) to give taraxasterol 4 (30 mg). Preparative TLC of fractions 36-50 (24 mg), eluted with toluene-acetone (98:2), developed with a mixture of toluene-acetone (96:4), give taraxasterol 4 (2.3 mg), and calenduladiol-3-O-myristate 5 (8 mg).A part of fraction 12, eluted with cyclohexane-EtOAc (8:2) was subjected to silica gel vacuum liquid chromatography (VLC)using a gradient of cyclohexane-EtOAc (9:1 to 6:4). Three fractions of 150 ml were collected and monitored by TLC in cyclohexane-EtOAc (8:2).

Fraction 2 was subjected to silica gel CC using petroleum ether-EtOAc (1:0 to 8:2) to give stigmasterol 6 (15 mg) and β-sitosterol7 (25 mg) in pure form.

The EtOAc soluble extract was fractionated by silica gel CC, using a gradient of CHCl3-MeOH (1:0 to 0:1). Fractions 7-9 (150 mg), eluted with were purified by silica gel CC eluted with EtOAc-MeOH (99:1 to 8:2) to provided apigenin8 (13 mg) and 7-O-methyl-apigenin 9 (10 mg). Fractions 13-16 (491 mg), eluted with CHCl3-MeOH (9:1), were fractionated by silica gel CC using a gradient of EtOAc-MeOH (1:0 to 6:4) to afforded fifteen fractions. Fractions eluted with EtOAc-MeOH (85:15) were submitted to reversed-phase (RP-18) CC, using a gradient of MeOH-H2O (6:4 to 1:0) as eluent, to yield conduritol C 2 (27 mg) and apigenin-7-Oglucoside 10 (30 mg). The purification of 25 mg of fraction 17, using the same conditions, afforded viburnitol 3 (4.4 mg). Fractions 18-21 (108 mg), eluted with CHCl3-MeOH (85:15), were purified by semi-preparative HPLC, using a linear gradient (15 to 20 % B during 20 min) at 4 ml/min to give kaempferol-3-O-glucoside 11 (tR 9.01 min, 3.5 mg), kaempferol-7-O-glucoside 12 (tR 12.34 min, 5.5 mg), and luteolin-3'-Oglucoside 13 (tR 11.24 min, 3.7 mg). Fraction 22 (25 mg) eluted with CHCl3-MeOH (8:2), gave 6-C-glucosyl-8-C-arabinosylapigenin 14 (tR 20.19 min, 7 mg), by semipreparative HPLC using a linear gradient (10 to 20 % B during 22 min) at 4 ml/min.

Compound 1

Colourless viscous liquid; 1 H and 13 C NMR, see Table 1; [α]D 20 +11.5° (c 0.76, CHCl3).

Conduritol C (2)

Colourless crystals; 1 H and 13 C NMR, see Table 2; ESIMS (positive-ion mode) m/z: 169 [M+ Na] + ; [α]D 20 209.5° (c 0.27, MeOH).

Viburnitol (3)

Colourless crystals; 1 H and 13 C NMR, see Table 2; ESIMS (positive-ion mode) m/z: 187

[M+ Na] + .

Antibacterial activity

American Type of Culture Collection (ATCC) standard against namely: Staphylococcus aureus (ATCC 25923), Enterococcus faecalis (ATCC 29212) and Gram negative Escherichia coli (ATCC 25922), Pseudomonas aeruginosa (ATCC 27853) and the clinically isolated strain Klebsiella pneumonia were used. The reference strains were obtained from the Pasteur Institute (Algiers). The clinical strain was obtained from the laboratory of bacteriology, Benbadis Hospital, Constantine, using conventional methods (clinical isolation).

Antibacterial screening

The disk assay was used with some modifications (Rasoanaivo and Ratsimamanga, 1992;NCCLS, 1997) to determine the growth inhibition of bacteria by the CHCl3 soluble fraction, EtOAc soluble fraction, n-BuOH soluble fractionand compounds 4 and 7of C. macrocarpum. The bacteria tested were maintained at 4 °Con nutrient agar (NA). Base plates were prepared by pouring 10 ml Mueller-Hinton (MH) agar into sterile petri dishes (9 cm) and allowed to set. Molten MH agar held at 48 °C was inoculated with a broth culture (10 6 -10 8 bacteria per ml) of the test organism and poured over the base plates forming a homogenous top layer. Paper discs (Whatman No.

3, 6mm diameter) were sterilized by autoclaving. 10 µl of the fractions or compounds 4 and 7(128 mg/10ml) were applied to paper disc so that each disc contains 128 µg of sample. After evaporation of the solvent in a sterile atmosphere, paper discs were placed in the top layer of the MH agar plates. The fractions were tested in triplicate, with ampicillin (10 µg ml -1 ) and ceftazidime (30 µg ml -1) disc as reference. The plates were evaluated after incubation at 37°C for 18 h. The results were recorded by measuring the diameter of inhibition zone at the end of 18 hours.

Thin-layer chromatography-direct bioautography

Thin-layer chromatography (TLC)-direct bioautographic method [START_REF] Shai | Four pentacyclictriterpenoids with antifungal and antibacterial activity fromCurtisiadentata(Burm.f) C.A. Sm. leaves[END_REF] was applied to identify and localize active compounds in the chloroform fraction against Pseudomonas aeruginosa, Escherichia coli (ATCC 25922), Staphylococcus aureus (ATCC 25923), Enterococcus faecalis (ATCC 29212) and Klebsiella pneumonia. The chloroform fraction or the compounds 4 and 7 were dissolved in chloroform at a concentration of 1 mg/ml. 100 µl of the chloroform fraction or 50 µl of 4 and 7 were spotted at 1 cm from the base of preparative Merck chromatography silica gel 60 F254 TLC plates. TLC plate loaded with the chloroform fraction and the pure compounds was developed in a hexane: ethyl acetate (8:2, v/v) solvent system. After migration, the solvent was evaporated. TLC plates were overlapped homogenously with 10 ml of nutrient agar (Mueller-Hinton) infected by nutrient broth containing bacteria (10 8 cfu/ml). After solidification of the medium, the TLC plates were incubated for 24 h at 37 °C. Subsequently, bioautograms developed were sprayed with 2 mg/ml of piodonitrotetrazolium bromide (INT) (Sigma). Clear zones on bioautograms indicated inhibition of growth after incubating for 2 h at 37 °C.

Determination of the minimums inhibitory concentrations (MICs).

Tests were performed by the agar dilution method according to the NCCLS with some modifications. The CHCl3 soluble fraction or compounds 4 and 7 were dissolved in ethanol to a final concentration of 512 µg/ml. This was serially diluted 2 fold with MH medium to obtain concentration of 0.5, 1, 2, 4, 8, 16, 32, 64 and 128 µg/ml. 2 ml of each concentration was added in Petri dished containing 18 ml of MH medium previously inoculated with suspension of bacteria (10 5 CFU/ml) for testing the samples. Control plates contain the culture medium and 2 % of ethanol. Plates were incubated at 37 °C for 18hours. The minimum inhibitory concentration (MIC) was defined as the concentration at which no colony was observed after incubation. All experiments were performed in triplicates.

Cell culture and cell proliferation assay (MTT)

Human colorectal cell line HT-29 and human prostate carcinoma PC3 cells were purchased from American Type Culture Collection (LGC Standards, Middlesex, UK).

Cells were seeded at 3.10 4 /cm 2 and grown in DMEM, supplemented with 10% foetal calf serum, 100 units/ml of penicillin, 100 mg/ml of streptomycin. Cells were incubated under 5% CO2 humidified atmosphere at 37 °C. Cell viability was determined by the trypan blue dye exclusion method.

The effect of the chloroform fraction of C. macrocarpum and the isolated taraxasterol (4) on cell viability was assessed using the [3-(4,5-dimethylthiazol-2-yl)-2,5diphenyltetrazolium bromide] (MTT) assay, as described previously [START_REF] Mosmann | Rapid colorimetric assay for cellular growth and survival: Application to proliferation and cytotoxicity assays[END_REF]. This test is based on the ability of the mitochondrial succinate deshydrogenase enzyme of living cells to reduce MTT to a purple-blue insoluble formazan precipitate. The HT-29 and PC3 cells were plated in 96-well plates at concentration of 10 4 cells/well and incubated for 24 h, then treated with various concentrations of chloroform fraction or 4 or ursolic acid used as positive control which were dissolved in dimethyl sulfoxide (DMSO) and diluted with culture medium to give the appropriate final concentrations of 20-140 µg/ml. Controls were done with the same final DMSO concentration in the medium as samples. After incubation of the cells with chloroform fraction for 24 or 48 h or with 4 for 48 hat 37 °C, 10 µl of 5 mg/ml MTT solution were added to each well and incubated for 4 h at 37 °C. Then, 100 µl of 10% sodium dodecyl sulfate (SDS) in 0.01 M HCl was added to each well. The degree of formazan formation, an indicator of living and metabolically active cells, was determined using ELISA reader at 550 nm. Six wells per dose and time point were measured in 3 different experiments.

  in 1 was supported by the following key HMBC correlations H-3/C-2, C-4, C-5, C-23, C-24; H-12/C-9, C-11, C-13, C-14, C-18; H-16/C-14, C-15, C-17, C-18, C-28; H-21/C-17, C-19, C-20, C-22, C-29, and C-30. The small coupling constant of H-21 (brt, J = 3.4 Hz, H-21) indicated an α-axial position for the hydroxyl group, and the large coupling constants of H-16 (J = 11.7 Hz) and H-3 (J = 9.3 Hz), indicated a β-equatorial position for the hydroxyl groups. In the ROESY experiment, the interaction observed between H-18/H-30 and between H-21/H-30 confirmed the βequatorial position for H-21.The α-axial orientation of H-3 and H-16 was also confirmed by the ROEs observed between H-3 and H-5α and H-23α, while H-16 showed a ROE correlation with H-27α. Thus, the aglycone of compound 1 was determined to be 3β,16β,21α-trihydroxy-olean-12 ene. The downfield shifts of H-3 and H-21 at δH 4.53 and 4.73, respectively in 1 suggested the esterification of the secondary hydroxyl groups at C-3 and C-21 by two acyloxy groups. This was confirmed by the HMBC correlations observed between H-3 (δH 4.53) and the carbonyl carbon signal of the first acyloxy side chain (δC 173.7), and between H-21 (δH 4.73) and the second carbonyl carbon signal (δC 173.3). The two acyloxy groups were identified by ESIMS as being palmitic acid esters. Subtraction of the triterpene unit from the molecular formula of the ion fragment at m/z 701 [M+Na-C16H32O2] + indicated that the second fatty acid ester was identical and correspond to a palmitic acid. Another ion fragment observed at m/z 685 [M+Na-HO-C16H32O2] + indicated the simultaneously loss of the palmitic acid in C-21 position and OH in C-16 position. Thus, compound 1 was identified as 3,21-dipalmitoyloxy-16β,21α-dihydroxy-β-amyrine.

  Optical rotations were measured in MeOH or CHCl3 with a Perkin-Elmer 241 polarimeter. 1 H and 13 C NMR spectra were recorded on a BrukerAvance DRX 500 NMR spectrometer ( 1 H at 500 MHz and 13 C at 125 MHz). 2D-NMR experiments were performed using standard Brukermicroprograms. Coupling constants were expressed in Hz, and chemical shifts were given on a δ (ppm) scale. ESIMS and HR-ESIMS experiments were performed using a Micromass Q-TOF micro instrument (Manchester, UK). The samples were introduced by direct infusion in a solution of MeOH at a rate of 5 µL min -1 .TLC was carried out on precoated silica gel 60 F254 (Merck), and spots were visualized by heating after spraying with 50% H2SO4. Kieselgel 60 (63-200 mesh) Merck or LiChroprep RP-18 (40-63 µm) Merck silica gel were used for Column chromatography (CC). HPLC was performed on a Dionex apparatus equipped with an ASI-100 autosampler, a P580 pump, a UVD 340S diode array detector, and Chromeleon ® software.A prepacked RP-18 column (201SP510, 250 × 10 mm, 5μm, 90 Å, Dionex, Vydac, France) was used for semi preparative HPLC of the flavonoids with a binary gradient elution (solvent A: H2O with 0.0025% TFA, solvent B: MeCN) at 25 °C and a flow rate of 3.5 mL min -1 ; the chromatogram was monitored at 254 and 365 nm. The following reagents were used for cell culture assay: DMEM and foetal calf serum (Gibco BRL, Cergy-Pontoise, France), 3-(4,5-dimethylthiazol-2-yl)-2,5diphenyltetrazolium bromide (MTT), and sodium dodecyl sulfate (SDS) (Sigma Chemical Co., St. Louis, MO).

  HRESIMS (positive-ion mode) m/z: 957.8634 [M+Na] + (calcd for C62H110O5Na: 957.8653); ESIMS (positiveion mode) m/z: 957 [M+Na] + , 701 [M+Na-C16H32O2] + , 685 [M+Na-OH-C16H32O2] + ;

  Fig. 1. Structures of compounds 1-14

Table 1 .

 1 1 H and 13 C NMR Spectroscopic Data (CDCl3) of compound 1

			1
	position	C	H (J in Hz)		C	H (J in Hz)
	1a	38.3	1.09, m	Acyl in C-3
	1b		1.65, m
	2a	23.6	1.66, m	1'	173.7
	2b		1.92, m
	3	80.5	4.53, dd (9.3, 5.5)	2'	34.9	2.32, q t d (7.1, 1.0)
	4	37.8		3'	25.2	1.65, m
	5	55.3	0.88, m	4'-13' 29.1-	1.28, brs
					29.9
	6a	18.2	1.44, m	14'	31.9	1.28, m
	6b		1.58, m
	7a	32.9	1.37, m	15'	22.7	1.32, m
	7b		1.56, m
	8	39.8		16'	14.2	0.90, t (7.4)
	9	46.8	1.58, m	Acyl in C-21
	10	36.8		1''	173.3
	11	23.6	1.97, m	2''	34.9	2.36, q t d
						(7.7, 2.7)
	12	122.	5.33, brt (3.5)	3''	25.2	1.67, m
		9
	13	142.		4''-	29.1-	1.28, brs
		9		13''	29.9
	14	43.9		14'	31.9	1.28, m
	15a	35.4	1.65, m	15''	22.7	1.32, m
	15b		1.29, m
	16	68.3	4.43, dd (11.7,	16''	14.2	0.90, t (7.4)
			4.7)
	17	37.4
	18	48.5	2.28, dd (14.1,
			4.2)
	19a	42.1	1.06, m
	19b		2.11, t (13.8)
	20	34.3
	21a	76.5	4.73, brt (3.4)
	21b
	22a	34.1	1. 39, dd (15.7,
	22b		3.3)
			2.17, dd (15.7,
			3.1)
	23	28.1	0.91, s
	24	16.8	0.89, s
	25	15.6	1.00, s
	26	16.8	1.01, s
	27	26.7	1.27, s
	28	21.4	0.81, s
	29	27.4	0.88, s
	30	24.8	1.02, s

Table 3 .

 3 Minimal Inhibitory Concentration (MIC) and Inhibition Zones of CHCl3 fraction and compounds 4 and 7 from Chrysanthemum macrocarpum against Gram positive and negative bacteria.

		Inhibition Zone diameters (mm)			MIC (µg/ml)
	Microorganisms	CHCl3	4	7	Ampicilline Ceftazidime	CHCl3	4	7	Ampicilline Ceftazidime
		fraction					fraction
	S. aureus	24	10	12	30		128	128	128	5
	E. faecalis	30	10	10	16		128	128	128	16
	E. coli	24	18	-	18		8	32	>128	10
										16
	P. aeruginosa	30	24	-	-	22	0.5	16	>128	-
	K. pneumoniae	18	12	-	14		4	32	>128	32
	-, No inhibition zone

Table 4 .

 4 IC50 of CHCl3 fraction of Chrysanthemum macrocarpum and compound 4 on the proliferation of human PC3 prostate cancer cells line and human colon cancer HT-

	29 cells line at 24 and 48h
	Sample	PC3 (IC50, µg/ml)	HT-29(IC50, µg/ml)
		24 h	48 h	24 h	48 h
	CHCl3 fraction	80.8	69.3	122.4	103.1
	Taraxasterol 4	nt	15.8 (37.1 µM)	nt	38.3 (89.7 µM)
	Ursolic acid	nt	25 µM	nt	25 µM
	nt : not tested at 24h

Acknowledgments

We wish to thank the Ministry of Higher Education and Scientific Research (Algeria) for financial support. Financial support by CNRS, Conseil Regional Champagne Ardenne, Conseil General de la Marne, Ministry of Higher Education and Research (MESR) and EU-programme FEDER to the PlANET CPER project is gratefully acknowledged. The authors are very grateful to Karen Ple (ICMR) for revising the English.


Y Pousset
email: pousset@sic.sp2mi.univ-poitiers.fr

R Vauzelle
email: vauzelle@sic.sp2mi.univ-poitiers.fr

P Combeau

OPTIMIZATION IN COMPUTATION TIME OF A RADIO COVERAGE PREDICTION FOR MICRO CELLULAR MOBILE SYSTEMS

This paper presents a method which optimises the computation time prediction of a radio coverage, whatever the propagation model used. The principle consists in reducing the number of application points of the propagation model in relation to a classical technique. The proposed method is based on a multi resolution analysis of measured signals carried out around 2 GHz, and on an electromagnetic analysis of the propagation environment. The performances of the method are evaluated in terms of reduction in computation time and of accuracy, in comparison with the classical technique.

1.Introduction:

Mobile cellular systems are based on a geographical distribution of base stations.

Each transmitter provides a radio coverage that must be predicted with accuracy. The final aim is to enable a satisfying quality of communication everywhere. Generally, the coverage prediction is computed by a propagation model applied with a regular step in the area studied [START_REF] Tan | A microcellular communications propagation model based on the uniform theory of diffraction and multiple image theory[END_REF]. This method leads to a considerable computation time which might become unexploitable in complex geometrical environments.

They have already been some searches studying how to reduce that computation time. They consist in simplifying the propagation model used. The process presented in this article is complementary to them: it is independent of the model. This work has been developed in collaboration with the France Telecom Research and Development centre.

The principle, based on an hypothesis explained in the section 2, consists in reducing the number of application points of the propagation model and of optimising their location. Moreover, this optimisation must be obtained without deteriorate the quality of the radio coverage prediction. This approach can be break down into two stages called "learning" and "application" stages which are developed in the section 3.

In the section 4, we compare the performances of our method with those of the classical technique.

2.Hypothesis:

For the coverage prediction, where the fast fading is not considered, our method is based on the following hypothesis: On a mobile route, the significant and slow fading of the received signal are due to changing of electromagnetic interactions combinations (reflection, diffraction and line of sight) undergone by the received waves.

Thus for a geographical area, the set of points associated to a same combination of electromagnetic interactions leads to delimit a spatial region. Each region is characterised by a stationary received signal: the variation of the mean power is small. So, to go from one region to another amounts to consider a significant and slow variation of the received signal.

The Figure 1 illustrates this hypothesis on a schematic example. The electromagnetic wave propagation laws leads to different regions whose, each of them is characterised by a combination of electromagnetic interactions (Figure 1 This hypothesis is the base of the method presented in the next section to optimise the computation of a coverage prediction.

3.Method

Our method is split up into two stages: a "learning" stage which verifies the validity of the hypothesis and an "application" stage which provides the coverage prediction optimised in computation time. The search of a strong correlation between the results of these two tools leads to the verification of our hypothesis and constitutes the learning stage.

Electromagnetic analysis:

A micro cell configuration is defined by a transmitter height below the average height of roof tops. It is the reason why the electromagnetic study is based in the horizontal plan (2D_H).

The electromagnetic analysis software is founded on the search of the influence regions of the electromagnetic interactions.

For the line of sight and the reflection phenomena, the geometrical optic theory is applied. In an environment, the regions induced by these phenomena are delimited by the optic boundaries.

The diffraction phenomenon is more complex to delimit than the previous ones because the diffracted field due to an obstacle is present throughout the space.

Nevertheless, the amplitude of the diffracted waves decreases as the diffraction angle increases [3][4]. Therefore, the electromagnetic analysis takes into account a diffraction angle α which is the limit angle from this the amplitude of the diffracted wave is negligible in comparison with that of the line of sight. To calculate α, we consider firstly the diffraction loss A dB for a perfectly conductive infinite half plane (Figure 2 a); its expression is given by (1) :

-1 for v v) ) 1 ² v (( 20log + 6,4 0 P P(v) log 10 (v) A 5 , 0 10 10 dB > + + =         = (1)
where :

P 0 is the received power without obstacle;

and P(v) is the received power with the obstacle where

v = h 2 λ 1 1 1 2 1 2 d d +             / .
The α angle is defined such that the diffraction loss A dB (v) is equal to -3dB (Figure 2 b).

For this value, the magnitude of the wave in line of sight is equal to that of the diffracted wave. On this base, a study generalised to wedges and based on the UTD [START_REF] Holm | A new heuristic UTD diffraction coefficient for nonperfectly conducting wedges[END_REF] has shown that, whatever the wedge angle, the length of link and the electrical nature of the wedge, α is roughly equal to one degree. Because, the α values are smaller than the geometrical parameters of the area studied, the diffraction phenomenon is considered only in the shadow zone in the electromagnetic analysis software.

To summarise, the electromagnetic analysis relies on the search of the optical boundaries and leads to a partition in regions based on the nature of the electromagnetic phenomena (diffraction, reflection, line of sight). A simple example of a partition is shown in the Figure 3 where the visualisation of each region is realised thanks to a grey scale. For the example given, with a maximum number of diffractions and reflections equal to one, five regions exit. To generalise this example, we underline that the complexity of a spatial cut out is directly connected to the number of the different interactions.

Signal segmentation:

This technique is based on a multiscale analysis which allows the measured signal variations thought to significant to be detected; each scale corresponds to a particular frequency band.

The wavelet transform of a signal s(r), where r is the curvilinear abscissa, is defined by (2): This relation indicates that

) ( * ) ( ) (
) ( 1 r s W e
is the first derivative of the signal followed by a smoothing at scale e. The rapid variations of s(r) disappear progressively as the scale becomes high. Thus the extrema are due to the significant and slow variations of the signal [START_REF] Carre | Segmentation of signals by wavelet maxima: application to the prediction of the radioelectric coverage zones[END_REF].

In our problematic, this wavelet analysis identifies automatically the three significant intervals associated with the significant variations of the signal of the Figure 1(b).

Principle:

The objective of the electromagnetic analysis software is to give a partition which will lead to a coverage computation ( §3.2). However, this calculation is built on the determination of the significant variations of received signals ( §2). So, the partition must be constitute only by the regions which lead to the such variations. So for each region of the partition, it is necessary to define a hierarchy of the influence of the different electromagnetic interactions in order to determine the predominant combinations, in terms of received energetic level. Thus for example, a region constituted by line of sight and reflected waves will be considered as a region in line of sight. On such considerations and after groupings of contiguous regions having the same characteristics, the significant partition of the Figure 3 (b) is obtained for the example of the Figure 3 (a).

Thus, the number of maximum diffraction and reflection taken into account in the electromagnetic analysis software must ensure that the limits of intervals detected by the "wavelet maxima" coincide with changing of regions. The search principle of these major interactions is described by the algorithm of the Figure 4. In view of the statistical validity of these numbers of interactions, they will be considered, whatever the micro cell environment studied, without apply the learning stage process. So, in the next section these values are used in the application stage to provide a satisfying coverage prediction.

Coverage prediction: "application stage"

To compute the radio coverage, it is necessary to apply a propagation model in each region of the spatial partition. To define the minimum number of application points per region, a rule has been considered. It relies on statistical results [START_REF] Parsons | The Mobile Radio Propagation Channel[END_REF] on non-fade duration of measured signals : in 90% of the time, the signal magnitude is above to its local average value minus 10dB. Thus, it has been shown that, statistically, the minimum number of application points of the propagation model per region can be equal to two [START_REF] Combeau | An acceleration technical for the prediction of radioelectric coverage zone[END_REF]. The average of these two estimated received powers provides a good estimation of the average signal in the considered region. Concerning the location of these two points, it is around the barycentre of the region.

In an operational context, on an area defined by geographical data basis, the application stage consists in carrying out the spatial partition software with the correct number of diffractions and reflections provided by the learning stage. On the computed partition, a propagation model is applied in each region according to the previous rule. This procedure leads to the optimised coverage prediction.

4.Results:

The interest of our method is evaluated by comparing its coverage prediction to the one provided by the classical approach. The comparison is realised on the micro cellular environment of the Figure 5. The performances of the two approaches have been tested with the "microG" scalar propagation model applied during the deployment stage of some mobile networks [START_REF] Lagrange | Les réseaux radiomobiles[END_REF]. The result of the prediction coverage zone due to our optimised method is presented on the figure 7.

For this example, the computation has been realised thank to a 1.6GHz Pentium computer. The classical computation of the coverage zone, with a constant spatial step equal to 5 meters, leads to a number of application points and a computation time equals to 30601 and 11 minutes respectively. Our optimised technique leads to a number of application points and a computation time equals to 3590 points and 1 minutes 8 seconds respectively. So, the comparison of these results shows that the reduction ratios in application points and computation time are equals to 88% and 9.7 respectively. Nevertheless, it is necessary to note that the previous reduction ratio in computation time must increase if we apply a 3D vectorial propagation model based on a ray tracing [START_REF] Escarieu | Outdoor and indoor channel characterization by a 3D simulation software[END_REF] or launching [START_REF] Kloch | Comparison of measured and predicted time dispersion and direction of arrival for multipath in a small cell environment[END_REF]. These models, although very accurate, have a computation time largely more important than this of the scalar propagation models. Thus, for a reduction ratio in application points equal to 88.3% the computation time ratio considerably increases (near 50).

Moreover, such we can observe it on the cumulative functions of the figure 8(a) and (b), the optimisation time is obtained without a significant degradation of the accuracy.

The Figure 8(a) shows that the classical and optimised repartition of the total losses are very closed. This result is confirmed by the repartition of the absolute deviation losses (Figure 8(b)). This representation indicates that for the probabilities equals to 0.5 and 0.9, the absolute deviation are equals to 1.46 dB and 5.53 dB respectively. These results are completely satisfying in an operational context because the operators require generally a standard deviation less than roughly 6dB.

5.Conclusion:

In conclusion, we have shown the interest of our optimised method of coverage prediction for micro cellular configurations. It provides a reduction factor of the computation time varying between 9 and 50 according to the propagation models used in comparison with the classical method. Moreover, this reduction is obtained without significant degradation of the accuracy.

The future works are dedicated to an extension of the presented method in vertical planes (2D_V) and in the combination of the 2D versions (2D_V and 2D_H) for a application in others types of configurations such as small and macro cells.

Figure captions

  (a):[START_REF] Tan | A microcellular communications propagation model based on the uniform theory of diffraction and multiple image theory[END_REF] visibility, (2) visibility plus reflection, (3) diffraction). Considering the hypothesis, the received signal on the mobile route of the Figure1(b) presents significant and slow variations at the points P 1 , P 2 . These points correspond to the limits of the different regions which are crossed by the mobile route.

3. 1 .

 1 Verification of the hypothesis: "Learning stage" This stage necessitates two complementary analysis. The first one, heart of the optimised prediction, is an electromagnetic analysis software developed in the laboratory. It implements the wave propagation mechanisms which lead to the determination of a spatial cut out according to a transmitter located in a geographical area. The second one is a segmentation technique of signals based on the wavelets maxima. It allows to detect the behaviour changing of measured signals on a mobile route.

  which smoothes the signal s(r) at the scale e.

  After a statistical verification on several measured signals in different micro cellular environments such as the signal of the Figure6measured on the mobile route in the district of Paris presented in the Figure5, the numbers of diffractions and reflections are equals to 4 and 0 respectively.

Figure 1 :

 1 Figure 1 : (a):Schematic studied area.

Figure 2 :

 2 Figure 2 : (a):Diffraction on a infinite half plane.

Figure 3 :

 3 Figure 3 : User interface of the spatial tool and visualisation of the regions produced by a

Figure 4 :

 4 Figure 4: Algorithm of the learning stage on a mobile route.

Figure 5 :

 5 Figure 5: Top view of a district of Paris.

Figure 6 :

 6 Figure 6: Signal received on the mobile route of the Figure 5.

Figure 7 :

 7 Figure 7: Radio coverage due to our optimised method, for 0 reflection and 4 diffractions, on

Figure 8 :

 8 Figure 8: Cumulative functions of: (a): The coverage zone losses:

  Figure 1

  Figure 3

Figure 7 TransmetterFigure 8 Deviation

 78 Figure 7

Acknowledgement: This work has benefited by the support of France Telecom R&D in the contract n°01 1B323.


María Pérez-Ortiz

Omar Rivasplata

Benjamin Guedj

Matthew Gleeson

Jingyu Zhang

John Shawe-Taylor

Miroslaw Bober

Josef Kittler

Learning PAC-Bayes Priors for Probabilistic Neural Networks

Recent works have investigated deep learning models trained by optimising PAC-Bayes bounds, with priors that are learnt on subsets of the data. This combination has been shown to lead not only to accurate classifiers, but also to remarkably tight risk certificates, bearing promise towards self-certified learning (i.e. use all the data to learn a predictor and certify its quality). In this work, we empirically investigate the role of the prior. We experiment on 6 datasets with different strategies and amounts of data to learn datadependent PAC-Bayes priors, and we compare them in terms of their effect on test performance of the learnt predictors and tightness of their risk certificate. We ask what is the optimal amount of data which should be allocated for building the prior and show that the optimum may be dataset dependent. We demonstrate that using a small percentage of the prior-building data for validation of the prior leads to promising results. We include a comparison of underparameterised and overparameterised models, along with an empirical study of different training objectives and regularisation strategies to learn the prior distribution.

Introduction

The majority of deep learning algorithms output weights of neural networks: in recent years, a growing body of works has investigated algorithms which rather output probability distributions over the connection weights of a network, with a number of advantages. This is evidenced e.g. by works inspired by Bayesian learning (see e.g. [START_REF] Blundell | Weight uncertainty in neural networks[END_REF][START_REF] Osawa | Practical deep learning with Bayesian principles[END_REF][START_REF] Gordon | Bayesian deep learning and a probabilistic perspective of generalization[END_REF] among many others) or by the frequentist PAC-Bayes bounds (see e.g. [START_REF] Gintare | Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data[END_REF][START_REF] Letarte | Dichotomize and generalize: PAC-Bayesian binary activated deep neural networks[END_REF][START_REF] Biggs | Differentiable pac-bayes objectives with partially aggregated neural networks[END_REF][START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]). In both cases, a probability distribution over neural network weights defines what can be called a Probabilistic Neural Network (PNN).

Our Contributions.

• We demonstrate consistency of the tightness of the risk certificates across 6 classification datasets.

• We contribute supporting evidence on the crucial role of learning the PAC-Bayes prior for achieving efficient neural network classifiers and tight risk certificates.

• We show that there is potential for tight risk certificates even for highly overparameterised architectures.

Related Literature

Bayesian Neural Networks (BNNs). BNNs are special cases of PNNs, in the sense that BNN methods also output distributions over network weights. Works from this front have shown very promising results, e.g. that BNNs provide an intuitive approach to uncertainty quantification and principled implementations of model pruning/distillation [START_REF] Blundell | Weight uncertainty in neural networks[END_REF]. Furthermore, training objectives inspired by the evidence lower bound (ELBO) and commonly used in BNNs have been seen to act as an implicit regulariser. The resemblance between the ELBO and PAC-Bayes bounds, which has been pointed out in the literature ( [START_REF] Alquier | On the properties of variational approximations of Gibbs posteriors[END_REF][START_REF] Achille | Emergence of invariance and disentanglement in deep representations[END_REF], among others), suggests that some of these properties may be shared by methods inspired by PAC-Bayes bounds. Bayes vs. PAC-Bayes. The 'prior' and 'posterior' distributions that appear in PAC-Bayes bounds should not be confused with their Bayesian counterparts. In PAC-Bayes bounds, what is called 'prior' is a reference distribution, and what is called 'posterior' is an unrestricted distribution, in the sense that there is no likelihood factor connecting them (cf. [START_REF] Guedj | A primer on PAC-Bayesian learning[END_REF][START_REF] Rivasplata | PAC-Bayes analysis beyond the usual bounds[END_REF]). Data-dependent PAC-Bayes priors. In this line of thought, a precursor work is that in [START_REF] Gintare | Data-dependent PAC-Bayes priors via differential privacy[END_REF]. The protocol for partitioning the data goes back to the works in [START_REF] Ambroladze | Tighter PAC-Bayes bounds[END_REF][START_REF] Lever | Distribution-dependent PAC-Bayes priors[END_REF][START_REF] Parrado-Hernández | PAC-Bayes bounds with data dependent priors[END_REF][START_REF] Lever | Tighter PAC-Bayes bounds through distribution-dependent priors[END_REF].

Elements of Statistical Learning

Supervised classification algorithms receive training data S = ((X 1 , Y 1 ), . . . , (X n , Y n )) consisting of pairs that encode inputs X i ∈ X ⊆ R d and their labels Y i ∈ Y. Classifiers h w : X → Y are mappings from input space X to label space Y, and we assume they are parametrised by 'weight vectors' w ∈ W ⊆ R p . The quality of h w is given by its risk L(w), which by definition is the expected classification error on a randomly chosen pair (X, Y ). However, L(w) is an inaccessible measure of quality, since the distribution that generates the data is unknown. An accessible measure of quality is given by the empirical risk functional LS (w) = n -1 n i=1 (h w (X i ), Y i ), defined in terms of a loss function : R × Y → [0, ∞). Indeed, the empirical risk minimisation (ERM) paradigm aims to find w ∈ W that minimises this functional for some choice of loss function (zero-one loss or a surrogate loss).

The outcome of training a PNN is a distribution Q over weight space and this distribution depends on the sample S. Then, given a fresh input X, the randomised classifier predicts its label by drawing a weight vector W at random from Q and applying the predictor h W to X. For the sake of simplicity, we identify the randomised predictor with the distribution Q that defines it. The quality of this randomised predictor is measured by the expected loss notions under the random draws of weights. Thus, the loss of Q is given by Q[L] = W L(w)Q(dw); and the empirical loss of Q is given by Q

[ LS ] = W LS (w)Q(dw).
The PAC-Bayes-quadratic bound in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] says that for any δ ∈ (0, 1), with probability of at least 1 -δ over size-n i.i.d. random samples S, simultaneously for all distributions Q over W we have:

Q[L] ≤   Q[ LS ] + KL(Q Q 0 ) + log( 2 √ n δ ) 2n + KL(Q Q 0 ) + log( 2 √ n δ ) 2n   2 .
In this case the prior Q 0 must be chosen without any dependence on the data S on which the empirical term LS is evaluated. In this work, we use a partitioning scheme for the training data S = S pri ∪ S cert such that the prior is trained on S pri , the posterior is trained on the whole set S and the risk certificate is evaluated on S cert . See Figure 1.

Learning and Certification Strategy

In a nutshell, the learning and certification strategy has three components: (1) choose/learn a prior; (2) learn a posterior; and (3) evaluate the risk certificate for the posterior.

Data-dependent PAC-Bayes priors

We experiment with Gaussian priors Q 0 with a diagonal covariance matrix centered at i) random weights (uninformed data-free priors) and ii) learnt weights (data-dependent priors) on a subset of the dataset which is independent of the subset used to compute the risk certificate (see Figure 1). In all cases, the posterior is initialised to the prior. Similar approaches have been considered before in the PAC-Bayesian literature (we refer to [21, 7, ?].

Training objectives to learn the prior

We evaluate several training strategies to learn the PAC-Bayes prior:

1. Deterministic training objectives: We compare learning the prior mean by nonregularised ERM (referred to as f erm ) to ERM with dropout (f erm + dropout) and Mixup [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF] (f mixup ), which has been shown to act as an implicit regulariser [START_REF] Zhang | How does mixup help with robustness and generalization?[END_REF]. Note that all of these training objectives learn a deterministic neural network with a fixed setting of weights. The prior distribution is then centered at the learnt weight vector, and the scale of the distribution is set as a hyper-parameter.

Probabilistic training objectives:

We compare learning the prior distribution via the Bayes by backprop (f bbb ) [START_REF] Blundell | Weight uncertainty in neural networks[END_REF] objective inspired by the evidence lower bound, to learning it via the objective inspired by the PAC-Bayes-quadratic bound (f quad ) [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]. Since these objectives are probabilistic, they learn the full prior distribution, and not only the mean, as it is the case with the deterministic objectives. For both of these objectives, we first need to set a pre-prior, for which we use uninformed priors centered at zero or at a random weights, setting the scale as a hyper-parameter. We then use this uninformed prior in the KL computation of the training objective, in order to learn the data-dependent prior, which is later used to train the posterior.

Validation of prior and posterior

The work in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] highlighted the difficulty of avoiding overfitting of the prior learnt by ERM and suggested the use of dropout. In this paper, we propose to use the following strategy to monitor the performance of prior and posterior and choose the best prior/posterior accordingly:

1. Prior validation: We propose to select a small percentage of the prior-building data to use as a validation set, with the objective of monitoring the validation loss while learning the prior predictor. Then, at the end of the prior-building stage we choose the one that performs the best in the validation set.

Posterior validation:

In this case we compute the risk certificate at different stages during posterior learning and choose the posterior with the best risk certificate, thus not needing any additional data partition.

A summary of the partitions used at the different stages of learning and certification is shown in Figure 1. Note that the prior validation set is still being used to train the posterior, hence not discarding any data from the learning process.

Posterior Optimisation & Certification

We now present the essential idea of training PNNs by minimizing a PAC-Bayes upper bound on the risk. We use a recently proposed PAC-Bayes inspired training objective [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF], derived from Eq. ( 1) in the context of neural network classifiers:

f quad (Q) =   Q[ Lx-e S ] + KL(Q Q 0 ) + log( 2 √ n δ ) 2n + KL(Q Q 0 ) + log( 2 √ n δ ) 2n   2 .
This objective is implemented using the cross-entropy loss, which is the standard surrogate loss commonly used on these problems. Since the PAC-Bayes bounds of Eq. ( 1) require the loss within [0,1], we construct a 'bounded cross-entropy' loss by lower-bounding the network probabilities by a value p min > 0 (cf. [START_REF] Gintare | Data-dependent PAC-Bayes priors via differential privacy[END_REF][START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]). The empirical risk term is Lx-e S (w) is calculated with the 'bounded' version of the cross-entropy loss.

Optimization of the objective in Eq. ( 1) entails minimizing over Q. By choosing Q in a parametric family of distributions, we can use the pathwise gradient estimator [START_REF] Price | A useful theorem for nonlinear devices having Gaussian inputs[END_REF][START_REF] Jankowiak | Pathwise derivatives beyond the reparameterization trick[END_REF] as done by [START_REF] Blundell | Weight uncertainty in neural networks[END_REF]. The details of the reparametrisation strategy are outlined in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]. Following [START_REF] Blundell | Weight uncertainty in neural networks[END_REF], the reparametrization we use is W = µ + σ V with Gaussian distributions for each coordinate of V . The optimization uses σ = log(1 + exp(ρ)), thus gradient updates are with respect to µ and ρ.

Evaluation of the Risk Certificates. After optimising the posterior distribution over network weights through the previously presented training objective, we compute a risk certificate on the error of the stochastic predictor. To do so, we follow the procedure outlined in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF], which was used before by [START_REF] Gintare | Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data[END_REF] and goes back to the work of [START_REF] Langford | Not) bounding the true error[END_REF]. This certification procedure uses the PAC-Bayes-kl bound. In particular, the procedure is based on numerical inversion of the binary KL divergence, as done by [START_REF] Gintare | Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data[END_REF][START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF].

Experiments

We conduct the following five experiments on the effect of PAC-Bayes priors on PNN predictors: E1 We study the relationship between the learnt PAC-Bayes prior predictive performance and the posterior risk certificate in a large set of experiments on MNIST.

E2 We test if the validation of prior and posterior leads to better performance, even when the amount of data used for prior learning is effectively reduced.

E3 We experiment with different trade-offs of the amount of data to learn the prior and certify the predictor, extending the results in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] to 5 additional datasets, which demonstrates the role of data-dependent priors on the tightness of the risk certificates.

E4 We study the role of the number of parameters in the architecture in the risk certificate and KL term.

E5 Finally, we compare several training objectives and regularisation strategies for learning the prior.

Table 1 shows the datasets used (all except for MNIST available at OpenML.org), selected so as to represent a wide range of characteristics (small vs large, low vs high dimensional and binary vs multiclass). For all datasets except MNIST we select 20% of the data as test set (stratified with class label). For MNIST, we use the standard data partitions.

Experimental setup

In all experiments the models are compared under the same conditions, i.e. weight initialisation and optimiser (vanilla SGD with momentum). The mean parameters µ 0 of the prior are initialised randomly from a truncated centered Gaussian distribution with standard deviation set to 1/ √ n in , where n in is the dimension of the inputs to a particular layer, truncating at ±2 standard deviations. All risk certificates are computed using the the PAC-Bayes-kl inequality, as explained in Section 6 of [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF], with δ = 0.025 and δ = 0.01 and m = 150000 Monte Carlo model samples. We also report the average 01 error of the stochastic predictor, where we randomly sample fresh model weights for each test example 10 times and compute the average 01 error. Input data was standardised for all datasets.

For all experiments we performed a grid search over all hyper-parameters and selected the run with the best risk certificate. We did a grid sweep over the prior distribution scale hyper-parameter (i.e. standard deviation σ 0 ) with values in [0.1, 0.05, 0.04, 0.03, 0.02, 0.01]. For SGD with momentum we performed a grid sweep over learning rate in [1e -3 , 5e -3 , 1e and momentum used for learning the prior (testing the same values as before). The dropout rate used for learning the prior was selected from [0.01, 0.05, 0.1, 0.2]. The grid sweep is done for experiment E2. In the subsequent experiments, we use the same best performing hyper-parameters from E2. We experiment with fully connected neural networks (FCN) with 2/3 layers (excluding the 'input layer') and 100 units per hidden layer (unless specified otherwise). ReLU activations are used in each hidden layer. For learning the prior we ran the training for 500 epochs (except for MNIST for which we ran 100). Posterior training was run for 100 epochs. We use a training batch size of 250. PyTorch code will be released anonymously at the repository associated to this project1 .

E1: The role of the PAC-Bayes prior in PNNs

The work in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] showed that PAC-Bayes bounds can be used not only as training objectives but also for model selection. Specifically, the authors compared the risk certificates obtained to the test stochastic 01 error in a large-scale hyper-parameter grid search experiment involving more than 600 runs on MNIST, showing a strong relationship which motivated using the risk certificates for model selection.

We begin by replicating this experiment to study the role of the prior on the risk certificate. To do this, we perform the same grid search with the architecture reported in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] over 6 hyper-parameters: prior scale, dropout rate, and the learning rate and momentum both for learning the prior and the posterior. These experiments use a reduced MNIST training set (i.e. 10%), while the test set is maintained. Figure 2 shows the relationship between three metrics of interest and the risk certificate. We start by comparing the test 01 loss of the stochastic predictor to the risk certificate as done in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] (right plot). We observe a similarly strong relationship. The middle plot shows the relationship between the deterministic test 01 loss of the prior mean learnt by ERM and the risk certificate. Interestingly, this plot shows that when the prior is competitive, then the risk certificate is as well. However, good posteriors and risk certificates can be achieved even if the prior is not very competitive. Studying the hyper-parameters in this experiment, we observe that both the standard deviation used in the prior and the use of dropout when learning the prior have a big impact on the posterior risk certificate, independently of the prior's test set 01 error. Finally, the right part of the plot shows the relationship between the KL term in the bound and the risk certificate. This plot demonstrates the significant influence of the KL term (and thus the prior) in the final risk certificate. with different hyper-parameters on MNIST as done in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]. We use a reduced subset of MNIST for these experiments (10% of training data).

E2: Validation of prior and posterior

We now test the use of the validation strategy to choose the best fitted prior and posterior. To do so, we use 50% of the training data to build the prior, and 5% of those 50% for validation. The results of this experiment can be seen in Table 2, where the column named Validation indicates whether we use the above mentioned strategy to select the best prior and posterior. We note that even when discarding a small part of the prior set, this strategy improves the risk certificate in most cases (4 out of 6 datasets). This seems to lead to much smaller values of the KL term. However, the posterior stochastic 01 error does not improve in all cases (worsens in 3 out of the 6 datasets and improves in the rest). All the subsequent experiments in the present paper use this validation strategy to choose the prior and posterior. Table 2 also includes a comparison of the posterior and prior in terms of the stochastic 01 error and the deterministic 01 error of the mean of the distribution. The results show a consistent improvement of the stochastic error of the posterior over the stochastic error of the prior, while often the error of the mean of the distribution is kept similar or slightly worsened.

E3: Percentage of data devoted to prior learning

Table 3 shows the results obtained when using different percentages of the dataset to learn the prior by ERM (the rest of the dataset being used to certify the predictor).

The first relevant conclusion is that data-free priors (represented in the table as 0%) are far from competitive, thus justifying that the tightness of the risk certificates are indeed due to the use of data-dependent priors. The table also shows that the amount of data for the prior that leads to the best risk certificates is dataset dependent, with 50% and 75% approaching the optimal trade-off. Interestingly, using more data for the prior (e.g. 90%) improves the performance of the predictor but often harms the risk certificates, as the number of data points used to evaluate the bound decreases. This is in line with the findings of [START_REF] Gintare | On the role of data in PAC-Bayes bounds[END_REF]. We note that the best trade-off on the amount of data for prior building and evaluation of the risk certificate may depend on the size of the

E4: Differently parametrised architectures

The experiments in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF] also showed that the KL term decreased with the depth of the architecture in CIFAR-10. This is a noteworthy observation and possibly non intuitive initially, as there are many more parameters in the computation of the KL for deep architectures and one would expect the KL term to increase significantly with the number of parameters. However, the authors conjectured that this may be because in a higherdimensional weight space, the weight updates have smaller euclidean norms (see the discussion in [START_REF] Zhang | Are all layers created equal?[END_REF]), hence the smaller KL. We further validate with our experiments that the KL term does not increase with the number of parameters in the architecture. The results are shown in Table 5, where we test multiple fully connected architectures architectures with different amount of hidden layers and neurons. Our results demonstrate that highly over parameterised models show a KL term of similar factor than those architectures with less parameters. This could be because over parameterised models have optimization properties qualitatively different from those that are under parameterised [START_REF] Belkin | Reconciling modern machine-learning practice and the classical bias-variance trade-off[END_REF] or that over parameterised priors are so efficient that posteriors do not need to travel far. Either way, we believe this to be a promising result, demonstrating that there is potential for tight risk certificates even for highly over parameterised architectures.

E5: Training objectives for learning the prior

The results with different training objectives to learn the prior distribution are gathered in Table 4. The first conclusion is that regularisation of the prior using dropout improves the performance of f erm consistently. However, mixup regularisation seems to worsen the risk certificates and the stochastic error, at least for tabular datasets and FCNs. Note that this objective has an additional hyper-parameter for which we do the same grid sweep as suggested in [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF] for MNIST. The Bayesian inspired learning objective f bbb , which learns the full prior distribution, improves the results over f erm , generally both in terms of the risk certificate and the stochastic 01 error. In this sense, uninformed priors centered at zero, as suggested by [START_REF] Blundell | Weight uncertainty in neural networks[END_REF], seem to perform better than random uninformed priors. Note that f bbb requires an additional sweep over a KL trade-off coefficient (which attenuates the effect of the KL in the training objective) which was done with values in [1e -6 , 1e -5 , . . . , 1e -2 ], see [START_REF] Blundell | Weight uncertainty in neural networks[END_REF]. The results using the original f quad objective for learning the prior led to vacuous bounds, so we experimented with a version of this objective that also attenuates the KL (see [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]). The results in Table 4 include the results of f quad with the same KL trade-off coefficient sweep as in f bbb . In this case, the risk certificates are not vacuous although not as tight as when learning the prior with f bbb . We conjecture that f quad still puts much more emphasis on the KL term than f bbb , which in this case measures the divergence to an uninformed prior, which restricts the learning significantly. Given that mixup regularisation was originally proposed for vision datasets, convolutional architectures and data starvation scenarios [START_REF] Kimura | Why mixup improves the model performance[END_REF], we set to experiment with its use in MNIST with a CNN with 4 layers where we remove training data at random, reducing the training set to 25% of the original. The results are in which can be seen as a type of data augmentation, leads to very promising results in this case. Specially, it can be seen that f mixup is data efficient, in the sense that we may need less data to build the prior to achieve tight risk certificates and accurate classifiers (compare the results of f mixup with 10% of the data to build the prior to f erm with 50% of the data for the prior).

Tightness across datasets. Risk certificates are consistently tight across multiple datasets, as shown in Figure 3, where we plot all results reported in this paper together.

Conclusion

This work empirically studies learning PAC-Bayes priors from data. Our results show that data-dependent priors lead to consistently tight risk certificates in 6 datasets for probabilistic neural classifiers and over parameterised networks, setting a stepping stone towards achieving self-certified learning. We compare a wide range of training objectives for learning the prior distribution, and show that regularisation of the prior is important and that Bayesian inspired learning objectives hold potential for learning appropriate priors, in this case learning the full prior distribution, as opposed to only the mean. Our results also demonstrate that data augmentation may be desirable during prior learning.

Figure 1 :

 1 Figure 1: Comparison of data partitions in traditional machine learning (where the model is certified with a point estimate) vs self-certified learning through PAC-Bayes bounds. Note that all n training data are used by the self-certified learning algorithm.

Figure 2 :

 2 Figure 2: Comparison of several metrics vs the risk certificate for more than 600 runswith different hyper-parameters on MNIST as done in[START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]. We use a reduced subset of MNIST for these experiments (10% of training data).

Figure 3 :

 3 Figure 3: Consistency of the tightness of the risk certificates across datasets.

Table 1 :

 1 Datasets used: n is the total number of data points, #f the number of features and #c the number of classes.

	Dataset	n	#f #c
	Spambase	4601 58 2
	Bioresponse	3751 1777 2
	Har	10299 562 6
	Electricity	45312 9	2
	Mammography 11183 7	2
	MNIST	70000 784 10

Table 2 :

 2 -2 ] and momentum in [0.95, 0.99]. We also performed a grid sweep over the learning rate Cert. S. 01 err. KL/ncert Post. mean 01 err. Prior mean 01 err. S. 01 err. Results with/without the validation set to choose the prior/posterior in an experiment using data-dependent priors. The stochastic 01 test set errors (S. 01 err.) for both prior and posterior are averaged over 10 samples. We do not include the standard deviation because they are low (around 0.001 in 6 cases and lower than 0.001 in the rest). The table also includes the 01 error of the posterior and prior mean. n cert indicates the number of data points used for the computation of the risk certificate.

	Posterior	Prior

  data for prior R. Cert. S. 01 err. S. 01 err. R. Cert. S. 01 err. S. 01 err. R. Cert. S. 01 err. S. 01 err.

	Dataset		Spambase			Bioresponse			Har
		Posterior	Prior	Posterior	Prior	Posterior	Prior
	Perc. of Data-free (0%)	0.485	0.406	0.545	0.531	0.494	0.505	0.789	0.758	0.759
	Data-depend. (10%)	0.213	0.153	0.182	0.344	0.315	0.315	0.072	0.063	0.077
	Data-depend. (25%)	0.197	0.133	0.100	0.314	0.270	0.266	0.042	0.036	0.038
	Data-depend. (50%)	0.127	0.065	0.067	0.291	0.258	0.257	0.037	0.024	0.025
	Data-depend. (75%)	0.144	0.056	0.062	0.305	0.258	0.258	0.035	0.023	0.024
	Data-depend. (90%)	0.182	0.062	0.065	0.345	0.255	0.255	0.037	0.018	0.018
	ERM baseline (01 err.)		0.045			0.229			0.015
	Dataset		Electricity			Mammography		MNIST
		Posterior	Prior	Posterior	Prior	Posterior	Prior
	Perc. of data for prior R. Cert. S. 01 err. S. 01 err. R. Cert. S. 01 err. S. 01 err. R. Cert. S. 01 err. S. 01 err.
	Data-free (0%)	0.441	0.427	0.427	0.038	0.023	0.214	0.321	0.220	0.891
	Data-depend. (10%)	0.244	0.234	0.235	0.024	0.020	0.019	0.082	0.075	0.076
	Data-depend. (25%)	0.234	0.224	0.232	0.024	0.018	0.018	0.047	0.046	0.047
	Data-depend. (50%)	0.233	0.212	0.227	0.023	0.017	0.017	0.030	0.027	0.031
	Data-depend. (75%)	0.028	0.023	0.208	0.027	0.018	0.018	0.028	0.023	0.029
	Data-depend. (90%)	0.223	0.201	0.202	0.031	0.017	0.017	0.026	0.022	0.026
	ERM baseline (01 err.)		0.192			0.017			0.022

Table 3 :

 3 Results when using different percentages of the dataset for prior building and evaluation of the risk certificate. Best risk certificates for each dataset are highlighted in bold face.

	Dataset	Spambase	Bioresponse	Har		Electricity	Mammography
	Prior objective R. Cert. ferm 0.132	0.088	0.291	0.270	0.036	0.024	0.222	0.215	0.022	0.017
	ferm + dropout	0.126	0.063	0.290	0.255	0.033	0.021	0.219	0.205	0.022	0.017
	fmixup	0.166	0.105	0.299	0.272	0.038	0.019	0.241	0.231	0.028	0.021
	fbbb (random prior) 0.123	0.068	0.290	0.249	0.031	0.020	0.222	0.209	0.022	0.017
	fbbb (zero prior)	0.120	0.060	0.288	0.254	0.031	0.020	0.214	0.204	0.022	0.017
	fquad (random prior) 0.186	0.129	0.309	0.272	0.045	0.033	0.247	0.237	0.027	0.018
	fquad (zero prior)	0.243	0.168	0.309	0.272	0.045	0.033	0.247	0.237	0.027	0.018

S. 01 err. R. Cert. S. 01 err. R. Cert. S. 01 err. R. Cert. S. 01 err. R. Cert. S. 01 err.

Table 4 :

 4 Results with different training objectives to learn the prior. f erm and f mixup based objectives are deterministic and only learn the mean of the prior distribution. f bbb and f quad are probabilistic and learn the full prior distribution.

Table 6 :

 6 note that dropout also helps in this case to learn better PAC-Bayes priors. Additionally, f mixup ,September 22, 2021 Perc. of data for prior Prior objective R. Cert. S. 01 err.

		ferm	0.110	0.074
	10%	ferm + dropout 0.106	0.067
		fmixup	0.065	0.041
		ferm	0.089	0.056
	25%	ferm + dropout 0.044	0.029
		fmixup	0.041	0.025
		ferm	0.061	0.042
	50%	ferm + dropout 0.026	0.020
		fmixup	0.025	0.019

Table 6 :

 6 Results with deterministic training objectives to learn the prior on MNIST in a data starvation experiment (reducing the training set in MNIST to 25% of the original).

	Stochastic 01 error (test)
	Risk certificate

https://anonymous.4open.science/r/pacbayespriors-F355/

Acknowledgments

We gratefully acknowledge support and funding from the U.S. Army Research Laboratory and the U. S. Army Research Office, and by the U.K. Ministry of Defence and the U.K. Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/R013616/1.

We warmly acknowledge the Department of Computer Science and the AI Centre at University College London, and the Department of Electric and Electronic Engineering at the University of Surrey, for providing friendly and stimulating work environments.

Omar Rivasplata gratefully acknowledges sponsorship from DeepMind for conducting research studies in machine learning at University College London.

Table 5: Results with architectures with different numbers of parameters (both under and over parametrised with respect to the dataset size [START_REF] Belkin | Reconciling modern machine-learning practice and the classical bias-variance trade-off[END_REF]).

dataset. For example, in the case of electricity, which is the second largest dataset, using 90% of the data for building the prior does not harm the risk certificate considerably (0.2%), as we will still be using a reasonable amount to evaluate the bound. It is also worth noting the tightness of the risk certificate in this case (2.2% risk certificate vs 2.1% stochastic test set 01 error). The same is applicable to MNIST, where the risk certificates improve when using 90% of the data for prior building (since we will still be using 6000 examples for computing the risk certificate). However, in the case of Bioresponse and Spambase (smallest datasets) the risk certificates worsen by 6.4% and 4.6% respectively when comparing 50% of the data for prior building to 90%. The table also includes the results of the ERM baseline. For ERM, we used the same range for optimising the learning rate, momentum and dropout rate. However, given that in this case we do not have a risk certificate we need to set aside some data for validation and hyper-parameter tuning. In all cases we set 5% of the data for validation. Note that this predictor is deterministic, which usually shows slightly better performance than the stochastic counterpart (as also discussed by [START_REF] Gintare | Data-dependent PAC-Bayes priors via differential privacy[END_REF][START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]). The results on these datasets show a slightly larger gap between PNNs and standard neural networks learnt by ERM when compared to the results on MNIST and CIFAR-10 in [START_REF] Pérez-Ortiz | Tighter risk certificates for neural networks[END_REF]. Nonetheless, the risk certificates are tight.